Elliptic canonical bases for toric hyper-Kahler manifolds
EElliptic canonical bases for toric hyper-K¨ahler manifolds
Tatsuyuki Hikita ∗ Abstract
Lusztig defined certain involutions on the equivariant K -theory of Slodowy varieties and gavea characterization of certain bases called canonical bases. In this paper, we give a conjecturalgeneralization of these involutions and K -theoretic canonical bases to conical symplectic resolutionswhich have good Hamiltonian torus actions and state several conjectures related to them which wecheck for toric hyper-K¨ahler manifolds. We also propose an elliptic analogue of these bar involutions.As a verification of our proposal, we explicitly construct elliptic lifts of K -theoretic canonical basesand prove that they are invariant under elliptic bar involutions for toric hyper-K¨ahler manifolds. Lusztig [20, 21] defined certain modules of affine Hecke algebras called periodic modules and definedthe notion of canonical bases in them. In [22, 23], Lusztig gave a geometric construction of thesemodules in terms of equivariant K -theory of Springer resolutions or Slodowy varieties, and gave ageometric characterization of (signed) canonical bases using certain involution called bar involution.Basic properties of canonical bases including their existence and relation to modular representationtheory of semisimple Lie algebras in large enough characteristic were also conjectured by Lusztig andproved by Bezrukavnikov-Mirkovi´c [6]. K -theoretic canonical bases One of the aim of this paper is to give an analogue of the notion of bar involutions and canonical basesto equivariant K -theory of conical symplectic resolutions which have Hamiltonian torus actions withfinitely many fixed points. For ADE type quiver varieties, analogous bar involutions and canonical baseswere proposed by Lusztig in [24], and constructed by Varagnolo-Vasserot in [42]. In all the previousworks, bar involutions are defined by composing several automorphisms and it seems complicated atleast to the author. Our main observation in this paper is that the bar involution for Springer resolutions(and conjecturally for other known cases satifying our assumptions) have a very simple characterizationby using the K -theoretic analogue of the stable bases introduced by Maulik-Okounkov [25].In this introduction, we give a rough definition of the K -theoretic bar involution and the K -theoreticcanonical bases. Let X be a conical symplectic resolution with conical action of S := C × . Throughoutthis paper, we always assume that the symplectic form has weight 2 and there exists a Hamiltoniantorus action H (cid:121) X commuting with the S -action such that the fixed point set X H is finite.The K -theoretic stable basis (see e.g. [32, 34]) depends on some data called chamber C ⊂ X ∗ ( H ) ⊗ Z R ,polarization T / ∈ K H × S ( X ), and slope s ∈ Pic( X ) ⊗ Z R . Associated with these data, one can givea characterization of certain element Stab K C ,T / ,s ( p ) ∈ K H × S ( X ) in the equivariant K -theory of X corresponding to each fixed point p ∈ X H . These elements form a basis in the localized equivariant K -theory K H × S ( X ) loc := K H × S ( X ) ⊗ K H × S (pt) Frac( K H × S (pt)).Let v ∈ K S (pt) ∼ = Z [ v, v − ] be the element corresponding to the natural representation of S . Wewant to define K H (pt)-linear and K S (pt)-antilinear (i.e. β K ( v · − ) = v − β K ( − )) map β KX,T / ,s = β K : ∗ [email protected] a r X i v : . [ m a t h . AG ] M a r H × S ( X ) loc → K H × S ( X ) loc by the formula β K (Stab K C ,T / ,s ( p )) = ± v ? Stab K − C ,T / ,s ( p ) (1)for any p ∈ X H . For a more precise formula about the normalization, we refer to the main body of thepaper. We only note here that in order to write down the formula, one needs the information about thetangent bundle of the symplectic dual X ! of X in the sense of Braden-Licata-Proudfoot-Webster [8].One of the drawback of this approach is that it is not clear if β K is actually an involution, andit preserves the integral form K H × S ( X ) ⊂ K H × S ( X ) loc . In fact, these requirements give a strongrestriction on the normalization in (1). One of the main conjecture in this paper is that in the explicitlyspecified normalization, these properties for β hold. Conjecture 1.1.
The above β K is an involution and preserves K H × S ( X ) ⊂ K H × S ( X ) loc . Moreover,this does not depend on the choice of C .We note that the K -theoretic bar involution does depend on the choice of polarization and slope.When X is a Springer resolution, we will check that this bar involution essentially coincides with thebar involution defined by Lusztig for a specific choice of polarization and slope.Once we have defined the bar involution, we can give a characterization of the (signed) canonicalbasis following Lusztig [22, 23]. Let D X : K H × S ( X ) → K H × S ( X ) be the Serre duality. We define aninner product ( − : − ) : K H × S ( X ) × K H × S ( X ) → Frac( K H × S (pt)) by( F : G ) := [ R Γ( X, F ⊗ L X G )] , (2)and another inner product ( −||− ) : K H × S ( X ) × K H × S ( X ) → Frac( K H × S (pt)) by( F||G ) := ( F : D X β K G ) . (3)Using this, we define B ± X,T / ,s := (cid:8) m ∈ K H × S ( X ) (cid:12)(cid:12) β K ( m ) = m, ( m || m ) ∈ v − K H (pt)[[ v − ]] (cid:9) . (4)We note that for any λ ∈ X ∗ ( H ) and m ∈ B ± X,T / ,s , we have ± λm ∈ B ± X,T / ,s . In Proposition 3.21, wegive a conjectural Kazhdan-Lusztig type algorithm to compute B ± X,T / ,s , and in particular, fix the signin it which gives a subset B X,T / ,s ⊂ B ± X,T / ,s such that B ± X,T / ,s = B X,T / ,s (cid:116) − B X,T / ,s . Anothermain conjecture in this paper is the following. Conjecture 1.2.
There exists an H × S -equivariant tilting bundle T T / ,s on X such that B X,T / ,s coincides with the set of K -theory classes of indecomposable summands of T T / ,s up to shifts by X ∗ ( H ).In particular, B X,T / ,s is a basis of K H × S ( X ) as a Z [ v, v − ]-module.When X is a Springer resolution and for the specific choice of the data as above, this conjecturefollows from a result of Bezrukavnikov-Mirkovi´c [6] combined with our comparison result Proposition 2.3.In this paper, we also check it for toric hyper-K¨ahler manifolds, see Corollary 5.36. We remark that thistilting bundle essentially coincides with the one constructed by McBreen-Webster [28] and ˇSpenko-Vanden Bergh [40].We remark that by the definition of K -theoretic canonical basis, the endomorphism ring A T / ,s :=End( T T / ,s ) of the conjectural tilting bundle has nonnegative grading with respect to S . In particular,this is Koszul by the Kaledin’s argument in [6]. We will formulate a conjecture on the highest weightcategory structure on the equivariant module category of the Koszul dual of A T / ,s , see Conjecture 3.40.A natural duality functor on it should lift the K -theoretic bar involution to an involution on the derivedcategory D b Coh H × S ( X ) of H × S -equivariant coherent sheaves on X , see Conjecture 3.44.We also note that by varying the slope parameters, we would obtain a family of tilting bundles andhence t -structures on D b Coh H × S ( X ). This should be a part of the data defining the real variations ofstability conditions in the sense of Anno-Bezrukavnikov-Mirkovi´c [2], see Conjecture 3.47. We will give aconjecture on the wall-crossing of K -theoretic canonical bases, which is also given by a Kazhdan-Lusztigtype algorithm, see Conjecture 3.49. 2 .2 Elliptic bar involutions The second and the main aim of this paper is to give an elliptic analogue of the K -theoretic barinvolution. Since Aganagic-Okounkov [1] defined the elliptic analogue of the stable basis, it seemsnatural to consider the elliptic analogue of (1) replacing K -theoretic stable bases by elliptic stablebases.Let Stab AO C ,T / ( p ) be the elliptic stable basis associated with p ∈ X H in the sense of Aganagic-Okounkov [1]. This is a section of some line bundle on equivariant elliptic cohomology of X extendedby adding K¨ahler parameters. In order to obtain an involution, it seems natural to shift the K¨ahlerparameters by v det T / and then multiply it by Θ( N ! p ! , − ), the Thom class of the negative part of thetangent bundle of X ! at the fixed point p ! ∈ ( X ! ) H ! corresponding to p ∈ X H under symplectic duality.In order to consider its K -theory limits, we also twist it by v ? · (cid:113) det T / · det T / , ! p ! − . Let us denoteby S X, C ( p ) the resulting renormalized elliptic stable basis. Here, we omit the polarization from thenotation but they depends on the choice.We want to define the elliptic bar involution β ellX = β ell by the formula β ell ( S X, C ( p )) = ( − dim X S X, − C ( p )for any p ∈ X H . Here, β ell should be considered as an involution on some space of meromorphicfunctions on Spec ( K H × S ( X ) ⊗ Z C [Pic( X ) ∨ ]). We also conjecture that β ell does not depend on thechoice of C , and hence it is an involution. We will check this for toric hyper-K¨ahler manifolds inCorollary 6.10.We remark that in the above normalization, following symmetry under the symplectic duality isexpected (cf. [36, 37]): S X, C ( p ) | p = ± S X ! , C ! ( p !2 ) | p !1 . Here, we identified the equivariant parameters for X and K¨ahler parameters for X ! and vice versa, aspredicted by symplectic duality. In some sense, this symmetry explains the naturality of the abovenormalization. By taking the K -theory limits, this also explains the appearance of symplectic dualityin the normalization of our K -theoretic bar involutions. For more detail, see section 4.3.The main problem toward a definition of the notion of elliptic canonical basis is to generalize otherconditions such as asymptotic norm one property or triangular property of K -theoretic canonical basesto the elliptic case. Although we do not know how to deal with this problem in general, we give acandidate for the elliptic canonical bases for toric hyper-K¨ahler manifolds by explicitly constructing anelliptic lift of K -theoretic canonical bases which are invariant under the elliptic bar involution. Let us explain the main result of this paper briefly. Let 1 → S → T → H → X ∗ ( T ) ∼ = Z n with its dual by taking the standardpairing ( − , − ) on Z n . The toric hyper-K¨ahler manifold X := µ − (0) ss /S is defined by the Hamiltonianreduction of T ∗ C n by S . Here, µ is the moment map for the S -action and the GIT stability parameteris taken generically. We assume that X is smooth.By considering the induced line bundles on the quotient, we obtain a natural homomorphism L : X ∗ ( T ) → Pic H × S ( X ). By considering X ∗ ( S ) ⊂ X ∗ ( T ) as a subset of X ∗ ( T ) by using the pairing, wedefine the provisional elliptic canonical basis by the following formula. Definition 1.3.
For each λ ∈ X ∗ ( T ), we setΘ X ( λ ) := ( q ; q ) − rk S ∞ (cid:88) β ∈ X ∗ ( S ) ( − ( κ,β ) q ( β,β )+( λ + κ,β ) L ( λ + β ) z β . (5)Here, ( q ; q ) ∞ := (cid:81) m ≥ (1 − q m ), κ = (1 , , . . . , ∈ Z n , and z β is the K¨ahler parameter correspondingto β . 3ne can easily check that for β ∈ X ∗ ( S ) and α ∈ X ∗ ( H ) ⊂ X ∗ ( T ), we haveΘ X ( λ + β ) = ( − ( κ,β ) q − ( β,β ) − ( λ + κ,β ) z − β Θ X ( λ ) , Θ X ( λ + α ) = a α Θ X ( λ ) . Here, a α is the equivariant parameter corresponding to α ∈ X ∗ ( H ). Therefore, the number of indepen-dent elements in { Θ X ( λ ) } λ ∈ X ∗ ( T ) is at most the number of elements of Ξ := X ∗ ( T ) / ( X ∗ ( S ) + X ∗ ( H )) ∼ = X ∗ ( S ) / X ∗ ( S ) ∼ = X ∗ ( H ) / X ∗ ( H ), which turns out to be equal to the rank of K ( X ). We also remark thatthe formula (5) has the form of the theta function associated with the lattice X ∗ ( S ), where the pairingis induced from X ∗ ( T ). We point out that if we specialize all the equivariant parameters to 1 in (5),we get the character of irreducible module corresponding to λ ∈ X ∗ ( S ) / X ∗ ( S ) for the lattice vertexoperator superalgebra V X ∗ ( S ) associated with X ∗ ( S ) for certain choice of conformal vector.One can also define the elliptic canonical basis Θ X ! ( λ ) for the symplectic dual X ! , which is definedanalogously by the dual exact sequence of algebraic tori 1 → H ∨ → T ∨ → S ∨ →
1. We note that weidentify X ∗ ( T ) ∼ = X ∗ ( T ∨ ) by using the pairing. We also remark that the choice of GIT parameter for X ! is given by the choice of chamber for X . The main result of this paper is the following, see Theorem 6.9and Corollary 6.10. Theorem 1.4.
In the above setting, we have β ellX (Θ X ( λ )) = Θ X ( λ ) and β ellX ! (Θ X ! ( λ )) = Θ X ! ( λ ) for any λ ∈ X ∗ ( T ). We also have the following expansion of the elliptic stable bases into the elliptic canonicalbases. ± S X, C ( p ) = (cid:88) λ ∈ Ξ ( − ( λ,κ ) q ( λ,λ + κ ) Θ X ! ( λ ) | p ! · Θ X ( λ ) . (6)We note that in the sum of (6), each term does not depend on the choice of representatives for λ . We remark that this formula resembles the irreducible decomposition of the lattice vertex operatorsuperalgebra V Z n as a module for the commuting action of vertex operator subalgebras V X ∗ ( S ) and V X ∗ ( H ) , which forms a dual pair in the sense that they are commutant to each other in V Z n . The authoris not sure if this kind of phenomenon happens in general or not, but we hope that the results of thispaper would give some hints for the investigation of elliptic canonical bases for other conical symplecticresolutions.The plan of this paper is as follows. In section 2, we check that our definition of K -theoreticbar involution essentially coincides with Lusztig’s definition for Springer resolutions. In section 3,we propose the definition of K -theoretic bar involutions and state several conjectures related to K -theoretic canonical bases. In section 4, we propose the definition of elliptic bar involutions. In section5, we specialize to the case of toric hyper-K¨ahler manifolds and calculate K -theoretic canonical bases.We also prove all the conjectures stated in section 3 in these cases. In section 6, we prove our mainresult on the elliptic canonical bases for toric hyper-K¨ahler manifolds. The author thanks Tomoyuki Arakawa, Akishi Ikeda, Hiroshi Iritani, Syu Kato, Hitoshi Konno, ToshiroKuwabara, Michael McBreen, Takahiro Nagaoka and Andrei Okounkov for valuable discussions relatedto this work. Especially, the author thanks Andrei Okounkov for his suggestion to consider the ellipticstable envelope. This work was supported by JSPS KAKENHI Grant Number 17K14163.
In this section, we reformulate Lusztig’s definition of the bar involution on the equivariant K -theoryof Springer resolutions. We should remark that the Lusztig’s definition works for Slodowy varietiesassociated with any nilpotent element. If the nilpotent element is regular in some Levi algebras, thenwe can use our approach to define K -theoretic bar involutions. The results of this section are included4or motivational purposes and will not be used elsewhere in this paper. Hence we restrict ourselves tothe case of Springer resolutions for simplicity. Comparison of our definition with Lusztig’s definition forother Slodowy varieties should be straight-forward once some formulas for the K -theoretic stable basesanalogous to Proposition 2.2 are available.We first list some standard notations used in this section. Let G be a semisimple algebraic groupover C of adjoint type. We fix B ⊂ G a Borel subgroup and H ⊂ B a Cartan subgroup. We willdenote by g , b , h their Lie algebras. We use the convention that the nonzero H -weights appearing in b isnegative. We denote by X ∗ ( H ) (resp. X ∗ ( H )) the character lattice (resp. cocharacter lattice) of H . Weset h ∗ R := X ∗ ( H ) ⊗ Z R and h R := X ∗ ( H ) ⊗ Z R . Let { α i } i ∈ I be the set of simple roots and { α ∨ i } i ∈ I be theset of simple coroots. Let W be the Weyl group of G and s i ∈ W be the simple reflection correspondingto i ∈ I . For w ∈ W , we denote by l ( w ) the length of w . Let w ∈ W be the longest element and e ∈ W be the identity element. We denote by R + (resp. R ∨ + ) the set of positive roots (resp. positive coroots)and ρ = (cid:80) α ∈ R + α ∈ h ∗ R the half sum of positive roots. We briefly recall Lusztig’s bar involution on the equivariant K -theory of Springer resolutions. For moredetails, we refer to [22, 23].Let B := G/B be the flag variety and X := T ∗ B ∼ = { ( gB, y ) ∈ B × g ∗ | Ad( g ) − ( y ) ∈ ( g / b ) ∗ } be theSpringer resolution. The torus T := H × S acts naturally on X by ( h, σ ) · ( gB, y ) = ( hgB, σ − Ad( h ) y )and this action preserves the subvariety B . By the pushforward along zero section, we obtain an inclusion K T ( B ) ⊂ K T ( X ). Let pr : X → B be the natural projection.Let H be the affine Hecke algebra associated with the Langlands dual of G . I.e., H is the Z [ v, v − ]-algebra generated by T w ( w ∈ W ) and θ λ ( λ ∈ X ∗ ( H )) with the following relations: • ( T s i − v )( T s i + v − ) = 0 ( ∀ i ∈ I ), • T w T w (cid:48) = T ww (cid:48) if l ( ww (cid:48) ) = l ( w ) + l ( w (cid:48) ) ( w, w (cid:48) ∈ W ), • θ λ T s i − T s i θ s i ( λ ) = ( v − v − ) θ [ λ ] − [ si ( λ )]1 − [ − αi ] ( ∀ i ∈ I , λ ∈ X ∗ ( H )), • θ λ θ λ (cid:48) = θ λ + λ (cid:48) ( ∀ λ, λ (cid:48) ∈ X ∗ ( H )), • θ = 1.Here, for λ ∈ X ∗ ( H ), we denote by [ λ ] ∈ Z [ X ∗ ( H )] ∼ = K H (pt) the corresponding element and for c = (cid:80) λ ∈ X ∗ ( H ) c λ [ λ ] ∈ K H (pt), we set θ c = (cid:80) λ ∈ X ∗ ( H ) c λ θ λ . It is known (see e.g. [9, 22]) that H actson K T ( B ) and K T ( X ), and the inclusion K T ( B ) ⊂ K T ( X ) is compatible with the H -actions. We donot recall its construction, but in order to fix the convention, we write down its action on the fixedpoint basis. Our convention mainly follows that of [22].The fixed points of B and X with respect to the H -action are parametrized by W . For each w ∈ W ,we also denote by w := wB ∈ B the corresponding fixed point. We denote by O w ∈ K T ( B ) ⊂ K T ( X )the K -theory class of the structure sheaf of the fixed point w . The H -action on the fixed point basisis given by • T s i O w = v − v − − [ − w ( α i )] O w + v − [ − w ( α i )] − v − [ − w ( α i )] O ws i ( ∀ i ∈ I ) • θ λ O w = [ w ( λ )] · O w ( ∀ λ ∈ X ∗ ( H ))We note that the action of θ λ is given by tensor product of an equivariant line bundle L λ := ( G × C λ ) /B on B (or its pullback to X ), where the action of B is given by b · ( g, x ) = ( gb − , λ ( b ) x ).We fix a Lie algebra automorphism (cid:36) : g → g such that (cid:36) ( h ) = − h for any h ∈ h . This induces anautomorphism of B and X , which is denoted by the same letter. Lusztig’s bar involution β L : K T ( X ) → K T ( X ) is then defined by β L := ( − v ) l ( w ) T − w (cid:36) ∗ D X . −||− ) L : K T ( X ) × K T ( X ) → Frac( K T (pt)) in the same way as (3)replacing β K by β L . In order to compute the image of K -theoretic stable bases by β L , the followinglemma proved in [23] is useful. Lemma 2.1 ([23], Lemma 8.13) . Let Z := pr − ( e ) ⊂ X . Then for any f, f (cid:48) ∈ X ∗ ( H ) and w, w (cid:48) ∈ W ,we have ( f T − w O Z || f (cid:48) T − w (cid:48) O Z ) L = v − l ( w ) f f (cid:48)− δ w,w (cid:48) . We remark that in [23], this formula is proved more generally for Slodowy varieties associated withnilpotent elements which are regular in some Levi subalgebras.
Next we recall a result of Su-Zhao-Zhong [41] describing the K -theoretic stable bases for Springerresolutions. For the definition and basic properties of K -theoretic stable bases used in this paper, werefer to section 3.2. In particular, the sign convention is slightly different from [32, 34].We first fix the data defining the K -theoretic stable basis. For the chamber, we take the negativeWeyl chamber C = { x ∈ h R | ∀ i ∈ I, (cid:104) x, α i (cid:105) < } . For the polarization T / , we take the pullback ofthe tangent bundle of B by X → B . For the slope, we take s ∈ ρ − A +0 ⊂ Pic( X ) ⊗ Z R ∼ = h ∗ R , where A +0 = { x ∈ h ∗ R | ∀ α ∨ ∈ R ∨ + , < (cid:104) x, α ∨ (cid:105) < } is the fundamental alcove. We note that ρ correspondsto an actual (nonequivariant) line bundle on X and we take an T -equivariant lift L ρ . The followingdescription of the K -theoretic stable basis is essentially proved in type A by Rim´anyi-Tarasov-Varchenko[38] and in general by Su-Zhao-Zhong [41]. Proposition 2.2 ([41], Theorem 0.1) . For any w ∈ W , we haveStab K C ,T / ,s ( w ) = ( − l ( w ) [ ρ − wρ ] · T − w O Z Proof . We only need to compare the convention in [41] with ours. First, we note that the choice of theBorel subgroup in [41] is opposite to ours, hence the fixed point corresponding to w in [41] is ww inour notation. We also note that the line bundle corresponding to λ ∈ X ∗ ( H ) is L w λ in our notation.Hence the chamber is the same and the slope in [41] is taken in − A +0 . Moreover, the polarization isopposite to ours. Finally, one can check that the affine Hecke algebra action denoted by T w in [41] isgiven by v l ( w ) L ρ T w ww L − ρ in our notation. This does not depend on the choice of L ρ .One can easily check that( − l ( w ) Stab K C ,T / , − ρ + s ( e ) = ( − v ) l ( w ) [2 ρ ] · O Z . Therefore, Theorem 0.1 of [41] is( − l ( w ) − l ( w ) Stab K C ,T / , − ρ + s ( w ) = L ρ T − w L − ρ · ( − v ) l ( w ) [2 ρ ] · O Z in our notation. Here, the sign in the left hand side comes from our convention on the sign of K -theoreticstable basis. By using Lemma 3.7 and Lemma 3.8, we obtainStab K C ,T / ,s ( w ) = v − l ( w ) (det T / w ) − i ∗ w L ρ · L − ρ ⊗ Stab K C ,T / , − ρ + s ( w )= ( − l ( w ) [ ρ − wρ ] · T − w O Z as required. We now prove the following formula which was our starting point of this work.6 roposition 2.3.
For any w ∈ W , we have β L (Stab K C ,T / ,s ( w )) = ( − v ) l ( w ) Stab K − C ,T / ,s ( w ) . (7) Proof . By Lemma 2.1 and Proposition 2.2, we obtain(Stab K C ,T / ,s ( w ) : D X β L Stab K C ,T / ,s ( w (cid:48) )) = v − l ( w ) δ w,w (cid:48) for any w, w (cid:48) ∈ W . On the other hand, Lemma 3.9 and Lemma 3.10 imply that(Stab K C ,T / ,s ( w ) : D X Stab K − C ,T / ,s ( w (cid:48) )) = ( − v ) l ( w ) δ w,w (cid:48) . By comparing the two formulas, the proposition follows since the pairing ( − : − ) is perfect afterlocalization and { Stab K C ,T / ,s ( w ) } w ∈ W forms a basis of localized equivariant K -theory.Our main observation in this paper is that except for the normalization, an analogue of the formula(7) makes sense if one can define K -theoretic stable basis in order to characterize certain antilinearmap. Therefore, we try to make this kind of formula into a definition of K -theoretic bar involution.Our remaining task is to fix the normalization in some way, which turns out to be related to the notionof symplectic duality introduced by Braden-Licata-Proudfoot-Webster in [8]. In this section, we propose a general definition of K -theoretic canonical bases for conical symplecticresolutions which have good Hamiltonian torus actions. We also formulate several conjectures aboutthem which will be proved for toric hyper-K¨ahler manifolds in section 5. First we briefly recall basic definitions on symplectic resolution and symplectic duality in the form weneed later. In this paper, we mainly follow the setting of [8] for symplectic resolution. For symplecticduality, we partly follow the definition of 3d mirror symmetry in [37]. This is designed to give certainsymmetry of elliptic stable bases under the duality, see Conjecture 4.4.Let X be a connected smooth algebraic variety over C with algebraic symplectic form ω and an S -action. We assume that the S -weight of ω is 2 and the S -action is conical, which means that S -weightsappearing in the coordinate ring C [ X ] are nonnegative and the weight 0 part consists only of constantfunctions. If the natural morphism π : X → Spec( C [ X ]) is a resolution of singularity, X = ( X, ω, S )is called conical symplectic resolution. We denote by o ∈ Spec( C [ X ]) the unique S -fixed point and L := π − ( o ) the central fiber of π .In this paper, we always assume for simplicity that there does not exist another conical symplecticresolution X (cid:48) and symplectic vector space V (cid:54) = 0 such that X ∼ = X (cid:48) × V . We also assume that thereexists an effective action of another algebraic torus H on X which is Hamiltonian and commute withthe S -action such that the fixed point set X H is finite. We always take maximal H among such torusactions. For p ∈ X H , we will denote by i p : { p } (cid:44) → X the natural inclusion. We set T := H × S .For any p ∈ X H , we denote by Φ( p ) the multiset of H -weights appearing in the tangent space T p X at p . We call it the multiset of equivariant roots at p . We also simply call an element in the union (as a set)Φ := ∪ p ∈ X H Φ( p ) equivariant root for X . An equivariant root α define a hyperplane in h R = X ∗ ( H ) ⊗ Z R by H α := { ξ ∈ h R | (cid:104) ξ, α (cid:105) = 0 } . A connected component C in the complement h R \ ∪ α ∈ Φ H α is calledchamber, and it gives a decomposition of the tangent space T p X = N p, + ⊕ N p, − into attracting andrepelling parts. We denote by Attr C ( p ) := { x ∈ X | lim t → ξ ( t ) · x = p } the attracting set of p withrespect to C , where ξ ∈ X ∗ ( H ) is a one parameter subgroup of H contained in C . We note that thisdoes not depend on the choice of ξ in C . The choice of chamber also gives a partial order (cid:22) C on X H generated by p ∈ Attr C ( p (cid:48) ) = ⇒ p (cid:22) C p (cid:48) . 7e set P := Pic( X ) and P ∨ := Hom Z ( P, Z ) its dual. We denote by P R := P ⊗ Z R and A ⊂ P R theample cone of X . We will also need the notion of K¨ahler roots at each fixed point, but unfortunately,we do not know intrinsic definition of this notion. Our temporary definition is to take the equivariantroots at the corresponding fixed point for dual conical symplectic resolution. In this paper, we considerthem as additional data and take a multiset Ψ( p ) of elements in P ∨ for each p ∈ X H . As a conditionthey should satisfy, we assume that the walls in the slope parameters where the K -theoretic stable basisStab K C ,T / ,s ( p ) recalled in the next section jump are contained in the walls of the form { s ∈ P R | (cid:104) s, β (cid:105) ∈ Z } for some β ∈ Ψ( p ) . We also assume that A is a connected component of the complement of linearhyperplanes in P R defined by the K¨ahler roots in the union Ψ := ∪ p ∈ X H Ψ( p ). Remark 3.1.
For quiver varieties, one can read off the information about Ψ( p ) without using symplecticduality by a conjecture of Dinkins-Smirnov [14]. However, this does depend on the presentation of X as a GIT quotient. Hence we need to allow some factor of symplectic vector spaces and modify thestatement of symplectic duality to include some information about how to present the conical symplecticresolutions.Since these differences only affect the overall constants in the K -theoretic bar involutions and canon-ical bases, we do not pursue this point further here. We only note that this freedom on the normalizationis important for example when one try to compare our definition to the notion of global crystal bases oflevel 0 extremal weight modules of quantum affine algebras defined by Kashiwara [19]. For the canonicalbases in equivariant K -theory of ADE type quiver varieties defined by Varagnolo-Vasserot [42], thiskind of comparison is given by Nakajima [31].We also take an equivariant lift L : P → Pic T ( X ), i.e., a section of the natural homomorphismPic T ( X ) → P given by forgetting the equivariant structures. In this paper, when we consider symplecticresolutions, they are always equipped with the above additional data such as C , Ψ( p ), and L . We nowformulate a notion of dual pair between conical symplectic resolutions X = ( X, C , A , Φ , Ψ , L ) and X ! = ( X ! , C ! , A ! , Φ ! , Ψ ! , L ! ) as follows. Definition 3.2.
We say that a pair of conical symplectic resolutions X and X ! forms a dual pair if • There exists an order reversing bijection ( X H , (cid:22) C ) ∼ = (( X ! ) H ! , (cid:23) C ! ). We denote by p ! ∈ ( X ! ) H ! the fixed point corresponding to p ∈ X H ; • There exist isomorphisms X ∗ ( H ) ∼ = P ! and P ∼ = X ∗ ( H ! ) such that under this identification, wehave C = A ! , A = C ! , Φ( p ) = Ψ( p ! ), and Ψ( p ) = Φ( p ! ); • For any λ ∈ P , λ ! ∈ P ! , and p ∈ X H , we have (cid:104) wt H i ∗ p L ( λ ) , λ ! (cid:105) = −(cid:104) wt H ! i ∗ p ! L ! ( λ ! ) , λ (cid:105) , (8)wt S i ∗ p L ( λ ) = −(cid:104) wt H ! det N ! p ! , − , λ (cid:105) , (9)wt S ! i ∗ p ! L ! ( λ ! ) = −(cid:104) wt H det N p, − , λ ! (cid:105) ; (10) • For any p ∈ X H , we havewt S det N p, − + 12 dim X = − (cid:18) wt S ! det N ! p ! , − + 12 dim X ! (cid:19) . (11)Let { a , . . . , a e } be a basis of X ∗ ( H ) considered as elements of K H (pt) and { c , . . . , c e } be the dualbasis of X ∗ ( H ) ∼ = P ! . Similarly, let { z , . . . , z r } be a basis of P ∨ ∼ = X ∗ ( H ! ) considered as elements of In this paper, the notations for the bar involutions are always equipped with some upper index, and simple β is usedto denote a K¨ahler root. We hope this notation does not cause any confusions. H ! (pt) and { l , . . . , l r } be the dual basis of P . We also identify S ! = S and their equivariant parametersby v ! = v . In these notations, the third condition in Definition 3.2 is equivalent to the following: i ∗ p L ( λ ) = e (cid:89) i =1 a −(cid:104) i ∗ p ! L ! ( c i ) ,λ (cid:105) i · v −(cid:104) det N ! p ! , − ,λ (cid:105) , (12) i ∗ p ! L ! ( λ ! ) = r (cid:89) i =1 z −(cid:104) i ∗ p L ( l i ) ,λ ! (cid:105) i · v −(cid:104) det N p, − ,λ ! (cid:105) . (13) Example 3.3.
Let X = T ∗ ( G/B ) be the Springer resolution as in section 2. We recall that G is ofadjoint type. Let G ∨ be the adjoint type semisimple algebraic group whose Lie algebra is the Langlandsdual of g . We claim that the pair X and X ! = T ∗ ( G ∨ /B ∨ ) with certain choice of data forms a dualpair in the above sense. Here, we take a Borel subgroup B ∨ ⊂ G ∨ as in section 2.In this case, it is well-known that the Picard group P of X is given by the weight lattice and theample cone is given by the positive Weyl chamber in our convention. For example, we take the chamber C to be the negative Weyl chamber as in section 2.2. Since the ample cone of X ! is the positive Weylchamber, we twist the isomorphism P ! ∼ = X ∗ ( H ) by − C and A ! corresponds to each other.We take C ! to be the positive Weyl chamber and the isomorphism P ∼ = X ∗ ( H ! ) to be the natural one.We take Ψ( w ) to be the set of all coroots for any w ∈ W . The data Ψ ! is chosen similarly. For L : P → Pic T ( X ), we take L ( λ ) = v (cid:104) ρ ∨ ,λ (cid:105) [ − λ ] · L λ for any λ ∈ P , where L λ is defined as in section 2.1and 2 ρ ∨ ∈ P ∨ is the sum of all positive coroots. We note that the shift [ − λ ] is needed to make them H -equivariant. Similarly, we take L ! ( λ ! ) = v (cid:104) ρ,λ ! (cid:105) [ − λ ! ] · L ! λ ! . Here, we consider 2 ρ as an element of( P ! ) ∨ .We identify X H ∼ = W and ( X ! ) H ! ∼ = W by w ↔ w − . One can check that the partial order on X H given by C is the Bruhat order and the partial order on ( X ! ) H ! given by C ! is the opposite Bruhat order.Therefore, this gives an order reversing bijection. The second condition in Definition 3.2 is obviousfrom our choice. We note that det N w, − = v − l ( w ) [2 ρ ] and det N ! w − , − = v − l ( w )+2 l ( w ) [ − ρ ∨ ]. Thiseasily implies (9), (10), and (11). Here, we note that 2 ρ ∈ X ∗ ( H ) is identified with − ρ ∈ ( P ! ) ∨ . Wenote that wt H i ∗ w L ( λ ) = wλ − λ ∈ X ∗ ( H ) and wt H ! i ∗ w − L ! ( λ ! ) = w − λ ! − λ ! ∈ X ∗ ( H ! ) for any λ ∈ P and λ ! ∈ P ! . Since λ ! ∈ P ! is identified with − λ ! ∈ X ∗ ( H ), (8) follows from the obvious equation (cid:104) wλ − λ, − λ ! (cid:105) = −(cid:104) w − λ ! − λ ! , λ (cid:105) .We conjecture that if X and X ! are symplectic dual in the sense of Braden-Licata-Proudfoot-Webster[8], then they form a dual pair in the sense of Definition 3.2. We will check this for toric hyper-K¨ahlermanifolds in Proposition 5.6. In this paper, we always assume that a symplectic resolution consideredin this paper is equipped with a dual symplectic resolution in the above sense. Assumption 3.4.
For any choice of C , the conical symplectic resolution X = ( X, C , A , . . . ) has a dualconical symplectic resolution X ! = ( X ! , C ! , A ! , . . . ) such that X and X ! forms a dual pair in the senseof Definition 3.2.We note that if L ∈
Pic T ( X ) is a T -equivariant line bundle and l ∈ P is its underlying line bundle,then L ⊗ L ( l ) − is a trivial as a non-equivariant line bundle on X . The Assumption 3.4 implies that w ( L ) := wt S i ∗ p L − (cid:88) β ∈ Ψ + ( p ) (cid:104) β, l (cid:105) does not depend on the choice of p ∈ X H . Here, Ψ + ( p ) is the sub-multiset of Ψ( p ) consisting of K¨ahlerroots at p which are positive with respect to the ample cone A . We note that w : Pic T ( X ) → Z gives ahomomorphism and the image of L lands in the kernel of w . K -theoretic standard bases In this section, we recall the definition of K -theoretic stable basis introduced in [32, 34]. In or-der to define this, we need to choose further data. For an element F ∈ K T ( X ), we denote by9 ∨ := [ R Hom( F , O X )] ∈ K T ( X ) its dual. For a T -equivariant vector bundle V , we set (cid:86) •− V := (cid:80) i ≥ ( − i (cid:86) i V ∈ K T ( X ), where (cid:86) i V is the i -th exterior product of V . We take an element T / ∈ K T ( X ) called polarization satisfying T / + v − ( T / ) ∨ = T X , where T X is the K -theory class of the tan-gent bundle of X . We note that for any G ∈ K T ( X ), T / G := T / − G + v − G ∨ and T / := v − ( T / ) ∨ are also polarization for X .We also take a generic element s ∈ P R \ ∪ β ∈ Ψ { s ∈ P R | (cid:104) s, β (cid:105) ∈ Z } called slope. The map L naturally extends to give a fractional line bundle L ( s ) ∈ Pic T ( X ) ⊗ Z R and each restriction at a fixedpoint i ∗ p L ( s ) gives an element of X ∗ ( T ) ⊗ Z R . For an element of the form m = (cid:80) µ ∈ h ∗ R m µ µ , we denoteby deg H ( m ) ⊂ h ∗ R the convex hull of { µ ∈ h ∗ R | m µ (cid:54) = 0 } . Definition 3.5 ([32, 34]) . A set of elements { Stab K C ,T / ,s ( p ) } p ∈ X H of K T ( X ) is called stable basis if itsatisfies the following conditions: • Supp(Stab K C ,T / ,s ( p )) ⊂ (cid:116) p (cid:48) (cid:22) C p Attr C ( p (cid:48) ); • i ∗ p Stab K C ,T / ,s ( p ) = (cid:114) det N p, − det T / p · (cid:86) •− N ∨ p, − ; • deg H (cid:16) i ∗ p (cid:48) Stab K C ,T / ,s ( p ) · i ∗ p L ( s ) (cid:17) ⊂ deg H (cid:16) i ∗ p (cid:48) Stab K C ,T / ,s ( p (cid:48) ) · i ∗ p (cid:48) L ( s ) (cid:17) for any p (cid:48) (cid:22) C p ∈ X H .Here, the square root appearing in the normalization is well-defined by [32, 34]. We note that ournormalization is different from [32, 34] by (cid:113) det T / p, =0 where T / p, =0 is the H -invariant part of T / p := i ∗ p T / . The existence of (cid:113) det T / p, =0 follows from the equation T / p, =0 + v − ( T / p, =0 ) ∨ = 0. If the slope s issufficiently generic so that wt H i ∗ p L ( s ) − wt H i ∗ p (cid:48) L ( s ) / ∈ X ∗ ( H ) for any p (cid:54) = p (cid:48) ∈ X H , then the K -theoreticstable basis is unique if it exists by [32, Proposition 9.2.2]. The existence is claimed in some generalityin [33] and proved for example when X is a toric hyper-K¨ahler manifold, quiver variety [1], or Springerresolution [41]. In this paper, we assume that the K -theoretic stable bases exist uniquely for any X and the additional data we consider. Assumption 3.6.
The slope s ∈ P R satisfies wt H i ∗ p L ( s ) − wt H i ∗ p (cid:48) L ( s ) / ∈ X ∗ ( H ) for any p (cid:54) = p (cid:48) ∈ X H .Moreover, Stab K C ,T / ,s ( p ) exists for any p ∈ X H .We next collect some basic results on the K -theoretic stable bases for our reference since we havechanged the convention slightly. Lemma 3.7 ([32], Exercise 9.1.2) . For any
G ∈ K T ( X ) and p ∈ X H , we haveStab K C ,T / G ,s ( p ) = v rk G det i ∗ p G ·
Stab K C ,T / ,s +det G ( p ) . Lemma 3.8.
For any l ∈ P and p ∈ X H , we haveStab K C ,T / ,s + l ( p ) = ( i ∗ p L ( l )) − · L ( l ) ⊗ Stab K C ,T / ,s ( p ) . Lemma 3.9 ([34]) . For any p ∈ X H , we haveStab K C ,T / ,s ( p ) ∨ = ( − v ) − dim X Stab K C ,T / , − s ( p ) . These formulas easily follow from the definition and the uniqueness of K -theoretic stable basis.Recall the inner product ( − : − ) defined in (2). The K -theoretic stable bases have the followingorthogonality property with respect to ( − : − ). We changed the sign and normalization from [32, 34]. We note that this only depends on the determinant of T / . emma 3.10 ([34], Proposition 1) . For any p, p (cid:48) ∈ X H , we have (cid:16) Stab K C ,T / ,s ( p ) : Stab K − C ,T / , − s ( p (cid:48) ) (cid:17) = δ p,p (cid:48) . In order to define K -theoretic bar involutions, we further renormalize the K -theoretic stable bases.For each p ∈ X H , we set a p ( T / , s ) := (cid:88) β ∈ Ψ + ( p ) (cid:18) (cid:98)(cid:104) s, β (cid:105)(cid:99) + 12 (cid:19) −
14 dim X + 12 w (det T / ) ∈ Z . By numerical experiments, we expect that a p ( T / , s ) ∈ Z . This is equivalent to the following assump-tion which is also assumed in this paper. Assumption 3.11.
There exists a polarization T / on X such that w (det T / ) ≡
12 dim X + 12 dim X ! mod 2 . We note that if T / satisfies the above condition, then T / G also satisfies the condition for any G ∈ K T ( X ). We set F := { ( λ, p ) | λ ∈ X ∗ ( H ) , p ∈ X H } . (14)This set will label the K -theoretic standard bases and K -theoretic canonical bases. Definition 3.12.
For any ( λ, p ) ∈ F , we set S C ,T / ,s ( λ, p ) := ( − dim X v a p ( T / ,s ) [ λ ] · Stab K C ,T / ,s ( p ) . We call the set B std C ,T / ,s := {S C ,T / ,s ( λ, p ) } ( λ,p ) ∈ F ⊂ K T ( X ) K -theoretic standard basis for X .We will simply write S C ,T / ,s ( p ) := S C ,T / ,s (0 , p ). The standard basis should be considered as a Z [ v, v − ]-basis for the equivariant K -theory of the full attracting set (cid:116) p ∈ X H Attr C ( p ). One explanationof the seemingly strange normalization in this definition will be given in section 4.3 by considering itselliptic analogue. Here, we list some basic properties of the K -theoretic standard bases for our reference. Lemma 3.13.
For any
G ∈ K T ( X ) and ( λ, p ) ∈ F , we have S C ,T / G ,s ( λ, p ) = S C ,T / ,s +det G ( λ + wt H det i ∗ p G , p ) . Proof . This follows from Lemma 3.7 and a p ( T / G , s ) = a p ( T / , s + det G ) − rk G − wt S det i ∗ p G . Lemma 3.14.
For any l ∈ P and ( λ, p ) ∈ F , we have S C ,T / ,s + l ( λ, p ) = L ( l ) ⊗ S C ,T / ,s ( λ − wt H i ∗ p L ( l ) , p ) . Proof . This follows from Lemma 3.8 and a p ( T / , s + l ) = a p ( T / , s ) + wt S det i ∗ p L ( l ). Lemma 3.15.
For any ( λ, p ) ∈ F , we have S C ,T / ,s ( λ, p ) ∨ = ( − v ) dim X S C ,T / , − s ( − λ, p ) . Proof . This follows from Lemma 3.9 and a p ( T / , − s ) = − a p ( T / , s ) − dim X . Lemma 3.16.
For any p, p (cid:48) ∈ X H , we have (cid:16) S C ,T / ,s ( p ) : S − C ,T / , − s ( p (cid:48) ) (cid:17) = v − dim X δ p,p (cid:48) . Proof . This follows from Lemma 3.10 and a p ( T / , − s ) = − a p ( T / , s ) − dim X .11 .3 K -theoretic bar involution Now we define the K -theoretic bar involution. Since {S C ,T / ,s ( p ) } p ∈ X H forms a basis of K T ( X ) loc over Frac( K T (pt)), we can define a K H (pt)-linear map β K = β K C ,T / ,s : K T ( X ) loc → K T ( X ) loc by thefollowing conditions: • β K ( vm ) = v − β K ( m ) for any m ∈ K T ( X ) loc ; • β K (cid:0) S C ,T / ,s ( p ) (cid:1) = ( − v ) dim X S − C ,T / ,s ( p ) for any p ∈ X H .We note that β K − C ,T / ,s ◦ β K C ,T / ,s = id . The following conjecture implies that β K is an involution, andhence we call it K -theoretic bar involution associated with the data C , T / , and s . Conjecture 3.17.
The K -theoretic bar involution β K C ,T / ,s does not depend on the choice of C .For toric hyper-K¨ahler manifolds, this conjecture is proved in Corollary 5.16. Assuming this conjec-ture, we will sometimes omit C from the notation and write β KT / ,s = β K C ,T / ,s . In this section, we provecertain triangular properties of β K with respect to the K -theoretic standard bases. We first define apartial order on the labeling set F . Definition 3.18.
For ( λ, p ) , ( λ (cid:48) , p (cid:48) ) ∈ F , we write ( λ, p ) ≤ C ,s ( λ (cid:48) , p (cid:48) ) if we have (cid:104) λ − wt H i ∗ p L ( s ) , ξ (cid:105) ≤(cid:104) λ (cid:48) − wt H i ∗ p (cid:48) L ( s ) , ξ (cid:105) for any ξ ∈ C . Lemma 3.19.
The relation ≤ C ,s defines a partial order on F . Moreover, the number of elements( λ, p ) ∈ F satisfying ( λ (cid:48) , p (cid:48) ) ≤ C ,s ( λ, p ) ≤ C ,s ( λ (cid:48)(cid:48) , p (cid:48)(cid:48) ) is finite for any ( λ (cid:48) , p (cid:48) ) , ( λ (cid:48)(cid:48) , p (cid:48)(cid:48) ) ∈ F , i.e., thepartial order ≤ C ,s is interval finite. Proof . If ( λ, p ) ≤ C ,s ( λ (cid:48) , p (cid:48) ) and ( λ (cid:48) , p (cid:48) ) ≤ C ,s ( λ, p ), then we have λ − wt H i ∗ p L ( s ) = λ (cid:48) − wt H i ∗ p (cid:48) L ( s ).This implies that wt H i ∗ p L ( s ) − wt H i ∗ p (cid:48) L ( s ) ∈ X ∗ ( H ) and hence p = p (cid:48) by Assumption 3.6. This provesthe antisymmetry. The other properties are trivial to check.The second claim follows from the compactness of the set (cid:84) ξ ∈ C { µ ∈ h ∗ R | (cid:104) µ (cid:48) , ξ (cid:105) ≤ (cid:104) µ, ξ (cid:105) ≤ (cid:104) µ (cid:48)(cid:48) , ξ (cid:105)} for any µ (cid:48) , µ (cid:48)(cid:48) ∈ h ∗ R .Using this partial order, we define a Z [ v, v − ]-module of formal sums M C ,s := (cid:88) ( λ,p ) ∈ F f λ,p ( v ) S λ,p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f λ,p ( v ) ∈ Z [ v, v − ] , ∃ ( λ , p ) , . . . , ( λ m , p m ) ∈ F s.t. if f λ,p (cid:54) = 0 , then ( λ, p ) ≥ C ,s ( λ i , p i ) for some i . For any
F ∈ K T ( X ), we have F = (cid:80) p ∈ X H v dim X (cid:16) F : S − C ,T / , − s ( p ) (cid:17) · S C ,T / ,s ( p ) by Lemma 3.16and the possible denominators appearing in the inner product are of the form (cid:86) •− ( T ∗ p X ) for some p ∈ X H . Therefore, by sending S C ,T / ,s ( λ, p ) to S λ,p and expanding the rational function appearingin the coefficients into formal series in the positive or negative direction with respect to C , we obtaintwo natural embeddings ι ± C ,T / ,s : K T ( X ) (cid:44) → M ± C ,s . The following lemma together with Lemma 3.19implies that one can extend the K -theoretic bar involution to M ± C ,s . Lemma 3.20.
For each ( λ, p ) ∈ F , we have ι ± C ,T / ,s (( − v ) ∓ dim X S − C ,T / ,s ( λ, p )) ∈ S λ,p + (cid:88) ( λ (cid:48) ,p (cid:48) ) > ± C ,s ( λ,p ) Z [ v, v − ] · S λ (cid:48) ,p (cid:48) . Proof . We first note that v dim X (cid:16) S − C ,T / ,s ( p ) : S − C ,T / , − s ( p (cid:48) ) (cid:17) = (cid:88) p (cid:48)(cid:48) ∈ X H i ∗ p (cid:48)(cid:48) Stab K − C ,T / ,s ( p ) · i ∗ p (cid:48)(cid:48) Stab K − C ,T / , − s ( p (cid:48) ) (cid:86) •− ( N ∨ p (cid:48)(cid:48) , + ) · (cid:86) •− ( N ∨ p (cid:48)(cid:48) , − ) (15)12f we write N p (cid:48)(cid:48) , + = (cid:80) i w i , w i ∈ X ∗ ( T ), then we have i ∗ p (cid:48)(cid:48) Stab K − C ,T / ,s ( p (cid:48)(cid:48) ) · i ∗ p (cid:48)(cid:48) Stab K − C ,T / , − s ( p (cid:48)(cid:48) ) (cid:86) •− ( N ∨ p (cid:48)(cid:48) , + ) · (cid:86) •− ( N ∨ p (cid:48)(cid:48) , − ) = v dim X det N p (cid:48)(cid:48) , + · (cid:86) •− ( N ∨ p (cid:48)(cid:48) , + ) (cid:86) •− ( N ∨ p (cid:48)(cid:48) , − )= ( − v ) dim X (cid:89) i − w i − v w i . If we expand this in positive (resp. negative) direction with respect to C , then the expansion start from( − v ) dim X (resp. ( − v ) − dim X ). This formula also implies that if we set ρ p (cid:48)(cid:48) := wt H det N p (cid:48)(cid:48) , + , then forany ξ ∈ C , we have (cid:68) ξ, deg H (cid:16) i ∗ p (cid:48)(cid:48) Stab K − C ,T / ,s ( p (cid:48)(cid:48) ) · i ∗ p (cid:48)(cid:48) Stab K − C ,T / , − s ( p (cid:48)(cid:48) ) (cid:17)(cid:69) = [ −(cid:104) ξ, ρ p (cid:48)(cid:48) (cid:105) , (cid:104) ξ, ρ p (cid:48)(cid:48) (cid:105) ]On the other hand, by the definition of K -theoretic stable basis, we havedeg H (cid:16) i ∗ p (cid:48)(cid:48) Stab K − C ,T / ,s ( p ) (cid:17) ⊂ deg H (cid:18) i ∗ p (cid:48)(cid:48) Stab K − C ,T / ,s ( p (cid:48)(cid:48) ) · i ∗ p (cid:48)(cid:48) L ( s ) i ∗ p L ( s ) (cid:19) , (16)deg H (cid:16) i ∗ p (cid:48)(cid:48) Stab K − C ,T / , − s ( p (cid:48) ) (cid:17) ⊂ deg H (cid:32) i ∗ p (cid:48)(cid:48) Stab K − C ,T / , − s ( p (cid:48)(cid:48) ) · i ∗ p (cid:48)(cid:48) L ( − s ) i ∗ p (cid:48) L ( − s ) (cid:33) . (17)Therefore, we obtain (cid:68) ξ, deg H (cid:16) i ∗ p (cid:48)(cid:48) Stab K − C ,T / ,s ( p ) · i ∗ p (cid:48)(cid:48) Stab K − C ,T / , − s ( p (cid:48) ) (cid:17)(cid:69) ⊂ (cid:20)(cid:28) ξ, − ρ p (cid:48)(cid:48) + wt H i ∗ p (cid:48) L ( s ) i ∗ p L ( s ) (cid:29) , (cid:28) ξ, ρ p (cid:48)(cid:48) + wt H i ∗ p (cid:48) L ( s ) i ∗ p L ( s ) (cid:29)(cid:21) This implies that if S λ (cid:48) ,p (cid:48) appears in the expansion of ι ± C ,T / ,s (( − v ) ∓ dim X S − C ,T / ,s ( λ, p )), then wehave (cid:104) ξ, λ (cid:48) − λ (cid:105) ≥ (cid:104) ξ, wt H i ∗ p (cid:48) L ( s ) − wt H i ∗ p L ( s ) (cid:105) for any ξ ∈ ± C , which means ( λ, p ) ≤ ± C ,s ( λ (cid:48) , p (cid:48) ).For the coefficient of S λ,p , we note that when p (cid:48)(cid:48) (cid:54) = p = p (cid:48) , the fractional shift appearing in (16) and(17) are opposite and not integral. Hence we have (cid:68) ξ, deg H (cid:16) i ∗ p (cid:48)(cid:48) Stab K − C ,T / ,s ( p ) · i ∗ p (cid:48)(cid:48) Stab K − C ,T / , − s ( p ) (cid:17)(cid:69) ⊂ ( −(cid:104) ξ, ρ p (cid:48)(cid:48) (cid:105) , (cid:104) ξ, ρ p (cid:48)(cid:48) (cid:105) ) . This implies that only p (cid:48)(cid:48) = p = p (cid:48) part in (15) contribute to the coefficient of S λ,p . K -theoretic canonical basis Once we have defined the bar involution, we can follow Lusztig [22, 23] to define the notion of signedcanonical basis by imposing bar invariance and asymptotic norm one property. Recall the inner product( −||− ) : K T ( X ) × K T ( X ) → Frac( K T (pt)) defined in (3). We note that this depends on the choice of C , T / , and s through β K C ,T / ,s , but we simply omit the dependence from the notation if there is nochance of confusion. Recall that L ⊂ X is the central fiber. We set B ± L,T / ,s := (cid:110) m ∈ K T ( L ) (cid:12)(cid:12)(cid:12) β KT / ,s ( m ) = v dim X m, ( m || m ) ∈ v − K H (pt)[ v − ] (cid:111) , B ± X,T / ,s := (cid:110) m ∈ K T ( X ) (cid:12)(cid:12)(cid:12) β KT / ,s ( m ) = m, ( m || m ) ∈ v − K H (pt)[[ v − ]] (cid:111) , where we expand the rational function in Laurent series of v − with coefficients in Frac( K H (pt)).We note that for any λ ∈ X ∗ ( H ) and m ∈ B ± X,T / ,s , we have ± [ λ ] · m ∈ B ± X,T / ,s . If we assumeConjecture 3.17, then this definition does not depend on the choice of C . We call B ± L,T / ,s and B ± X,T / ,s signed K -theoretic canonical bases for L and X .If we assume that β K is an involution, we can formally construct a family of bar invariant andasymptotic norm one elements by Kazhdan-Lusztig type algorithm. More precisely, we can prove thefollowing, whose proof provides such an algorithm.13 roposition 3.21. Assume that β K C ,T / ,s is an involution.1. For any ( λ, p ) ∈ F , there exists unique C T / ,sλ,p ∈ S λ,p + (cid:80) ( λ (cid:48) ,p (cid:48) ) > C ,s ( λ,p ) v − Z [ v − ] · S λ (cid:48) ,p (cid:48) ⊂ M C ,s such that v − dim X · β K C ,T / ,s ( C T / ,sλ,p ) = C T / ,sλ,p .2. For any ( λ, p ) ∈ F , there exists unique E T / ,sλ,p ∈ S λ,p + (cid:80) ( λ (cid:48) ,p (cid:48) ) < C ,s ( λ,p ) v − Z [ v − ] · S λ (cid:48) ,p (cid:48) ⊂ M − C ,s such that β K C ,T / ,s ( E T / ,sλ,p ) = E T / ,sλ,p . Proof . We only prove the first statement since the second statement can be proved similarly. We firstprove the existence. We take any total order on F refining ≤ C ,s and prove inductively that for any( λ (cid:48) , p (cid:48) ) ≥ C ,s ( λ, p ), there exists an element of the form C ≤ ( λ (cid:48) ,p (cid:48) ) λ,p = (cid:88) ( λ,p ) ≤ C ,s ( λ (cid:48)(cid:48) ,p (cid:48)(cid:48) ) ≤ C ,s ( λ (cid:48) ,p (cid:48) ) f λ (cid:48)(cid:48) ,p (cid:48)(cid:48) λ,p ( v ) S λ (cid:48)(cid:48) ,p (cid:48)(cid:48) with f λ,pλ,p ( v ) = 1 and f λ (cid:48)(cid:48) ,p (cid:48)(cid:48) λ,p ( v ) ∈ v − Z [ v − ] for any ( λ (cid:48)(cid:48) , p (cid:48)(cid:48) ) (cid:54) = ( λ, p ) such that v − dim X · β K C ,T / ,s (cid:16) C ≤ ( λ (cid:48) ,p (cid:48) ) λ,p (cid:17) − C ≤ ( λ (cid:48) ,p (cid:48) ) λ,p ∈ (cid:88) ( λ (cid:48)(cid:48) ,p (cid:48)(cid:48) ) > C ,s ( λ (cid:48) ,p (cid:48) ) Z [ v, v − ] · S λ (cid:48)(cid:48) ,p (cid:48)(cid:48) . For ( λ (cid:48) , p (cid:48) ) = ( λ, p ), we can take C λ,pλ,p = S λ,p by Lemma 3.20. Assume that we have constructed C < ( λ (cid:48) ,p (cid:48) ) λ,p and write v − dim X · β K C ,T / ,s (cid:16) C < ( λ (cid:48) ,p (cid:48) ) λ,p (cid:17) − C < ( λ (cid:48) ,p (cid:48) ) λ,p ∈ g ( v ) S λ (cid:48) ,p (cid:48) + (cid:88) ( λ (cid:48)(cid:48) ,p (cid:48)(cid:48) ) > C ,s ( λ (cid:48) ,p (cid:48) ) Z [ v, v − ] · S λ (cid:48)(cid:48) ,p (cid:48)(cid:48) . for some g ( v ) ∈ Z [ v, v − ]. Since β K C ,T / ,s is an involution, we also obtain v − dim X · β K C ,T / ,s (cid:16) C < ( λ (cid:48) ,p (cid:48) ) λ,p (cid:17) − C < ( λ (cid:48) ,p (cid:48) ) λ,p ∈ − g ( v − ) S λ (cid:48) ,p (cid:48) + (cid:88) ( λ (cid:48)(cid:48) ,p (cid:48)(cid:48) ) > C ,s ( λ (cid:48) ,p (cid:48) ) Z [ v, v − ] · S λ (cid:48)(cid:48) ,p (cid:48)(cid:48) . by Lemma 3.20. By comparing the coefficient, we obtain g ( v ) = − g ( v − ). Hence there exists an element f λ (cid:48) ,p (cid:48) λ,p ( v ) ∈ v − Z [ v − ] such that g ( v ) = f λ (cid:48) ,p (cid:48) λ,p ( v ) − f λ (cid:48) ,p (cid:48) λ,p ( v − ). Then it suffices to take C ≤ ( λ (cid:48) ,p (cid:48) ) λ,p = C < ( λ (cid:48) ,p (cid:48) ) λ,p + f λ (cid:48) ,p (cid:48) λ,p ( v ) S λ (cid:48) ,p (cid:48) . This proves the existence of C T / ,sλ,p .For the uniqueness, we assume that there is another C (cid:48) λ,p satisfying the above conditions. Then wecan expand C (cid:48) λ,p = C T / ,sλ,p + (cid:88) ( λ (cid:48) ,p (cid:48) ) > C ,s ( λ,p ) g λ (cid:48) ,p (cid:48) λ,p ( v ) C T / ,sλ (cid:48) ,p (cid:48) with g λ (cid:48) ,p (cid:48) λ,p ( v ) ∈ v − Z [ v − ]. By v − dim X · β K C ,T / ,s ( C (cid:48) λ,p ) = C (cid:48) λ,p , we also obtain C (cid:48) λ,p = C T / ,sλ,p + (cid:88) ( λ (cid:48) ,p (cid:48) ) > C ,s ( λ,p ) g λ (cid:48) ,p (cid:48) λ,p ( v − ) C T / ,sλ (cid:48) ,p (cid:48) . Hence we have g λ (cid:48) ,p (cid:48) λ,p ( v ) = g λ (cid:48) ,p (cid:48) λ,p ( v − ) ∈ v − Z [ v − ] ∩ v Z [ v ] = { } . This proves C (cid:48) λ,p = C T / ,sλ,p . Conjecture 3.22.
The K -theoretic bar involution β K C ,T / ,s is an involution. For any ( λ, p ) ∈ F , thereexists C C ,T / ,s ( λ, p ) ∈ K T ( L ) (resp. E C ,T / ,s ( λ, p ) ∈ K T ( X )) such that ι + C ,T / ,s ( C C ,T / ,s ( λ, p )) = C T / ,sλ,p (resp. ι − C ,T / ,s ( E C ,T / ,s ( λ, p )) = E T / ,sλ,p ). Moreover, B L,T / ,s := {C C ,T / ,s ( λ, p ) } ( λ,p ) ∈ F (resp. B X,T / ,s := {E C ,T / ,s ( λ, p ) } ( λ,p ) ∈ F ) forms a basis of K T ( L ) (resp. K T ( X )) as a Z [ v, v − ]-module.14or toric hyper-K¨ahler manifolds, this conjecture is proved in Proposition 5.15, Lemma 5.17, andCorollary 5.37. We note that C C ,T / ,s ( λ, p ) = [ λ ] · C C ,T / ,s ( p ) and E C ,T / ,s ( λ, p ) = [ λ ] · E C ,T / ,s ( p ) forany ( λ, p ) ∈ F , where we set C C ,T / ,s ( p ) := C C ,T / ,s (0 , p ) and E C ,T / ,s ( p ) := E C ,T / ,s (0 , p ). We call B L,T / ,s and B X,T / ,s K -theoretic canonical bases for L and X associated with the data T / and s .Conjecture 3.22 implies that the K -theoretic bar involutions preserve K T ( L ) and K T ( X ), which arealready quite nontrivial from our definition. We should remark that C C ,T / ,s ( λ, p ) and E C ,T / ,s ( λ, p )does depend on the choice of C , but as we will show, the sets B L,T / ,s and B X,T / ,s will not dependon it (possibly up to sign). If we change C , then the parametrization of the canonical basis by the fixedpoints will change.We also note that since L is contained in the full attracting sets, the conjecture implies that thesum in the definition of C T / ,sλ,p is actually a finite sum. In particular, the above Kazhdan-Lusztig typealgorithm gives a way to calculate K -theoretic canonical bases for L if we know some formula for the K -theoretic stable bases. The sum in the definition of E T / ,sλ,p is always an infinite sum except forthe trivial cases, but as Proposition 3.31 shows, B L,T / ,s and B X,T / ,s are dual basis with respect to( −||− ), hence one can also calculate B X,T / ,s if we know B L,T / ,s .We also conjecture the following positivity property for the expansion of the K -theoretic canonicalbases in terms of the K -theoretic standard bases. This is an analogue of the positivity of the Kazhdan-Lusztig polynomials. Conjecture 3.23.
For any ( λ, p ) ∈ F , we have C T / ,sλ,p ∈ (cid:88) ( λ (cid:48) .p (cid:48) ) ≥ C ,s ( λ,p ) Z ≥ [( − v ) − ] · S λ (cid:48) ,p (cid:48) ,E T / ,sλ,p ∈ (cid:88) ( λ (cid:48) .p (cid:48) ) ≤ C ,s ( λ,p ) Z ≥ [ v − ] · S λ (cid:48) ,p (cid:48) . For toric hyper-K¨ahler manifolds, this conjecture is proved in Corollary 5.44.
In this section, we prove some basic properties of the K -theoretic canonical bases assuming Conjec-ture 3.22. We first check that B ± L,T / ,s = B L,T / ,s (cid:116) − B L,T / ,s and B ± X,T / ,s = B X,T / ,s (cid:116) − B X,T / ,s .The asymptotic norm one property follows from the following lemma. Lemma 3.24.
For any p, p (cid:48) ∈ X H , we have (cid:0) S C ,T / ,s ( p ) ||S C ,T / ,s ( p (cid:48) ) (cid:1) = δ p,p (cid:48) . Proof . By Lemma 3.15 and Lemma 3.16, we have (cid:0) S C ,T / ,s ( p ) ||S C ,T / ,s ( p (cid:48) ) (cid:1) = (cid:16) S C ,T / ,s ( p ) : D X β K C ,T / ,s ( S C ,T / ,s ( p (cid:48) )) (cid:17) = ( − v ) dim X (cid:0) S C ,T / ,s ( p ) : S − C ,T / ,s ( p (cid:48) ) ∨ (cid:1) = v dim X · (cid:16) S C ,T / ,s ( p ) : S − C ,T / , − s ( p (cid:48) ) (cid:17) = δ p,p (cid:48) . We denote by † : K T (pt) → K T (pt) the involution induced from the inverse for H . Corollary 3.25.
Assume Conjecture 3.22. For any p, p (cid:48) ∈ X H , we have( C C ,T / ,s ( p ) ||C C ,T / ,s ( p (cid:48) )) ∈ δ p,p (cid:48) + v − K H (pt)[ v − ] , ( E C ,T / ,s ( p ) ||E C ,T / ,s ( p (cid:48) )) ∈ δ p,p (cid:48) + v − K H (pt)[[ v − ]] . roof . For the first statement, this follows from Lemma 3.24 since the expansion of C C ,T / ,s ( p ) interms of the standard basis is a finite sum. For the second statement, let us write E C ,T / ,s ( p ) = (cid:88) p (cid:48)(cid:48) ∈ X H f p,p (cid:48)(cid:48) · S C ,T / ,s ( p (cid:48)(cid:48) )for some f p,p (cid:48)(cid:48) ∈ Frac( K T (pt)). The formal construction of E T / ,s ,p implies that if we expand f p,p (cid:48)(cid:48) in v − , we have f p,p (cid:48)(cid:48) ∈ δ p,p (cid:48)(cid:48) + v − Frac( K H (pt))[[ v − ]]. On the other hand, since X S = L S is smooth andproper, we have f p,p (cid:48)(cid:48) ∈ K H (pt)(( v − )). Hence we obtain f p,p (cid:48)(cid:48) ∈ δ p,p (cid:48)(cid:48) + v − K H (pt)[[ v − ]] and( E C ,T / ,s ( p ) ||E C ,T / ,s ( p (cid:48) )) = (cid:88) p (cid:48)(cid:48) f p,p (cid:48)(cid:48) · f † p (cid:48) ,p (cid:48)(cid:48) ∈ δ p,p (cid:48) + v − K H (pt)[[ v − ]]by Lemma 3.24.We denote by ( − ) : K T (pt) → K T (pt) the involution induced from the inverse map for S Corollary 3.26.
Assume Conjecture 3.22. We have B ± L,T / ,s = B L,T / ,s (cid:116) − B L,T / ,s , B ± X,T / ,s = B X,T / ,s (cid:116) − B X,T / ,s . Proof . We only prove the second statement since the proof is the same for the first. By Corollary 3.25,we have B X,T / ,s (cid:116) − B X,T / ,s ⊂ B ± X,T / ,s . Hence we only need to prove the other inclusion.Let E ∈ B ± X,T / ,s . Since B X,T / ,s is a basis of K T ( X ), one can write E = (cid:80) p ∈ X H f p E C ,T / ,s ( p ) forsome f p ∈ K T (pt). By β C ,T / ,s -invariance, we obtain f p = f p . Let N be the maximal degree in v of f p , p ∈ X H , and write f p = (cid:80) Ni = − N f p,i v i for some f p,i ∈ K H (pt). By Corollary 3.25, we obtain( E||E ) = (cid:88) p ∈ X H f p,N f † p,N v N + · · · . By the asymptotic norm one property of E , we have N = 0 and (cid:80) p ∈ X H f p, f † p, = 1. This implies that f p = ± [ λ ] · δ p,p (cid:48) for some λ ∈ X ∗ ( H ) and p (cid:48) ∈ X H , hence E ∈ B X,T / ,s (cid:116) − B X,T / ,s .We note that Corollary 3.26 together with Conjecture 3.17 implies that the K -theoretic canonicalbases B L,T / ,s and B X,T / ,s does not depend on the choice of C up to sign. Recall that under the as-sumption of Conjecture 3.22, we have β K C ,T / ,s = β K − C ,T / ,s and hence the signed K -theoretic canonicalbases are the same for C and − C . In particular, for each ( λ, p ) ∈ F , there exists ( λ − , p − ) ∈ F such that C C ,T / ,s ( λ, p ) = ±C − C ,T / ,s ( λ − , p − ). This implies that C C ,T / ,s ( λ, p ) ∈ ±S − C ,T / ,s ( λ − , p − ) + (cid:88) ( λ (cid:48) ,p (cid:48) ) < C ,s ( λ − ,p − ) v − Z [ v − ] · S − C ,T / ,s ( λ (cid:48) , p (cid:48) ) . By the bar invariance, we also obtain C C ,T / ,s ( λ, p ) ∈ ± ( − v ) − dim X S C ,T / ,s ( λ − , p − ) + (cid:88) ( λ (cid:48) ,p (cid:48) ) < C ,s ( λ − ,p − ) v − dim X +1 Z [ v ] · S C ,T / ,s ( λ (cid:48) , p (cid:48) ) . If we further assume Conjecture 3.23, then the sign above must be positive. This proves the followingstatement. 16 roposition 3.27.
Assume Conjecture 3.22. For each ( λ, p ) ∈ F , there exists ( λ − , p − ) ∈ F such that C C ,T / ,s ( λ, p ) = (cid:88) ( λ,p ) ≤ C ,s ( λ (cid:48) ,p (cid:48) ) ≤ C ,s ( λ − ,p − ) P λ (cid:48) ,p (cid:48) λ,p ( v ) · S C ,T / ,s ( λ (cid:48) , p (cid:48) ) , where P λ,pλ,p ( v ) = 1, P λ − ,p − λ,p ( v ) = ± ( − v ) − dim X , and P λ (cid:48) ,p (cid:48) λ,p ( v ) ∈ v − Z [ v − ] ∩ v − dim X +1 Z [ v ]for ( λ, p ) < C ,s ( λ (cid:48) , p (cid:48) ) < C ,s ( λ − , p − ). If we further assume Conjecture 3.23, then we have P λ − ,p − λ,p ( v ) =( − v ) − dim X .Using this, one can also give a characterization of B L,T / ,s which is similar to Kashiwara’s theoryof global crystal bases. Corollary 3.28.
Assume Conjecture 3.22. We have K T ( L ) ∩ S C ,T / ,s ( λ, p ) + (cid:88) ( λ (cid:48) ,p (cid:48) ) ∈ F (cid:16) v − Z [ v − ] ∩ v − dim X Z [ v ] (cid:17) · S C ,T / ,s ( λ (cid:48) , p (cid:48) ) = (cid:8) C C ,T / ,s ( λ, p ) (cid:9) . Proof . Let C be an element of the LHS. Since B L,T / ,s is a Z [ v, v − ]-basis of K T ( L ), we can write C = (cid:88) ( λ (cid:48) ,p (cid:48) ) ∈ F f λ (cid:48) ,p (cid:48) ( v ) · C C ,T / ,s ( λ (cid:48) , p (cid:48) )for some f λ (cid:48) ,p (cid:48) ( v ) ∈ Z [ v, v − ]. By Proposition 3.27, the maximal and minimal degree of f λ (cid:48) ,p (cid:48) ( v ) must be0. By comparing the constant term of the coefficient of S C ,T / ,s ( λ (cid:48) , p (cid:48) ), we obtain f λ (cid:48) ,p (cid:48) ( v ) = δ ( λ,p ) , ( λ (cid:48) ,p (cid:48) ) and hence C = C C ,T / ,s ( p ).We next compute the pairing of B L,T / ,s and B X,T / ,s . We first list some basic properties ofthe pairing. Recall that † : K T (pt) → K T (pt) is the involution induced from the inverse for H and( − ) : K T (pt) → K T (pt) is the involution induced from the inverse for S . Lemma 3.29.
For any F , G ∈ K T ( X ) loc , we have ( G||F ) = (
F||G ) † . If we denote by ( −||− ) opp theinner product defined by β K C ,T / , − s , then we have ( F ∨ ||G ∨ ) opp = v dim X · ( F||G ) ∨ Proof . We only need to check the formulas for F = S C ,T / ,s ( p ) and G = S C ,T / ,s ( p (cid:48) ). By Lemma 3.24,this is obvious for the first one and the second one follows from Lemma 3.15. Lemma 3.30.
Assume that β K C ,T / ,s is an involution. For any F , G ∈ K T ( X ) loc , we have ( β K ( F ) || β K ( G )) = v dim X ( F||G ). Proof . We only need to check the formula for F = S C ,T / ,s ( p ) and G = S C ,T / ,s ( p (cid:48) ). By the assump-tion and Lemma 3.24 applied to the opposite chamber, we obtain ( S − C ,T / ,s ( p ) ||S − C ,T / ,s ( p (cid:48) )) = δ p,p (cid:48) .Then the statement follows from the definition of β K . Proposition 3.31.
Assume Conjecture 3.22. We have ( C C ,T / ,s ( p ) ||E C ,T / ,s ( p (cid:48) )) = δ p,p (cid:48) for any p, p (cid:48) ∈ X H . Proof . By Conjecture 3.22, ( C C ,T / ,s ( p ) ||E C ,T / ,s ( p (cid:48) )) ∈ K T (pt) since the support of C C ,T / ,s ( p )is proper. By Proposition 3.21 and Lemma 3.24, we obtain ( C C ,T / ,s ( p ) ||E C ,T / ,s ( p (cid:48) )) ∈ δ p,p (cid:48) + v − K H (pt)[ v − ]. By Lemma 3.30, we also obtain ( C C ,T / ,s ( p ) ||E C ,T / ,s ( p (cid:48) )) = ( C C ,T / ,s ( p ) ||E C ,T / ,s ( p (cid:48) )),hence we need to have ( C C ,T / ,s ( p ) ||E C ,T / ,s ( p (cid:48) )) = δ p,p (cid:48) .17f we define the map ∂ : K T (pt) → Z [ v, v − ] by ∂ ( (cid:80) µ ∈ X ∗ ( H ) f µ ( v ) · [ µ ]) = f ( v ), then Proposition 3.31implies that B L,T / ,s and B X,T / ,s forms a dual basis with respect to ∂ ( −||− ). As a corollary, we alsoobtain the following result on the expansion of B std C ,T / ,s in terms of B X,T / ,s . Corollary 3.32.
Assume Conjecture 3.22. For any ( λ (cid:48) , p (cid:48) ) ∈ F , we have S C ,T / ,s ( λ (cid:48) , p (cid:48) ) = (cid:88) ( λ,p ) ≤ C ,s ( λ (cid:48) ,p (cid:48) ) P λ (cid:48) ,p (cid:48) λ,p ( v ) · E C ,T / ,s ( λ, p ) , where P λ (cid:48) ,p (cid:48) λ,p ( v ) is the same as Proposition 3.27 and P λ (cid:48) ,p (cid:48) λ,p ( v ) = 0 if ( λ (cid:48) , p (cid:48) ) (cid:2) C ,s ( λ − , p − ). Proof . This follows from Proposition 3.27 and Proposition 3.31.If we change the polarization or shift the slope, then the K -theoretic canonical bases change asfollows. Lemma 3.33.
Assume Conjecture 3.22. We have C C ,T / G ,s ( λ, p ) = C C ,T / ,s +det G ( λ + wt H det i ∗ p G , p )and E C ,T / G ,s ( λ, p ) = E C ,T / ,s +det G ( λ + wt H det i ∗ p G , p ) for any G ∈ K T ( X ) and ( λ, p ) ∈ F . In particular,we have B L,T / G ,s = B L,T / ,s +det G and B X,T / G ,s = B X,T / ,s +det G . Proof . By Lemma 3.13, we obtain β K C ,T / G ,s = β K C ,T / ,s +det G . Since ( λ (cid:48) , p (cid:48) ) > C ,s ( λ, p ) is equivalent to( λ (cid:48) + wt H det i ∗ p (cid:48) G , p (cid:48) ) > C ,s +det G ( λ + wt H det i ∗ p G , p ) and C C ,T / G ,s ( λ, p ) ∈ S C ,T / G ,s ( λ, p ) + (cid:88) ( λ (cid:48) ,p (cid:48) ) > C ,s ( λ,p ) v − Z [ v − ] · S C ,T / G ,s ( λ (cid:48) , p (cid:48) )= S C ,T / ,s +det G ( λ + wt H det i ∗ p G , p ) + (cid:88) ( λ (cid:48) ,p (cid:48) ) > C ,s ( λ,p ) v − Z [ v − ] · S C ,T / ,s +det G ( λ (cid:48) + wt H det i ∗ p (cid:48) G , p (cid:48) ) , C C ,T / G ,s ( λ, p ) also satisfies the characterizing properties of C C ,T / ,s +det G ( λ +wt H det i ∗ p G , p ). The proofof E C ,T / G ,s ( λ, p ) = E C ,T / ,s +det G ( λ + wt H det i ∗ p G , p ) is similar. Lemma 3.34.
Assume Conjecture 3.22. For any l ∈ P and ( λ, p ) ∈ F , we have C C ,T / ,s + l ( λ, p ) = L ( l ) ⊗C C ,T / ,s ( λ − wt H i ∗ p L ( l ) , p ) and E C ,T / ,s + l ( λ, p ) = L ( l ) ⊗E C ,T / ,s ( λ − wt H i ∗ p L ( l ) , p ). In particular,we have B L,T / ,s + l = L ( l ) ⊗ B L,T / ,s and B X,T / ,s + l = L ( l ) ⊗ B X,T / ,s . Proof . By Lemma 3.14, we obtain β K C ,T / ,s + l = L ( l ) ◦ β K C ,T / ,s ◦ L ( l ) − , where L ( l ) is identified with theautomorphism of K T ( X ) given by the tensor product of L ( l ). Since ( λ (cid:48) , p (cid:48) ) > C ,s + l ( λ, p ) is equivalentto ( λ (cid:48) − wt H i ∗ p (cid:48) L ( l ) , p (cid:48) ) > C ,s ( λ − wt H i ∗ p L ( l ) , p ) and C C ,T / ,s + l ( λ, p ) ∈ S C ,T / ,s + l ( λ, p ) + (cid:88) ( λ (cid:48) ,p (cid:48) ) > C ,s + l ( λ,p ) v − Z [ v − ] · S C ,T / ,s + l ( λ (cid:48) , p (cid:48) )= L ( l ) ⊗ S C ,T / ,s ( λ − wt H i ∗ p L ( l ) , p ) + (cid:88) ( λ (cid:48) ,p (cid:48) ) > C ,s + l ( λ,p ) v − Z [ v − ] · S C ,T / ,s ( λ (cid:48) − wt H i ∗ p (cid:48) L ( l ) , p (cid:48) ) , L ( l ) − ⊗ C C ,T / ,s + l ( λ, p ) also satisfies the characterizing properties of C C ,T / ,s ( λ − wt H i ∗ p L ( l ) , p ). Theproof of E C ,T / ,s + l ( λ, p ) = L ( l ) ⊗ E C ,T / ,s ( λ − wt H i ∗ p L ( l ) , p ) is similar.For the behavior of K -theoretic canonical bases under the duality, we have the following. Lemma 3.35.
Assume Conjecture 3.22 and Conjecture 3.23. For any ( λ, p ) ∈ F , we have v − dim X C C ,T / ,s ( λ, p ) ∨ = C C ,T / , − s ( − λ − , p − ) and E C ,T / ,s ( λ, p ) ∨ = E C ,T / , − s ( − λ − , p − ). Here, ( λ − , p − ) is as in Proposition 3.27.In particular, we have v − dim X · B ∨ L,T / ,s = B L,T / , − s and B ∨ X,T / ,s = B X,T / , − s .18 roof . By Lemma 3.15 and Proposition 3.27, one can check that v − dim X C C ,T / ,s ( λ, p ) ∨ ∈ K T ( L )is contained in S C ,T / , − s ( − λ − , p − ) + (cid:80) ( λ (cid:48) ,p (cid:48) ) ∈ F (cid:16) v − dim X +1 Z [ v ] ∩ v − Z [ v − ] (cid:17) · S C ,T / , − s ( λ (cid:48) , p (cid:48) ). ByCorollary 3.28, this implies v − dim X C C ,T / ,s ( λ, p ) ∨ = C C ,T / , − s ( − λ − , p − ). The second statement fol-lows from the first by Lemma 3.29 and Proposition 3.31.Since the fixed point basis often has some representation theoretic and combinatorial meaning, wealso give a formula for the transition matrix between the fixed point basis and B L,T / ,s . For each p ∈ X H , we denote by O p ∈ K T ( X ) the K -theory class of the skyscraper sheaf at p . Proposition 3.36.
Assume Conjecture 3.22. For any p ∈ X H , we have O p = v dim X · (cid:88) p (cid:48) ∈ X H (cid:0) i ∗ p E C ,T / ,s ( p (cid:48) ) (cid:1) ∨ · C C ,T / ,s ( p (cid:48) ) . Proof . This follows from Proposition 3.31 and( O p ||E C ,T / ,s ( p (cid:48) )) = ( O p : D X E C ,T / ,s ( p (cid:48) )) = v dim X · (cid:0) i ∗ p E C ,T / ,s ( p (cid:48) ) (cid:1) ∨ . Remark 3.37.
For example, when X is the Hilbert scheme of points in the affine plane and the slopeis sufficiently close to 1, it turns out that E C ,T / ,s ( p )’s are given by the indecomposable summandsof the Procesi bundle. If we identify K T ( L ) and the space of symmetric functions as in [17], then C C ,T / ,s ( p )’s corresponds to the Schur functions and O p ’s corresponds to the modified Macdonaldpolynomials. Transition matrix of these bases are given by the q, t -Kostka polynomials which are givenby the characters of fibers of indecomposable summands of the Procesi bundle. In this section, we state several conjectures about the categorical meaning of canonical and standardbases. We assume all the conjectures and assumptions stated in the previous sections without anycomment.We first recall the notion of tilting bundle on X . Let T be a vector bundle on X . We say that T isa tilting bundle on X if it satisfies • T is a weak generator for D (QCoh( X )), i.e., R Hom( T , F ) = 0 implies F ∼ = 0 for
F ∈ D (QCoh( X )). • Ext i ( T , T ) = 0 for i (cid:54) = 0.When X is a Slodowy variety, Bezrukavnikov-Mirkovi´c [6] proved that there exists an T -equivarianttilting bundle on X such that Lusztig’s K -theoretic canonical basis for X consists of indecomposablesummands of the tilting bundle up to equivariant shifts. Moreover, the structure sheaf of X is containedin the canonical basis. We also expect in general that B X,T / ,s is given by the classes of indecomposablesummands of some tilting bundle on X . Conjecture 3.38.
For each p ∈ X H , there exists an T -equivariant vector bundle lifting E C ,T / ,s ( p )(denoted by the same letter) such that T C ,T / ,s := ⊕ p ∈ X H E C ,T / ,s ( p ) is a tilting bundle on X . Moreover,at least one of E C ,T / ,s ( p ) is a line bundle.For toric hyper-K¨ahler manifolds, this conjecture is proved in Corollary 5.36. In particular, thisconjecture implies that B X,T / ,s consists of K -theory classes of actual vector bundles on X and henceour choice of the sign is geometrically natural. Corollary 3.39.
If we assume Conjecture 3.38, then the ring A C ,T / ,s := End( T C ,T / ,s ) opp is non-negatively graded with respect to the S -action and its degree 0 part is semisimple.19 roof . By the definition of tilting bundle, we have( E C ,T / ,s ( p ) ||E C ,T / ,s ( p (cid:48) )) = [ R Γ( E C ,T / ,s ( p ) ⊗ L D X E C ,T / ,s ( p (cid:48) ))]= [ R Γ( D X R H om ( E C ,T / ,s ( p ) , E C ,T / ,s ( p (cid:48) )))]= [Hom( E C ,T / ,s ( p ) , E C ,T / ,s ( p (cid:48) )) ∨ ]as rational functions in equivariant parameters. Since the last term is also contained in K H (pt)(( v − )),we obtain [Hom( E C ,T / ,s ( p ) , E C ,T / ,s ( p (cid:48) ))] ∈ δ p,p (cid:48) + vK H (pt)[[ v ]] by Lemma 3.25. This implies that itis non-negatively graded with respect to S and the degree zero part is semisimple.By Kaledin’s argument in [6] section 5.5, this also implies that A C ,T / ,s is Koszul. We denoteby B C ,T / ,s its Koszul dual. We note that A C ,T / ,s and B C ,T / ,s has a natural H -action. By theindependence of B X,T / ,s on C , A C ,T / ,s and B C ,T / ,s does not depend on the choice of C if we forgetthe H -actions. Since X is smooth, A C ,T / ,s has finite global dimension and hence B C ,T / ,s is finitedimensional.By the standard properties of tilting bundle, there is a derived equivalence ψ C ,T / ,s : D b Coh T ( X ) ∼ = D b ( A C ,T / ,s -gmod H ) (18)given by the functor R Hom( T C ,T / ,s , − ). Here, we denote by A C ,T / ,s -gmod H the category of finitelygenerated graded H -equivariant modules of A C ,T / ,s . We denote the t -structure on D b Coh T ( X ) cor-responding to the standard t -structure on D b ( A C ,T / ,s -gmod H ) by τ T / ,s . By construction, F ∈ D b Coh T ( X ) is contained in the heart if and only if R (cid:54) =0 Hom( T C ,T / ,s , F ) = 0.For ( λ, p ) ∈ F , we will write E C ,T / ,s ( λ, p ) := [ λ ] · E C ,T / ,s ( p ) as in the K -theoretic one. Each ψ C ,T / ,s ( E C ,T / ,s ( λ, p )) defines a graded H -equivariant indecomposable projective module of A C ,T / ,s denoted by P A C ,T / ,s ( λ, p ). It has a unique one-dimensional simple quotient denoted by L A C ,T / ,s ( λ, p ).We have ∂ ( ψ − C ,T / ,s ( L A C ,T / ,s ( λ, p )) ||E C ,T / ,s ( λ (cid:48) , p (cid:48) )) = (cid:104) R Hom( E C ,T / ,s ( λ (cid:48) , p (cid:48) ) , v dim X ψ − C ,T / ,s ( L A C ,T / ,s ( λ, p ))[dim X ]) H (cid:105) = v dim X · δ ( λ,p ) , ( λ (cid:48) ,p (cid:48) ) . Hence the K -theory class of v − dim X · ψ − C ,T / ,s ( L A C ,T / ,s ( λ, p )) coincides with C C ,T / ,s ( λ, p ) by Propo-sition 3.31. Using this lift, we sometimes regard K -theoretic canonical bases of K T ( L ) as objects in D b Coh T ( X ).By the Koszul duality (cf. [4, 26]), we also obtain the following derived equivalence K : D b ( A C ,T / ,s -gmod H ) ∼ = D b ( B C ,T / ,s -gmod H ) . (19)We note that since B C ,T / ,s is finite dimensional, the Koszul duality equivalence preserves the bound-edness by [4, Theorem 2.12.6]. The standard t -structure on D b ( B C ,T / ,s -gmod H ) induces a t -structureon D b ( A C ,T / ,s -gmod H ) and its heart consists of linear complex of projective modules, that is, objectquasi-isomorphic to a complex of the form0 → v N P N → v N − P N − → · · · → v M P M → , where each P i sits in the ( − i )-th term and it is a direct sum of projective modules of the form P A C ,T / ,s ( λ, p ). By the construction of K , we have K ◦ v [1] = v − ◦ K and L B C ,T / ,s ( λ, p ) := K ( P A C ,T / ,s ( λ, p )) is a graded H -equivariant one-dimensional simple module of B C ,T / ,s . We note thatany simple object in B C ,T / ,s -gmod H is of the form v m L B C ,T / ,s ( λ, p ) for some m ∈ Z and ( λ, p ) ∈ F .Since A C ,T / ,s is Koszul, L A C ,T / ,s ( λ, p ) is quasi-isomorphic to a linear complex of projective modulesand hence we have I B C ,T / ,s ( λ, p ) := K ( L A C ,T / ,s ( λ, p )) ∈ B C ,T / ,s -gmod H . This is the injective hull of L B C ,T / ,s ( λ, p ).For a categorical lift of standard basis, we conjecture the following.20 onjecture 3.40. For each ( λ, p ) ∈ X H , there exists an object ∇ B C ,T / ,s ( λ, p ) ∈ B C ,T / ,s -gmod H suchthat ∆ A C ,T / ,s ( λ, p ) := K − ( ∇ B C ,T / ,s ( λ, p )) is contained in the standard heart of D b ( A C ,T / ,s -gmod H )and the K -theory class of ∇ C ,T / ,s ( λ, p ) := ψ − C ,T / ,s (∆ A C ,T / ,s ( λ, p )) ∈ D b Coh T ( X ) coincides with β K C ,T / ,s ( S C ,T / ,s ( λ, p )). If we set ∆ C ,T / ,s ( λ, p ) := v − dim X ∇ − C ,T / ,s ( λ, p )[ − dim X ] ∈ D b Coh T ( X ),then we haveHom D b Coh T ( X ) (cid:0) v j ∆ C ,T / ,s ( λ, p ) , ∇ C ,T / ,s ( λ (cid:48) , p (cid:48) )[ i ] (cid:1) = (cid:40) C if i = j = 0 and ( λ, p ) = ( λ (cid:48) , p (cid:48) ) , . (20)For toric hyper-K¨ahler manifolds, this conjecture is proved in Theorem 5.45. We note that the K -theory class of ∆ C ,T / ,s ( λ, p ) coincides with S C ,T / ,s ( λ, p ) and the equation (20) lifts the orthonormalityof K -theoretic standard bases in Lemma 3.24. We expect that ∇ C ,T / ,s ( λ, p ) and ∆ C ,T / ,s ( λ, p ) areessentially given by the theory of categorical stable envelope (c.f. [33]). Since ∆ A C ,T / ,s ( λ, p ) is a Koszulmodule of A C ,T / ,s , we have a resolution of the form0 → v dim X P dim X → · · · → vP → P A C ,T / ,s ( λ, p ) → ∆ A C ,T / ,s ( λ, p ) → A C ,T / ,s -gmod H by Corollary 3.32, where each P i ( i = 1 , . . . , dim X ) is a direct sum of projectivemodules of the form P A C ,T / ,s ( λ (cid:48) , p (cid:48) ) satisfying ( λ (cid:48) , p (cid:48) ) < C ,s ( λ, p ). By the construction of K , it followsthat ∇ B C ,T / ,s ( λ, p ) is non-positively graded and we have an inclusion L B C ,T / ,s ( λ, p ) (cid:44) → ∇ B C ,T / ,s ( λ, p )such that any composition factor v j L B C ,T / ,s ( λ (cid:48) , p (cid:48) ) of the quotient ∇ B C ,T / ,s ( λ, p ) /L B C ,T / ,s ( λ, p ) satis-fies j < λ (cid:48) , p (cid:48) ) < C ,s ( λ, p ).Since the shift v [1] preserves the linear complex of projective modules, ψ C ,T / ,s (∆ C ,T / ,s ( λ, p )) isquasi-isomorphic to a complex of the form0 → P A C ,T / ,s ( λ, p ) → v − P → · · · → v − dim X P dim X → P A C ,T / ,s ( λ, p ) sits in degree 0 and each P i is the same as (21). If we write ∆ B C ,T / ,s ( λ, p ) := K ( ψ C ,T / ,s (∆ C ,T / ,s ( λ, p ))) ∈ B C ,T / ,s -gmod H , then this is non-negatively graded and we have aprojection ∆ B C ,T / ,s ( λ, p ) (cid:16) L B C ,T / ,s ( λ, p ) such that any composition factor v j L B C ,T / ,s ( λ (cid:48) , p (cid:48) ) of thekernel satisfies j > λ (cid:48) , p (cid:48) ) < C ,s ( λ, p ). By (20), we obtainExt i B C ,T / ,s - gmod H (cid:16) v j ∆ B C ,T / ,s ( λ, p ) , ∇ B C ,T / ,s ( λ (cid:48) , p (cid:48) ) (cid:17) = (cid:40) C if i = j = 0 and ( λ, p ) = ( λ (cid:48) , p (cid:48) )0 otherwise . (22)Next we show that B C ,T / ,s -gmod H has a structure of graded highest weight category assumingConjecture 3.40. We first recall the definition of graded highest weight category following [10, 11].Let C be a C -linear abelian category with a free Z -action which is locally Artinian, contains enoughinjectives, and satisfies Grothendieck’s condition AB5. The action of j ∈ Z on an object M ∈ C is denoted by M (cid:55)→ M (cid:104) j (cid:105) . For each M, N ∈ C , we set hom C ( M, N ) := ⊕ j ∈ Z Hom C ( M (cid:104) j (cid:105) , N ) andext i C ( M, N ) := ⊕ j ∈ Z Ext i C ( M (cid:104) j (cid:105) , N ). For a simple object L , we denote by [ M : L ] the multiplicity of L in the composition series of M . Definition 3.41 ([10, 11]) . A category C as above is called graded highest weight category if thereexists an interval finite poset Λ satisfying the following conditions: • For each λ ∈ Λ, we have a simple object L ( λ ) such that { L ( λ ) (cid:104) j (cid:105)} j ∈ Z ,λ ∈ Λ is a complete set ofnon-isomorphic simple objects of C . • For each λ ∈ Λ, there is an object ∇ ( λ ) (called costandard object ) with an inclusion L ( λ ) (cid:44) → ∇ ( λ )such that any composition factor L ( µ ) (cid:104) j (cid:105) of the quotient ∇ ( λ ) /L ( λ ) satisfies j < µ < λ .Moreover, for each λ, µ ∈ Λ, dim C hom C ( ∇ ( λ ) , ∇ ( µ )) and (cid:80) j ∈ Z [ ∇ ( λ ) : L ( µ ) (cid:104) j (cid:105) ] are finite.21 For each λ ∈ Λ, injective hull I ( λ ) of L ( λ ) has an increasing filtration 0 = F ( λ ) ⊂ F ( λ ) ⊂ · · · with ∪ i F i ( λ ) = I ( λ ) such that F ( λ ) ∼ = ∇ ( λ ) and F i ( λ ) /F i − ( λ ) ∼ = ∇ ( µ i ) (cid:104) j (cid:105) for some j < µ i > λ if i >
1. Moreover, the set { i | µ i = µ } is finite for any µ ∈ Λ. Proposition 3.42.
Assume Conjecture 3.40. Then the category B C ,T / ,s -gmod H has a structureof graded highest weight category with poset ( F , ≤ C ,s ) and the costandard object parametrized by( λ, p ) ∈ F is given by ∇ B C ,T / ,s ( λ, p ). Proof . Since B C ,T / ,s is finite dimensional over C , the category B C ,T / ,s -gmod H is Artinian. The free Z -action on B C ,T / ,s -gmod H is given by the grading shifts M (cid:104) j (cid:105) := v j M . The poset F is interval finiteby Lemma 3.19. For each ( λ, p ) ∈ F , we associate the simple module L B C ,T / ,s ( λ, p ). Then the first twocondition in Definition 3.41 has been already checked above.On the level of K -theory, we have[ I B C ,T / ,s ( λ, p )] = (cid:88) ( λ,p ) ≤ C ,s ( λ (cid:48) ,p (cid:48) ) P λ (cid:48) ,p (cid:48) λ,p ( − v ) · [ ∇ B C ,T / ,s ( λ (cid:48) , p (cid:48) )]by Proposition 3.27. Since P λ,pλ,p ( − v ) = 1 and P λ (cid:48) ,p (cid:48) λ,p ( − v ) ∈ v − Z [ v − ] if ( λ (cid:48) , p (cid:48) ) (cid:54) = ( λ, p ), it is enough toprove that I B C ,T / ,s ( λ, p ) has a costandard filtration with F ( λ, p ) = ∇ B C ,T / ,s ( λ, p ). This follows fromthe following lemma and its proof. Lemma 3.43.
An object M ∈ B C ,T / ,s -gmod H has a costandard filtration if and only ifext B C ,T / ,s - gmod H (cid:16) ∆ B C ,T / ,s ( λ, p ) , M (cid:17) = 0 (23)for any ( λ, p ) ∈ F . Proof . The only if part follows from (22). Let M be an object satisfying (23). Since M has finitelength, we may prove the statement by induction on the length of M . If M (cid:54) = 0, then one can takea minimal ( λ, p ) ∈ F such that hom (cid:16) L B C ,T / ,s ( λ, p ) , M (cid:17) (cid:54) = 0. For any ( λ (cid:48) , p (cid:48) ) < C ,s ( λ, p ), let K bethe kernel of ∆ B C ,T / ,s ( λ (cid:48) , p (cid:48) ) (cid:16) L B C ,T / ,s ( λ (cid:48) , p (cid:48) ). By the discussion above, any composition factor v j L B C ,T / ,s ( λ (cid:48)(cid:48) , p (cid:48)(cid:48) ) of K satisfies ( λ (cid:48)(cid:48) , p (cid:48)(cid:48) ) < C ,s ( λ (cid:48) , p (cid:48) ) < C ,s ( λ, p ) and hence we have hom( K, M ) = 0.By the exact sequencehom(
K, M ) → ext (cid:16) L B C ,T / ,s ( λ (cid:48) , p (cid:48) ) , M (cid:17) → ext (cid:16) ∆ B C ,T / ,s ( λ (cid:48) , p (cid:48) ) , M (cid:17) we obtain ext (cid:16) L B C ,T / ,s ( λ (cid:48) , p (cid:48) ) , M (cid:17) = 0 for any ( λ (cid:48) , p (cid:48) ) < C ,s ( λ, p ). Since the composition factorsof ∇ B C ,T / ,s ( λ, p ) /L B C ,T / ,s ( λ, p ) are of the form v j L B C ,T / ,s ( λ (cid:48) , p (cid:48) ) for ( λ (cid:48) , p (cid:48) ) < C ,s ( λ, p ), we obtainext (cid:16) ∇ B C ,T / ,s ( λ, p ) /L B C ,T / ,s ( λ, p ) , M (cid:17) = 0 and hom (cid:16) ∇ B C ,T / ,s ( λ, p ) /L B C ,T / ,s ( λ, p ) , M (cid:17) = 0. Thisimplies that hom (cid:16) ∇ B C ,T / ,s ( λ, p ) , M (cid:17) ∼ = hom (cid:16) L B C ,T / ,s ( λ, p ) , M (cid:17) , hence we can lift the inclusion L B C ,T / ,s ( λ, p ) (cid:44) → M to a homomorphism f : ∇ B C ,T / ,s ( λ, p ) → M .We claim that f is injective. Otherwise, there is a simple submodule v j L B C ,T / ,s ( λ (cid:48) , p (cid:48) ) in Ker( f ) ⊂∇ B C ,T / ,s ( λ, p ). Then there is a nontrivial homomorphism between v j ∆ B C ,T / ,s ( λ (cid:48) , p (cid:48) ) and ∇ B C ,T / ,s ( λ, p ),hence we must have j = 0 and ( λ (cid:48) , p (cid:48) ) = ( λ, p ) by (22). This implies f ( L B C ,T / ,s ( λ, p )) = 0 which con-tradicts the choice of f .Therefore, we obtain an inclusion ∇ B C ,T / ,s ( λ, p ) (cid:44) → M . Let N be the cokernel of this inclusion.Then N also satisfies the condition (23) by (22) and the length of N is smaller than M . By inductionhypothesis, N has a costandard filtration and hence M does.Finally, we also conjecture the following statement which lifts the K -theoretic bar involutions to thederived category. Let B C ,T / ,s = ⊕ λ ∈ X ∗ ( H ) B λ C ,T / ,s be the H -weight space decomposition.22 onjecture 3.44. There exists an anti-involution ι on B C ,T / ,s which is identity on degree 0 part,compatible with the grading, and satisfies ι ( B λ C ,T / ,s ) = B − λ C ,T / ,s for any λ ∈ X ∗ ( H ).For toric hyper-K¨ahler manifolds, this conjecture is proved in Corollary 5.40. Let M = ⊕ i ∈ Z ,λ ∈ X ∗ ( H ) M λi be a graded H -equivariant module of B C ,T / ,s , where M λi has degree i and H -weight λ . We de-fine D M = ⊕ i ∈ Z ,λ ∈ X ∗ ( H ) ( D M ) λi ∈ B C ,T / ,s -gmod H by setting ( D M ) λi := Hom C ( M λ − i , C ) equippedwith a left B C ,T / ,s -module structure through ι . The properties of ι in Conjecture 3.44 implies that D L B C ,T / ,s ( λ, p ) ∼ = L B C ,T / ,s ( λ, p ). In particular, D induces an involution on D b Coh T ( X ) which lifts the K -theoretic bar involution. By varying the slope parameter, we obtain a family of t -structures τ T / ,s on D := D b Coh L ( X ), thederived category of coherent sheaves on X set-theoretically supported in L . It may be natural toexpect that this is part of a data defining real variations of stability condition in the sense of Anno-Bezrukavnikov-Mirkovi´c [2]. We first recall the definition of real variations of stability conditions in oursituation.Let Alc K be the set of connected components of P R \ ∪ β ∈ Ψ { s ∈ P R | (cid:104) s, β (cid:105) ∈ Z } . We call an elementof Alc K K¨ahler alcove. For two K¨ahler alcoves A − (cid:54) = A + ∈ Alc K sharing a codimension one facecontained in w β,n := { s ∈ P R | (cid:104) s, β (cid:105) = m } for some m ∈ Z and β ∈ Ψ which is positive with respectto A , we say that A + is above A − if A − ⊂ { s ∈ P R | (cid:104) s, β (cid:105) < m } and A + ⊂ { s ∈ P R | (cid:104) s, β (cid:105) > m } .Let Z : P R → Hom Z ( K ( D ) , R ) be a polynomial map and τ be a map from Alc to the set of bounded t -structures on D . For A ∈ Alc K , let C A be the heart of the t -structure τ ( A ) on D . For a hyperplane w ⊂ P R and n ∈ Z ≥ , let C n A ,w ⊂ C A be the full subcategory consisting of objects M ∈ C A such thatthe polynomial function on P R defined by s (cid:55)→ (cid:104) Z ( s ) , [ M ] (cid:105) has zero of order at least n on w . This isa Serre subcategory of C A and let D n A ,w := {F ∈ D | ∀ j, H jτ ( A ) ( F ) ∈ C n A ,w } be a thick subcategoryof D . Here, H jτ ( A ) is the j -th cohomology functor with respect to the t -structure τ ( A ). We setgr n A ,w ( D ) := D n A ,w / D n +1 A ,w and gr nw ( C A ) := C n A ,w / C n +1 A ,w . Definition 3.45 ([2]) . A data ( Z , τ ) as above is called real variation of stability conditions on D if itsatisfies the following conditions: • For any A ∈ Alc K and nonzero M ∈ C A , we have (cid:104) Z ( s ) , [ M ] (cid:105) > s ∈ A . • For any A − (cid:54) = A + ∈ Alc K sharing a codimension one face contained in a hyperplane w with A + being above A − , we have D n A − ,w = D n A + ,w and gr nw ( C A − ) = gr nw ( C A + )[ n ] in gr n A − ,w ( D ) =gr n A + ,w ( D ) for any n ∈ Z ≥ . The polynomial function Z is called central charge for the real variation of stability conditions. Remark 3.46.
A part of the conjecture of Bezrukavnikov-Okounkov stated in [2] claims that thereexists a real variation of stability conditions on D as above (we do not need to assume that the fixed pointset X H is finite). Moreover, the t -structures τ are given by quantization of X in positive characteristic.Let (cid:96) be a prime number and consider our conical symplectic resolutions over an algebraically closedfield of characteristic (cid:96) temporarily. For λ ∈ P , one can consider Frobenius constant quantization of O X with quantization parameter λ which gives a sheaf of Azumaya algebras A λ on the Frobenius twist X (1) of X . Then the conjecture says that when (cid:96) is sufficiently large and − λ(cid:96) ∈ A , the Azumaya algebra A λ splits on the formal neighborhood of L (1) and if the splitting bundles are chosen in a compatible way,their S -equivariant lifts to X ∼ = X (1) gives a tilting bundle and its dual gives a t -structure compatiblewith τ ( A ) under base change to positive characteristic. See also [43] for another approach to this tiltingbundle. We change the sign here from [2] since we mainly use the slope parameters, which are essentially opposite to thequantization parameters.
23s an analogue of Lusztig’s conjecture on modular representation theory, we expect that the set ofvector bundles {E C ,T / ,s ( p ) } p ∈ X H (considered in positive characteristic) give the set of indecomposablesummands of the dual of a splitting bundle of A λ up to equivariant parameter twists when s = − λ(cid:96) and (cid:96) is sufficiently large. We note that if we change the polarization T / by T / G for some G ∈ K T ( X ),then the K -theoretic canonical bases will change by tensor product of some line bundle by Lemma 3.33and Lemma 3.34. Since a line bundle twist of a splitting bundle is also a splitting bundle, this changecan be absorbed into the choice of splitting bundles.Let P be a vector bundle on X and Z P : P R → Hom Z ( K ( D ) , R ) be a polynomial function satisfying (cid:104) Z P ( L ) , F(cid:105) = 1rk
P · χ ( X, F ⊗ P ⊗ L − )for any line bundle L ∈ P and F ∈ D . We note that Z P ⊕ n = Z P for any n ∈ Z > . In this paper, weonly conjecture the following weaker statement. We will check this for toric hyper-K¨ahler manifolds inCorollary 5.49. Conjecture 3.47.
There exists a vector bundle P on X such that ( Z P , τ ) gives a real variation ofstability conditions on D , where τ is given by τ ( A ) = τ T / ,s for s ∈ A . Remark 3.48.
The vector bundle P should be a line bundle when X is a Slodowy variety by the resultof [2], but it should be a vector bundle of higher rank in general. This vector bundle is expected tobe given by looking at the asymptotic behavior under (cid:96) → ∞ of the multiplicity of indecomposablesummands of the splitting bundle above for a fixed λ . More precisely, fix a generic λ and let m p ( (cid:96) )be the multiplicity of E C ,T / , − λ(cid:96) ( p ) ∨ in the splitting bundle of A λ (as non-equivariant vector bundles).We set m p := lim (cid:96) →∞ (cid:96) dim X/ m p ( (cid:96) ) and take m ∈ Z > such that m · m p is an integer for any p ∈ X H .Let A ∈ Alc K be the alcove containing − λ(cid:96) for any sufficiently large (cid:96) and take s ∈ A . Then we expectthat one can take P = (cid:88) p ∈ X H m · m p · E C ,T / ,s ( p ) ∨ . We note that the central charge Z P does not depend on the choice of m . We also expect that this doesnot depend on the choice of λ if we forget the equivariant structures.We now describe the behavior of K -theoretic canonical bases under the wall-crossing of the slopeparameters. As in the previous section, we expect that the information on the equivariant parameter v has some information on the cohomological shifts appearing in Definition 3.45. The following conjecturecomes from numerical experiments. Conjecture 3.49.
Let A − (cid:54) = A + ∈ Alc K be two K¨ahler alcoves sharing a codimension one facecontained in a hyperplane w with A + being above A − . For s − ∈ A − and s + ∈ A + , there existsa sequence of integers 0 ≤ n < n < · · · < n l and decompositions B X,T / ,s − = (cid:116) li =0 B is − ,w and B X,T / ,s + = (cid:116) li =0 B is + ,w stable under equivariant parameter shifts for H such that for any E (cid:48) ∈ B is + ,w ,there exists E ∈ B is − ,w satisfying E (cid:48) = ( − n i − n − i v n i E + (cid:88) j , we define a line bundle O ( D ) on S r E , the r -th symmetric product of E , by the factors ofautomorphy of the symmetric function ( x , . . . , x r ) (cid:55)→ (cid:81) ri =1 ϑ ( x i ).Let V be a T -equivariant vector bundle on X . Its characteristic classes give us a morphism c V : Ell T ( X ) → S r E and Θ( V ) := c ∗ V O ( D ) ∈ Pic(
Ell T ( X )) is called the Thoms class of V . Inparticular, by considering the characteristic classes of line bundles, we obtain a morphism Ell T ( X ) → Hom Z (Pic T ( X ) , E ) ∼ = ( E T P ) ∨ and hence (cid:101) E ( X ) → E T P × ( E T P ) ∨ . We denote by U X the line bundle on (cid:101) E ( X ) defined by pulling back the Poincar´e line bundle on E T P × ( E T P ) ∨ .For λ ∈ Pic T ( X ) ∼ = Hom( E, E T P ) and µ ∈ X ∗ ( T ) ∼ = Hom( E T , E ), let τ ( λ, µ ) : (cid:101) B X → (cid:101) B X be the shiftof K¨ahler parameters ( t, z ) (cid:55)→ ( t, z + λ ( µ ( t ))), where t ∈ E T and z ∈ E T P . We denote by the same letterfor the shift of K¨ahler parameters on (cid:101) E ( X ).For each fixed point p ∈ X H , we obtain a natural morphism i p : (cid:101) B X ∼ = Ell T ( p ) × E T P → (cid:101) E ( X ) comingfrom the inclusion i p : { p } (cid:44) → X and the functoriality of elliptic cohomology. We set U p := i ∗ p U X . Recallthat we take a chamber C and a polarization T / . Take a sufficiently generic ξ ∈ C and decompose T / p = ind p +ind − p + T / p, =0 into attracting, repelling, and fixed parts with respect to ξ , where we assumethat T / p, =0 coincides with the H -fixed part of T / p . We note that this decomposition might depends onthe choice of ξ , but the definition of elliptic stable basis does not depend on this choice. Definition 4.1 ([1]) . For each p ∈ X H , the elliptic stable basis Stab AO C ,T / ( p ) is a section of some linebundle on (cid:101) E ( X ) characterized by the following conditions: • Stab AO C ,T / ( p ) is a section of U X ⊗ Θ( T / ) ⊗ (cid:101) π ∗ ( τ (det ind p , v − ) ∗ U − p ⊗ Θ( T / p, =0 ) − ) ⊗ . . . , where . . . is a certain line bundle pulled back from (cid:101) B X /E H and the section is allowed to be meromorphicon this factor. Here, E H acts on (cid:101) B X by the translation on the factor E T . • The support of Stab AO C ,T / ( p ) is contained in (cid:116) p (cid:48) (cid:22) C p Attr C ( p (cid:48) ). • We have i ∗ p Stab AO C ,T / ( p ) = ϑ ( N p, − ) ∈ Γ( (cid:101) B X , Θ( N p, − )), where ϑ ( N p, − ) = (cid:81) i ϑ ( w i ) if we write N p, − = (cid:80) i [ w i ] ∈ K T ( p ), w i ∈ X ∗ ( T ).By [1], this is unique if it exists and the existence is proved for the case where X is a toric hyper-K¨ahler manifold or a quiver variety. We assume the existence for the conical symplectic resolutions weconsider in this paper. Moreover, this is constant on the E ∨ T -orbits, hence defines a section of some linebundle on E ( X ) which is also denoted by Stab AO C ,T / ( p ).As in the case of K -theory, it might be better to change the normalization of the elliptic stablebases slightly for our purpose. Recall that we always assume the existence of dual conical symplecticresolution X ! = ( X ! , C ! , A ! , . . . ) for X = ( X, C , A , . . . ). Definition 4.2.
For each p ∈ X H , we define the elliptic standard basis Stab ellX ( p ) byStab ellX ( p ) = ϑ ( N ! p ! , − ) · τ (det T / , v ) ∗ (Stab AO C ,T / ( p )) . Next we describe the line bundle of which Stab ell C ( p ) defines a section. More precisely, we givea formula for the factors of automorphy of the restriction S X,p (cid:48) ,p = S p (cid:48) ,p := i ∗ p (cid:48) Stab ellX ( p ) for every p, p (cid:48) ∈ X H . We will consider S p (cid:48) ,p as a multivalued meromorphic function on B KX := ( X ∗ ( T ) × P ) ⊗ Z C × .As in section 3.1, we take a basis { a , . . . , a e , v } of X ∗ ( T ) which will be considered as a system ofcoordinates on X ∗ ( T ) ⊗ Z C × . Similarly, we take a basis { z , . . . , z r } of P ∨ which will be considered26s a system of coordinates on P ⊗ Z C × . For each γ ∈ X ∗ ( T ) × P , we denote by θ X,p ( γ ) the factorof automorphy of the function ϑ ( N p, − ) under the translation by q γ := γ ⊗ q ∈ B KX . We also set θ X,p (cid:48) ,p ( γ ) = θ p (cid:48) ,p ( γ ) := θ X,p (cid:48) ( γ ) · θ X ! ,p ! ( γ ). Assuming the existence of the dual pair in the sense ofDefinition 3.2, we prove the following. Proposition 4.3.
In the above situation, S p (cid:48) ,p satisfies the following: • For each c ∈ X ∗ ( H ), we have S p (cid:48) ,p ( a (cid:55)→ q c a ) = i ∗ p ! L ! ( c ) − · i ∗ p (cid:48) ! L ! ( c ) · θ p (cid:48) ,p ( c ) · S p (cid:48) ,p . (24) • For each l ∈ P , we have S p (cid:48) ,p ( z (cid:55)→ q l z ) = i ∗ p L ( l ) · i ∗ p (cid:48) L ( l ) − · θ p (cid:48) ,p ( l ) · S p (cid:48) ,p . (25) • For δ ∈ X ∗ ( S ) satisfying (cid:104) δ, v (cid:105) = 1, we have S p (cid:48) ,p ( v (cid:55)→ qv ) = det N p, − det N p (cid:48) , − det N ! p (cid:48) ! , − det N ! p ! , − · q wt S det Np, − det Np (cid:48) , − · θ p (cid:48) ,p ( δ ) · S p (cid:48) ,p . (26) • (cid:113) det N p (cid:48) , − · det N ! p ! , −− · S p (cid:48) ,p is single valued on B KX . Proof . By Proposition 3.1 in [1] and T / p (cid:48) = N p (cid:48) , − + ind p (cid:48) − v − ind ∨ p (cid:48) + T / p (cid:48) , =0 , S p (cid:48) ,p is a meromorphicsection of the line bundle τ (det T / , v ) ∗ U p (cid:48) τ (det T / · det ind − p , v ) ∗ U p · Θ(ind p (cid:48) )Θ( v ) rk ind p (cid:48) Θ( v − · ind ∨ p (cid:48) ) · Θ( T / p (cid:48) , =0 )Θ( v ) − rk ind p (cid:48) Θ( T / p, =0 )Θ( v ) − rk ind p · Θ( N p (cid:48) , − )Θ( N ! p ! , − ) . (27)In particular, this implies the single valuedness of (cid:113) det N p (cid:48) , − · det N ! p ! , −− · S p (cid:48) ,p . We now describe thefactors of automorphy for each factors in (27).Let { l , . . . , l r } ⊂ P be the dual basis of { z , . . . , z r } . We set L i := L ( l i ) ∈ Pic T ( X ). Then { L , . . . , L r , a , . . . , a e , v } is a basis of Pic T ( X ). Let { z , . . . , z r , z a , . . . , z a e , z v } ⊂ Pic T ( X ) ∨ be its dualbasis. By definition, the line bundle U p on (cid:101) B X is characterized by the factors of automorphy of thefunction r (cid:89) i =1 ψ ( i ∗ p L i , z i ) · e (cid:89) i =1 ψ ( a i , z a i ) · ψ ( v, z v )for each p ∈ X H . Therefore, for L = (cid:81) i L n i i · (cid:81) i a n ai i · v n v ∈ Pic T ( X ), the line bundle τ ( L , v ) ∗ U p ischaracterized by the factor of automorphy of the function r (cid:89) i =1 ψ ( i ∗ p L i , z i v n i ) · e (cid:89) i =1 ψ ( a i , z a i v n ai ) · ψ ( v, z v v n v ) . By using the formula ψ ( q m x, q n y ) = q − mn x − n y − m ψ ( x, y ) , one can check that the factors of automorphy for τ ( L , v ) ∗ U p are given as follows: • For a (cid:55)→ q c a , it is given by r (cid:89) i =1 z −(cid:104) i ∗ p L i ,c (cid:105) i · e (cid:89) i =1 z −(cid:104) a i ,c (cid:105) a i · v −(cid:104) i ∗ p L ,c (cid:105) ;27 For z (cid:55)→ q l z , it is given by i ∗ p L ( l ) − ; • For v (cid:55)→ qv , it is given by z − v r (cid:89) i =1 z − wt S i ∗ p L i i · ( qv ) − wt S i ∗ p L · i ∗ p L − . Hence the factors of automorphy for the first factor in (27) is given as follows: • For a (cid:55)→ q c a , it is given by r (cid:89) i =1 z (cid:104) i ∗ p L i ,c (cid:105)−(cid:104) i ∗ p (cid:48) L i ,c (cid:105) i · v (cid:104) det T / p ,c (cid:105)−(cid:104) det T / p (cid:48) ,c (cid:105)− (cid:104) det ind p ,c (cid:105) ; • For z (cid:55)→ q l z , it is given by i ∗ p L ( l ) · i ∗ p (cid:48) L ( l ) − ; • For v (cid:55)→ qv , it is given by r (cid:89) i =1 z wt S i ∗ p L i − wt S i ∗ p (cid:48) L i i · ( qv ) wt S det T / p − wt S det T / p (cid:48) − S det ind p · det T / p det T / p (cid:48) · (det ind p ) − . Since the second and the third factor in (27) does not depend on the K¨ahler parameters, the equation(25) follows.One can also check that the factor of automorphy for the second factor in (27) is given as follows: • For a (cid:55)→ q c a , it is given by v (cid:104) det ind p (cid:48) ,c (cid:105) ; • For v (cid:55)→ qv , it is given by ( qv ) S det ind p (cid:48) · (det ind p (cid:48) ) .Since we have det N p, − = v − p · det T / p · (det ind p ) − · (det T / p, =0 ) − (28)and the third factor in (27) does not depend on the equivariant parameters, the factor of automorphyof S p (cid:48) ,p under a (cid:55)→ q c a is given by r (cid:89) i =1 z (cid:104) i ∗ p L i ,c (cid:105)−(cid:104) i ∗ p (cid:48) L i ,c (cid:105) i · v (cid:104) det N p, − ,c (cid:105)−(cid:104) det N p (cid:48) , − ,c (cid:105) · θ p (cid:48) ,p ( c ) . This and (13) imply (24).Since T / p, =0 + v − ( T / p, =0 ) ∨ = 0, we have T / p, =0 = (cid:80) i ( v m i − v − − m i ) for some m i ∈ Z . Using this,one can check that the factor of automorphy for the line bundle Θ( T / p, =0 ) under v (cid:55)→ qv is given by( qv ) wt S det T / p, =0 and hence the factor of automorphy of the third factor in (27) under v (cid:55)→ qv is givenby ( qv ) wt S det T / p (cid:48) , =0 +2 rk ind p (cid:48) − wt S det T / p, =0 − p After some simplification using (28), one can check that the factor of automorphy of S p (cid:48) ,p under v (cid:55)→ qv is given by r (cid:89) i =1 z wt S i ∗ p L i − wt S i ∗ p (cid:48) L i i · ( qv ) wt S det Np, − det Np (cid:48) , − · det N p, − det N p (cid:48) , − · θ p (cid:48) ,p ( δ ) . By (11), we obtain wt S det N p, − − wt S det N p (cid:48) , − = wt S det N ! p (cid:48) ! , − − wt S det N ! p ! , − and hence (12) implies r (cid:89) i =1 z wt S i ∗ p L i − wt S i ∗ p (cid:48) L i i · v wt S det Np, − det Np (cid:48) , − = det N ! p (cid:48) ! , − det N ! p ! , − . This proves (26). 28n particular, if we set S ! p ! ,p (cid:48) ! := i ∗ p ! Stab ellX ! ( p (cid:48) ! ), then the line bundle on B X defined by the factorsof automorphy of S p (cid:48) .p is the same as the line bundle on B X ! defined by S ! p ! ,p (cid:48) ! under the identification B X ∼ = B X ! . Following [1, 36, 37], we conjecture that elliptic standard bases have certain symmetryunder the symplectic duality. Conjecture 4.4 ([1, 36, 37]) . For any p, p (cid:48) ∈ X H , S p (cid:48) .p is holomorphic and S p (cid:48) .p = ± S ! p ! ,p (cid:48) ! .For toric hyper-K¨ahler manifolds, this conjecture is proved in Corollary 6.3. Before considering an elliptic analogue of the bar involution, we state some conjectures about theelliptic stable bases for the maximal flop of X . Recall that our conical symplectic resolution is alwaysequipped with some additional data as in section 3.1. For X = ( X, C , A , Φ , Ψ , L ), we define − X :=( X, − C , A , Φ , Ψ , L ). We also assume that − X ! has a dual conical symplectic resolution in the sense ofDefinition 3.2, which is denoted by X flop = ( X flop , C flop , A flop , Φ flop , Ψ flop , L flop ) and called a maximalflop of X . By definition, we have an identification X ∗ ( H ) ∼ = P ! ∼ = X ∗ ( H flop ), P ∼ = X ∗ ( H ! ) ∼ = P flop , and( X H , (cid:22) C ) ∼ = ( X H flop flop , (cid:23) C flop ). We simply denote by p ∈ X H flop the fixed point corresponding to p ∈ X H .Under this identification, we also have C flop = C , A flop = − A , Φ flop ( p ) = Φ( p ), and Ψ flop ( p ) = Ψ( p ).By (12), we also obtain i ∗ p L flop ( λ ) = i ∗ p L ( λ ) for any λ ∈ P . The relation Φ flop ( p ) = Φ( p ) implies thatthe tangent spaces T p X flop and T p X have the same multiset of H -weights, and in particular, we havedim X flop = dim X . For the S -weights, we expect the following. Conjecture 4.5.
For any p ∈ X H , we have T p X flop = v − · T p X as T -modules.For toric hyper-K¨ahler manifolds, this conjecture is checked in Corollary 5.7. In particular, thisconjecture implies N flop p, − = v − · N p, − , where N flop p, − is the repelling part of T p X flop . This is compatiblewith the relation wt S det N flop p, − = − wt S det N p, − − dim X which comes from (11). We note that from ourdefinition, the relation det N flop p, − = v − dim X · det N p, − always holds without assuming Conjecture 4.5.Let (cid:102) M X be the field of multivalued meromorphic functions on B KX . We set S X := ( S X,p,p (cid:48) ) p,p (cid:48) ∈ X H .This is a matrix whose entries are elements in (cid:102) M X . By the triangular property of elliptic stable bases, S X is invertible. As in K -theory, the inverse for S induces an involution ( − ) : (cid:102) M X → (cid:102) M X . Weconjeture the following formula expressing S X geometrically. Conjecture 4.6. S X = ( − dim X + dim X !2 · S − X flop .For toric hyper-K¨ahler manifolds, this conjecture is proved in Corollary 6.4. If we assume Conjec-ture 4.5, then we have N p, − = v · N flop p, − = ( N flop p, + ) ∨ and hence for the diagonal entries, we obtain S X,p,p = ϑ ( N p, − ) ϑ ( N ! p ! , − ) = ( − dim X + dim X !2 · ϑ ( N flop p, + ) ϑ ( N ! , flop p ! , + ) = ( − dim X + dim X !2 · S − X flop ,p,p . As another evidence for Conjecture 4.6, one can also check that each corresponding entry has the samefactors of automorphy.
Proposition 4.7.
Assume Conjecture 4.5. For any p, p (cid:48) ∈ X H , both S X,p (cid:48) ,p and S − X flop ,p (cid:48) ,p can beconsidered as a section of the same line bundle on B X ∼ = B − X flop . Proof . As above, one can check θ X,p (cid:48) ,p ( γ ) = θ − X flop ,p (cid:48) ,p ( γ ) for any γ ∈ X ∗ ( H ) × P and θ X,p (cid:48) ,p ( − δ ) = θ − X flop ,p (cid:48) ,p ( δ ) for δ ∈ X ∗ ( S ). By Proposition 4.3, the coincidence of factors of automorphy under a (cid:55)→ q c a and z (cid:55)→ q l z follows from i ∗ p L !flop ( c ) = i ∗ p L ! ( c ) and i ∗ p L flop ( l ) = i ∗ p L ( l ).For v (cid:55)→ qv , we note that by (11) and (26), we have S X,p (cid:48) ,p ( v (cid:55)→ qv ) = S X,p (cid:48) ,p ( v (cid:55)→ q − v ) = det N p (cid:48) , − det N p, − det N ! p ! , − det N ! p (cid:48) ! , − · q wt S det Np, − det Np (cid:48) , − · θ X,p (cid:48) ,p ( − δ ) · S X,p (cid:48) ,p . v (cid:55)→ qv together with the coincidence ofmonodromies follows from N p, − = ( N flop p, + ) ∨ .We now state the main conjecture in this section. This conjecture together with Conjecture 4.6 willimply that the elliptic bar involution defined in the next section is actually an involution. Conjecture 4.8.
The matrix M X := S X flop · S − X does not depend on the choice of the chamber C .For toric hyper-K¨ahler manifolds, this conjecture is proved in Corollary 6.12. As another evidencefor this conjecture, we prove that each entry of M X can be considered as a section of some line bundleon B X which does not depend on the choice of C . We write U p,p (cid:48) and S flop p,p (cid:48) the ( p, p (cid:48) )-entry of S − X and S X flop respectively. We first calculate the factors of automorphy for U p,p (cid:48) . Lemma 4.9. U p,p (cid:48) satisfies the following: • For each c ∈ X ∗ ( H ), we have U p,p (cid:48) ( a (cid:55)→ q c a ) = i ∗ p ! L ! ( c ) · i ∗ p (cid:48) ! L ! ( c ) − · θ p (cid:48) ,p ( c ) − · U p,p (cid:48) ; • For each l ∈ P , we have U p,p (cid:48) ( z (cid:55)→ q l z ) = i ∗ p L ( l ) − · i ∗ p (cid:48) L ( l ) · θ p (cid:48) ,p ( l ) − · U p,p (cid:48) ; • For δ ∈ X ∗ ( S ) satisfying (cid:104) δ, v (cid:105) = 1, we have U p,p (cid:48) ( v (cid:55)→ qv ) = det N p (cid:48) , − det N p, − det N ! p ! , − det N ! p (cid:48) ! , − · q wt S det Np (cid:48) , − det Np, − · θ p (cid:48) ,p ( δ ) − · U p,p (cid:48) ; • (cid:113) det N p (cid:48) , − · det N ! p ! , − · U p,p (cid:48) is single valued on B KX . Proof . Since S p,p (cid:48) = U p,p (cid:48) = 0 unless p (cid:22) C p (cid:48) , we prove the statements by induction on the number of p (cid:48)(cid:48) ∈ X H satisfying p (cid:22) C p (cid:48)(cid:48) (cid:22) C p (cid:48) . We note that by Proposition 4.3, the factor of automorphy of S p (cid:48) ,p under the translation by q γ is of the form f p ( γ ) f p (cid:48) ( γ ) − θ p (cid:48) ,p ( γ ) for any γ ∈ X ∗ ( T ) × P . It is enough tocheck that the factor of automorphy for U p,p (cid:48) is given by f p ( γ ) − f p (cid:48) ( γ ) θ p (cid:48) ,p ( γ ) − . If p = p (cid:48) , then wehave U p,p = ( S p,p ) − and the claim follows immediately.If p ≺ C p (cid:48) , then we have U p,p (cid:48) = − ( S p (cid:48) ,p (cid:48) ) − · (cid:88) p (cid:22) C p (cid:48)(cid:48) ≺ C p (cid:48) U p,p (cid:48)(cid:48) · S p (cid:48)(cid:48) ,p (cid:48) . By the induction hypothesis, the factor of automorphy of each term in the RHS is given by θ p (cid:48) ,p (cid:48) ( γ ) − · f p ( γ ) − f p (cid:48)(cid:48) ( γ ) θ p (cid:48)(cid:48) ,p ( γ ) − · f p (cid:48) ( γ ) f p (cid:48)(cid:48) ( γ ) − θ p (cid:48)(cid:48) ,p (cid:48) ( γ ) = f p ( γ ) − f p (cid:48) ( γ ) θ p (cid:48) ,p ( γ ) − . This proves the first three statements. The fourth statement can be proved similarly.
Proposition 4.10.
Each entry of M X is a section of a line bundle on B X which does not depend onthe choice of C . Proof . By Proposition 4.3 and Lemma 4.9, the factors of automorphy of S flop p,p (cid:48)(cid:48) · U p (cid:48)(cid:48) ,p (cid:48) for p (cid:23) C p (cid:48)(cid:48) (cid:22) C p (cid:48) are given as follows: • For a (cid:55)→ q c a , it is given by i ∗ p ! L ! ( c ) · θ X flop ,p ( c ) i ∗ p (cid:48) ! L ! ( c ) · θ X,p (cid:48) ( c ) ; (29)30 For z (cid:55)→ q l z , it is given by i ∗ p (cid:48) L ( l ) i ∗ p L flop ( l ) · i ∗ p (cid:48)(cid:48) L flop ( l ) i ∗ p (cid:48)(cid:48) L ( l ) · θ − X ! ,p (cid:48)(cid:48) ! ( l ) θ X ! ,p (cid:48)(cid:48) ! ( l ) ; (30) • For v (cid:55)→ qv , it is given bydet N p (cid:48) , − det N flop p, − · det N ! p ! , + det N ! p (cid:48) ! , − · q wt S det Np (cid:48) , − det N flop p, − · θ X flop ,p ( δ ) θ X,p (cid:48) ( δ ) · det N flop p (cid:48)(cid:48) , − det N p (cid:48)(cid:48) , − · det N ! p (cid:48)(cid:48) ! , − det N ! p (cid:48)(cid:48) ! , + · q wt S det N flop p (cid:48)(cid:48) , − det Np (cid:48)(cid:48) , − · θ − X ! ,p (cid:48)(cid:48) ! ( δ ) θ X ! ,p (cid:48)(cid:48) ! ( δ ) . (31)We first consider the case of a (cid:55)→ q c a . Since (29) does not depend on p (cid:48)(cid:48) , M X,p,p (cid:48) = (cid:80) p (cid:48)(cid:48) S flop p,p (cid:48)(cid:48) · U p (cid:48)(cid:48) ,p (cid:48) has (29) as the factor of automorphy. In order to check the independence on C , it is sufficient to checkthat i ∗ p ! L ! ( c ) · θ X,p ( c ) does not depend on the choice of C . By (13), the H -weight of i ∗ p ! L ! ( c ) does notdepend on the choice of C and the S -weight is given by −(cid:104) det N p, − , c (cid:105) . If we write N p, − = (cid:80) i w i v m i for some w i ∈ X ∗ ( H ) and m i ∈ Z , then it is enough to check the independence of v −(cid:104) det N p, − ,c (cid:105) · θ X,p ( c ) = (cid:89) i ( − (cid:104) w i ,c (cid:105) q − (cid:104) wi,c (cid:105) w −(cid:104) w i ,c (cid:105) i v − ( m i +1) (cid:104) w i ,c (cid:105) on the choice of C . If we change C , then for some i , w i is replaced by w − i and m i is replaced by − m i − N p, − . Since this replacement does not change each factor of the RHS, theindependence follows.We next consider the case of z (cid:55)→ q l z . By (12), we have i ∗ p (cid:48)(cid:48) L flop ( l ) i ∗ p (cid:48)(cid:48) L ( l ) · θ − X ! ,p (cid:48)(cid:48) ! ( l ) θ X ! ,p (cid:48)(cid:48) ! ( l ) = v (cid:104) det N ! p (cid:48)(cid:48) ! , − ,l (cid:105) · v − (cid:104) det N ! p (cid:48)(cid:48) ! , − ,l (cid:105) = 1 . Therefore, (30) does not depend on p (cid:48)(cid:48) and hence it is the factor of automorphy of M X,p,p (cid:48) . Itsindependence on C is clear.Now we consider the case of v (cid:55)→ qv . We first check that (31) does not depend on p (cid:48)(cid:48) . One can checkthat θ − X ! ,p (cid:48)(cid:48) ! ( δ ) θ X ! ,p (cid:48)(cid:48) ! ( δ ) = ( qv ) − S det N ! p (cid:48)(cid:48) ! , − − dim X ! · v − dim X ! · (det N ! p (cid:48)(cid:48) ! , − ) − . On the other hand, we havedet N flop p (cid:48)(cid:48) , − det N p (cid:48)(cid:48) , − = v − dim X − S det N p (cid:48)(cid:48) , − = v dim X ! +2 wt S det N ! p (cid:48)(cid:48) ! , − by (11) and det N ! p (cid:48)(cid:48) ! , − det N ! p (cid:48)(cid:48) ! , + = v dim X ! · (det N ! p (cid:48)(cid:48) ! , − ) . These equations imply that (31) does not depend on p (cid:48)(cid:48) . In order to prove the independence of (31) onthe choice of C , it is enough to check thatdet N ! p ! , − det N p, − · q − wt S det N p, − · θ X,p ( δ )31oes not depend on C . We note that the H ! -weight of det N ! p ! , − does not depend on C and the S -weightis − dim X ! − dim X − wt S det N p, − by (11). If we write N p, − = (cid:80) i w i v m i for some w i ∈ X ∗ ( H ) and m i ∈ Z , then we have v wt S det N ! p ! , − det N p, − · q − wt S det N p, − · θ X,p ( δ ) = v − dim X !2 − dim X · (cid:89) i ( − m i ( qv ) − mi ( mi +2)2 w − m i − i . Since each factor in the RHS does not change under w i (cid:55)→ w − i and m i (cid:55)→ − m i −
2, the independenceon C follows.Finally, independence of monodromies of (cid:112) det N p, − on C easily implies the independence of mon-odromies of M X,p,p (cid:48) . Remark 4.11.
We note that the matrix M X is closely related to the monodromy operator appearingin [1, Proposition 6.5] which intertwines the vertex function for X and X flop . We expect that thedependence on C in [1, Proposition 6.5] is eliminated by our choice of the normalization on the ellipticstandard bases. Since the vertex functions do not depend on the choice of C , one may hope that thesekind of relations would imply Conjecture 4.8. We plan to investigate this approach in the future. In this section, we give a proposal for a definition of elliptic bar involution. Since our approach givesinvolution only after localization, we set K ( X ) loc := ⊕ p ∈ X H M X , where M X is the field of (single-valued) meromorphic functions on B KX , and we will only work on K ( X ) loc in this paper. By restrictionto the fixed points, we obtain a natural map K T ( X ) ⊗ Z Z [ P ∨ ] → K ( X ) loc . In order to consider Stab ellX ( p )as an element of K ( X ) loc similarly, we need to kill the multi-valuedness. We use the data of polarizationto fix this modification.We choose a splitting Pic H ( X ) ∼ = P ⊕ X ∗ ( H ) by using the data L : P → Pic T ( X ) and take apolarization T / satisfying Assumption 3.11. Let κ ∈ P ⊕ X ∗ ( H ) be the element corresponding todet T / ∈ Pic H ( X ). We also denote by L : P ⊕ X ∗ ( H ) → Pic T ( X ) the natural extension of L . We notethat L ( κ ) = v − w (det T / ) · det T / . We also take similar data T / , ! and κ ! ∈ P ! ⊕ X ∗ ( H ! ) on the dualconical symplectic resolution X ! . Definition 4.12.
For any p ∈ X H , we set S κ,κ ! X ( p ) := (cid:113) L ( κ ) · i ∗ p ! L ! ( κ ! ) − · Stab ellX ( p ) . Lemma 4.13.
We have (cid:16) i ∗ p (cid:48) S κ,κ ! X ( p ) (cid:17) p (cid:48) ∈ X H ∈ K ( X ) loc . Proof . By Assumption 3.11 for T / and T / , ! , we obtain w (det T / ) ≡ w (det T / , ! ) mod 2. Sincewe have (cid:113) i ∗ p (cid:48) L ( κ ) · i ∗ p ! L ! ( κ ! ) = v − w (det T / − w (det T / , !)2 (cid:113) det T / p (cid:48) · det T / , ! p ! , the statement follows fromProposition 4.3.We will identify S κ,κ ! X ( p ) and (cid:16) i ∗ p (cid:48) S κ,κ ! X ( p ) (cid:17) p (cid:48) ∈ X H ∈ K ( X ) loc . By the triangular property of theelliptic stable bases, { S κ,κ ! X ( p ) } p ∈ X H forms a basis of K ( X ) loc over M X . Definition 4.14.
We define the M X -semilinear map β ellX = β ellX,κ : K ( X ) loc → K ( X ) loc by β ellX ( S κ,κ ! X ( p )) = ( − dim X S κ,κ ! − X ( p ) (32)for any p ∈ X H . Here, M X -semilinear means β ellX ( f · m ) = ¯ f · β ellX ( m ) for any f ∈ M X and m ∈ K ( X ) loc .We also identified P !flop ⊕ X ∗ ( H !flop ) and P ! ⊕ X ∗ ( H ) in order to take κ ! for X !flop in the RHS.32e note that β ellX does not depend on the choice of κ ! because of the relation L !flop ( κ ! ) = L ! ( κ ! ). Wealso conjecture that β ellX does not depend on the data C . As an evidence for this conjecture, we checkit by assuming Conjecture 4.6 and Conjecture 4.8. Proposition 4.15.
Assume Conjecture 4.6 and Conjecture 4.8. The map β ellX does not depend on thechoice of C . Proof . In this proof, we denote by β C := β ellX . We take another chamber C (cid:48) and define β C (cid:48) similarlyby using this chamber. We also set S C := S X and define S C (cid:48) similarly by using C (cid:48) instead of C . Let usdefine the matrix S C := ( i ∗ p S κ,κ ! X ( p (cid:48) )) p,p (cid:48) ∈ X H and S C (cid:48) similarly. If we identify L ( κ ) and the diagonalmatrix diag(( i ∗ p L ( κ )) p ∈ X H ), then we have S C = L ( κ ) − · S C · L ! C ( κ ! ) − . Here, L ! C ( κ ! ) is the diagonal matrix defined similarly as L ( κ ) by using L ! ( κ ), but since it depends onthe choice of C , we put the index to indicate the dependence. We note that L ! − C ( κ ! ) = L ! C ( κ ! ). Bydefinition, we have β C ( S C ) = ( − dim X · S − C , where we understand that β C is applied column by column. Since β C (cid:48) ( S C ) = β C (cid:48) ( S C (cid:48) · S − C (cid:48) S C ) = ( − dim X · S − C (cid:48) · S − C (cid:48) S C , the statement is equivalent to S C · S − − C = S C (cid:48) · S − − C (cid:48) . By (13) and Conjecture 4.6, we have S C · S − − C = L ( κ ) − · S C · L ! C ( κ ! ) − · L ! − C ( κ ! ) · S − − C · L ( κ ) = ( − dim X + dim X !2 · L flop ( κ ) − · S flop − C · S − − C · L ( κ ) , where we set S flop − C = S − X flop . Therefore, the statement follows from Conjecture 4.8.We now explain the relation between elliptic and K -theoretic bar involutions. Take a generic slope s ∈ P R . By [1, Proposition 3.6], elliptic stable bases and K -theoretic stable bases are related by certainlimit under q →
0. More precisely, we havelim q → (cid:18)(cid:112) det T / − · Stab AO C ,T / ( p ) | z = q − s (cid:19) = Stab K C ,T / ,s ( p ) , (33)where z = q − s means we specialize the K¨ahler parameter at q − s ∈ P ⊗ Z C × . We note that (32) isequivalent to β ellX (cid:18)(cid:112) det T / − · Stab AO C ,T / ( p ) (cid:48) (cid:19) = ( − dim X v w (det T / ) · ϑ ( N ! , flop p ! , − ) ϑ ( N ! p ! , − ) · (cid:112) det T / − · Stab AO − C ,T / ( p ) (cid:48) , where we set Stab AO C ,T / ( p ) (cid:48) := τ (det T / , v ) ∗ Stab AO C ,T / ( p ). By using Conjecture 4.6 andlim q → ϑ ( q α x ) ϑ ( q α ) = x −(cid:98) α (cid:99)− α ∈ R \ Z , we obtain lim q → (cid:32) ϑ ( N ! , flop p ! , − ) | z = q − s ϑ ( N ! p ! , − ) | z = q − s (cid:33) = (cid:89) β ∈ Ψ + ( p ) v (cid:98)(cid:104) s,β (cid:105)(cid:99) +1 . Since the limit in (33) does not change if we replace Stab AO C ,T / ( p ) by Stab AO C ,T / ( p ) (cid:48) , the K -theory limit β KT / ,s of β ellX satisfies β KT / ,s (Stab K C ,T / ,s ( p )) = ( − v ) dim X · v a p ( T / ,s ) · Stab K − C ,T / ,s ( p ) . This is equivalent to our definition of K -theoretic bar involution and explains seemingly ad hoc nor-malizations in our definition of the K -theoretic standard bases. As an evidence for the main conjectures, we check all the conjectures stated in section 3 for the torichyper-K¨ahler manifolds in this section. The conjectures stated in section 4 will be proved in the nextsection.
In this section, we briefly recall basic facts about toric hyper-K¨ahler manifolds introduced by Bielawski-Dancer [7]. For more detail, see for example [7, 18, 30, 35]. We first prepare some notations used in thefollowing sections. We fix an integer n ∈ Z ≥ and set T := ( C × ) n . We consider an exact sequence ofalgebraic tori of the form 1 → S → T → H → , (34)where S ∼ = ( C × ) r and H ∼ = ( C × ) d for some r, d ∈ Z ≥ with r + d = n . Let0 → X ∗ ( S ) t b −→ X ∗ ( T ) a −→ X ∗ ( H ) → X ∗ ( T ) ∼ = Z n and take the standard basis { ε i } ni =1 . We set a i := a ( ε i ) and assume that a i (cid:54) = 0 for any i = 1 , . . . , n . We also assume that a isunimodular, i.e., if { a i , . . . , a i d } is linearly independent, then they always generate X ∗ ( H ) over Z . Weset B := { I ⊂ { , . . . , n } | { a i } i ∈ I is a basis of X ∗ ( H ) } . We also consider the dual of the above exact sequence0 → X ∗ ( H ) t a −→ X ∗ ( T ) b −→ X ∗ ( S ) → . (36)Let { ε ∗ i } ni =1 ⊂ X ∗ ( T ) be the dual basis of { ε i } ni =1 and set b i := b ( ε ∗ i ). We also assume that b i (cid:54) = 0for any i = 1 , . . . , n . We note that the unimodularity of a is equivalent to the unimodularity of b and { b j } j ∈ J is a basis of X ∗ ( S ) if and only if J c := { , . . . , n } \ J ∈ B .A subset equipped with a decomposition C = C + (cid:116) C − ⊂ { , . . . , n } is called signed circuit if { a i } i ∈ C is a minimal linearly dependent subset of { a , . . . , a n } and (cid:80) i ∈ C + a i − (cid:80) i ∈ C − a i = 0. We note that bythe unimodularity assumption, any minimal linear relations between a i ’s can be written in this form upto scalar multiplication. For a signed circuit C , we denote by β C = ( β , . . . , β n ) ∈ X ∗ ( T ) the elementdefined by β i = i / ∈ C i ∈ C + − i ∈ C − .
34y definition, we have β C ∈ Ker( a ) = X ∗ ( S ). We similarly define the notion of signed cocircuit using b i instead of a i . For a signed cocircuit C ∨ , we define α C ∨ ∈ X ∗ ( H ) in the same way as β C .We consider the natural T -action on T ∗ C n given by( t , . . . , t n ) · ( x , . . . , x n , y , . . . , y n ) = ( t x , . . . , t n x n , t − y , . . . , t − n y n ) , where ( t , . . . , t n ) ∈ T , ( x , . . . , x n ) is a point in C n , and ( y , . . . , y n ) is a point in the cotangent fiber.This action is Hamiltonian with respect to the standard symplectic structure on T ∗ C n and its momentmap is given by µ n ( x, y ) = (cid:80) ni =1 x i y i . Let µ ( x, y ) := (cid:80) ni =1 x i y i b i ∈ s ∗ be the moment map for the S -action given by restriction, where s is the Lie algebra of S . We note that in the coordinate ring C [ T ∗ C n ] = C [ x , . . . , x n , y , . . . , y n ], the S -weights of x i and y i are given by − b i and b i respectively.We fix a generic element η ∈ X ∗ ( S ), where generic means for any circuit C , we have (cid:104) η, β C (cid:105) (cid:54) = 0.A point ( x, y ) ∈ T ∗ C n is called η -semistable if there exists a positive integer m and a polynomial f ∈ C [ T ∗ C n ] such that f has S -weight − mη and f ( p ) (cid:54) = 0. Associated with these data, the torichyper-K¨ahler manifold X is defined by X := µ − (0) η − ss /S , where the superscript means the subsetconsisting of η -semistable points. We also define the Lawrence toric variety by X := ( T ∗ C n ) η − ss /S .By the unimodularity assumption, these varieties are smooth and it is known that X is the universalPoisson deformation of X in the sense of Namikawa, see [30].For I ∈ B and j ∈ I c , we denote by C Ij the unique signed circuit contained in I ∪ { j } and j ∈ C Ij, + and we set β Ij := β C Ij ∈ X ∗ ( S ). We similarly define α Ii ∈ X ∗ ( H ) for I ∈ B and i ∈ I using signedcocircuit contained in I c ∪ { i } . We note the following identity for any i ∈ I and j ∈ I c : (cid:104) α Ii , a j (cid:105) = −(cid:104) β Ij , b i (cid:105) . (37)We decompose I = I + (cid:116) I − and I c = I c + (cid:116) I c − , where we set I ± := { i ∈ I | ±(cid:104) ξ, α Ii (cid:105) > } and I c ± := { j ∈ I c | ±(cid:104) η, β Ij (cid:105) > } . We note that these decompositions depend on the choice of ξ and η , butwe omit the dependence from the notation. Lemma 5.1.
A point ( x, y ) ∈ T ∗ C n is η -semistable if and only if there exists I ∈ B such that x j (cid:54) = 0( ∀ j ∈ I c + ) and y j (cid:54) = 0 ( ∀ j ∈ I c − ). Proof . Let p = ( x, y ) be a η -semistable point and f ∈ C [ x, y ] be a polynomial with S -weight − mη and f ( p ) (cid:54) = 0 for some m ∈ Z > . We may assume that f = (cid:81) i x m i i y n i i is a monomial. Then we have (cid:80) i ( m i − n i ) b i = mη and Lemma 5.2 implies that there exists I ∈ B such that ± ( m j − n j ) > j ∈ I c ± . This implies that x j (cid:54) = 0 for any j ∈ I c + and y j (cid:54) = 0 for any j ∈ I c − . Conversely, if p ∈ T ∗ C n satisfies the latter condition, then f = (cid:81) j ∈ I c + x (cid:104) η,β Ij (cid:105) j · (cid:81) j ∈ I c − y −(cid:104) η,β Ij (cid:105) j has S -weight − η and f ( p ) (cid:54) = 0. Lemma 5.2.
If we have (cid:80) i m i b i = η for some m i ∈ R , then there exists an I ∈ B such that ± m j > j ∈ I c ± . Proof . We set I := { i | m i = 0 } . If { a i } i ∈ I is linearly dependent, then there exists a circuit C ⊂ I and we obtain (cid:104) η, β C (cid:105) = 0, which contradicts the genericity of η . Hence { a i } i ∈ I is linearly independent.If I ∈ B , then we have m j = (cid:104) η, β I j (cid:105) and the statement follows. If I / ∈ B , then there exists nonzero λ ∈ h ∗ R such that (cid:104) λ, a i (cid:105) = 0 for any i ∈ I . Since we have (cid:80) i (cid:104) λ, a i (cid:105) b i = 0, we may replace m i by m tj := m i + t (cid:104) λ, a i (cid:105) for sufficiently small t ∈ R without changing the sign of m j for j ∈ I c . By taking aminimal t such that some m tj becomes 0, we obtain t (cid:48) ∈ R such that ± m t (cid:48) j ≥ j with ± m j > I := { i | m t (cid:48) i = 0 } (cid:41) I . By continuing this process, we obtain I ∈ B such that I ⊃ I and ± m j > j ∈ I c ± .For I ∈ B , we set (cid:101) U I := { ( x, y ) ∈ T ∗ C n | x j (cid:54) = 0 ( ∀ j ∈ I c + ) , y j (cid:54) = 0 ( ∀ j ∈ I c − ) } , U I := (cid:101) U I /S ⊂ X , and U I := ( µ − (0) ∩ (cid:101) U I ) /S ⊂ X . By Lemma 5.1, {U I } I ∈ B and { U I } I ∈ B are Zariski open affine coverings of X and X respectively. For i ∈ I , we set x Ii := x i (cid:89) j ∈ I c + x (cid:104) α Ii , a j (cid:105) j (cid:89) j ∈ I c − y −(cid:104) α Ii , a j (cid:105) j , y Ii := y i (cid:89) j ∈ I c + x −(cid:104) α Ii , a j (cid:105) j (cid:89) j ∈ I c − y (cid:104) α Ii , a j (cid:105) j . C [ U I ] = C [ (cid:101) U I ] S . It is easy to see that C [ U I ] = C [ x j y j ( j ∈ I c ) , x Ii , y Ii ( i ∈ I )]and C [ U I ] = C [ x Ii , y Ii ( i ∈ I )]. In particular, we have U I ∼ = C d and dim X = 2 d . Let µ X : X → s ∗ bethe morphism induced from the moment map µ . This is well-defined since µ is S -invariant. The abovedescription of open coverings gives the following. Lemma 5.3.
The morphism µ X is flat and X ∼ = µ − X (0) as schemes.Since H -weights of x Ii and y Ii (as functions on U I ) are − α Ii and α Ii respectively, U I has a unique H -fixed point which is denoted by p I . In particular, we obtain a one-to-one correspondence between B and X H given by I (cid:55)→ p I .We consider the action of S := C × on X or X induced by σ · ( x, y ) = ( σ − x, σ − y ), ( σ ∈ S ,( x, y ) ∈ T ∗ C n ). With this S -action, it is known that X is a conical symplectic resolution. As inprevious sections, we set T := H × S . We denote by L the central fiber of X → Spec( C [ X ]).We regard λ ∈ X ∗ ( T ) as a 1-dimensional representation of T × S with trivial S -action and writethe associated T -equivariant line bundle on the quotients X or X as L ( λ ) or (cid:101) L ( λ ). We note thatif λ ∈ X ∗ ( H ), then L ( λ ) is a trivial line bundle if we forget H -equivariant structure and the H -action is given by λ . This map gives an isomorphism Pic( X ) ∼ = X ∗ ( S ) under our assumption that b i (cid:54) = 0. We also fix a splitting ι : X ∗ ( S ) → X ∗ ( T ) of the natural surjection X ∗ ( T ) (cid:16) X ∗ ( S ) and set L ( l ) := L ( ι ( l )) ∈ Pic T ( X ) for any l ∈ X ∗ ( S ). We note that L ( η ) is an ample line bundle relative to theprojective morphism X → Spec( C [ X ]). For any λ ∈ X ∗ ( H ), we write a λ ∈ K H (pt) the K -theory classcorresponding to λ . The following result can be checked easily. Lemma 5.4.
For any λ ∈ X ∗ ( T ) and I ∈ B , we have the following identity in K T ( p I ): i ∗ p I L ( λ ) = v (cid:80) j ∈ Ic + (cid:104) λ,β Ij (cid:105)− (cid:80) j ∈ Ic − (cid:104) λ,β Ij (cid:105) · a λ − (cid:80) j ∈ Ic (cid:104) λ,β Ij (cid:105) ε ∗ j , where λ − (cid:80) j ∈ I c (cid:104) λ, β Ij (cid:105) ε ∗ j is considered as an element of Ker( b ) ∼ = X ∗ ( H ).By using (37), we obtain the following corollary of Lemma 5.4. Corollary 5.5.
For j ∈ I c ± , we have i ∗ p I L ( ε ∗ j ) = v ± . For i ∈ I , we have i ∗ p I L ( ε ∗ i ) = v −(cid:104) α Ii , (cid:80) j ∈ Ic + a j − (cid:80) j ∈ Ic − a j (cid:105) a α Ii . The equivariant K -theory class of the tangent bundle T X of X is given by the following formula. T X = n (cid:88) i =1 v − L ( ε ∗ i ) + n (cid:88) i =1 v − L ( − ε ∗ i ) − r · O X − rv − · O X . By using Corollary 5.5, we obtain i ∗ p I T X = (cid:88) i ∈ I v − −(cid:104) α Ii , (cid:80) j ∈ Ic + a j − (cid:80) j ∈ Ic − a j (cid:105) · a α Ii + (cid:88) i ∈ I v − (cid:104) α Ii , (cid:80) j ∈ Ic + a j − (cid:80) j ∈ Ic − a j (cid:105) · a − α Ii . (38)This implies that in the notation of section 3, the multiset of equivariant roots at p I is given by Φ( p I ) = {± α Ii } i ∈ I and Φ = { α C ∨ | C ∨ : signed cocircuit } . We fix ξ ∈ X ∗ ( H ) satisfying (cid:104) ξ, α C ∨ (cid:105) (cid:54) = 0 for anycocircuit C ∨ and take the chamber C to be the connected component of h R \ ∪ α ∈ Φ { x ∈ h R | (cid:104) x, α (cid:105) = 0 } containing ξ . With respect to this choice of chamber, we obtain N p I , − = (cid:88) i ∈ I − v − −(cid:104) α Ii , (cid:80) j ∈ Ic + a j − (cid:80) j ∈ Ic − a j (cid:105) · a α Ii + (cid:88) i ∈ I + v − (cid:104) α Ii , (cid:80) j ∈ Ic + a j − (cid:80) j ∈ Ic − a j (cid:105) · a − α Ii . (39)For the multiset of K¨ahler roots, we take Ψ( p I ) := {± β Ij } j ∈ I c . We have Ψ = { β C | C : signed circuit } .It is known (see for example [30]) that the ample cone A ⊂ Pic( X ) ⊗ Z R ∼ = s ∗ R is given by the connectedcomponent of s ∗ R \∪ β ∈ Ψ { x ∈ s ∗ R | (cid:104) x, β (cid:105) = 0 } containing η . As in previous sections, X is always equippedwith these additional data ( X, C , A , Φ , Ψ , L ). 36 .2 Dual pairs In this section, we give a dual pair ( X ! , C ! , A ! , Φ ! , Ψ ! , L ! ) for ( X, C , A , Φ , Ψ , L ). For X ! , this is given bya symplectic dual of ( X, C ) in the sense of Braden-Licata-Proudfoot-Webster [8]. I.e., X ! is the torichyper-K¨ahler manifolds defined by the exact sequence of algebraic tori which is dual to (34):1 → H ∨ → T ∨ → S ∨ → . (40)Here, the GIT parameter η ! ∈ X ∗ ( H ∨ ) ∼ = X ∗ ( H ) for X ! is taken to be ξ . We note that the exactsequence of cocharacter lattices associated with (40)0 → X ∗ ( H ∨ ) t a −→ X ∗ ( T ∨ ) b −→ X ∗ ( S ∨ ) → . is naturally isomorphic to the exact sequence (36) and the exact sequence of character lattices0 → X ∗ ( S ∨ ) t b −→ X ∗ ( T ∨ ) a −→ X ∗ ( H ∨ ) → . is isomorphic to (35). In particular, the roles of a i and b i are exchanged. Hence the set parametrizingthe H ! := S ∨ -fixed points of X ! is given by B ! := { J ⊂ { , . . . , n } | { b j } j ∈ J is a basis of X ∗ ( S ) } . The map I (cid:55)→ I c gives a natural bijection B ∼ = B ! and hence gives a bijection X H ∼ = ( X ! ) H ! . We denoteby p ! I the fixed point of X ! corresponding to I c ∈ B ! for any I ∈ B . Under the natural identification X ∗ ( H ! ) ∼ = X ∗ ( S ), we obtain Φ ! ( p ! I ) = {± β Ij } j ∈ I c . For the chamber C ! ⊂ s ∗ R , we take C ! := A . For theK¨ahler roots, we take Ψ ! ( p ! I ) := {± α Ii } i ∈ I . By our choice of GIT parameter, the ample cone A ! ⊂ h R isgiven by C . Therefore, the second condition of Definition 3.2 is satisfied. The order reversing propertyof the bijection X H ∼ = ( X ! ) H ! will be checked in the next section, see Corollary 5.10.As in the case of X , we have a natural map L ! : X ∗ ( T ∨ ) → Pic T ! ( X ! ). In order to define the lift L ! , we need to take a splitting ι ! : Pic( X ! ) ∼ = X ∗ ( H ) → X ∗ ( T ) which is compatible with the splitting ι : X ∗ ( S ) → X ∗ ( T ). Here, the compatibility means that for any λ ∈ X ∗ ( S ) and λ ! ∈ X ∗ ( H ), we have (cid:104) ι ( λ ) , ι ! ( λ ! ) (cid:105) = 0 . (41)Existence of such a splitting is clear. We define L ! : X ∗ ( H ) → Pic T ! ( X ! ) by L ! ( λ ! ) = L ! ( ι ! ( λ ! )). Proposition 5.6.
The pair ( X, C , A , Φ , Ψ , L ) and ( X ! , C ! , A ! , Φ ! , Ψ ! , L ! ) forms a dual pair in the senseof Definition 3.2. Proof . We need to check (8), (9), (10), and (11) in our situation. By (39), we obtaindet N p I , − = v − d + (cid:104) (cid:80) i ∈ I + α Ii − (cid:80) i ∈ I − α Ii , (cid:80) j ∈ Ic + a j − (cid:80) j ∈ Ic − a j (cid:105) · a − (cid:80) i ∈ I + α Ii + (cid:80) i ∈ I − α Ii . In particular, we havewt S det N p I , − + dim X (cid:104) (cid:88) i ∈ I + α Ii − (cid:88) i ∈ I − α Ii , (cid:88) j ∈ I c + a j − (cid:88) j ∈ I c − a j (cid:105) . Similarly, we obtainwt S det N ! p ! I , − + dim X ! (cid:104) (cid:88) j ∈ I c + β Ij − (cid:88) j ∈ I c − β Ij , (cid:88) i ∈ I + b i − (cid:88) i ∈ I − b i (cid:105) . Hence the equation (11) follows from (37). 37ince we have wt H ! det N ! p ! I , − = − (cid:88) j ∈ I c + β Ij + (cid:88) j ∈ I c − β Ij , the equation (9) follows from Lemma 5.4. The equation (10) can be proved similarly. Now we check(8). By Lemma 5.4 applied for both X and X ! , we have to check (cid:104) ι ( λ ) − (cid:88) j ∈ I c (cid:104) λ, β Ij (cid:105) ε ∗ j , λ ! (cid:105) = −(cid:104) ι ! ( λ ! ) − (cid:88) i ∈ I (cid:104) λ ! , α Ii (cid:105) ε i , λ (cid:105) (42)for any λ ∈ X ∗ ( S ), λ ! ∈ X ∗ ( H ), and I ∈ B . Since { a i } i ∈ I is a basis of X ∗ ( H ), it suffices to check (42)for any λ ! = a i , i ∈ I . Since we have ι ( λ ) − (cid:80) j ∈ I c (cid:104) λ, β Ij (cid:105) ε ∗ j ∈ X ∗ ( H ), we can calculate LHS of (42) byusing any lift of a i to X ∗ ( T ). Therefore, we obtain (cid:104) ι ( λ ) − (cid:88) j ∈ I c (cid:104) λ, β Ij (cid:105) ε ∗ j , a i (cid:105) = (cid:104) ι ( λ ) − (cid:88) j ∈ I c (cid:104) λ, β Ij (cid:105) ε ∗ j , ε i (cid:105) = (cid:104) ι ( λ ) , ε i (cid:105) . On the other hand, we may replace λ by ι ( λ ) in the RHS of (42) since ι ! ( λ ! ) − (cid:80) i ∈ I (cid:104) λ ! , α Ii (cid:105) ε i ∈ X ∗ ( S ).Therefore, we obtain −(cid:104) ι ! ( a i ) − (cid:88) i (cid:48) ∈ I (cid:104) a i , α Ii (cid:48) (cid:105) ε i (cid:48) , λ (cid:105) = −(cid:104) ι ! ( a i ) , ι ( λ ) (cid:105) + (cid:104) ε i , ι ( λ ) (cid:105) . Hence the equation (42) follows from (41). This proves (12). The proof of (13) is similar.Proposition 5.6 implies that a maximal flop X flop in the sense of section 4.2 is obtained in the sameway as X by replacing η by − η . Since this exchanges I c + and I c − , the formula (38) implies Conjecture 4.5. Corollary 5.7.
Conjecture 4.5 holds for toric hyper-K¨ahler manifolds. K -theoretic standard bases In this section, we recall the description of K -theoretic stable bases for toric hyper-K¨ahler manifolds.In this paper, we always take the following polarization for the toric hyper-K¨ahler manifold X : T / := n (cid:88) i =1 v − L ( ε ∗ i ) − r · O X . (43)In particular, we have det T / = v − n L ( ε ∗ + · · · + ε ∗ n ) and hence we obtain w (det T / ) = − n . Therefore,Assumption 3.11 is satisfied. Since we will not use other polarization below, we will omit T / from thenotations. We set s ∗ reg := { x ∈ s ∗ R | (cid:104) x, β C (cid:105) / ∈ Z , ∀ C : circuit } and we take a slope parameter s ∈ s ∗ reg . As in section 3.2, we consider the fractional line bundle L ( s ).By Lemma 5.4, we obtainwt H i ∗ p I L ( s ) − wt H i ∗ p J L ( s ) = (cid:88) i ∈ J c (cid:104) s, β Ji (cid:105) ε ∗ i − (cid:88) j ∈ I c (cid:104) s, β Ij (cid:105) ε ∗ j = − (cid:88) j ∈ I c ∩ J (cid:104) s, β Ij (cid:105) α Jj (44)for any I, J ∈ B . Here, we have used β Ji = (cid:80) j ∈ I c (cid:104) β Ji , b j (cid:105) β Ij and α Jj = ε ∗ j − (cid:80) i ∈ I c (cid:104) β Ji , b j (cid:105) ε ∗ i in thesecond equality. In particular, this is not contained in X ∗ ( H ) if I (cid:54) = J . This implies the first part ofAssumption 3.6 and hence the uniqueness of K -theoretic stable bases. The existence of K -theoreticstable bases is proved in Proposition 5.11.We note that the coordinate function x i (resp. y i ) can be considered as a section of v − L ( ε ∗ i ) (resp. v − L ( − ε ∗ i )) on X . For I ∈ B , let L I be the subvariety of X defined by the equations x i = 0 ( i ∈ I − )and y i = 0 ( i ∈ I + ). One can check that these defining equations form a regular sequence and theKoszul resolution gives the following. 38 emma 5.8. Let V I := (cid:76) i ∈ I − v − L ( ε ∗ i ) ⊕ (cid:76) i ∈ I + v − L ( − ε ∗ i ) be a vector bundle on X . We have thefollowing exact sequence 0 → (cid:86) d V ∨ I → . . . → (cid:86) V ∨ I → V ∨ I → O X → O L I → . In particular, we have O L I = (cid:89) i ∈ I − (1 − v L ( − ε ∗ i )) (cid:89) i ∈ I + (1 − v L ( ε ∗ i )) (45)in the equivariant K -theory of X . Moreover, one can easily check the following. Lemma 5.9.
For any I ∈ B , we have L I = Attr C ( p I ). In particular, p J ∈ Attr C ( p I ) is equivalent to I + ∩ J c − = I − ∩ J c + = ∅ for any I, J ∈ B . Corollary 5.10.
For any
I, J ∈ B , p J (cid:22) C p I is equivalent to p ! I (cid:22) C ! p ! J . Proof . Lemma 5.9 implies that p J ∈ Attr C ( p I ) is equivalent to p ! I ∈ Attr C ! ( p ! J ).Now we give an explicit formula for the K -theoretic stable bases for X . This is an explicit versionof Exercise 9.1.15 in [32]. Proposition 5.11.
For any I ∈ B , we haveStab K C ,s ( p I ) = v (cid:80) j ∈ Ic − (cid:100)(cid:104) s,β Ij (cid:105)(cid:101)− (cid:80) j ∈ Ic + (cid:98)(cid:104) s,β Ij (cid:105)(cid:99) · L − (cid:88) i ∈ I + ε ∗ i + (cid:88) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j ⊗ O L I . (46) Proof . Let us denote by Stab I the RHS of (46). We check the three conditions in Definition 3.5 forStab I . Since we have Supp(Stab I ) = L I = Attr C ( p I ) by Lemma 5.9, the first condition in Definition 3.5is satisfied.We note that by Corollary 5.5, we have T / p I = (cid:88) i ∈ I v − −(cid:104) α Ii , (cid:80) j ∈ Ic + a j − (cid:80) j ∈ Ic − a j (cid:105) · a α Ii + | I c − | ( v − − . Hence by (39) and Corollary 5.5, we obtain (cid:115) det N p I , − det T / p I = v | I c − | · (cid:89) i ∈ I + v (cid:104) α Ii , (cid:80) j ∈ Ic + a j − (cid:80) j ∈ Ic − a j (cid:105) · a − α Ii = v | I c − | · i ∗ p I L − (cid:88) i ∈ I + ε ∗ i . On the other hand, we have (cid:86) •− ( N ∨− ,p I ) = i ∗ p I O L I by (45). Therefore, the second condition in Defini-tion 3.5 follows from Corollary 5.5.Finally, we check the third condition of Definition 3.5. For I, J ∈ B , let us assume that i ∗ p J Stab I (cid:54) = 0.By Lemma 5.9, we have I + ∩ J c − = I − ∩ J c + = ∅ . Hence up to the factor of v and sign, we obtain i ∗ p J Stab I = ± v ? (1 − v ) | I ± ∩ J c ± | (cid:89) j ∈ I c ∩ J a (cid:98)(cid:104) s,β Ij (cid:105)(cid:99) α Jj · (cid:89) i ∈ I ± ∩ J (cid:18) − v ∓ (cid:104) α Ji , (cid:80) j ∈ Jc + a j − (cid:80) j ∈ Jc − a j (cid:105) · a − α Ji (cid:19) . By (44), we obtaindeg H (cid:0) i ∗ p J Stab I · i ∗ p I L ( s ) · i ∗ p J L ( s ) − (cid:1) = (cid:88) j ∈ I c ∩ J ( (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) − (cid:104) s, β Ij (cid:105) ) · α Jj + (cid:88) i ∈ I ∩ J deg H (1 − a − α Ji ) . Here, the sum means the Minkowski sum. On the other hand, we havedeg H (cid:0) i ∗ p J Stab J (cid:1) = (cid:88) i ∈ J deg H (1 − a − α Ji ) . Therefore, the third condition in Definition 3.5 follows from ( (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) − (cid:104) s, β Ij (cid:105) ) · α Jj ∈ deg H (1 − a − α Jj )for each j ∈ I c ∩ J . 39e next determine the K -theoretic standard bases. We note that in the notation of section 3.1, wehave Ψ + ( p I ) = { β Ij } j ∈ I c + ∪ {− β Ij } j ∈ I c − . Since we have w (det T / ) = − n = − r − d , we obtain a p I ( s ) = (cid:88) j ∈ I c + (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) − (cid:88) j ∈ I c − (cid:100)(cid:104) s, β Ij (cid:105)(cid:101) − d. Hence we obtain the following corollary of Proposition 5.11.
Corollary 5.12.
For any I ∈ B , the standard basis S C ,s ( p I ) is given by S C ,s ( p I ) = ( − v ) − d · L − (cid:88) i ∈ I + ε ∗ i + (cid:88) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j ⊗ O L I . By using (45), we also obtain S C ,s ( p I ) = (cid:88) K ⊂ I ( − v ) −| K | · L − (cid:88) i ∈ I + ∩ K ε ∗ i − (cid:88) i ∈ I − ∩ K c ε ∗ i + (cid:88) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j . (47)Here, the sum runs over all subsets of I . By exchanging I + and I − , we also obtain S − C ,s ( p I ) = (cid:88) K ⊂ I ( − v ) −| K | · L − (cid:88) i ∈ I − ∩ K ε ∗ i − (cid:88) i ∈ I + ∩ K c ε ∗ i + (cid:88) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j . (48)We note that these formulas depend only on the K¨ahler alcove containing the slope s . In this section, we reformulate Corollary 5.12 by using certain combinatorics of alcoves in h ∗ R anddetermine the K -theoretic canonical bases for toric hyper-K¨ahler manifolds.We write ι ( s ) = ( s , . . . , s n ) ∈ X ∗ ( T ) ⊗ Z R ∼ = R n . Using this data, we consider a periodic hyperplanearrangement in h ∗ R defined by H si,m = H i,m := { x ∈ h ∗ R | (cid:104) x, a i (cid:105) + s i = m } for any i = 1 , . . . , n and m ∈ Z . We note that if we use a different choice of the lift ι , then the resulting hyperplane arrangementis given by a translation of the original one. We also remark that by the condition s ∈ s ∗ reg , we have ∩ i ∈ C H i,m i = ∅ for any circuit C and any choice of m i ∈ Z . We denote by Alc s the set of connectedcomponents of h ∗ R \ ∪ i,m H i,m . We simply call an element of Alc s alcove.By using the data C , one can give a bijection between Alc s and F , where F is defined in (14). For any A ∈ Alc s , the closure A is a polytope and the linear form ξ : h ∗ R → R restricted to A takes its minimumat a vertex x A by the genericity of ξ . We set H s, ± i,m = H ± i,m := { x ∈ h ∗ R | ± ( (cid:104) x, a i (cid:105) + s i − m ) > } . If wewrite { x A } = ∩ i ∈ I H i,m i for some m i ∈ Z , then we have I ∈ B and A ⊂ (cid:84) i ∈ I + H + i,m i ∩ (cid:84) i ∈ I − H − i,m i . Wedefine a map ϕ C ,s : Alc s → F by ϕ C ,s ( A ) := ( (cid:80) i ∈ I m i α Ii , p I ). We note that this does not depend onthe choice of ξ ∈ C . It is easy to check that ϕ C ,s is a bijection by the genericity of s . We denote by ≤ C the partial order on Alc s induced from the partial order ≤ C ,s on F via ϕ C ,s . By using Lemma 5.4, weobtain (cid:88) i ∈ I m i α Ii − wt H i ∗ p I L ( s ) = x A . (49)This implies the following lemma. Lemma 5.13.
For any
A, B ∈ Alc s , A ≤ C B if and only if (cid:104) x A , ξ (cid:105) ≤ (cid:104) x B , ξ (cid:105) for any ξ ∈ C .40or any A ∈ Alc s , we set µ A := n (cid:88) i =1 (cid:98)(cid:104) x, a i (cid:105) + s i (cid:99) ε ∗ i ∈ X ∗ ( T ) (50)for some x ∈ A and consider the T -equivariant line bundle E ( A ) := L ( µ A ). Note that this definition doesnot depend on the choice of x ∈ A and also the choice of chamber C . If ϕ C ,s ( A ) = ( (cid:80) i ∈ I m i α Ii , p I ), thenwe have A ⊂ (cid:84) i ∈ I + H + i,m i ∩ (cid:84) i ∈ I − H − i,m i . Hence, we obtain E ( A ) = L (cid:16) − (cid:80) i ∈ I − ε ∗ i + (cid:80) nj =1 (cid:98)(cid:104) x A , a j (cid:105) + s j (cid:99) ε ∗ j (cid:17) .Since (cid:104) x A , a i (cid:105) + s i = m i ∈ Z for any i ∈ I and a j = − (cid:80) i ∈ I (cid:104) β Ij , b i (cid:105) a i for any j ∈ I c , we have (cid:98)(cid:104) x A , a j (cid:105) + s j (cid:99) = (cid:98)− (cid:88) i ∈ I (cid:104) x A , a i (cid:105)(cid:104) β Ij , b i (cid:105) + s j (cid:99) = (cid:98)− (cid:88) i ∈ I m i (cid:104) β Ij , b i (cid:105) + s j + (cid:88) i ∈ I s i (cid:104) β Ij , b i (cid:105)(cid:99) = (cid:88) i ∈ I m i (cid:104) α Ii , a j (cid:105) + (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) . Therefore, we obtain E ( A ) = (cid:89) i ∈ I a m i α Ii · L − (cid:88) i ∈ I − ε ∗ i + (cid:88) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j . (51)We set E C ,s ( p I ) := L (cid:16) − (cid:80) i ∈ I − ε ∗ i + (cid:80) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j (cid:17) . Moreover, if B ∈ Alc s is an alcove such that x A ∈ B and K ⊂ I is a subset such that {H k,m k } k ∈ K is the set of hyperplanes separating A and B ,then we have E ( B ) = (cid:89) i ∈ I a m i α Ii · L − (cid:88) i ∈ I + ∩ K ε ∗ i − (cid:88) i ∈ I − ∩ K c ε ∗ i + (cid:88) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j . (52)In particular, the RHS of (52) is of the form E C ,s ( p I (cid:48) ) for some I (cid:48) ∈ B up to some H -equivariantparameter shift.For any A, B ∈ Alc s , we write (cid:96) ( A, B ) the number of hyperplanes separating A and B . We set N ( A ) := { B ∈ Alc s | x A ∈ B } and define S ( A ) ∈ K T ( X ) by the following formula: S ( A ) := (cid:88) B ∈ N ( A ) ( − v ) − (cid:96) ( A,B ) E ( B ) . (53)We note that by Lemma 5.13, we have B ≤ C A for any B ∈ N ( A ). By (47) and (52), we obtain thefollowing formula expressing S C ,s ( ϕ C ,s ( A )) defined in Definition 3.12. Lemma 5.14.
For any A ∈ Alc s , we have S ( A ) = S C ,s ( ϕ C ,s ( A )).We now prove E ( A ) = E C ,s ( ϕ C ,s ( A )) in the notation of Conjecture 3.22. By (53), we obtain E ( A ) ∈ S ( A ) + (cid:88) B< C A v − Z [ v − ] · S ( B ) (54)under certain completion as in section 3.3. Therefore, it is enough to check the following. Proposition 5.15.
For any A ∈ Alc s , we have β K C ,s ( E ( A )) = E ( A ). Proof . Since {S C ,s ( p I ) } I ∈ B is a basis of K T ( X ) loc over Frac( K T (pt)), {E C ,s ( p I ) } I ∈ B is also a basis of K T ( X ) loc over Frac( K T (pt)) by (51) and (53). We define Frac( K H (pt))-linear involution β (cid:48) on K T ( X ) loc by 41 β (cid:48) ( vm ) = v − β (cid:48) ( m ) for any m ∈ K T ( X ) loc , • β (cid:48) ( E C ,s ( p I )) = E C ,s ( p I ) for any I ∈ B .By (51), we have β (cid:48) ( E ( A )) = E ( A ) for any A ∈ Alc s and hence β (cid:48) ( S C ,s ( p I )) = (cid:88) K ⊂ I ( − v ) | K | L − (cid:88) i ∈ I − ∩ K c ε ∗ i − (cid:88) i ∈ I + ∩ K ε ∗ i + (cid:88) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j = ( − v ) d (cid:88) K ⊂ I ( − v ) −| K | L − (cid:88) i ∈ I − ∩ K ε ∗ i − (cid:88) i ∈ I + ∩ K c ε ∗ i + (cid:88) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j = ( − v ) d S − C ,s ( p I )for any I ∈ B by (47), (48), and (52). This implies that β (cid:48) = β K C ,s and hence E ( A ) is bar invariant.Since the set {E ( A ) } A ∈ Alc s does not depend on the choice of C , we obtain Conjecture 3.17 for torichyper-K¨ahler manifolds. Corollary 5.16.
The K -theoretic bar involution β K C ,s does not depend on the choice of C .Next, we determine C C ,s ( ϕ C ,s ( A )) for any A ∈ Alc s . Recall the map ∂ : K T (pt) → Z [ v, v − ] definedin section 3.5. By Lemma 3.24 and Lemma 5.14, we have ∂ ( S ( A ) ||S ( B )) = δ A,B for any
A, B ∈ Alc s .By induction, this and (54) implies that for any A ∈ Alc s , there exists a unique element C ( A ) ∈ S ( A ) + (cid:88) B> C A v − Z [ v − ] · S ( B )such that ∂ ( C ( A ) ||E ( B )) = δ A,B for any
A, B ∈ Alc s . Since we have ∂ ( C ( A ) ||S ( B )) = (cid:88) C ∈ N ( B ) ( − v ) (cid:96) ( C,B ) ∂ ( C ( A ) ||E ( C ))= (cid:40) ( − v ) (cid:96) ( A,B ) if A ∈ N ( B )0 otherwise,we obtain C ( A ) = (cid:88) B ∈ N − ( A ) ( − v ) − (cid:96) ( A,B ) S ( B ) (55)where we set N − ( A ) := { B ∈ Alc s | A ∈ N ( B ) } . We note that N − ( A ) is a finite set. Lemma 5.17.
For any A ∈ Alc s , we have β K C ,s ( C ( A )) = v dim X · C ( A ). Proof . By Lemma 3.30 and Proposition 5.15, we have ∂ ( v − dim X · β K C ,s ( C ( A )) ||E ( B )) = v − dim X ∂ ( β K C ,s ( C ( A )) || β K C ,s ( E ( B )))= ∂ ( C ( A ) ||E ( B ))= δ A,B for any
A, B ∈ Alc s . Hence we obtain β K C ,s ( C ( A )) = v dim X · C ( A ).Following the notation in section 3.4, we set B X,s := {E ( A ) } A ∈ Alc s and B L,s := {C ( A ) } A ∈ Alc s andcall them K -theoretic canonical bases for K T ( X ) and K T ( L ). We will prove later (Corollary 5.37) that B X,s (resp. B L,s ) is actually a basis of K T ( X ) (resp. K T ( L )).42 .5 Wall-crossings In this section, we study the behavior of K -theoretic canonical bases under the variation of s ∈ s ∗ reg .We fix a signed circuit C = C + (cid:116) C − satisfying (cid:104) η, β C (cid:105) > w C := { x ∈ s ∗ R | (cid:104) x, β C (cid:105) ∈ Z } . We take a generic element s ∈ w C such that s does not lie in any w C (cid:48) for some circuit C (cid:48) (cid:54) = C . We consider two slopes s − , s + ∈ s ∗ reg sufficiently close to s such that they lie inthe same connected component of s ∗ R \ ∪ C (cid:48) (cid:54) = C w C (cid:48) as s and satisfy (cid:104) s − , β C (cid:105) < (cid:104) s , β C (cid:105) < (cid:104) s + , β C (cid:105) . Westudy the difference between K -theoretic canonical bases B X,s − and B X,s + . We fix a path γ connecting s − and s + in a neighborhood of s and passing through s .For ( λ, p I ) ∈ F , we consider the vertex x λ,I ( s ) corresponding to ϕ − C ,s ( λ, p I ) ∈ Alc s as in the pre-vious section. By (49), we have x λ,I ( s ) = λ − wt H i ∗ p I L ( s ). If s goes to s , then it can happenthat lim s → s x λ,I ( s ) = lim s → s x µ,J ( s ) for some ( µ, p J ) (cid:54) = ( λ, p I ) ∈ F . If this does not happen, thenhyperplanes other than H si,m i for i ∈ I will be away from x λ,I ( s ) along s ∈ γ . Hence we obtain E C ,s − ( λ, p I ) = E C ,s + ( λ, p I ). Lemma 5.18.
For I (cid:54) = J ∈ B , lim s → s x λ,I ( s ) = lim s → s x µ,J ( s ) for some λ, µ ∈ X ∗ ( H ) if and only if | I c ∩ C | = | J c ∩ C | = 1 and I ∩ C c = J ∩ C c . Proof . Assume that lim s → s x λ,I ( s ) = lim s → s x µ,J ( s ) for some ( µ, p J ) (cid:54) = ( λ, p I ) ∈ F . By (44), thisimplies that (cid:80) j ∈ I c ∩ J (cid:104) s , β Ij (cid:105) α Jj ∈ X ∗ ( H ) and hence (cid:104) s , β Ij (cid:105) ∈ Z for any j ∈ I c ∩ J . By the choice of s , we must have | I c ∩ J | = 1 and β Ij = ± β C for j ∈ I c ∩ J . In particular, we have j ∈ C ⊂ I ∪ { j } andhence we obtain | I c ∩ C | = 1. By exchanging the role of I and J , we also obtain I ∩ J c = { i } for some i and i ∈ C ⊂ J ∩ { i } . This implies that I ∩ C c = ( I \ { i } ) ∩ C c = ( J \ { j } ) ∩ C c = J ∩ C c .Conversely, we assume that I (cid:54) = J ∈ B satisfy | I c ∩ C | = | J c ∩ C | = 1 and I ∩ C c = J ∩ C c =: K . Ifwe set I c ∩ C = { j } and J c ∩ C = { i } , then we obtain I = ( C \ { j } ) ∪ K and J = ( C \ { i } ) ∪ K and hence I c ∩ J = { j } and I ∩ J c = { i } . Since we have j ∈ C ⊂ I ∪ { j } and i ∈ C ⊂ J ∪ { i } , we obtain β Ij = ± β C and β Ji = ± β C . By (44), we obtain lim s → s x λ,I ( s ) = lim s → s x µ,J ( s ) for some λ, µ ∈ X ∗ ( H ).Now we study the behavior of E C ,s ( λ, p I ) under the wall-crossing, where I satisfies | I c ∩ C | = 1and I ∩ C c = K for a fixed subset K ⊂ C c . We may assume that { a i } i ∈ K is linearly independentand { a i } i ∈ C ∪ K spans X ∗ ( H ). In this case, the number of such I ∈ B is given by | C | . We fix m i ∈ Z for each i ∈ C ∪ K such that ∩ i ∈ C ∪ K H s i,m i =: { x } is not empty. We note that this implies that (cid:80) i ∈ C + m i − (cid:80) i ∈ C − m i = (cid:104) s , β C (cid:105) . We consider all ( λ, p I ) ∈ F such that lim s → s x λ,I ( s ) = x . Thenumber of such ( λ, p I ) ∈ F is also given by | C | . By choosing s + , s − sufficiently close to s , we mayassume that there exists a convex neighborhood U of x ∈ h ∗ R containing all x λ,I ( s ) such that anyhyperplane of the form H si,m does not intersect U unless i ∈ C ∪ K and m = m i for any s ∈ γ . It isenough to consider the alcoves which intersect with U . We note that such an alcove A is characterizedby the sign (cid:15) : C ∪ K → {±} such that A ⊂ ∩ i ∈ C ∪ K H s,(cid:15) ( i ) i,m i .Let h C be the R -span of { a i } i ∈ C and h K be the R -span of { a i } i ∈ K . By the choice of K , we obtaina decomposition h R ∼ = h C × h K and this induces a decomposition h ∗ R ∼ = h ∗ C × h ∗ K such that (cid:104) h C , h ∗ K (cid:105) = 0and (cid:104) h K , h ∗ C (cid:105) = 0. We may also assume that U ∼ = U C × U K for some convex open subsets U C ⊂ h ∗ C and U K ⊂ h ∗ K . Since we have H si,m i = ( H si,m i ∩ h ∗ C ) × h ∗ K for each i ∈ C and H si,m i = h ∗ C × ( H si,m i ∩ h ∗ K ) foreach i ∈ K , they induce hyperplane arrangements on h ∗ C and h ∗ K . For each alcove A with A ∩ U (cid:54) = ∅ ,there is an alcove A C in h ∗ C and an alcove A K in h ∗ K such that A ∩ U ∼ = ( A C ∩ U C ) × ( A K ∩ U K ). Inparticular, we can consider alcoves for h ∗ C and h ∗ K separately in U .Since { a i } i ∈ K is linearly independent, ∩ i ∈ K H s,(cid:15) ( i ) i,m i ∩ U K is not empty and its volume is away from0 along s ∈ γ for any sign (cid:15) : K → {±} . For each sign (cid:15) : C → {±} , we set ∆ (cid:15) ( s ) := ∩ i ∈ C H s,(cid:15) ( i ) i,m i ∩ h ∗ C .We define (cid:15) ± : C → {±} by (cid:15) ( i ) = ± for any i ∈ C + and (cid:15) ( i ) = ∓ for any i ∈ C − . Lemma 5.19. If ±(cid:104) s − s , β C (cid:105) >
0, then ∆ (cid:15) ( s ) (cid:54) = ∅ unless (cid:15) = (cid:15) ∓ . Moreover, the volume of ∆ (cid:15) ( s ) ∩ U C isaway from 0 along s ∈ γ unless (cid:15) = (cid:15) ± and the volume of ∆ (cid:15) ± ( s ) ∩ U C is proportional to |(cid:104) s − s , β C (cid:105)| | C |− .43 roof . By definition, x ∈ ∆ (cid:15) ( s ) if and only if (cid:15) ( i ) · ( (cid:104) x, a i (cid:105) + s i − m i ) > i ∈ C . Since we have (cid:88) i ∈ C + ( (cid:104) x, a i (cid:105) + s i − m i ) − (cid:88) i ∈ C − ( (cid:104) x, a i (cid:105) + s i − m i ) = (cid:104) s − s , β C (cid:105) , such an x does not exist for (cid:15) = (cid:15) ∓ . If (cid:15) (cid:54) = (cid:15) ± and (cid:15) (cid:54) = (cid:15) ∓ , then ∆ (cid:15) ( s ) is also not empty and the volumeof ∆ (cid:15) ( s ) ∩ U C is positive. This implies the second statement.If (cid:15) = (cid:15) ± , then ∆ (cid:15) ± ( s ) is a ( | C | − U , every vertex of∆ (cid:15) ± ( s ) is contained in U C and hence ∆ (cid:15) ± ( s ) ⊂ U C . Since each edge of ∆ (cid:15) ± ( s ) has length proportionalto |(cid:104) s − s , β C (cid:105)| , its volume is proportional to |(cid:104) s − s , β C (cid:105)| | C |− .For each sign (cid:15) : C ∪ K → {±} satisfying ∩ i ∈ C ∪ K H s,(cid:15) ( i ) i,m i (cid:54) = ∅ , we denote by A (cid:15),x ( s ) ∈ Alc s theunique alcove which is contained in ∩ i ∈ C ∪ K H s,(cid:15) ( i ) i,m i and intersect with U . By definition, we obtain E ( A (cid:15),x ( s )) = L − (cid:88) i ∈ (cid:15) − ( − ) ε ∗ i ⊗ L x for any s ∈ γ , where we set L x := L (cid:32) (cid:88) i ∈ C ∪ K m i ε ∗ i + (cid:88) i/ ∈ C ∪ K (cid:98)(cid:104) x , a i (cid:105) + ι ( s ) i (cid:99) ε ∗ i (cid:33) . Lemma 5.19 implies that for any subset C (cid:48) ⊂ C and K (cid:48) ⊂ K , L ( − (cid:80) i ∈ C (cid:48) ∪ K (cid:48) ε ∗ i ) ⊗ L x is contained in B X,s ± if and only if C (cid:48) (cid:54) = C ± . In particular, L ( − (cid:80) i ∈ C − ∪ K (cid:48) ε ∗ i ) ⊗ L x is contained in B X,s + but not in B X,s − , and L ( − (cid:80) i ∈ C + ∪ K (cid:48) ε ∗ i ) ⊗ L x is contained in B X,s − but not in B X,s + . In summary, we obtainedthe following. Proposition 5.20.
For any element
E ∈ B X,s + , E is not contained in B X,s − if and only if the volumeVol( A ( s )) of A ( s ) vanishes under the limit s → s , where A ( s ) ∈ Alc s is the alcove satisfying E ( A ( s )) = E for any s ∈ s ∗ R , reg contained in the same K¨ahler alcove as s + . If this holds, then the order of vanishingof Vol( A ( s )) at s = s is given by | C | − L (cid:16)(cid:80) i ∈ C − ε ∗ i − (cid:80) i ∈ C + ε ∗ i (cid:17) ⊗ E is contained in B X,s − butnot in B X,s + . Moreover, L (cid:16)(cid:80) i ∈ C (cid:48) ∩ C − ε ∗ i − (cid:80) i ∈ C (cid:48) ∩ C + ε ∗ i (cid:17) ⊗ E is contained in both B X,s + and B X,s − for any subset ∅ (cid:54) = C (cid:48) (cid:40) C .Now the wall-crossing formula relating B X,s + and B X,s − follows from the following lemma. Lemma 5.21.
For any signed circuit C satisfying (cid:104) η, β C (cid:105) >
0, there exists an exact sequence0 → W | C | → · · · → W → W → X , where W k := (cid:77) C (cid:48) ⊂ C | C (cid:48) | = k v k L (cid:88) i ∈ C (cid:48) ∩ C − ε ∗ i − (cid:88) i ∈ C (cid:48) ∩ C + ε ∗ i . Proof . Since we have (cid:104) η, β C (cid:105) = (cid:80) j ∈ C ∩ I c (cid:104) η, β Ij (cid:105)(cid:104) b j , β C (cid:105) >
0, we must have C + ∩ I c + (cid:54) = ∅ or C − ∩ I c − (cid:54) = ∅ for any I ∈ B . This implies that the subvariety of X defined by x i = 0 for any i ∈ C + and y i = 0for any i ∈ C − is empty. By considering the Koszul complex for these equations, we obtain an exactsequence of the form (56). 44n particular, Proposition 5.20 and Lemma 5.21 imply the first part of Conjecture 3.49 by taking l = 1, n = 0, n = | C | , B s ± ,w = B X,s − ∩ B X,s + , and B s ± ,w = B X,s ± \ B s ± ,w . In section 5.11, we willshow that the central charge of C ( A ( s )) is given by Vol( A ( s )) for any A ( s ) ∈ Alc s . This implies thesecond part of Conjecture 3.49.As another application of these results, we obtain the following. Corollary 5.22.
For any s ∈ s ∗ reg , the vector bundle T C ,s := ⊕ I ∈ B E C ,s ( p I ) weakly generate D (QCoh( X )). Proof . We note that any line bundle of the form L ( λ ) for λ ∈ X ∗ ( T ) is contained in B X,s (cid:48) for some s (cid:48) ∈ s ∗ reg by Lemma 3.34. By connecting s and s (cid:48) by a generic path and applying Proposition 5.20and Lemma 5.21 each time when the path crosses a wall, we obtain that L ( λ ) is contained in the fulltriangulated subcategory of D b Coh( X ) generated by {E C ,s ( p I ) } I ∈ B for any λ ∈ X ∗ ( T ). In particular, R Hom( T C ,s , F ) = 0 implies that R Γ( F ⊗ L ) = 0 for any sufficiently ample line bundles L . This implies F ∼ = 0.
In this section, we collect some results about linear programming which will be used in the proof ofConjecture 3.38 and Conjecture 3.40 for toric hyper-K¨ahler manifolds. Our reference for the theory oflinear programming is [3].We first prepare some notations about sign vectors. For x ∈ R , we set σ ( x ) := + if x > − if x <
00 if x = 0and for x = ( x , . . . , x n ) ∈ R n , we set σ ( x ) := ( σ ( x ) , . . . , σ ( x n )) ∈ { + , − , } n . We will be interested inthe sign patterns σ ( V ) := { σ ( x ) | x ∈ V } ⊂ { + , − , } n for a vector subspace V ⊂ R n .We set E := { , . . . , n } . For a sign vector Y ∈ { + , − , } E , we define Y + := { i ∈ E | Y i = + } , Y − := { i ∈ E | Y i = −} , and Y := { i ∈ E | Y i = 0 } . We define its support by Supp( Y ) := Y + ∪ Y − .For a subset I ⊂ E , we write Y I ≥ Y i ∈ { + , } , Y I ≤ Y i ∈ {− , } , and Y I = 0 if Y i = 0 for any i ∈ I . Definition 5.23.
Two sign vectors
Y, Z ∈ { + , − , } E are called orthogonal if( Y + ∩ Z + ) ∪ ( Y − ∩ Z − ) (cid:54) = ∅ ⇔ ( Y + ∩ Z − ) ∪ ( Y − ∩ Z + ) (cid:54) = ∅ . This is denoted by Y ⊥ Z . For a subset F ∈ { + , − , } E , the set F ⊥ := { Y ∈ { + , − , } E | Y ⊥ Z for any Z ∈ F} is called the orthogonal complement of F .For a vector subspace V ⊂ R n , we denote by V ⊥ ⊂ R n the orthogonal complement of V ⊂ R n withrespect to the standard inner product on R n . Proposition 5.24.
For any vector subspace V ⊂ R n , we have σ ( V ) ⊥ = σ ( V ⊥ ). Proof . See Corollary 5.42 in [3].
Definition 5.25.
For a subset
F ⊂ { + , − , } E and disjoint subsets I, J ⊂ E , we set F \
I/J := { Y ∈ { + , − , } E \ ( I ∪ J ) | ∃ Z ∈ F s.t. Z i = 0 for i ∈ I and Z e = Y e for e ∈ E \ ( I ∪ J ) } . This is called the minor of F obtained by deleting I and contracting J .45 emma 5.26. For any vector subspace V ⊂ R n and disjoint subsets I, J ⊂ E , we have( σ ( V ) \ I/J ) ⊥ = σ ( V ) ⊥ \ J/I.
Proof . This follows from Proposition 5.24. See also Lemma 5.51 and Lemma 5.52 in [3].
Definition 5.27.
For a subset
F ⊂ { + , − , } E , nonzero Y ∈ F is called elementary sign vector of F if ∅ (cid:54) = Supp( Z ) ⊂ Supp( Y ) implies Supp( Z ) = Supp( Y ) for any Z ∈ F . The set of all elementary signvectors of F is denoted by elem( F ). Proposition 5.28.
For any vector subspace V ⊂ R n , we have elem( σ ( V )) ⊥ = σ ( V ) ⊥ . Proof . See Corollary 5.37 in [3].
Definition 5.29.
For
Y, Z ∈ { + , − , } E , we write Y (cid:22) Z and say that Y conforms to Z if Y + ⊂ Z + and Y − ⊂ Z − . This relation defines a partial order (cid:22) on { + , − , } E . Lemma 5.30.
For any vector subspace V ⊂ R n , elem( σ ( V )) coincides with the set of minimal nonzeroelements of σ ( V ) with respect to the partial order (cid:22) . Proof . See Lemma 5.30 in [3].
Proposition 5.31 (Minty’s Lemma) . For any vector subspace V ⊂ R n and every partition E = R (cid:116) G (cid:116) B (cid:116) W with e ∈ R (cid:116) G , exactly one of the following holds: • There exists Y ∈ σ ( V ) such that e ∈ Supp( Y ), Y R ≥ Y G ≤
0, and Y W = 0. • There exists Z ∈ σ ( V ⊥ ) such that e ∈ Supp( Z ), Z R ≥ Z G ≤
0, and Z B = 0. Proof . See Proposition 5.12 in [3].In the below, we will consider the case V = Ker( b ) ⊗ Z R ⊂ X ∗ ( T ) ⊗ Z R ∼ = R n , where the identification X ∗ ( T ) ⊗ Z R ∼ = R n is given by the fixed basis { ε ∗ , . . . , ε ∗ n } . We note that the sign vectors σ ( V ) doesnot change without tensoring R . In this case, we have V ⊥ = Ker( a ) ⊗ Z R and hence elem( σ ( V ⊥ )) = { σ ( β C ) | C = C + (cid:116) C − : signed circuit } . In this section, we recall a description of cohomology of line bundles on toric varieties. Since we onlyconsider semi-projective toric varieties in this paper, we restrict our attention to these cases. Let0 ≤ r ≤ N be nonnegative integers and consider an exact sequence of tori1 → T r → T m → T m − r → , where T k := ( C × ) k for k ∈ Z ≥ . Let0 → X ∗ ( T r ) t B −→ X ∗ ( T m ) A −→ X ∗ ( T m − r ) → → X ∗ ( T m − r ) t A −→ X ∗ ( T m ) B −→ X ∗ ( T r ) → a i := A ( ε i ) and b i := B ( ε ∗ i ) for a fixed basis { ε , . . . , ε m } of X ∗ ( T m ) and its dual basis { ε ∗ , . . . , ε ∗ m } of X ∗ ( T m ). We fix ageneric element η ∈ (cid:80) mi =1 Z ≥ b i such that if η is contained in a cone of the form R ≥ b i + · · · R ≥ b i l for { i , . . . , i l } ⊂ { , . . . , m } , then { b i , . . . , b i l } generates X ∗ ( T r ) ⊗ Z R . We setΩ η := { I ⊂ { , . . . , m } | η ∈ (cid:88) i ∈ I R ≥ b i } X ∗ ( T m − r ) ⊗ Z R by Σ := { σ I | I ∈ Ω η } , where σ I := (cid:80) j ∈ I c R ≥ a j . Let X (Σ)be the toric variety associated with the fan Σ. By Theorem 2.4 in [18], X (Σ) is isomorphic to theGIT quotient ( C m ) η − ss //T r . For simplicity, we assume that X (Σ) is smooth, i.e., { a j } j ∈ I c generates X ∗ ( T m − r ) over Z for any I ∈ Ω η such that | I | = r . In this case, the action of T r on ( C m ) η − ss is free andhence one can define a T m − r -equivariant line bundle L ( λ ) associated with each character λ ∈ X ∗ ( T m ).We set R (Σ) := C [ x , . . . , x m ] with an X ∗ ( T m )-grading given by deg( x i ) = ε ∗ i . Let B (Σ) = (cid:0)(cid:81) i ∈ I x i | I ∈ Ω η (cid:1) be a monomial ideal of R (Σ). Since B (Σ) is generated by homogeneous elements,the local cohomology H iB (Σ) ( R (Σ)) of R (Σ) with supports in B (Σ) is also X ∗ ( T m )-graded. We denoteby R (Σ) λ and H iB (Σ) ( R (Σ)) λ the weight λ parts of R (Σ) and H iB (Σ) ( R (Σ)) for any λ ∈ X ∗ ( T m ). Onecan relate them and the cohomology of the line bundle L ( λ ) as follows. Lemma 5.32.
For any λ ∈ X ∗ ( T m ) and i ≥
1, we have H i ( X (Σ) , L ( λ )) T m ∼ = H i +1 B (Σ) ( R (Σ)) λ . Proof . See for example Theorem 9.5.7 in [13].Next we recall a description of H iB (Σ) ( R (Σ)) λ in terms of simplicial cohomology due to Mustat¸ˇa[29]. For each i = 1 , . . . , m , let ∆ i := {I ⊂ Ω η | i / ∈ ∪ I ∈I I } be a simplicial complex on Ω η . Fora subset M ⊂ { , . . . , m } , we set ∆ M := ∪ i ∈ M ∆ i . Here, we understand that if M = ∅ , then ∆ M is the void complex which has trivial reduced cohomology. For λ = ( λ , . . . , λ m ) ∈ Z m , we defineneg( λ ) := { i ∈ { , . . . , m } | λ i < } . For a simplicial complex ∆, we denote by ˜ H i (∆) the i -th reducedcohomology group of ∆. Lemma 5.33 ([29]) . For each λ ∈ Z m and i ∈ Z ≥ , we have H iB (Σ) ( R (Σ)) λ ∼ = ˜ H i − (∆ neg( λ ) ). Proof . See Theorem 2.1 in [29].We will also need the following special case of Demazure vanishing theorem.
Lemma 5.34.
We have H > ( X (Σ) , O ) = 0. Proof . Since | Σ | = (cid:80) ni =1 R ≥ a i is convex, this follows from Demazure vanishing theorem, see forexample Theorem 9.2.3 in [13].For example, we may apply the above results for Lawrence toric varieties. For this, we take m = 2 n and change the index set { , . . . , m } by E n := {± , . . . , ± n } . We take b ± i = ± b i for i = 1 , . . . , n . Inthis case, we can take a i = ( a i , e i ) ∈ Z d ⊕ Z n and a − i = (0 , e i ) ∈ Z d ⊕ Z n , where { e , . . . , e n } is a basisof Z n . We also take the same η . By definition, the toric variety X (Σ) associated with these data is theLawrence toric variety X . By our assumption that a i (cid:54) = 0 for any i = 1 , . . . , n , we have E n \{ e } ∈ Ω η forany e ∈ E n . In particular, the set of one dimensional cones Σ(1) in Σ is given by Σ(1) = { R ≥ a e } e ∈ E and hence the ring R (Σ) = C [ x , . . . , x n , y , . . . , y n ] coincides with the Cox’s homogeneous coordinatering [12] for the Lawrence toric variety X , where we write y i for the variable corresponding to − i ∈ E n .Therefore, we obtain a ring isomorphism R (Σ) ∼ = ⊕ λ ∈ X ∗ ( T n ) Γ( X , (cid:101) L ( λ )) T n (57)by Proposition 1.1 in [12]. If we restrict the torus action of T n on R (Σ) to T via T → T n givenby t (cid:55)→ ( t, t − ), then the weight µ ∈ X ∗ ( T ) part of R (Σ) is given by C [ u , . . . , u n ] · x µ , where we set u i := x i y i and x µ := (cid:81) i,µ i > x µ i i (cid:81) i,µ i < y − µ i i . Combined with (57), we obtainΓ( X , (cid:101) L ( µ )) T ∼ = C [ u , . . . , u n ] · x µ . (58)We note that the degree of x µ with respect to S -action is given by (cid:80) i | µ i | . For each i = 1 , . . . , n ,the section x i ∈ Γ( X , (cid:101) L ( ε ∗ i )) T gives a C [ u , . . . , u n ]-module homomorphism x i · : Γ( X , (cid:101) L ( µ − ε ∗ i )) T → Γ( X , (cid:101) L ( µ )) T of degree 1. In terms of the isomorphism (58), we have x i · x µ − ε ∗ i = (cid:40) x µ if µ i > ,u i · x µ if µ i ≤ . (59)47imilarly, the section y i ∈ Γ( X , (cid:101) L ( − ε ∗ i )) T gives a C [ u , . . . , u n ]-module homomorphism y i · : Γ( X , (cid:101) L ( µ + ε ∗ i )) T → Γ( X , (cid:101) L ( µ )) T given by y i · x µ + ε ∗ i = (cid:40) x µ if µ i < ,u i · x µ if µ i ≥ . (60) In this section, we check Conjecture 3.38 for toric hyper-K¨ahler manifolds, i.e., T C ,s := ⊕ I ∈ B E C ,s ( p I ) is atilting bundle on X . The main result of this section is proved via different methods by McBreen-Webster[28] and ˇSpenko-Van den Bergh [40] independently. We give still another direct proof of it.We note that for any t, t (cid:48) ∈ R , we have (cid:98) t (cid:99) + (cid:98) t (cid:48) (cid:99) ≤ (cid:98) t + t (cid:48) (cid:99) ≤ (cid:98) t (cid:99) + (cid:98) t (cid:48) (cid:99) + 1 , (cid:98) t (cid:99) − (cid:98) t (cid:48) (cid:99) − ≤ (cid:98) t − t (cid:48) (cid:99) ≤ (cid:98) t (cid:99) − (cid:98) t (cid:48) (cid:99) . These inequalities easily imply that for any A ∈ Alc s and signed circuit C , we have (cid:98)(cid:104) s, β C (cid:105)(cid:99) − | C + | + 1 ≤ (cid:104) µ A , β C (cid:105) ≤ (cid:98)(cid:104) s, β C (cid:105)(cid:99) + | C − | , where µ A is defined as in (50). In particular, we have −| C | + 1 ≤ (cid:104) µ A − µ A (cid:48) , β C (cid:105) ≤ | C | − A, A (cid:48) ∈ Alc s .Recall that one can associate a T -equivariant line bundle (cid:101) L ( λ ) on the Lawrence toric variety X foreach λ ∈ X ∗ ( T ). We define π : X ∗ ( T n ) → X ∗ ( T ) by π ( λ , . . . , λ n , λ − , . . . , λ − n ) = (cid:80) ni =1 ( λ i − λ − i ) ε ∗ i .By Lemma 5.32 and Lemma 5.33, we obtain H > ( X , (cid:101) L ( λ )) T ∼ = ⊕ ˜ λ ∈ π − ( λ ) ˜ H ≥ (∆ neg(˜ λ ) ) . (62)We first prove the vanishing of higher extensions on the level of Lawrence toric variety. Proposition 5.35.
For any
A, A (cid:48) ∈ Alc s , we have H > ( X , (cid:101) L ( µ A − µ A (cid:48) )) = 0. Proof . It is enough to prove H > ( X , (cid:101) L ( µ A − µ A (cid:48) )) T = 0 since we have H > ( X , (cid:101) L ( µ A − µ A (cid:48) )) ∼ = ⊕ α ∈ X ∗ ( H ) H > ( X , (cid:101) L ( µ A + α − µ A (cid:48) )) T We set µ := µ A − µ A (cid:48) ∈ X ∗ ( T ) and Y := σ ( µ ) ∈ { + , − , } n . We note that if λ ∈ X ∗ ( T ) satisfies b ( λ ) = 0, then Lemma 5.34 and (62) imply that ˜ H ≥ (∆ neg(˜ λ ) ) = 0 for any lift ˜ λ ∈ π − ( λ ). Therefore,it is enough to prove that for any lift ˜ µ ∈ π − ( µ ), there exists λ ∈ Ker( b ) and ˜ λ ∈ π − ( λ ) such thatneg(˜ µ ) = neg(˜ λ ). If ± µ i >
0, then the possibilities for the set neg(˜ µ i , ˜ µ − i ) is ∅ , {∓ i } , or { i, − i } . If µ i = 0, then the possibilities for the set neg(˜ µ i , ˜ µ − i ) is ∅ or { i, − i } , which is contained in the possibilitiesfor µ i (cid:54) = 0. This implies that it is enough to prove the existence of λ ∈ Ker( b ) such that σ ( µ i ) = σ ( λ i )if µ i (cid:54) = 0, i.e., Y | Supp( Y ) ∈ σ (Ker( b )) /Y . By Proposition 5.24, Lemma 5.26, and Proposition 5.28, wehave σ (Ker( b )) /Y = elem( σ (Ker( a )) \ Y ) ⊥ . Since elem( σ (Ker( a )) \ Y ) = { σ ( β C ) | C : signed circuit , C ⊂ Supp( Y ) } , if Y | Supp( Y ) / ∈ σ (Ker( b )) /Y ,then there exists a signed circuit C ⊂ Supp( Y ) such that Y is not orthogonal to σ ( β C ). We may assumethat ( Y + ∩ C − ) ∪ ( Y − ∩ C + ) = ∅ . This and C ⊂ Supp( Y ) imply that for each i ∈ C ± , we have ± µ i ≥ (cid:104) µ, β C (cid:105) ≥ | C | which contradicts (61). Corollary 5.36.
The vector bundle T C ,s is a tilting bundle on X .48 roof . By Corollary 5.22, it is enough to prove Ext > ( T C ,s , T C ,s ) = 0, i.e., H > ( X, L ( µ A − µ A (cid:48) )) = 0for any A, A (cid:48) ∈ Alc s . Let i : { } (cid:44) → s ∗ be the inclusion and recall the morphism µ X : X → s ∗ definedbefore Lemma 5.3. Since µ X is flat, i and µ X are Tor independent and hence the base change formulaimplies R Γ( X, L ( µ A − µ A (cid:48) )) ∼ = Li ∗ Rµ X ∗ (cid:101) L ( µ A − µ A (cid:48) ) . (63)Since Rµ X ∗ (cid:101) L ( µ A − µ A (cid:48) ) is concentrated on cohomological degree 0 by Proposition 5.35, the RHS of(63) has vanishing cohomology at positive degree. This proves H > ( X, L ( µ A − µ A (cid:48) )) = 0.Combined with Proposition 5.15 and Lemma 5.17, we obtain Conjecture 3.22 for toric hyper-K¨ahlermanifolds. Corollary 5.37. B X,s (resp. B L,s ) is a Z [ v, v − ]-basis of K T ( X ) (resp. K T ( L )). Proof . Corollary 5.36 implies that B X,s generates K T ( X ) over Z [ v, v − ] and hence it is a basis. Sincethe pairing ( − : − ) defined in (2) induces a perfect pairing between K T ( X ) and K T ( L ), the pairing( −||− ) also gives a perfect pairing between K T ( X ) and K T ( L ). Therefore, the dual basis B L,s of B X,s is a Z [ v, v − ]-basis of K T ( L ).As a module over C [ u , . . . , u n ] ∼ = C [ t ∗ ], we have R (Σ) ∼ = ⊕ µ ∈ X ∗ ( T ) C [ t ∗ ] · x µ and the C [ s ∗ ]-modulestructure coming from the morphism µ X : X → s ∗ is the one induced from the natural inclusion C [ s ∗ ] (cid:44) → C [ t ∗ ]. We also denote by x µ ∈ Γ( X, L ( µ )) H the section coming from x µ ∈ Γ( X , (cid:101) L ( µ )) T . As acorollary of (63), we obtain the following. Lemma 5.38. If H > ( X , (cid:101) L ( µ )) = 0 for µ ∈ X ∗ ( T ), then we have Γ( X, L ( µ )) H ∼ = C [ h ∗ ] · x µ .As in section 3.6, we set A C ,s := End( T C ,s ) opp . By Corollary 5.36, we obtain a derived equivalence ψ C ,s : D b Coh T ( X ) ∼ = D b ( A C ,s -gmod H ) (64)given by ψ C ,s ( F ) = R Hom( T C ,s , F ) for F ∈ D b Coh T ( X ).We next give a presentation of the ring A C ,s . We set µ I := − (cid:88) i ∈ I − ε ∗ i + (cid:88) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j (65)so that T C ,s = ⊕ I ∈ B L ( µ I ) for any I ∈ B . Let e I ∈ A C ,s be the idempotent corresponding to the identitymap in Hom( L ( µ I ) , L ( µ I )). For I, J ∈ B , the H -weight space of e I A C ,s e J of weight α ∈ X ∗ ( H ) is givenby Hom( L ( µ I + α ) , L ( µ J )) H ∼ = C [ h ∗ ] · x µ J − µ I − α by Lemma 5.38. We set m αIJ := x µ J − µ I − α ∈ e I A C ,s e J .We note that e I = m II and C [ h ∗ ] = C [ X ] H is contained in the center of A C ,s . For I, J, J (cid:48) , K ∈ B and α, α (cid:48) ∈ X ∗ ( H ), we obtain m αIJ · m α (cid:48) J (cid:48) K = δ J,J (cid:48) (cid:89) i : µ i µ (cid:48) i < u min {| µ i | , | µ (cid:48) i |} i · m α + α (cid:48) IK , (66)where we set µ = µ J − µ I − α and µ (cid:48) = µ K − µ J (cid:48) − α (cid:48) . In summary, we obtained the following. Lemma 5.39.
The C [ h ∗ ]-algebra A C ,s is isomorphic to (cid:77) I,J ∈ B α ∈ X ∗ ( H ) C [ h ∗ ] · m αIJ , where the multiplication rule is given by (66). Moreover, h ⊂ C [ h ∗ ] have H -weight 0 and degree 2, and m αIJ has H -weight α and degree (cid:80) i | µ i | , where µ = µ J − µ I − α .49or another presentation of the algebra A C ,s which is apparently quadratic and a presentation of itsKoszul dual B C ,s , see [28]. This presentation is enough to check Conjecture 3.44. Corollary 5.40.
The algebra A C ,s and B C ,s has an anti-involution which is identity on degree 0 part,compatible with the grading, and reversing the H -weights. Proof . For the algebra A C ,s , we define the C [ h ∗ ]-algebra anti-involution by sending m αIJ to m − αJI . Itis easy to check that this preserves the relation (66) and satisfies the required conditions. It is alsoeasy to check that this anti-involution induces a similar anti-involution on its quadratic dual which isisomorphic to B C ,s . In this section, we prove Conjecture 3.23 and the first part of Conjecture 3.40 for toric hyper-K¨ahlermanifolds. For A ∈ Alc s with ϕ C ,s ( A ) = ( λ, p I ), we define a categorical lift ∆ C ,s ( A ) ∈ D b Coh T ( X ) of S ( A ) by the formula∆ C ,s ( A ) := v − d a λ · L − (cid:88) i ∈ I + ε ∗ i + (cid:88) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j ⊗ O L I [ − d ] . We set ∇ C ,s ( A ) := v d ∆ − C ,s ( A )[ d ]. By definition and (51), we have ∇ C ,s ( A ) = E ( A ) ⊗ O L − I , where L − I is the subvariety of X defined by x i = 0 ( i ∈ I + ) and y i = 0 ( i ∈ I − ). For any subset K ⊂ I ,we set µ A,K := µ A − (cid:80) j ∈ K ∩ I + ε ∗ j + (cid:80) j ∈ K ∩ I − ε ∗ j . We note that by (52), we have L ( µ A,K ) ∈ B X,s forany K ⊂ I . By Lemma 5.8, we have the following exact sequence:0 → v d L ( µ A,I ) → (cid:77) K ⊂ I | K | = d − v d − L ( µ A,K ) → · · · → (cid:77) i ∈ I v L ( µ A, { i } ) → L ( µ A ) → ∇ C ,s ( A ) → . (67)We recall that this is given by Koszul resolution. Proposition 5.41.
For any
A, B ∈ Alc s , we have R (cid:54) =0 Hom( E ( B ) , ∇ C ,s ( A )) = 0 andHom( E ( B ) , ∇ C ,s ( A )) T = (cid:40) C · x µ if ± µ i ≤ i ∈ I ± , , (68)where µ = µ A − µ B and x µ is the image of x µ ∈ Γ( X, L ( µ )) T under the natural map Γ( X, L ( µ )) T → Γ( X, ∇ C ,s ( p I ) ⊗ L ( − µ B )) T coming from (67). Proof . By Proposition 5.35, we have H > ( X , (cid:101) L ( µ A,K − µ B )) = 0 for any K ⊂ I . Therefore, Lemma 5.38and (67) imply that R Hom( E ( B ) , ∇ C ,s ( A )) T is given by0 → C [ h ∗ ] · x µ A,I − µ B → · · · → (cid:77) i ∈ I + C [ h ∗ ] · x µ − ε ∗ i ⊕ (cid:77) i ∈ I − C [ h ∗ ] · x µ + ε ∗ i → C [ h ∗ ] · x µ → , where the complex is given by Koszul type complex for x i ( i ∈ I + ) and y i ( i ∈ I − ). If ± µ i ≤ i ∈ I ± , then by (59) and (60), this complex is isomorphic to the Koszul complex of C [ h ∗ ] withrespect to the regular sequence { u i } i ∈ I . This implies that its 0-th cohomology is one dimensional andother cohomologies vanish. If ± µ i > i ∈ I ± , then this complex is isomorphic to the Koszulcomplex of C [ h ∗ ] with respect to a sequence containing 1. Therefore, all the cohomologies vanish in thiscase. 50 orollary 5.42. For any A ∈ Alc s , ψ C ,s ( ∇ C ,s ( A )) ∈ D b ( A C ,s -gmod H ) is a Koszul module of A C ,s . Proof . Proposition 5.41 implies that ψ C ,s ( ∇ C ,s ( A )) is contained in the standard heart of D b ( A C ,s -gmod H ).The exact sequence (67) implies that ψ C ,s ( ∇ C ,s ( A )) is a Koszul module of A C ,s .Next we give a formula expressing S ( A ) (resp. E ( A )) in terms of {C ( B ) } B ∈ Alc s (resp. {S ( B ) } B ∈ Alc s ).For A ∈ Alc s , let M ( A ) be the set of alcove B such that for any hyperplane H i,m passing through x A , A and B are on the same side with respect to H i,m . We note that B ∈ M ( A ) implies B ≥ C A . Wealso set M − ( A ) := { B ∈ Alc s | A ∈ M ( B ) } . In terms of the combinatorics of alcoves, the conditionappearing in (68) can be written as B ∈ M ( A ) or not. Moreover, the degree of x µ is given by (cid:96) ( A, B ). Corollary 5.43.
For any A ∈ Alc s , we have S ( A ) = (cid:88) B ∈ M ( A ) v − (cid:96) ( A,B ) C ( B ) , (69) E ( A ) = (cid:88) B ∈ M − ( A ) v − (cid:96) ( A,B ) S ( B ) . (70) Proof . As in the proof of Corollary 3.39, Proposition 5.41 implies that ∂ ( E ( B ) ||S ( A )) = (cid:2) Hom( E ( B ) , ∇ C ,s ( A )) T (cid:3) ∨ = (cid:40) v − (cid:96) ( A,B ) if B ∈ M ( A )0 if B / ∈ M ( A ) . This implies (69) and (70).
Corollary 5.44.
Conjecture 3.23 holds for toric hyper-K¨ahler manifolds.
Proof . This follows from (55) and (70).
In this section, we prove the second half of Conjecture 3.40.
Theorem 5.45.
For any
A, B ∈ Alc s , we have R Hom(∆ C ,s ( B ) , ∇ C ,s ( A )) T ∼ = (cid:40) C if A = B A (cid:54) = B as S -modules, where C is considered as a trivial S -module sitting in cohomological degree 0. In particular,Conjecture 3.40 holds for toric hyper-K¨ahler manifolds. Proof . Let
A, B ∈ Alc s be two alcoves with ϕ C ,s ( A ) = ( λ A , p I ) and ϕ C ,s ( B ) = ( λ B , p J ). We recallthat µ A = λ A − (cid:80) i ∈ I − ε ∗ i + (cid:80) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j and µ B = λ B − (cid:80) i ∈ J − ε ∗ i + (cid:80) j ∈ J c (cid:98)(cid:104) s, β Jj (cid:105)(cid:99) ε ∗ j . We set µ := µ A − µ B and K := { k ∈ I ∩ J | x A , x B ∈ H k,m k for some m k ∈ Z } . We note that we have µ k = k ∈ K ∩ I + ∩ J − , − k ∈ K ∩ I − ∩ J + , k ∈ K ∩ (( I + ∩ J + ) ∪ ( I − ∩ J − )) . By Lemma 5.8, ∆ C ,s ( B ) is quasi-isomorphic to0 → E ( B ) → (cid:77) B (cid:48) ∈ N ( B ) (cid:96) ( B,B (cid:48) )=1 v − E ( B (cid:48) ) → (cid:77) B (cid:48) ∈ N ( B ) (cid:96) ( B,B (cid:48) )=2 v − E ( B (cid:48) ) → · · · , E ( B ) sits in cohomological degree 0. If Hom( E ( B (cid:48) ) , ∇ C ,s ( A )) T (cid:54) = 0 for B (cid:48) ∈ N ( B ), then we have B (cid:48) ∈ M ( A ) by Proposition 5.41. If ± µ i > i ∈ I ± ∩ K c , then we have M ( A ) ∩ N ( B ) = ∅ andhence R Hom(∆ C ,s ( B ) , ∇ C ,s ( A )) T = 0. Therefore, we may assume ± µ i ≤ i ∈ I ± ∩ K c .Let K (cid:48) ⊂ J be the subset satisfying µ B (cid:48) = µ B − (cid:80) j ∈ K (cid:48) ∩ J + ε ∗ j + (cid:80) j ∈ K (cid:48) ∩ J − ε ∗ j . We set µ K (cid:48) := µ A − µ B (cid:48) .The condition B (cid:48) ∈ M ( A ) implies that we must have K (cid:48) ⊃ K := K ∩ (( I − ∩ J + ) ∪ ( I + ∩ J − )) and K (cid:48) ∩ K ∩ (( I + ∩ J + ) ∪ ( I − ∩ J − )) = ∅ , i.e., K (cid:48) ⊂ K := ( J ∩ K c ) ∪ K . Conversely, the condition K ⊂ K (cid:48) ⊂ K implies B (cid:48) ∈ M ( A ) by the assumption that ± µ i ≤ i ∈ I ± ∩ K c . Therefore, R Hom(∆ C ,s ( B ) , ∇ C ,s ( A )) T is quasi-isomorphic to a Koszul type complex0 → C · x µ K → · · · → (cid:77) j ∈ K ∩ K c C · x µ K ∪{ j } → C · x µ K → . (71)Here, each map is induced from x j · : C [ h ∗ ] · x µ K (cid:48) − ε ∗ j → C [ h ∗ ] · x µ K (cid:48) for j ∈ J − ∩ K c and y j · : C [ h ∗ ] · x µ K (cid:48) + ε ∗ j → C [ h ∗ ] · x µ K (cid:48) for j ∈ J + ∩ K c . By (59) and (60), the map x j · : C · x µ (cid:48) − ε ∗ j → C · x µ (cid:48) isgiven by x j · x µ (cid:48) − ε ∗ j = (cid:40) x µ (cid:48) if µ (cid:48) j >
00 if µ (cid:48) j ≤ y j · : C · x µ (cid:48) + ε ∗ j → C · x µ (cid:48) is given by y j · x µ (cid:48) + ε ∗ j = (cid:40) x µ (cid:48) if µ (cid:48) j <
00 if µ (cid:48) j ≥ ν := µ K . If ν j > j ∈ J − ∩ K c or ν j < j ∈ J + ∩ K c , then the complex (71)is isomorphic to a Koszul complex of C with respect to a sequence containing 1 and hence it is acyclic.Therefore, if R Hom(∆ C ,s ( B ) , ∇ C ,s ( A )) T (cid:29)
0, then we must have ± ν j ≥ j ∈ J ± ∩ K c . We alsonote that ν k = 0 for any k ∈ K and ± ν i ≤ i ∈ I ± ∩ K c . We claim that these conditions imply I = J . If I = J , then we have ν i = 0 for any i ∈ I and K = ∅ . This implies that µ i = ν i = 0 for any i ∈ I and hence A = B . In this case, K = ∅ and the complex (71) is isomorphic to C which sits incohomological degree 0.Now we assume I (cid:54) = J . We first refine the inequalities (61). We note that ν = λ A − λ B − (cid:88) i ∈ I − ε ∗ i + (cid:88) i ∈ J − ε ∗ i + (cid:88) i ∈ K ∩ J + ε ∗ i − (cid:88) i ∈ K ∩ J − ε ∗ i + (cid:88) j ∈ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ε ∗ j − (cid:88) j ∈ J c (cid:98)(cid:104) s, β Jj (cid:105)(cid:99) ε ∗ j Since K ∩ J + = K ∩ I − ∩ J + and K ∩ J − = K ∩ I + ∩ J − , we have − (cid:88) i ∈ I − ε ∗ i + (cid:88) i ∈ J − ε ∗ i + (cid:88) i ∈ K ∩ J + ε ∗ i − (cid:88) i ∈ K ∩ J − ε ∗ i = − (cid:88) i ∈ I − ∩ ( J + (cid:116) J c ) ∩ K c ε ∗ i + (cid:88) J − ∩ ( I + (cid:116) I c ) ∩ K c ε ∗ i For any signed circuit C = C + (cid:116) C − , we have β C = (cid:80) j ∈ C + ∩ I c β Ij − (cid:80) j ∈ C − ∩ I c β Ij . This implies as in(61) that −| C + ∩ I c | + 1 + (cid:98)(cid:104) s, β C (cid:105)(cid:99) ≤ (cid:88) j ∈ C + ∩ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) − (cid:88) j ∈ C − ∩ I c (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) ≤ | C − ∩ I c | + (cid:98)(cid:104) s, β C (cid:105)(cid:99) . Therefore, we obtain (cid:104) ν, β C (cid:105) ≤ −| C + ∩ I − ∩ ( J + (cid:116) J c ) ∩ K c | + | C − ∩ I − ∩ ( J + (cid:116) J c ) ∩ K c | + | C − ∩ I c | + | C + ∩ J − ∩ ( I + (cid:116) I c ) ∩ K c | − | C − ∩ J − ∩ ( I + (cid:116) I c ) ∩ K c | + | C + ∩ J c | − ≤ −| C + ∩ I − ∩ J c | + | C − ∩ I − ∩ J + | + | C − ∩ I − ∩ J c | + | C − ∩ I c | + | C + ∩ J − ∩ I + | + | C + ∩ J − ∩ I c | − | C − ∩ J − ∩ I c | + | C + ∩ J c | − | C − ∩ ( I − \ J − ) | + | C − \ ( I ∪ J − ) | + | C + ∩ ( J − \ I − ) | + | C + \ ( J ∪ I − ) | − .
52n particular, if we assume the existence of a signed circuit C such that C + ⊂ I − ∪ J + and C − ⊂ I + ∪ J − ,then the above inequality implies (cid:104) ν, β C (cid:105) ≤ −
1. On the other hand, we have ν i ≥ i ∈ I − ∪ J + and ν i ≤ i ∈ I + ∪ J − . This implies that (cid:104) ν, β C (cid:105) ≥ V = Ker( b ) ⊗ R and E = { , . . . , n } as in section 5.6. We also define a partition E = R (cid:116) G (cid:116) B (cid:116) W by W =( I − ∪ J + ) ∩ ( I + ∪ J − ) = ( I + ∩ J + ) ∪ ( I − ∩ J − ), R = ( I − ∪ J + ) \ W = ( I − \ J − ) ∪ ( J + \ I + ), G =( I + ∪ J − ) \ W = ( I + \ J + ) ∪ ( J − \ I − ), and B = I c ∩ J c . We note that the assumption I (cid:54) = J implies R (cid:116) G (cid:54) = ∅ . By Lemma 5.30, it is enough to prove the existence of nonzero Z ∈ σ ( V ⊥ ) such that Z R ≥ Z G ≤
0, and Z B = 0. If we assume that such Z does not exists, then Proposition 5.31 impliesthat there exists a nonzero Y ∈ σ ( V ) such that Y R ≥ Y G ≤
0, and Y W = 0. By Lemma 5.30 again,there exists signed cocircuit C ∨ = C ∨ + (cid:116) C ∨− such that C ∨ + ⊂ R (cid:116) B and C ∨− ⊂ G (cid:116) B . Since we have I ∩ C ∨ + ⊂ I − , I ∩ C ∨− ⊂ I + , and I ∩ C ∨ (cid:54) = ∅ , we obtain (cid:104) ξ, α C ∨ (cid:105) = (cid:88) i ∈ I ∩ C ∨ + (cid:104) ξ, α Ii (cid:105) − (cid:88) i ∈ I ∩ C ∨− (cid:104) ξ, α Ii (cid:105) < . On the other hand, J ∩ C ∨ + ⊂ J + , J ∩ C ∨− ⊂ J − , and J ∩ C ∨ (cid:54) = ∅ imply that (cid:104) ξ, α C ∨ (cid:105) = (cid:88) i ∈ J ∩ C ∨ + (cid:104) ξ, α Ji (cid:105) − (cid:88) i ∈ J ∩ C ∨− (cid:104) ξ, α Ji (cid:105) > . This is a contradiction and hence the required circuit exists. This completes the proof of Theorem 5.45and hence Conjecture 3.40 for toric hyper-K¨ahler manifolds.
In this section, we prove Conjecture 3.47 and the second part of Conjecture 3.49. In order to provethem, we need to construct the central charge Z : s ∗ R → Hom Z ( K ( L ) , R ). We claim that for a K¨ahleralcove A and s ∈ A , the central charge of canonical bases C ( A ( s )) corresponding to A ( s ) ∈ Alc s isgiven by the volume of the polytope A ( s ) which is a polynomial function in s . In order to check thatthese polynomial functions do not depend on the choice of A , we consider their equivariant lifts.Recall that we write ι ( s ) = ( s , . . . , s n ) for s ∈ s ∗ R . For I ∈ B , we set (cid:3) I ( s ) := { x ∈ h ∗ R | ≤ (cid:104) x, a i (cid:105) + s i ≤ i ∈ I } . Take a generic c ∈ h such that (cid:104) c, α (cid:105) / ∈ Z for any equivariant root α ∈ X ∗ ( H ). We consider C asa module over K H (pt) by a λ (cid:55)→ e π √− (cid:104) c,λ (cid:105) and define a map Z c : s ∗ R → Hom K H (pt) ( K H ( L ) , C ) byassigning (cid:104) Z c ( s ) , O p I (cid:105) = (cid:90) (cid:3) I ( s ) e π √− (cid:104) c,x (cid:105) dx = (cid:89) i ∈ I e π √− (cid:104) c,α Ii (cid:105) − π √− (cid:104) c, α Ii (cid:105) · e − π √− s i (cid:104) c,α Ii (cid:105) for each I ∈ B . Here, O p I is the skyscraper sheaf at p I ∈ X H . Since {O p I } I ∈ B is a basis of K H ( L ) afterlocalization, the genericity of c implies that this extends to a K H (pt)-linear map Z c ( s ) : K H ( L ) → C .We note that these functions are analytic in s . Lemma 5.46.
For any s ∈ s ∗ reg and A ( s ) ∈ Alc s , we have (cid:104) Z c ( s ) , C ( A ( s )) (cid:105) = (cid:90) A ( s ) e π √− (cid:104) c,x (cid:105) dx. (72) Proof . We note that if we change A ( s ) by A ( s ) + α for some α ∈ X ∗ ( H ), then both sides of (72) aremultiplied by e π √− (cid:104) c,α (cid:105) . For any A ( s ) ∈ Alc s , there exists unique α ∈ X ∗ ( H ) such that A ( s ) + α ⊂ (cid:3) I { α Ii } i ∈ I is a basis of X ∗ ( H ). If A ( s ) ⊂ (cid:3) I ( s ), then we have wt H i ∗ p I E ( A ( s )) = 0 by Corollary 5.5.By Proposition 3.36, we obtain O p I = (cid:88) A ( s ) ∈ Alc s A ( s ) ⊂ (cid:3) I ( s ) C ( A ( s )) (73)as elements in K H ( L ) for any I ∈ B , i.e., after specializing v = 1. On the other hand, we obviouslyhave (cid:104) Z c ( s ) , O p I (cid:105) = (cid:88) A ( s ) ∈ Alc s A ( s ) ⊂ (cid:3) I ( s ) (cid:90) A ( s ) e π √− (cid:104) c,x (cid:105) dx (74)for any I ∈ B . One can solve (73) to express C ( A ( s )) in terms of O p I . Since c is generic, one canalso solve (74) to express (cid:82) A ( s ) e π √− (cid:104) c,x (cid:105) dx in terms of (cid:104) Z c ( s ) , O p I (cid:105) in the same way. Therefore, (cid:104) Z c ( s ) , C ( A ( s )) (cid:105) and (cid:82) A ( s ) e π √− (cid:104) c,x (cid:105) dx should coincide for any A ( s ) ⊂ (cid:3) I ( s ). This proves (72). Remark 5.47.
One can also consider an equivariant lift of Z to K T ( L ) by replacing the integral(72) by certain Euler type integrals [16] appearing in the theory of Gelfand-Kapranov-Zelevinsky’shypergeometric differential equations. We note that these differential equations come from quantumdifferential equations of toric hyper-K¨ahler manifolds by [27]. We do not pursue this direction furtherhere since we do not need it. Corollary 5.48.
There exists a polynomial map Z : s ∗ R → Hom Z ( K ( L ) , R ) such that (cid:104) Z ( s ) , C ( A ( s )) (cid:105) =Vol( A ( s )) for any s ∈ s ∗ reg and A ( s ) ∈ Alc s . Moreover, there exists a vector bundle P such that Z = Z P . Proof . Since the RHS of (72) is holomorphic in c and {C ( A ( s )) } A ( s ) ∈ Alc s forms a basis of K H ( L ), (cid:104) Z c ( s ) , C(cid:105) is holomorphic in c for any C ∈ K H ( L ). Therefore, one can substitute c = 0 to obtain amap Z := Z c =0 : s ∗ R → Hom Z ( K ( L ) , R ). Lemma 5.46 implies that for any A ( s ) ∈ Alc s , we have (cid:104) Z ( s ) , C ( A ( s )) (cid:105) = Vol( A ( s )). Since this is a polynomial function in s , Z is also a polynomial function.We next consider the value of the central charges at s = 0. We take s o ∈ s ∗ reg in a neighborhoodof 0 and consider P := m (cid:80) A ( s o ) ∈ Alc so / X ∗ ( H ) Vol( A (0)) · E ( A ( s o )) ∨ , where A (0) is the limit of A ( s o )as s o → m ∈ Z > is taken so that m Vol( A (0)) ∈ Z for any A ( s o ) ∈ Alc s o . We note thatthis does not depend on the choice of s o by Proposition 5.20. We note that rk P = m since we have (cid:80) A ( s ) ∈ Alc s / X ∗ ( H ) Vol( A ( s )) = Vol( (cid:3) I ( s )) = 1.We note that by Proposition 3.31, we have χ ( X, C ( A ) ⊗E ( A (cid:48) ) ∨ ) = δ A,A (cid:48) for any
A, A (cid:48) ∈ Alc s / X ∗ ( H ).This implies (cid:104) Z P (0) , C ( A ( s o )) (cid:105) = Vol( A (0)) = (cid:104) Z (0) , C ( A ( s o )) (cid:105) for any A ( s o ) ∈ Alc s o and hence Z P (0) = Z (0). For any l ∈ Pic( X ) ∼ = X ∗ ( S ), the periodic hyperplane arrangements in h ∗ R defined insection 5.4 does not change if we change s by s + l . This and Lemma 3.34 imply that for any A ( s o ) ∈ Alc s o , there exists A ( s o + l ) ∈ Alc s o + l such that Vol( A ( s o )) = Vol( A ( s o + l )) and C ( A ( s o + l )) = L ( l ) ⊗C ( A ( s o )). Therefore, we obtain (cid:104) Z P ( l ) , C ( A ( s o + l )) (cid:105) = Vol( A (0)) = Vol( A ( l )) = (cid:104) Z ( l ) , C ( A ( s o + l )) (cid:105) for any A ( s o + l ) ∈ Alc s o + l . This implies that Z P ( l ) = Z ( l ) for any l ∈ Pic( X ). Since both of themare polynomial functions in s , we obtain Z P ( s ) = Z ( s ) for any s ∈ s ∗ R .We now prove Conjecture 3.47 for toric hyper-K¨ahler manifolds. Recall that for any A ∈ Alc K , weassociate a t -structure τ ( A ) on D := D b Coh L ( X ) ⊂ D b Coh( X ) defined by the tilting bundle T C ,s for s ∈ A . Corollary 5.49.
The pair ( Z , τ ) gives a real variation of stability conditions on D . Proof . The first condition in Definition 3.45 follows from Corollary 5.48 since the volume of a fulldimensional polytope is positive. Hence it is enough to check the second condition in Definition 3.45.For any s ∈ A ∈ Alc K and a wall w = { x ∈ s ∗ R | (cid:104) x, β C (cid:105) = m } of A , we set Alc s,w := { A ( s ) ∈ Alc s | Vol( A ( s )) does not vanish on w } and Alc s,w := Alc s \ Alc s,w . By Proposition 5.20 and Corollary 5.48,54he Serre subcategories C n A of the heart C A of τ ( A ) defined in section 3.7 are given by C A if n = 0,generated by objects {C ( A ) | A ∈ Alc s,w } if 0 < n < | C | , and 0 if n ≥ | C | . Here, we identified C ( A ) ∈ K T ( X ) as an object of D as in section 3.6 by forgetting equivariant structures. We note thatwe have C A = C | C |− A = {F ∈ C A | Hom( E ( A ) , F ) = 0 for any A ∈ Alc s,w } . This easily implies that D A ,w = D | C |− A ,w = {F ∈ D | R Hom( E ( A ) , F ) = 0 for any A ∈ Alc s,w } .Let A + (cid:54) = A − ∈ Alc K be two K¨ahler alcove sharing the same wall w such that A + is above A − . We take s ± ∈ A ± . Since {E ( A ) | A ∈ Alc s + ,w } = {E ( A ) | A ∈ Alc s − ,w } by Proposition 5.20,we obtain D n A + ,w = D n A − ,w for any n ∈ Z ≥ . By the definition of τ ( A ± ), we have C | C |− A ± ,w = {F ∈ D | C |− A ± ,w | RHom (cid:54) =0 ( E ( A ) , F ) = 0 for any A ∈ Alc s ± ,w } . By Proposition 5.20 and Lemma 5.21, wehave R Hom( E ( A + ) , F ) ∼ = R Hom( E ( A − )[ | C | − , F ) for any F ∈ D | C |− A ± ,w and A ± ∈ Alc s ± such that E ( A − ) ∼ = L (cid:16)(cid:80) i ∈ C − ε ∗ i − (cid:80) i ∈ C + ε ∗ i (cid:17) ⊗ E ( A + ). This implies that gr | C |− w ( C A − ) = gr | C |− w ( C A + )[ | C | − | C |− A ± ,w D . Similarly, we have gr w ( C A ± ) = {F ∈ gr A ± ,w ( D ) | RHom (cid:54) =0 ( E ( A ) , F ) = 0 for any A ∈ Alc s ± ,w } and hence gr w ( C A − ) = gr w ( C A + ) in gr A ± ,w D . This proves the second condition in Defni-tion 3.45 and hence ( Z , τ ) gives a real variation of stability conditions. In this final section, we define what we call elliptic canonical bases for toric hyper-K¨ahler manifolds andprove some basic properties of them. As a corollary, we prove Conjecture 4.8 for toric hyper-K¨ahlermanifolds. We will follow the notations of section 4 and 5.
First we recall the description of elliptic stable bases for toric hyper-K¨ahler manifolds given in [1, 39].Recall that we fix the polarization T / in (43) which satisfies det T / = v − n L ( κ ), where we set κ := ε ∗ + · · · + ε ∗ n ∈ X ∗ ( T ). We also take similar polarization for X ! and κ ! := ε + · · · + ε n ∈ X ∗ ( T ). Proposition 6.1 ([1, 39]) . For any I ∈ B , we haveStab AO C ,T / ( p I ) = (cid:89) i ∈ I + ϑ ( v − L ( − ε ∗ i )) · (cid:89) i ∈ I − ϑ ( v − L ( ε ∗ i )) · (cid:89) j ∈ I c + ϑ ( v l j − z β Ij L ( ε ∗ j )) ϑ ( v l j z β Ij ) · (cid:89) j ∈ I c − ϑ ( v − l i − z − β Ij L ( − ε ∗ j )) ϑ ( v − l i z − β Ij ) , where l j := −(cid:104) β Ij , κ + (cid:80) i ∈ I + b i − (cid:80) i ∈ I − b i (cid:105) ± j ∈ I c ± . Proof . By Theorem 5 in [39], we haveStab AO C ,T / ( p I ) = (cid:89) i ∈ I + ϑ ( v − L ( − ε ∗ i )) · (cid:89) i ∈ I − ϑ ( v − L ( ε ∗ i )) · (cid:89) j ∈ I c + ϑ ( v l j − z β j L ( ε ∗ j )) ϑ ( v l j z β j ) · (cid:89) j ∈ I c − ϑ ( v − l i − z − β j L ( − ε ∗ j )) ϑ ( v − l i z − β j )(75)for some β j ∈ X ∗ ( S ) and l j ∈ Z for j ∈ I c . We only need to determine β j and l j . For any l ∈ X ∗ ( S ), thefactor of automorphy of i ∗ p J Stab AO C ,T / ( p I ) under z (cid:55)→ q l z is given by i ∗ p I L ( l ) · i ∗ p J L ( l ) − by (25). On theother hand, the factor of automorphy of the RHS of (75) is given by (cid:81) j ∈ I c i ∗ p I L ( ε ∗ j ) (cid:104) l,β j (cid:105) · i ∗ p J L ( ε ∗ j ) −(cid:104) l,β j (cid:105) by Corollary 5.5. This implies that (cid:80) j ∈ I c (cid:104) l, β j (cid:105) b j = l = (cid:80) j ∈ I c (cid:104) l, β Ij (cid:105) b j for any l ∈ P . Hence we have β j = β Ij for any j ∈ I c .We next consider the factor of automorphy of τ (det T / , v ) ∗ ( i ∗ p J Stab AO C ,T / ( p I )) under a (cid:55)→ q c a for c ∈ X ∗ ( H ). By (24) and Corollary 5.5, this is given by v − (cid:80) i ∈ I + (cid:104) c,α Ii (cid:105) + (cid:80) i ∈ I − (cid:104) c,α Ii (cid:105) · z − (cid:80) j ∈ J (cid:104) c,α Jj (cid:105) ε j + (cid:80) i ∈ I (cid:104) c,α Ii (cid:105) ε i · i ∗ p J L − (cid:88) j ∈ J (cid:104) c, α Jj (cid:105) ε ∗ j .
55n the other hand, (75) implies that this should be equal to v −(cid:104) c, (cid:80) i ∈ I + α Ji (cid:105) + (cid:104) c, (cid:80) i ∈ I − α Ji (cid:105)−(cid:104) c, (cid:80) j ∈ Ic + ( l (cid:48) j − α Jj (cid:105)−(cid:104) c, (cid:80) j ∈ Ic − ( l (cid:48) j +1) α Jj (cid:105) · z − (cid:80) j ∈ Ic (cid:104) c,α Jj (cid:105) β Ij · i ∗ p J L − (cid:88) j ∈ J (cid:104) c, α Jj (cid:105) ε ∗ j for any J ∈ B , where we set l (cid:48) j = l j + (cid:104) β Ij , κ (cid:105) . By comparing the exponent of v , we obtain − (cid:88) i ∈ I + α Ji + (cid:88) i ∈ I − α Ji − (cid:88) j ∈ I c + ( l (cid:48) j − α Jj − (cid:88) j ∈ I c − ( l (cid:48) j + 1) α Jj = − (cid:88) i ∈ I + α Ii + (cid:88) i ∈ I − α Ii for any J ∈ B . For each j ∈ I c ± , by taking J ∈ B such that j ∈ J and considering the pairing with a j ,we obtain l (cid:48) j = (cid:104) (cid:80) i ∈ I + α Ii − (cid:80) i ∈ I − α Ii , a j (cid:105) ±
1. This implies l j = −(cid:104) β Ij , κ + (cid:80) i ∈ I + b i − (cid:80) i ∈ I − b i (cid:105) ± Corollary 6.2.
For any I ∈ B , we haveStab ellX ( p I ) = ( − | I + | + | I c + | · n (cid:89) i =1 ϑ ( L ( ε ∗ i ) · i ∗ p ! I L ! ( ε i )) . Proof . We note that by Corollary 5.5 applied for X ! , we have i ∗ p ! I L ! ( ε j ) = v −(cid:104) β Ij , (cid:80) i ∈ I + b i − (cid:80) i ∈ I − b i (cid:105) z β Ij for any j ∈ I c and i ∗ p ! I L ! ( ε i ) = v ± for i ∈ I ± . This and Proposition 6.1 imply that τ (det T / , v ) ∗ Stab AO C ,T / ( p I ) = ( − | I + | n (cid:89) i =1 ϑ ( L ( ε ∗ i ) · i ∗ p ! I L ! ( ε i )) · (cid:89) j ∈ I c + ϑ ( v · i ∗ p ! I L ! ( ε i )) − · (cid:89) i ∈ I c − ϑ ( v − i ∗ p ! I L ! ( ε i )) − . Now the statement of the corollary follows from ϑ ( N ! p ! I , − ) = (cid:89) i ∈ I c + ϑ ( v − i ∗ p ! I L ! ( − ε i )) · (cid:89) i ∈ I c − ϑ ( v − i ∗ p ! I L ! ( ε i ))obtained by applying (39) to X ! .In particular, this implies Conjecture 4.4 for toric hyper-K¨ahler manifolds. Recall that we wrote S X,p J ,p I := i ∗ p J Stab ellX ( p I ). Corollary 6.3.
For any
I, J ∈ B , we have ( − | I + | + | I c + | · S X,p J ,p I = ( − | J c + | + | J + | S X ! ,p ! I ,p ! J and theyare holomorphic sections of the line bundle on B X ∼ = B X ! described in Proposition 4.3.Moreover, Corollary 6.2 also implies Conjecture 4.6 for toric hyper-K¨ahler manifolds. Corollary 6.4.
For any
I, J ∈ B , we have S X,p J ,p I = ( − n · S − X flop ,p J ,p I . Proof . Let L flop ( λ ) be the T -equivariant line bundle on X flop associated with λ ∈ X ∗ ( T ). By Corol-lary 6.2 and i ∗ p I L flop ( λ ) = i ∗ p I L ( λ ), we obtain( − | I − | + | I c − | · S − X flop ,p J ,p I = n (cid:89) i =1 ϑ ( i ∗ p J L flop ( ε ∗ i ) · i ∗ p ! I L !flop ( ε i ))= n (cid:89) i =1 ϑ ( i ∗ p J L ( ε ∗ i ) · i ∗ p ! I L ! ( ε i ))= ( − | I + | + | I c + | · S X,p J ,p I . .2 Elliptic canonical bases Now we construct the elliptic canonical bases for toric hyper-K¨ahler manifolds. From now on, weidentify X ∗ ( T ) ∼ = Z n and X ∗ ( T ) ∼ = Z n by using the standard inner product ( − , − ) : Z n × Z n → Z given by (( λ , . . . , λ n ) , ( µ , . . . , µ n )) = (cid:80) ni =1 λ i µ i . We note that under this identification, we have κ ! = κ . Using this identification, we may consider X ∗ ( S ) ⊂ X ∗ ( T ) and X ∗ ( H ) ⊂ X ∗ ( T ). We recall that( q ; q ) ∞ = (cid:81) m ≥ (1 − q m ). Definition 6.5.
For any λ ∈ Z n , we defineΘ X ( λ ) := ( q ; q ) − r ∞ (cid:88) β ∈ X ∗ ( S ) ( − ( κ,β ) q ( β,β + κ )+( λ,β ) L ( λ + β ) z β , Θ X ! ( λ ) := ( q ; q ) − d ∞ (cid:88) α ∈ X ∗ ( H ) ( − ( κ ! ,α ) q ( α,α + κ ! )+( λ,α ) L ! ( λ + α ) a α . We can consider { Θ X ( λ ) } λ ∈ Z n and { Θ X ! ( λ ) } λ ∈ Z n as elements of K ( X ) loc ∼ = K ( X ! ) loc and call them elliptic canonical bases for X and X ! respectively.We note that for any α ∈ X ∗ ( H ) and β ∈ X ∗ ( S ), we have ( α, β ) = 0. Using this, one can easilycheck the following relations. Lemma 6.6.
For any α ∈ X ∗ ( H ) and β ∈ X ∗ ( S ), we haveΘ X ( λ + α ) = a α Θ X ( λ ) , Θ X ( λ + β ) = ( − ( κ,β ) q − ( β,β + κ ) − ( λ,β ) z − β Θ X ( λ ) . In particular, the number of linearly independent elements in { Θ X ( λ ) } λ ∈ Z n over M X is less that orequal to the number of elements of Ξ := Z n / ( X ∗ ( H ) + X ∗ ( S )) ∼ = X ∗ ( S ) / X ∗ ( S ) ∼ = X ∗ ( H ) / X ∗ ( H ). Theunimodularity of a and b implies the following. Lemma 6.7.
We have | Ξ | = | B | . Proof . By taking a basis of X ∗ ( H ), we consider each a i ∈ X ∗ ( H ) as a column vector and a = ( a , . . . , a n )as a ( d × n )-matrix. We note that | X ∗ ( H ) / X ∗ ( H ) | = | det( a · t a ) | . For each subset I = { i , . . . , i d } ⊂{ , . . . , n } with | I | = d , we denote by a I = ( a i , . . . , a i d ) the ( d × d )-matrix obtained by removing certaincolumns from a . By the unimodularity of a , we have det( a I ) = ± I ∈ B and det( a I ) = 0 otherwise.Therefore, we obtain det( a · t a ) = (cid:88) I ⊂{ ,...,n }| I | = d det( a I · t a I ) = (cid:88) I ∈ B det( a I ) = | B | . Corollary 6.8.
For any data C and s ∈ s ∗ reg , the map I (cid:55)→ µ I gives a bijection B ∼ = Ξ, where µ I isdefined as in (65). Proof . It is enough to prove that the map is injective. If there exists α ∈ X ∗ ( H ), β ∈ X ∗ ( S ) and I (cid:54) = J ∈ B such that µ J = µ I + α + β , then (61) implies that ( β, β C ) ≤ | C | − C . Since {L ( µ I ) } I ∈ B forms a basis of K T ( X ) over K T (pt), we must have β (cid:54) = 0. By Lemma 5.30, thereexists a signed circuit C = C + (cid:116) C − such that ± β i > i ∈ C ± . This implies that ( β, β C ) ≥ | C | and hence gives a contradiction.Now we prove the main result of this paper. Recall the Jacobi triple product formula:( q ; q ) ∞ ϑ ( x ) = (cid:88) m ∈ Z ( − m q m ( m +1)2 x m + . (76)We also recall that S X ( p I ) := (cid:113) L ( κ ) · i ∗ p ! I L ! ( κ ! ) − · Stab ellX ( p I ).57 heorem 6.9. For any I ∈ B , we have( − | I + | + | I c + | · S X ( p I ) = (cid:88) λ ∈ Ξ ( − ( κ,λ ) q ( λ,λ + κ ) i ∗ p ! I Θ X ! ( λ ) · Θ X ( λ ) . (77)Here, we fix a lift Ξ → Z n and consider λ ∈ Ξ as an element of Z n . Proof . We first note that by Lemma 6.6, each term in the RHS of (77) does not depend on the choiceof a lift of λ ∈ Ξ to Z n . By Corollary 6.2 and (76), we have( − | I + | + | I c + | · ( q ; q ) n ∞ Stab ellX ( p I ) = n (cid:89) i =1 ( q ; q ) ∞ ϑ ( L ( ε ∗ i ) · i ∗ p ! I L ! ( ε i ))= (cid:88) µ ∈ Z n ( − ( µ,κ ) q ( µ,µ + κ ) i ∗ p ! I L ! ( µ + 12 κ ! ) · L ( µ + 12 κ )We note that any element of Z n can be uniquely written as λ + α + β for λ ∈ Ξ, α ∈ X ∗ ( H ), and β ∈ X ∗ ( S ). By using L ( α ) = a α , L ! ( β ) = z β , and ( α, β ) = 0 for α ∈ X ∗ ( H ) and β ∈ X ∗ ( S ), we obtain( − | I + | + | I c + | · S X ( p I ) = ( q ; q ) − n ∞ (cid:88) λ ∈ Ξ α ∈ X ∗ ( H ) β ∈ X ∗ ( S ) ( − ( λ + α + β,κ ) q ( λ + α + β,λ + α + β + κ ) i ∗ p ! I L ! ( λ + α + β ) · L ( λ + α + β )= (cid:88) λ ∈ Ξ ( − ( κ,λ ) q ( λ,λ + κ ) · ( q ; q ) − d ∞ (cid:88) α ∈ X ∗ ( H ) ( − ( κ ! ,α ) q ( α,α + κ ! )+( λ,α ) i ∗ p ! I L ! ( λ + α ) a α × ( q ; q ) − r ∞ (cid:88) β ∈ X ∗ ( S ) ( − ( κ,β ) q ( β,β + κ )+( λ,β ) L ( λ + β ) z β = (cid:88) λ ∈ Ξ ( − ( κ,λ ) q ( λ,λ + κ ) i ∗ p ! I Θ X ! ( λ ) · Θ X ( λ ) . Since { S X ( p I ) } I ∈ B is a basis of K ( X ) loc over M X , Lemma 6.7 and Theorem 6.9 implies that { Θ X ( λ ) } λ ∈ Ξ is also a a basis of K ( X ) loc over M X . By applying Theorem 6.9 for − X , we obtain( − | I − | + | I c + | · S − X ( p I ) = (cid:88) λ ∈ Ξ ( − ( κ,λ ) q ( λ,λ + κ ) i ∗ p ! I Θ X !flop ( λ ) · Θ − X ( λ )= (cid:88) λ ∈ Ξ ( − ( κ,λ ) q ( λ,λ + κ ) i ∗ p ! I Θ X ! ( λ ) · Θ X ( λ ) . This and (77) implies that if we define M X -semilinear map β (cid:48) X : K ( X ) loc → K ( X ) loc by β (cid:48) X (Θ X ( λ )) = Θ X ( λ ) (78)for any λ ∈ Ξ, then we have β (cid:48) X ( S X ( p I )) = ( − d S − X ( p I ) for any I ∈ B i.e., β (cid:48) X = β ellX . This provesthe following result which is the main observation in this paper and partly justify our definition ofelliptic canonical bases for toric hyper-K¨ahler manifolds. Corollary 6.10.
For each λ ∈ Z n , we have β ellX (Θ X ( λ )) = Θ X ( λ ) and β ellX ! (Θ X ! ( λ )) = Θ X ! ( λ ). More-over, the elliptic bar involution β ellX does not depend on the choice of chamber C . Proof . For the independence on C , it is enough to note that Θ X ( λ ) does not depend on the choice of C and (78) uniquely characterize the map β ellX . Remark 6.11.
If one try to prove β ellX (Θ X ( λ )) = Θ X ( λ ) directly from the definition of β ellX , then certainnontrivial identities of various theta functions will be needed. Our proof mimics that of Proposition 5.15and does not involve any nontrivial calculations. In fact, our definition of elliptic canonical bases isdesigned so that this kind of proof works nicely. 58 orollary 6.12. Conjecture 4.8 holds for toric hyper-K¨ahler manifolds.
Proof . This follows from Corollary 6.10 by reversing the argument of the proof of Proposition 4.15.This completes the proof of all the conjectures stated in section 3 and 4 in the case of toric hyper-K¨ahler manifolds. K -theory limits In this section, we check that the elliptic canonical bases for toric hyper-K¨ahler manifolds lift K -theoretic canonical bases for any slopes. This result gives another justification of our definition ofelliptic canonical bases.Let λ ∈ Z n and s ∈ s ∗ reg . Since Θ X ( λ ) | z = q − s might not have well-definded limit under q → s (Θ X ( λ )) := LT(Θ X ( λ ) | z = q − s ). Here, we write LT( f ) := f t for f = (cid:80) t ∈ R f t q t and t := min { t | f t (cid:54) = 0 } if it exists. In order to calculate LT s (Θ X ( λ )), we need toknow when the function ( β, β + κ ) + ( λ − s, β ) on β ∈ X ∗ ( S ) takes its minimum. We note thatLT s (Θ X ( λ + α )) = a α · LT s (Θ X ( λ )) and LT s (Θ X ( λ + β )) = ( − ( κ,β ) LT s (Θ X ( λ )) for any α ∈ X ∗ ( H )and β ∈ X ∗ ( S ). In particular, we only need to consider LT s (Θ X ( λ )) for λ ∈ Ξ. We will identifyΞ = { µ I } I ∈ B by using Corollary 6.8.We note that for any t ∈ R , the function from Z to R defined by m ( m + 1) − tm for m ∈ Z takesits minimum at m = (cid:98) t (cid:99) if t / ∈ Z and m = t, t − t ∈ Z . Therefore, the function from Z n to R givenby 12 ( µ, µ + κ ) − ( s, wt H ! i ∗ p ! I L ! ( µ )) = n (cid:88) i =1 µ i ( µ i + 1) − (cid:88) j ∈ I c µ j (cid:104) s, β Ij (cid:105) (79)for µ ∈ Z n takes its minimum when µ j = (cid:98)(cid:104) s, β Ij (cid:105)(cid:99) for any j ∈ I c and µ i = 0 , − i ∈ I . Weremark that for any such µ , we have L ( µ ) ∈ B X,s by (52). In particular, (79) takes its minimum when µ = µ I . Therefore, we have12 ( µ I , µ I + κ ) − ( s, wt H ! i ∗ p ! I L ! ( µ I )) ≤
12 ( µ I + β, µ I + β + κ ) − ( s, β + wt H ! i ∗ p ! I L ! ( µ I ))= 12 ( µ I , µ I + κ ) − ( s, wt H ! i ∗ p ! I L ! ( µ I )) + 12 ( β, β + κ ) + ( µ I − s, β )for any β ∈ X ∗ ( S ). I.e., the function ( β, β + κ ) + ( µ I − s, β ) on X ∗ ( S ) takes its minimum at β = 0.On the other hand, if this function takes its minimum at 0 (cid:54) = β ∈ X ∗ ( S ), then the function (79) takesits minimum at µ I + β and hence we obtain L ( µ I ) , L ( µ I + β ) ∈ B X,s . This contradicts the inequality(61) as in the proof of Corollary 6.8. Therefore, we obtain LT s (Θ X ( µ I )) = L ( µ I ) ∈ B X,s . In summary,we obtained the following formula.
Proposition 6.13.
For any λ ∈ Z n , there exists unique A ∈ Alc s and β ∈ X ∗ ( S ) such that λ = µ A + β ,where µ A is defined as in (50). Moreover, we have LT s (Θ X ( λ )) = ( − ( κ,β ) E ( A ).Let us take s + , s − ∈ s ∗ reg as in section 5.5 which are separated by a wall w = { x ∈ s ∗ R | (cid:104) x, β C (cid:105) = m } for some signed circuit C and m ∈ Z with (cid:104) η, β C (cid:105) >