Localized Chern Characters for 2-periodic complexes
aa r X i v : . [ m a t h . AG ] J u l LOCALIZED CHERN CHARACTERS FOR 2-PERIODICCOMPLEXES
BUMSIG KIM AND JEONGSEOK OH
Abstract.
For a two-periodic complex of vector bundles, Polishchukand Vaintrob have constructed its localized Chern character. We exploresome basic properties of this localized Chern character. In particular, weshow that the cosection localization defined by Kiem and Li is equivalentto a localized Chern character operation for the associated two-periodicKoszul complex, strengthening a work of Chang, Li, and Li. We applythis equivalence to the comparison of virtual classes of moduli of ε -stablequasimaps and moduli of the corresponding LG ε -stable quasimaps, infull generality. Contents
1. Localized Chern Characters 12. Koszul Complexes 33. Comparisons of virtual classes 8References 171.
Localized Chern Characters
Fix a base field k . Let Y be a finite type Deligne-Mumford stack over k and let X i ÝÑ Y be the inclusion of a closed substack X of Y . Let E ‚ be a2-periodic complex of vector bundles, which is exact off X : r E ´ d ´ / / E ` d ` o o s “ ... d ` ÝÝÑ E ´ d ´ ÝÝÑ E ` d ` ÝÝÑ E ´ d ´ ÝÝÑ ...E ` is in even degree and E ´ is in odd degree. Suppose that Ker d ´ andKer d ` restricted to Y ´ X are vector bundles.Polishchuck and Vaintrob [19] define a bivariant classch YX p E ‚ q P A ˚ p X i ÝÑ Y q Q generalizing the localized Chern characters developed in Baum, Fulton, andMacPherson [1]. For each r V s P A ˚ p Y q Q , this assigns a classch YX p E ‚ q X r V s P A ˚ p X q Q whose image in A ˚ p Y q Q is ch p E ` q X r V s ´ ch p E ´ q X r V s . Polishchuck and Vaintrob [19] use a localized Chern character to defineWitten’s top Chern class. This is a particular case of pure Landau-Ginzburgsides in gauged linear sigma model. H.-L. Chang, J. Li and W.-P. Li alsodefine Witten’s top Chern class via cosection localization. They show thatboth constructions coincide; see [4, Proposition 5.10]. It turns out to bea special case of the equivalence that a cosection localization of Kiem-Li[13] is the localized Chern character for the associated 2-periodic Koszulcomplex. We prove the equivalence; see Theorem 2.6. This equivalence willbe applied to the comparison of virtual classes in §
3. Also by this approach,we may define the virtual structure sheaves and study the comparisons ofthose defined by [15] and [14], respectively. This will be left to [18].Let V be a vector space with the standard diagonal action by the multi-plicative group G m so that P V “ r V ´ t u{ G m s , the space of 1-dimensionalsubspaces of V , and let V be a G m -space. Consider a G m -invariant element w of p Sym V _ q b V _ . Let E “ rp V ´ t u ˆ V q{ G m s be a vector bundleon P V . E has a cosection associated to w . This cosection amounts to aregular function w : | E | Ñ A which is linear in fiber coordinates of thetotal space | E | of E . H.-L. Chang and J. Li [3] introduce a moduli space LGQ g p E, d q of genus g , degree d , stable maps to a complex projective space P V with p -fields and construct a cosection dw LGQ of the obstruction sheafand a virtual class via cosection localization. This is a particular case ofgeometric sides in gauged linear sigma model. Let Z p dw q Ă | E | denote thecritical locus of w . When E is O P p´ q with Z p dw q a smooth quintic hyper-surface, they show that for d ‰
0, deg r LGQ g p E, d q s vir dw LGQ coincides with,up to sign, the degree deg r Q g p Z p dw q , d qs vir of the virtual class of modulispace Q g p Z p dw q , d q of genus g , degree d , stable maps to the quintic. Weprove its generalization that for any geometric gauged linear sigma model p V “ V ‘ V , G, w q and any positive rational number ε , the cosection lo-calized virtual class of moduli space LGQ εg,k p V {{ G, d q of ε -stable quasimapsto V {{ G with p -fields coincides, up to sign, with the virtual class of modulispace Q εg,k p Z p dw q , d q of ε -stable quasimaps to Z p dw q ; see Theorem 3.2. Acknowledgments
B. Kim would like to thank Yongbin Ruan for draw-ing his attention to the comparison question of virtual classes and AndreiOkounkov for stimulating discussions. The authors would like to thankIonut¸ Ciocan-Fontanine, Tom Graber and Taejung Kim for helpful com-ments in shaping the paper. This material is based upon work supported byNSF grant DMS-1440140 while the first author was in residence at MSRI inBerkeley during Spring 2018 semester. J. Oh would like to thank SanghyeonLee for useful discussions and University of California, Berkeley for excellentworking conditions.
OCALIZED CHERN CHARACTERS FOR 2-PERIODIC COMPLEXES 3 Koszul Complexes
Definition.
We recall the definition of localized Chern characters fora 2-period complex; see [19]. For a cycle j : V Ñ Y defined by an integralclosed substack of Y (see Gillet [12] and Vistoli [20] for the definition of theChow group A p Y q Q of Y with rational coefficients), we may letch YX p E ‚ q X r V s “ j p ch VV ˆ Y X p j ˚ E ‚ q X r V sq , where j : V ˆ Y X Ñ X be the induced inclusion. Hence, it is enough todefine the localized Chern character with assumption that V “ Y and Y isirreducible. When X “ Y , we define ch YX p E ‚ q “ ch p E ` q ´ ch p E ´ q .When X ‰ Y , we consider a graph construction for the homomorphism p d ` , d ´ q as follows. Let r be the rank of E ` . Note that the rank of E ´ isalso r . Denote by G the Grassmann bundle Gr r p E ` ‘ E ´ q of r -planes in E ` ‘ E ´ . Consider the projection π : G ˆ Y G ˆ A Ñ Y ˆ A and an its section ϕ : Y ˆ A Ñ G ˆ Y G ˆ A p y, λ q ÞÑ p graph p λd ` p y qq , graph p λd ´ p y qq . Let Γ be the closure of ϕ p Y ˆ A q in G ˆ Y G ˆ P . Let i : G ˆ Y G ˆ t8u ã Ñ G ˆ Y G ˆ P be the inclusion. There is a distinguished component Γ ,dist of Γ : “ i ˚8 r Γ s which birationally projects to Y . Note that Γ ,dist restricted to π ´ p Y ´ X q is p Ker d ` | Y ´ X , Ker d ´ | Y ´ X q ˆ Y p Ker d ` | Y ´ X , Ker d ´ | Y ´ X q . The remained components project into X . Let ξ ` , ξ ´ be tautological sub-bundles on G ˆ Y G ˆ P from each component G . Note that ξ | Γ ,dist “ YX p E ‚ q X r Y s “ η ˚ p ch p ξ q X p Γ ´ Γ ,dist qq , where η is the restriction of the projection G ˆ Y G Ñ Y to the inverse imageof X under the projection.It is clear from the definition that i ˚ ch YX p E ‚ q “ ch p E ` q ´ ch p E ´ q .2.2. Koszul Complex.
Koszul complexes yield ample examples of 2-periodiccomplexes. Let E be a vector bundle on Y with sections α P H p Y, E _ q , β P H p Y, E q such that x α, β y “ O Y β ÝÑ E α ÝÑ O Y . Let t α, β u denote the 2-periodic complex ‘ k Ź k ´ E _^ α ` ι β / / ‘ k Ź k E _^ α ` ι β o o KIM AND OH of vector bundles. Here ι β is the interior product by β . Let Z p α, β q : “ Z p α q X Z p β q be the zero substack of Y defined by the ideal sheaf generatedby Im p α q and Im p β _ q . The complex is exact off X : “ Z p α, β q and thekernels restricted to Y ´ X are vector bundles. The latter follows from thatlocally Ker d ˘ is direct summand of E ˘ ; see [7, Proposition 2.3.3] and itsproof. This can be regarded as a refined version of the usual Koszul complexgiven only by β .It will be useful to note that t α, β u “ t β _ , α _ u b p rank E ľ E _ qr rank E s (2.1)due to the duality of wedge product and interior product.2.3. Tautological Koszul complex.
Let M be a DM stack and F be avector bundle on M . Let σ P H p M, F _ q , which gives a regular function onthe total space | F | of F : w σ : | F | Ñ A . Denote by p the projection | F | Ñ M . Then there is a tautological section t F P H p| F | , p ˚ F q such that x p ˚ σ, t F y “ w σ . We obtain a matrix factoriza-tion t p ˚ σ, t F u for w σ . It becomes a 2-periodic complex when it is restrictedto the zero locus Z p w σ q of w σ .Starting from the setup in § M “ Z p β q , F “ E | Z p β q , σ “ α | Z p β q . Note that Z p σ q “ X . Let C V X Z p β q V denote the normal cone to V X Z p β q in V . It is naturally a closedsubstack of F . Lemma 2.1.
For each integral substack V of Y with nonempty V X Z p β q , ch YX pt α, β uq X r V s “ ch Z p w σ q Z p σ q pt p ˚ σ, t F uq X r C V X Z p β q V s . Proof.
By the restriction, we may assume that Y “ V . Now the statementfollows from the deformation of Y to the normal cone C Z p β q Y Ă F . Thatis, first consider the graph Γ β of a section λβ of F ˆ A on Y ˆ A and itsclosure Γ β in F ˆ P , whose fiber at P is C Z p β q Y . If ˜ p denotes theprojection of F ˆ P to Y , then the vector bundle ˜ p ˚ F | Γ β with the diagonalsection and ˜ p ˚ α realizes the deformation of F with β, α to p p ˚ F q| C Z p β q Y with t F , p ˚ σ . (cid:3) Let j : X ã Ñ Z p β q be the inclusion. Then the following corollary showsthat ch YX t α, β u X r Y s after pushforward by j is nothing but the localized topChern class of E up to Todd correction. Corollary 2.2.
For arbitrary section β : Y Ñ E , j ˚ p ch YX pt α, β uq X r V sq “ p td E | Z p β q q ´ ¨ ! E | Z p β q pr C V X Z p β q V sq OCALIZED CHERN CHARACTERS FOR 2-PERIODIC COMPLEXES 5 where V is an integral substack V of Y with V X Z p β q ‰ H . Furthermoreif β is regular, then j ˚ ch YX pt α, β uq “ p td E | Z p β q q ´ ¨ i ! Z p β q where i Z p β q denotes the regular immersion of Z p β q in Y .Proof. Apply Lemma 2.1 and then deform p ˚ σ Ñ
0. Now Proposition 2.3(vi) of [19] completes the proof. (cid:3)
Let V be an integral substack of Y . For the pair p E | V , β q , there is thenotion of the localized top Chern class of E with respect to β ; see [11, § ! E | Z p β q pr C V X Z p β q V sq P A ˚ p Z p β qq Q . This eventually yieldsa bivariant class in A p Z p β q Ñ Y q Q , which we call the localized top Chernclass operation of E with respect to β . Corollary 2.3.
The class td p E | Z p β q q¨ ch YZ p β q pt , β uq agrees with the localizedtop Chern class operation of E with respect to β . Splitting Principle.
The splitting principle shows that essentially lo-calized Chern character operation for a 2-periodic Koszul complex is a com-position of localized top Chern class operations, one given by a section andthe other given by a cosection, up to Todd correction.2.4.1. Consider the situation of § α is fac-tored through as a cosection α Q of a quotient vector bundle Q of E . Let f : K ã Ñ E be the kernel of the quotient map E Ñ Q . Let β K be a sectionof K .We consider the vector bundles on Y ˆ A by the pullback of E , K underthe projection map. They will be denoted by same symbols E , K abusingnotation. If µ denotes the standard coordinate of A , we have a commutingdiagram of homomorphisms of vector bundles on Y ˆ A : O Y ˆ A p µβ, p ´ µ q β K q (cid:15) (cid:15) β P ' ' ❖❖❖❖❖❖❖ / / K p f,µ id K q / / E ‘ K / / p α, q (cid:15) (cid:15) P α P w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ / / O Y ˆ A and the induced section β P and cosecton α P of P . Here P is defined to bethe cokernel of p f, µ id K q .Note that P restricted to µ “ Q ‘ K ; and P restricted to any nonzero µ is canonically isomorphic to E . Suppose that Z p α P , β P q Ă X ˆ A for some closed substack X of Y .Let E be a vector bundle on a DM stack B and let A be the zero locusof a section of E . We denote by Sp C A B the specialization homomorphism KIM AND OH A ˚ p B q Q Ñ A ˚ p C A B q Q followed by the pushforward to A ˚ p E | A q Q under theinclusion C A B Ă E | A . Lemma 2.4.
The following equality holds: ch YX pt α, β uq X r V s “ td p E | X q ´ p´ q rank Q i ˚ ! Q _ | Z p βK,αQ q p Sp C Z p βK,αQ q Z p β K q p ! K | Z p βK q r C V X Z p β K q V sqq where i is the inclusion Z p β K , α Q q ã Ñ X .Proof. Let p be the projection | K | Ñ Y . Note thatch YX pt α, β uqr V s“ ch YX pt α Q , u b t , β K uqr V s“ ch | K | Z p βK q | X pt p ˚ α Q , u b t , t K uqr C V X Z p β K q V s“p td K | X q ´ ch Z p β K q X pt α Q , uq ¨ ! K | Z p βK q r C V X Z p β K q V s . The first equality is from the homotopy invariance. Since a bivariant classis compatible with Gysin maps (or the intersection products), we have thehomotopy invariance of the generalized localized Chern characters; see [11,Corollary 18.1.1]. The second equality is the deformation to the normalcone. The last equality is Proposition 2.3 (vi) of [19]. Finally using (2.1)and the fact that td E “ td E _ ¨ p ch p Λ rank E E _ qq ´ for a vector bundle E , weconclude the proof. (cid:3) Corollary 2.5. If α , β are regular and X “ Z p α Q , β K q , then ch YX pt α, β uq “ p´ q rank Q td p E | X q ´ i ! Z p α Q ,β K q i ! Z p β K q , where i Z p α Q ,β K q , i Z p β K q are regular immersions of Z p α Q , β K q , Z p β K q into Z p β K q , Y respectively.Proof. This immediately follows from the definition of refined Gysin homo-morphisms; see § (cid:3) § W be an integral closed substack of | F | which is factored through p ´ p Z p σ qq , i.e. W Ñ p ´ p Z p σ qq Ă | F | . Note that this means that p ˚ σ | W “
0. Hence by Proposition 2.3 (vi) of [19]ch Z p w σ q Z p σ q pt p ˚ σ, t F uq X r W s “ td ´ p F | Z p σ q q ¨ ! F | Z p σ q pr W sq . (2.2)Let W be an irreducible cycle of | F | which is not factored through p ´ p Z p σ qq .Following [13, 4], we consider the blow-up M of M along Z p σ q . Let F bethe pullback of F to M , and let D be the exceptional divisor. On M we OCALIZED CHERN CHARACTERS FOR 2-PERIODIC COMPLEXES 7 obtain a short exact sequence of two-term complexes0 / / K / / (cid:15) (cid:15) F / / σ (cid:15) (cid:15) O M p´ D q s D (cid:15) (cid:15) / / / / / / O M O M / / , where σ is the pullback of σ and s D is the inclusion map of the idealsheaf O M p´ D q of O M . This shows that locally p F , w σ q is isomorphic to p| K ‘ O M p´ D q| , w s D q . Therefore we note that the proper transform W of W is contained in | K | since it is the case for generic points of W . By thecompatibility with proper pushforward Z p w σ q Ñ Z p w σ q , we havech Z p w σ q Z p σ q pt p ˚ σ, t F uq X r W s “ b ˚ ch | K | Z p σ q ptp p q ˚ σ , t F uq X r W s where b : Z p σ q “ D Ñ Z p σ q and p : | F | Ñ M are the projections.By Corollary 2.5 we conclude thatch Z p w σ q Z p σ q pt p ˚ σ, t F uq X r W s “ ´p td E | Z q ´ ¨ b ˚ p i ! D ¨ i ! M pr W sqq (2.3)where i D : D Ñ M is the inclusion and i M : M Ñ | K | is the inclusion asthe zero section.2.5. Cosection Localization.
Consider the setup in § ! F,σ : A ˚ p Z p w σ qq Q Ñ A ˚ p Z p σ qq Q , for an algebro-geometric understanding of a work of Lee andParker [16]. Theorem 2.6. ! F,σ “ td F | Z p σ q ¨ ch Z p w σ q Z p σ q pt p ˚ σ, t F uq . Proof.
The equations (2.2) and (2.3) exactly match with basic constructiondetermining the cosection localized Gysin map; see § (cid:3) Let d : A Ñ F be a complex of vector bundles on a DM stack M . Supposethat its dual gives rise to a perfect obstruction theory relative to a pure-dimensional stack M . Supposed that M Ñ M is representable. Consider acosection of F such that σ ˝ d “
0. Let C be the cone in F associated to therelative intrinsic normal cone of M over M . Assume that the cosection hasa lift as a cosection of absolute obstruction sheaf. Then C is as a cycle, i.e.set-theoretically, supported in Z p w σ q by Kiem - Li [13, Proposition 4.3].As the immediate consequence of Theorem 2.6 we obtain the followingcorollary. Corollary 2.7.
The following holds: r M s vir σ : “ ! F,σ r C s “ td F | Z p σ q ¨ ch Z p w σ q Z p σ q pt p ˚ σ, t F uq X r C s . Corollary 2.8. (Chang, Li, and Li [4, Proposition 5.10])
Consider the setupin § Y is smooth. Let F “ E | Z p β q and σ “ α | Z p β q . Then ! F,σ r C Z p β q Y s “ td E | Z p α,β q ¨ ch YZ p α,β q pt α, β uq X r Y s . KIM AND OH
Remark 2.9.
The difference between Theorem 2.6 and Corollary 2.8 is thatthe latter assumes that Y is smooth. In section §
3, we will need Theorem2.6. 3.
Comparisons of virtual classes
We apply the bivariant property of localized Chern characters to a com-parison of virtual classes. In this section, let the base field k be the field ofcomplex numbers.3.1. Conjecture.
Let V , V be vector spaces over k and let a reductivealgebraic group G act on V and V linearly. Fix a character θ of G suchthat V ss p θ q “ V s p θ q , i.e. there is no strictly semistable points of V withrespect to θ . Let E : “ rp V ss p θ q ˆ V q{ G s , ¯ E : “ r V ˆ V { G s , V ˆ V whichare vector bundles on stack quotients r V ss p θ q{ G s , r V { G s , V respectively.Fix w P pp
Sym ě V _ q b V _ q G . The polynomial w induces sections s , ¯ s of E _ , ¯ E _ and also regular morphisms f , w below:¯ E _ (cid:15) (cid:15) E _ ? _ o o (cid:15) (cid:15) r V { G s ¯ s ] ] r V ss p θ q{ G s , ? _ o o s ] ] V f / / V _ ,E w / / A . This is so-called a geometric side of a hybrid gauged linear sigma model; see[7]. Let r be the dimension of V , i.e. the rank of E . We require that thecritical locus Z p dw q of the function w is a smooth closed locus in the zerosection locus r V ss p θ q{ G s Ă E _ with codimension r . Note that canonically Z p dw q “ Z p s q .3.1.1. Tangent Complex of Z p f q . Let ε P Q ą . We consider the modulispace Q εg,k p Z p s q , d q of ε -stable quasimaps to Z p s q with type p g, k, d q where g is genus, k is the number of markings, d P Hom p ˆ G, Q q is a fixed curve class(see [17, 9]). Here ˆ G is the character group of G . The stable quasimapsto Z p s q are certain maps to the Artin stack Z p ¯ s q , not necessarily to Z p s q .The moduli space is a separated DM stack proper over the affine quotientSpec p Sym V _ q G . It comes with a canonical virtual fundamental class de-noted by r Q εg,k p Z p s q , d qs vir ; see [9, 6]. Conventions : Let M g,k p BG, d q be the moduli space of principal G -bundles P on genus g , k -marked prestable orbi-curves C with degree d such that theassociated classifying map C Ñ BG is representable. The algebraic k -stack M g,k p BG, d q is smooth; see [9, 6]. Let P be the universal G -bundle on C and let u : C Ñ P ˆ G V be the universal section. Let π be the universalcurve map M g,k p BG, d q . Let V : “ P ˆ G V and V : “ P ˆ G V . By abusingnotation, π will also denote the universal curve on various moduli spacesover M g,k p BG, d q . For example, π : C Ñ Q εg,k p Z p s q , d q denotes also theuniversal curve and P denotes also the universal bundle on this C . Therefore, OCALIZED CHERN CHARACTERS FOR 2-PERIODIC COMPLEXES 9 we may consider also V i as a vector bundle on the universal curve over C Ñ Q εg,k p Z p s q , d q .Consider a complex of cotangent bundles V ˆ V “ f ˚ Ω V _ df ÝÑ Ω V “ V ˆ V _ , (3.1)Its dual restricted to the affine scheme Z p f q Ă V is the tangent complex of Z p f q since it is a complete intersection scheme. By pulling back the dual of(3.1) to the universal curve over Q εg,k p Z p s q , d q and then pushforward by π we obtain R π ˚ p u ˚ p P ˆ G df _ qq : R π ˚ V Ñ R π ˚ V _ ;(3.2)see the proof of Proposition 4.4.1 of [9]. The dual of (3.2) is the canonicalperfect obstruction theory for Q εg,k p Z p s q , d q relative to M g,k p BG, d q , defining r Q εg,k p Z p s q , d qs vir .3.1.2. LG quasimaps.
On the other hand, we may consider the moduli space
LGQ εg,k p E, d q , for short LGQ , of genus g , k -pointed, degree d , ε -stablequasimaps to V {{ G with p -fields; see [3, 10, 7]. Here by a p -field we meanan element in H p C, V | C b ω C q , where C is a domain curve. Note herethat we use ω instead of ω log which is used usually in gauged linear sigmamodel ([10]). For simplicity, let us call LGQ the moduli of LG quasimapsto E . (Due to twisting by ω C , an LG quasimap to E is not a map to E evenfor larger enough ε .) Let L ‚ LGQ { M g,k p BG,d q denote the cotangent complex of LGQ relative to M g,k p BG, d q . By the same idea of [6], it is clear that LGQ comes with a perfect obstruction theory R ‚ π ˚ p V ‘ V b ω C q _ Ñ L ‚ LGQ { M g,k p BG,d q relative to M g,k p BG, d q . By [3, 10, 7] there is a cosection dw LGQ : R π ˚ p V ‘ V b ω C q Ñ O LGQ , where, by abusing notation, V i : “ P ˆ G V i with P the universal G -bundleon the universal curve C on LGQ .We recall the definition of dw LGQ . From the differential of w we mayconsider k -linear map dw : Sym V b V Ñ V _ , which induces R π ˚ dw : Sym R π ˚ V b R π ˚ V Ñ R π ˚ V _ . Then for p u, p q P LGQ , p u , p q P R π ˚ p V ‘ V b ω C q , we define dw LGQ | p u,p q p u , p q : “ Res p H p R π ˚ dw qp exp p u q b u q b p ` p b f p u qq (3.3)Here the residue map Res is the Grothendieck-Verdier duality pairing R π ˚ V _ b R π ˚ p V b ω C q ‘ R π ˚ p V b ω C q b R π ˚ V _ Ñ O . Example 1.
For G “ G m the multiplicative group, dw LGQ has the fol-lowing explicit description: for p u , p q “ p u i , p j q i,j P R π ˚ p V ‘ V b ω C q at p u, p q “ p u i , p j q i,j P R π ˚ p V ‘ V b ω C q , dw LGQ | p u,p q p u , p q “ Res p ÿ i,j u i B f j p u qB u i b p j ` ÿ j p j b f j p u qq where i, j run for 1 , ..., dim V , 1 , ..., dim V , respectively, and f “ p f j q j .Let a global vector bundle complex r F Ñ F s represent R π ˚ p V ‘ V b ω C q . Then induced from dw LGQ there is a cosection of F , which will bedenoted also by dw LGQ . The zero locus of the cosection dw LGQ , i.e. thelocus defined by the ideal sheaf Im p dw LGQ q , coincides with Q εg,k p Z p s q , d q .We can check this by considering (3.3) at test families of LG quasimaps to E .Let p : F Ñ LGQ be the projection. It can be shown that p ˚ dw LGQ ˝ t F “ support of the relative intrinsic normal cone of LGQ over M g,k p BG, d q ; see [10]. In fact it will be shown later in Corollary 3.5 that p ˚ dw LGQ ˝ t F “ t p ˚ dw LGQ , t F u to the obstruction cone in F (see Corollary 2.7),we obtain a virtual class r LGQ s vir dw LGQ supported in Z p p ˚ dw LGQ , t F q “ Z p dw LGQ q “ Q εg,k p Z p s q , d q . The latter space is proper over the affine quo-tient Spec k r V s G ; see [9].According to Chang and Li [3]; and Fan, Jarvis, and Ruan [10], we expectthe following. Conjecture 3.1. In A ˚ p Q εg,k p Z p s q , d qq Q , r Q εg,k p Z p s q , d qs vir “ p´ q χ p V _ q r LGQ εg,k p E, d q s vir dw LGQ (3.4) where χ p V _ q is the virtual rank of R π ˚ V _ . For a smooth quintic Z p s q in P , a pioneering work of Chang and Li[3, Theorem 1.1] shows that Conjecture 3.1 with k “ ε ąą
0, and d ą A p Q g, p Z p s q , d qq Q Ñ H p Q g, p Z p s q , d q , Q q “ Q for d ą Proof of Conjecture.
We prove Conjecture 3.1.
Theorem 3.2.
Conjecture 3.1 holds true.
Remark 3.3.
After an announcement of the above result, F. Janda toldthe first author that he, Q. Chen, and R. Webb are working on a proof ofthe conjecture using torus localization for cosection localized virtual classes[2]. Later a paper of H. Chang and M. Li [5] appeared, showing the aboveresult when r V ss { G s is the projective space with G “ G m and Z p s q is a OCALIZED CHERN CHARACTERS FOR 2-PERIODIC COMPLEXES 11 hypersurface. Their proof uses, among other things from the original proofof H. Chang and J. Li [3] in a special case, the degeneration of ‘target’ E tothe normal cone C Z p s q E . A similar degeneration appears also in our proof,too; see ˜ U P Q and β in § B : “ M g,k p BG, d q , X : “ Z p s q , Q εX : “ Q εg,k p Z p s q , d q , Q εV : “ Q εg,k p V {{ G, d q ; and let π : C Ñ B be the universal curve.3.2.1. Construction of φ A , ¯ φ B . For simple notation, let V be the dualvector space of V . Let Sym ‹ denote a fixed range ‘ d f a “ Sym a for somepositive integer d f such that w P ‘ d f a “ p Sym a V _ qb V . Let f : Sym ‹ V Ñ V be a linear map induced from w P Sym ‹ V _ b V . Combining with the naturalhomomorphism nat we getSym ‹ R π ˚ V nat ÝÝÑ R π ˚ Sym ‹ V R π ˚ f ÝÝÝÑ R π ˚ V (3.5)on B where V : “ V _ . Here for the definition of Sym ‹ : “ ‘ d f a “ Sym a operator (in particular for two-term complexes), see § m , let O p q : “ p ω log C b p P ˆ G C θ q ε q m whosepullback to LGQ is π -ample. We take an open substack of B ˝ of B suchthat the map LGQ Ñ B is factored through B ˝ and where O p q is still π -ample. We do the following construction over B ˝ .We first take a π -acyclic, locally free resolutions of V for large enough l Ñ V h ÝÑ A “ π ˚ p π ˚ p V _ b O p l qqq _ b O p l q Ñ B Ñ B is defined to the cokenel. This is an exact sequence of vectorbundles on the universal curve on B ˝ . There are the indued homomorphismsSym ‹ h : Sym ‹ V Ñ Sym ‹ A and f V : “ P ˆ G f : Sym ‹ V Ñ V . Wewant to construct a π -acyclic resolution V Ñ A Ñ B Ñ V with ahomomorphism Sym ‹ A Ñ A compatible with Sym ‹ h, f V (see (3.6)). Forthis construction, we consider the cokernel V of p f V , ´ h q : Sym ‹ V Ñ V ‘ Sym ‹ A . Note that the induced map V Ñ V is an inclusion of vector bundles.The locally free sheaf V of finite rank has π -acyclic locally free resolution0 Ñ V Ñ A Ñ B Ñ π -acyclic resolutionof V . Let us take A : “ A . Consider the map V Ñ A which is given bythe composition of inclusions V Ñ V Ñ A . This gives rise to a π -acyclic,locally free resolutions of V Ñ V Ñ A Ñ B Ñ B is defined to be the cokernel. Note that B is π -acyclic since A is π -acyclic. Now combining those two resolutions of V , V , we have a natural chain map of exact sequences0 / / Sym ‹ V ‹ h / / f V (cid:15) (cid:15) Sym ‹ A / / f A (cid:15) (cid:15) Coker f B | Coker1 (cid:15) (cid:15) / / / / V / / A / / B / / f A is the composition Sym ‹ A Ñ V Ñ A ; f B | Coker is determinedby f A ; and Coker is defined as the the quotient Sym ‹ A { Sym ‹ V .Furthermore let us take A such that the natural map π ˚ π ˚ A Ñ A issurjective. This makes sure that R π ˚ Sym ‹ V „ ÝÑ π ˚ Sym ‹ r A Ñ B s . Thederived map (3.5) is realized as each individual natural map as below exceptthe dashed arrow.Sym ‹ π ˚ A / / nat (cid:15) (cid:15) Sym ‹´ π ˚ A b π ˚ B / / nat (cid:15) (cid:15) Sym ‹´ π ˚ A b Λ π ˚ B / / nat (cid:15) (cid:15) ¨ ¨ ¨ π ˚ Sym ‹ A / / “ (cid:15) (cid:15) π ˚ p Sym ‹´ A b B q π ˚ B / / (cid:15) (cid:15) ✤✤✤ π ˚ p Sym ‹´ A b Λ B q / / (cid:15) (cid:15) ¨ ¨ ¨ π ˚ Sym ‹ A / / π ˚ f A (cid:15) (cid:15) π ˚ p Coker “ Ker pB qq π ˚ f B | Coker1 (cid:15) (cid:15) / / / / (cid:15) (cid:15) ¨ ¨ ¨ π ˚ A / / π ˚ B / / / / ¨ ¨ ¨ where B : Sym ‹´ A b B Ñ Sym ‹´ A b Λ B is the differential and ‹ ´ ‹ ´ r , d f ´ s , r , d f ´ s , respectively. Here nat is thenatural map followed by projection. In below we will suppress projectionswhen those are obvious. Note that canonically Coker – Ker B . By taking π ˚ f B | Coker ˝ nat , we obtain a O B ˝ -homomorphism ϕ B | π ˚ V : Sym ‹´ π ˚ V b B Ñ B . Here B i : “ π ˚ B i . We want to find a lift of ϕ B | π ˚ V ϕ locB : Sym ‹´ A b B Ñ B locally on B ˝ . Here A i : “ π ˚ A i . Note that π ˚ Ker pB q Ñ π ˚ p Sym ‹´ A b B q is a locally split monomorphism since its cokernel is locally free. Hencelocally there is a dotted arrow making a quasi-isomorphism between themiddle two complexes.We found that locally on B ˝ there is ϕ locB fitting in a cochain realizationof Sym ‹ R π ˚ V Ñ R π ˚ V :(3.7) Sym ‹ A / / ϕ A (cid:15) (cid:15) Sym ‹´ A b B / / ϕ locB (cid:15) (cid:15) Sym ‹´ A b Λ B / / (cid:15) (cid:15) ¨ ¨ ¨ A / / B / / / / ¨ ¨ ¨ OCALIZED CHERN CHARACTERS FOR 2-PERIODIC COMPLEXES 13 where ϕ A is the global ϕ A : “ π ˚ f A ˝ nat restricted to the local chart.We take a DM stack U ε which is an open substack of tot A and Q εV : “ Q εg,k p V {{ G, d q is a closed substack of U ε .In below for a vector bundle S on B ˝ and a B ˝ -stack M , we denote by S | M the pullback of S to M . For each non-negative integer m , there is anon-degnerate pairingSym m A b Sym m p A _ q Ñ O B ˝ , r v b ... b v m s b r v ˚ b ... b v ˚ m s ÞÑ ÿ σ P S m v ˚ p v σ p q q ...v ˚ m p v σ p m q q and hence an identification p Sym m A q _ “ Sym m p A _ q . This givesHom O B ˝ p O B ˝ , A b O B ˝ p Sym ‹ A q _ q“ Hom O B ˝ p O B ˝ , A b O B ˝ p Sym ‹ A _ qqÑ Hom O B ˝ p O B ˝ , A b O B ˝ p Sym A _ qq“ Hom O Spec p Sym A _ q p Sym A _ , A b O B ˝ p Sym A _ qq so that ϕ A P Hom O B ˝ p O B ˝ , A b O B ˝ p Sym ‹ A q _ q will give rise to a section ϕ A p exp p t A qq P H p U ε , A | U ε q . Therefore (by a similar method) we maydefine a section and a homomorphism φ A : “ ϕ A p exp p t A qq P H p U ε , A | U ε q ;¯ φ B : “ ϕ B | π ˚ V p exp p t V q b ´q : B | Q εV Ñ B | Q εV . (3.8)Define φ locB by ϕ locB p exp a b ´q for a P A at the local chart, extending ¯ φ B . Lemma 3.4.
The equality d A ˝ φ A ´ φ locB ˝ d A ˝ t A “ holds.Proof. In the statement, the morphism d A is defined by d A : π ˚ A “ A Ñ π ˚ B “ B . The morphism d A is defined in the same way. The equalitymeans that d A ˝ ϕ A p exp a q “ ϕ locB p exp a b d A p a qq for every a P A . Thelatter follows from the commutativity of the first square in diagram (3.7). (cid:3) Paring and Residue map.
From now on in this section we will alsouse notation that Q : “ A and P : “ B _ . Let U P : “ U ε ˆ B ˝ tot P, U
P Q : “ U P ˆ B ˝ tot Q Ă tot A ˆ B ˝ tot P ˆ B ˝ tot Q. Note that r P d P ÝÝÑ Q _ s represents R π ˚ p V b ω C q . Here d P : “ ´ d _ Q due toshifting. This yields the cochain map realization of Grothendieck duality R π ˚ p V b ω C qr s „ ÝÑ R H om p R π ˚ V _ , O B ˝ q , which in turn gives rise to acochain map realization of R π ˚ p V b ω C qr s b R π ˚ V _ R es ÝÝÑ O B ˝ as follows P b Q (cid:15) (cid:15) / / P _ b P ‘ Q _ b Q pairing (cid:15) (cid:15) / / Q _ b P _ (cid:15) (cid:15) / / O B ˝ / / . Note that for q P Q with d Q p q q “ p P P , then the paring x d P p p q , q y “´x p, d Q p q qy “
0. Hence taking cohomology level map of R es above weconclude that the parings restricted to Ker d P , Ker d Q are the residue parings,i.e. the following diagram commute P _ b Ker d P ‘ Q _ b Ker d Q paring / / (cid:15) (cid:15) O B ˝ R π ˚ V _ b R π ˚ p V b ω C q ‘ R π ˚ p V b ω C q b R π ˚ V _ ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ (3.9)3.2.3. Set-up 1.
Let p : F : “ p B ‘ Q _ ‘ Q q| U PQ Ñ U P Q be a vector bundle on U P Q , defined via the natural map U P Q Ñ B ˝ .Let ˜ p : ˜ F Ñ ˜ U P Q be the pullback of F by the projection map ˜ U P Q : “ U P Q ˆ A Ñ U P Q . We consider a section β of ˜ F and a cosection σ of˜ F | p Q εV ˆ B ˝ tot P ˆ B ˝ tot Q qˆ A defined by β : “ p d A ˝ t A , λd P ˝ t P , φ A ´ λt Q q ; σ : “ x ¯ φ B ˝ pr B , t P y P ` x id Q _ ˝ pr Q _ , t Q y Q ` x´ d Q ˝ pr Q , t P y P “ x ¯ φ B ˝ pr B ´ d Q ˝ pr Q , t P y P ` x id Q _ ˝ pr Q _ , t Q y Q (3.10)where λ is the coordinate of A , x , y P denotes the pairing P _ | ˜ U PQ b O ˜ UPQ P | ˜ U PQ Ñ O ˜ U PQ and the similar one is for x , y Q . Corollary 3.5. ˜ p ˚ σ ˝ t ˜ F “ on the stack C Z p β q ˜ U P Q
Proof.
Consider the local extension α loc of σ by replacing ¯ φ B with φ locB .Note that α loc ˝ β “ x φ locB ˝ d A ˝ t A ´ d Q ˝ φ A , t P y P which is zero by Lemma3.4. Applying α loc ˝ β “ C Z p β q ˜ U P Q as in Lemma 2.1, we conclude the proof. (cid:3)
We consider the composite dβ of maps T ˜ U PQ { B | Z p β q Ñ T ˜ F { B | Z p β q Ñ ˜ F | Z p β q of the differential of β relative to B and the natural projection. Similarlywe have the composite d k β : T ˜ U PQ { B | Z p β q Ñ T ˜ F { B | Z p β q Ñ ˜ F | Z p β q using thedifferential of β relative to k and the natural projection. In the above proofwe have shown that α loc ˝ β “
0. Applying the chain rule to α loc ˝ β “
0, wenote that σ ˝ d k β “
0. Therefore σ restricted to ˜ F | Z p β q is factored by some OCALIZED CHERN CHARACTERS FOR 2-PERIODIC COMPLEXES 15 ¯ σ, ˜ σ as in a following commuting diagram: T ˜ U PQ { B | Z p β q dβ / / (cid:15) (cid:15) ˜ F | Z p β q / / “ (cid:15) (cid:15) Coker dβ ¯ σ { { (cid:15) (cid:15) T ˜ U PQ { k | Z p β q d k β / / ˜ F | Z p β q / / σ % % ❑❑❑❑❑❑❑❑❑❑ Coker d k β ˜ σ (cid:15) (cid:15) O Z p β q . (3.11)3.2.4. Surjectivity and K . On the other hand, as shown [8], we can see thatthe perfect obstruction theory R π ˚ p u ˚ p P ˆ G df _ qq of (3.2) has an explicitdescription on Q εX : “ Q εg,k p Z p s q , d q as follows. For a vector bundle S on B ˝ ,denote by S | Q εX the pullback of S to Q εX . Then the obstruction theory is athree-term complex at r , , s A | Q εX p d A ,π ˚ p f A ˝ prod qp exp t V b´q ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ B | Q εX ‘ Q | Q εX ¯ φ B ´ d Q ÝÝÝÝÝÑ P _ | Q εX which is quasi-isomorphic to r A | Q εX Ñ K s where K is the kernel of the surjection ¯ φ B ´ d Q on Q εX . We explain the reason. First there is a naturalcommuting diagram on the universal curve on B ˝ Sym ‹´ V b V prod (cid:15) (cid:15) / / Sym ‹´ V b A prod (cid:15) (cid:15) id b d A / / Sym ‹´ V b B (cid:15) (cid:15) (cid:22) v ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ Sym ‹ V / / f V (cid:15) (cid:15) Sym ‹ A / / f A (cid:15) (cid:15) Coker f B | Coker1 (cid:15) (cid:15) (cid:31) (cid:127) / / Sym ‹´ A b B u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ V / / A d A / / B where ‹ ´ d f ´ prod is the quotient map. Notethat the compositions gives rise to a π -acyclic resolution of the complex P ˆ G df _ : 0 / / V | C P ˆ G df _ (cid:15) (cid:15) / / A | C (cid:15) (cid:15) / / B | C (cid:15) (cid:15) / / / / V | C / / A | C / / B | C / / C is the universal curve in Q εX .3.2.5. Set-up 2.
We consider the Koszul complex t ˜ p ˚ σ, t ˜ F u on C Z p β q ˜ U P Q .Note that t ˜ p ˚ σ, t ˜ F u is exact off the zero locus of σ and β . This zero locusis Q εX ˆ A because: t Q “ σ in (3.10); φ A “ the third term of β and t Q “ t P “ σ in (3.10) andthe surjectivity of ¯ φ B ´ t Q on Q εX (see § λ P A , let j λ : C Z p β λ q U P Q
Ă p C Z p β q ˜ U P Q q| λ be the natural closedimmersion. Let ˜ p : | ˜ F | Z p β q | Ñ Z p β q and p λ : | F | Z p βλ q | Ñ Z p β λ q be projections.Now we have λ ! ch C Z p β q ˜ U PQ Q εX ˆ A t ˜ p ˚ σ, t ˜ F u X r C Z p β q ˜ U P Q s“ ch C Z p β q ˜ U PQ | λ Q εX t p ˚ λ σ, t F u X λ ! r C Z p β q ˜ U P Q s“ ch C Z p β q ˜ U PQ | λ Q εX t p ˚ λ σ, t F u X j λ ˚ r C Z p β λ q U P Q s“ ch C Z p βλ q U PQ Q εX t p ˚ λ σ, t F u X r C Z p β λ q U P Q s . (3.13)Here the first equality follows by the fact that ch C Z p β q ˜ U PQ Q ε p X qˆ A t ˜ p ˚ σ, t ˜ F u is a bi-variant class so that it commutes with the refined Gysin homomorphism λ ! ;the second equality follows from Lemma 3.6 of [8]; and the third equalityfollows from the projection formula.3.2.6. Proof.
Case λ “
1: First note that Z p β | λ “ q – ÝÑ LGQ by a projec-tion. Under this isomorphism, C Z p β | λ “ q U P Q – C LGQ U P ˆ B ˝ Q as conesover LGQ ; and ¯ σ | λ “ in (3.11) coincides with dw LGQ ‘ R π ˚ of (3.12).Therefore(3.13) | λ “ “ ch C LGQ U P ˆ B ˝ QQ εX pt p ˚ dw LGQ , t F u b B ˝ t , t Q uqr C LGQ U P ˆ B ˝ Q s“ p td Q | Q εX q ´ ch C LGQ LGU ε Q εX pt p ˚ dw LGQ , t F uq X r C LGQ U P s“ p td F | Q εX q ´ X r
LGQ s vir dw LGQ where F : “ p B ‘ Q _ q| LGQ . Here the first equality is explained above, thesecond equality is by Proposition 2.3 (vi) of [19], and the third equality isexplained in § λ “ p´ q χ p V _ q td F becomes LHS of (3.4).Case λ “
0: We have Z p β q “ Q εX ˆ B ˝ | P | ˆ B ˝ | Q | C Z p β q U P Q “ C Q εX U ε ˆ B ˝ | P | ˆ B ˝ | Q | Ă F | Z p β q . (3.14)There is no constraint by β on the part | P | ˆ B ˝ | Q | . Hence this part landsin the zero section of F | Z p β q .By Lemma 3.4 the inclusion above (3.14) is the composition of C Q εX U ε ˆ B ˝ | P | ˆ B ˝ | Q | Ă q ˚ K Ă F where q : Z p β q Ñ Q εX is the projection. OCALIZED CHERN CHARACTERS FOR 2-PERIODIC COMPLEXES 17
Let m : C Z p β q U P Q Ñ C Q εX U ε and p : C Z p β q U P Q Ñ Z p β q be pro-jections. On C Z p β q U P Q , we obtain a commuting diagram of locally freesheaves O C Z p β q U PQ m ˚ t K w w ♥♥♥♥♥♥♥♥♥♥♥ t F (cid:15) (cid:15) p ˚ q ˚ K / / p ˚ p B ‘ Q ‘ Q _ q p ¯ φ B ´ d Q , id Q _ q / / p ˚ σ (cid:15) (cid:15) p ˚ p P _ ‘ Q _ q p ˚ taut s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ O C Z p β q U PQ where taut is the sum of the tautological paring of dual pairs p| P | , P _ q and p| Q | , Q _ q . On C Z p β q U P Q , we deform the complex t p ˚ σ, t F u supported on Q εX to t , m ˚ t K u b t p ˚ taut, u supported also on Q εX . By this deformation(3.13) | λ “ becomes the following:ch C Z p β q U PQ Q εX pt , m ˚ t K u b t p ˚ taut, uq X r C Z p β q U P Q s“ ch C Z p β q U PQ Q εX pt , m ˚ t K u b Λ ‚ p P _ ‘ Q _ q b Λ top p P ‘ Q qr top sq X r C Z p β q U P Q s“p´ q χ p V _ q td p P ‘ Q q| ´ Q εX ¨ ch p Λ top p P ‘ Q q| Q εX q ¨ ch C QεX U ε Q εX t , t K u X r i spr C Z p β q U P Q sq“p´ q χ p V _ q p td F | Q εX q ´ ¨ td K ¨ ch C QεX U ε Q εX t , t K u X r i spr C Z p β q U P Q sq“p´ q χ p V _ q p td F | Q εX q ´ ¨ td K ¨ ch C QεX U ε Q εX t , t K u X r C Q εX U ε s“p´ q χ p V _ q p td F | Q εX q ´ ¨ r Q εX s vir where Λ ‚ p P _ ‘ Q _ q is the 2-periodic Koszul-Thom complex, top means therank of P ‘ Q , and r i s is the canonical orientation of i : C Q εX U ε Ă C Z p β q U P Q the inclusion. The first equality is from (2.1). The second equality is from[19, Proposition 2.3 (vi)]. The third equality is from the easy fact thattd E “ td E _ ¨ p ch det E _ q ´ for a vector bundle E . The fourth equality fromthat r i spr C Z p β q U P Q sq “ r C Q εX U ε s . The last equality is from Corollary 2.2. References [1] P. Baum, W. Fulton and R. MacPherson,
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