DDISCRIMINANTS AND SEMI-ORTHOGONALDECOMPOSITIONS
ALEX KITE AND ED SEGAL
Abstract.
The derived categories of toric varieties admit semi-orthogonal de-compositions coming from wall-crossing in GIT. We prove that these decom-positions satisfy a Jordan-H¨older property: the subcategories that appear, andtheir multiplicities, are independent of the choices made.For Calabi-Yau toric varieties wall-crossing instead gives derived equiva-lences and autoequivalences, and mirror symmetry relates these to monodromyaround the GKZ discriminant locus. We formulate a conjecture equating in-tersection multiplicities in the discriminant with the multiplicities appearingin certain semi-orthogonal decompositions. We then prove this conjecture insome cases.
Contents
1. Introduction 1
62. Toric background 6
73. Semi-orthogonal decompositions for toric varieties 8 Introduction
Let X be a toric variety, constructed as a GIT quotient of a vector space V by atorus T . There is a well-established theory [Kaw, Seg, BFK, HL] that tells us howto produce semi-orthogonal decompositions of the derived category D b ( X ). Wedo it by considering other birational models of X , i.e. crossing walls in the GIT a r X i v : . [ m a t h . AG ] F e b ALEX KITE AND ED SEGAL problem T (cid:121) V . If we cross to a quotient X (cid:48) , and K X (cid:48) is ‘more negative’ than K X , then D b ( X ) decomposes as D b ( X ) = (cid:10) D b ( X (cid:48) ) , D b ( Z ) , ..., D b ( Z ) (cid:11) (1.1)where Z is another toric variety of smaller dimension. We do this repeatedly untilwe arrive at a ‘minimal’ chamber. Since the extra pieces are always equivalent tothe derived category of a toric variety they themselves can be decomposed by thesame procedure, and we get a recursive algorithm which terminates after a finitenumber of steps.If X is projective then the result of this algorithm is a full exceptional collectionfor X , i.e. every piece of the final decomposition is equivalent to D b ( C ). But forquasi-projective varieties there will usually be many different categories occuring,each one with some multiplicity. Moreover the decomposition is not unique; at eachstep of the algorithm one may have a choice about which wall to cross through andthese choices result in different decompositions. The main technical result of thispaper is the following Jordan-H¨older type theorem: Theorem A (Theorem 3.12) . Let X be a toric variety. If we decompose D b ( X )using the wall-crossing algorithm then the subcategories occuring in the final de-composition, and their multiplicities, are independent of all choices.This result is not particularly hard to prove and neither is it an abstract result;we prove it by analysing the algorithm. But it is notable that the Jordan-H¨olderproperty does not hold for semi-orthogonal decompositions in general [BBS, Kuz].Our real motivation for proving the theorem above was to be able to understanda conjecture appearing in a physics paper by Aspinwall–Plesser–Wang [APW]. Partof what they state is already understood in the mathematical literature but thereremains a significant unsolved problem which we are able to formulate precisely us-ing our theorem (Conjecture 4.17). This generalizes a conjecture made by Halpern-Leistner–Shipman [HLSh].We will use the remainder of this introduction to explain the motivation andcontext for this conjecture1.1. Spherical functors from wall-crossing.
Our conjecture concerns the spe-cial case when the torus action T (cid:121) V is through the subgroup SL ( V ). In this caseall the GIT quotients X will be Calabi-Yau, meaning K X ∼ = O X , and not projective.In this situation the wall-crossing theory does not provide any decompositions of D b ( X ), instead it proves that all the GIT quotients are derived equivalent since thedecomposition (1.1) just becomes D b ( X ) = D b ( X (cid:48) ). However the category D b ( Z )still has an important role.The derived equivalence between X and X (cid:48) is not unique, the theory gives usmultiple equivalences for every wall-crossing, and by composing them we get au-toequivalences of D b ( X ). From work of Halpern-Leistner–Shipman [HLSh] it isknown that each of these autoequivalences can be described as a twist T F arounda spherical functor F : D b ( Z ) → D b ( X )where Z is the same toric variety that appears in (1.1).By combining these, and the Picard groups of each GIT quotient, we can getmany autoequivalences of D b ( X ). So the interesting problem becomes to under-stand this large group of autoequivalences.1.2. FI parameter spaces.
Now we explain some heuristics from physics andmirror symmetry. In string theory the data of T acting on V determines an abeliangauged linear sigma model, a widely studied class of N = (2 ,
2) superconformal field
ISCRIMINANTS AND SEMI-ORTHOGONAL DECOMPOSITIONS 3 theories. In this theory there are certain important parameters called complexifiedFayet-Iliopoulos parameters, they take values in a complex manifold which we callthe FI Parameter Space (FIPS). They are related to stability conditions in the GITproblem and in certain limiting regions of the FIPS the theory reduces to a sigmamodel whose target is one of the quotients X . In physical terminology X is a phase of the model. In this region we can identify the FI parameters with the complexifiedK¨ahler moduli of X so the FIPS is closely related to the extended or stringy K¨ahlermoduli space of X . Under mirror symmetry the FI parameters become complex parameters, so theFIPS is the base of the mirror family. Since toric mirror symmetry has a math-ematically precise formulation this gives us a rigorous definition of the FIPS: it’sthe complement of the GKZ discriminant locus ∇ inside the dual torus T ∨ (Section4.2). It is helpful to think of T ∨ as an open subset of the secondary toric variety F and to take the closure ∇ ⊂ F , because then the phases correspond to the toricfixed points in F . From this point-of-view the FIPS is obtained by deleting ∇ andthe toric boundary from F .The mirror family is a locally-trivial family of symplectic manifolds over theFIPS with fibre ˇ X . The monodromy of this family gives an action of π ( F IP S ) onˇ X as symplectomorphisms, and hence as autoequivalences of the Fukaya categoryFuk( ˇ X ). On the mirror side this predicts an action: π ( F IP S ) (cid:121) D b ( X )This is the ‘B-brane monodromy’. Examples and physical calculations suggestthat this is essentially the group of autoequivalences that arise via wall-crossing asdescribed in Section 1.1. This prediction appears in many places in the maths andphysics literature ( e.g. [HHP, HW, HLSam, HLSh]) and has been verified for someexamples [DS, Kit]. It seems to be a difficult problem to verify it in general, mainlybecause it is hard to understand π ( F IP S ).1.3.
The rank 1 case.
The case where T = C ∗ is quite well-known and easy tounderstand directly. In this case there are two possible phases which we denote by X ± . If we split V by weights as V + ⊕ V ⊕ V − then it’s easy to see that X ± is avector bundle over P V ± × V , where P V ± is a weighted projective space.In this rank 1 case the discriminant locus is always a single point δ so the FIPSis C ∗ \ δ (see Example 4.4). Or we can say that the secondary toric variety F is a P and that the FIPS is obtained from it by deleting the two toric fixed points andone more non-fixed point. The phase X + corresponds to the region near one of thetoric fixed points, and the loop around that fixed point simply acts as ⊗O (1) on D b ( X + ).More interesting is the loop around the non-fixed point δ - often called the conifold point - which corresponds to wall-crossing to X − and back again. If thereare no zero weights then the resulting autoequivalence is the twist T S around aspherical object S = O P V + given by the sky-scraper sheaf along the zero section in X + . If there are zero weightswe upgrade this to a twist around the spherical functor F : D b ( V ) → D b ( X + ) The FIPS is not quite the same as the SKMS, the latter should be intrinsic to X whereas theformer depends on its presentation as V//T . Also note that the SKMS is expected to be a complexsubmanifold of the space of Bridgeland stability conditions; on the mirror side this is the differencebetween small and big quantum cohomology. The FIPS is easier to compute than either the SKMSor the space of stability conditions.
ALEX KITE AND ED SEGAL given by pulling-up to P V + × V and then pushing-forward along the inclusion into X + . In the notation of Section 1.1 the variety Z is V . Remark . If there is only one positive weight then X + is an affine orbifold andPic( X + ) is a finite cyclic group Z /k . In this case it’s sensible to allow that toricfixed point as part of the FIPS. The reason is that F is (if we’re careful) an orbifold P and this fixed point has isotropy group Z /k , so we get an action of the orbifoldfundamental group.This subtlety is interesting in the rank 1 case since it is occurs in the well-known‘Calabi-Yau/Landau-Ginsburg correspondence’. In higher rank it happens veryrarely and is of no significance for this paper. For us the FIPS will contain none ofthe toric boundary and hence we can ignore any orbifold structure on F .1.4. Components of the discriminant.
Suppose we have a higher rank torus T ∼ = ( C ∗ ) r . The discrimant locus ∇ is now some hypersurface in ( C ∗ ) r and it isusually the union of several irreducible components: ∇ = ∇ ∪ ... ∪ ∇ k Aspinwall–Plesser–Wang [APW] observed that there is a correspondence betweenthese components ∇ i and certain toric varieties Z i , built from subsets of the originaltoric data. They conjecture that for each phase X there should be a sphericalfunctor F i : D b ( Z i ) → D b ( X ) (1.3)and that T F i corresponds to the monodromy around the component ∇ i . Thereis some deliberate ambiguity here; there is no canonical loop around ∇ i (evenup to homotopy), so the functors F i are at best defined up to composition byautoequivalences.1.5. Factorizations and multiplicities.
To understand this conjecture of [APW]more clearly we pick two adjacent chambers of the secondary fan, separated by awall W . This is the situation we discussed in Section 1.1. The two chambers givetwo phases X ± which are derived equivalent, and we get an autoequivalence of D b ( X + ) which is the twist around a spherical functor F : D b ( Z ) → D b ( X + ) (1.4)for some smaller toric variety Z .In the secondary toric variety F our wall W corresponds to a rational curve C W connecting the toric fixed points corresponding to our two phases. It turns out thatthe discriminant locus ∇ always intersects C W in a single point δ (Corollary 4.12).This is the same picture that we saw in Section 1.3, and the reason for this is thatby focusing on a single wall-crossing we are essentially reducing to a rank 1 GITproblem. There is a the 1-parameter subgroup λ W ⊂ T normal to the wall and it isonly stability with respect to λ W that is changing. So, just as in the rank 1 case, aloop in C W that goes around the point δ should correspond to the autoequivalence T F .However, C W is not part of the FIPS since it lies in the toric boundary of F . Toget an actual element of π ( F IP S ) we have to perturb C W (or an open subset ofit) off the toric boundary, and take a loop in the perturbed curve.When we do this peturbation the point δ may split into several points because ∇ typically meets C W with some multiplicity. This means that our element of π ( F IP S ) is naturally a composite of several loops, one around each of our newmissing points. In fact each component ∇ i might meet C W with multiplicity, andwe can group the new missing points according to these components (see Figure 1). ISCRIMINANTS AND SEMI-ORTHOGONAL DECOMPOSITIONS 5
Figure 1. (L) A real picture of C W as the straight line connectingthe two points marked by X ± . (R) A complex picture of a 2-spherenear to the rational curve C W , where the point δ has split intothree. A loop from X + to X − and back again will factor into twoloops around ∆ and one loop around ∆ .So the loop around δ naturally factors into several loops around the differentcomponents of ∇ , with each component possibly appearing multiple times. Thissuggests that we should look for a corresponding factorization of the autoequiva-lence T F .This factorization does indeed exist. The toric variety Z is not usually a Calabi-Yau, which means that D b ( Z ) (unlike D b ( X )) can be can be decomposed using thewall-crossing algorithm. Moreover, the subcategories that appear in this decompo-sition are always equivalent to D b ( Z i ) where Z i is one of the varieties considered byAspinwall–Plesser–Wang (Section 1.4). So we get a semi-orthogonal decomposition D b ( Z ) = (cid:10) D b ( Z ) , D b ( Z ) , ..., D b ( Z k ) , D b ( Z k ) (cid:11) (1.5)where each D b ( Z i ) occurs some number of times (possibly zero). The order of thefactors here depends on the choices made in the algorithm, but by our Theorem3.12 the multiplicities do not.Halpern-Leistner–Shipman [HLSh] observed that this decomposition gives us afactorization of the autoequivalence T F . If we restrict the spherical functor F (1.4) to each piece of D b ( Z ) then we again get a spherical functor, and T F is thecomposition of all the corresponding twists. This provides the spherical functors F i required by Aspinwall–Plesser–Wang and matches with our discussion of loopsin the FIPS.However, for this story to make sense there is one essential numerical condition: Conjecture B (Conjecture 4.17) . The multiplicity of D b ( Z i ) in the decomposition(1.5) agrees with the intersection multiplicity of ∇ i with C W .We finish by proving our conjecture in some special cases, the strongest of whichis: Theorem C (Theorem 4.22) . If the torus T has rank 2 then Conjecture B holds. Remark . A significant part of this story was already understood by Halpern-Leistner–Shipman. They only consider the case when Z is projective, meaningthat the decomposition of D b ( Z ) is actually a full exceptional collection, and they ALEX KITE AND ED SEGAL conjecture that the number of exceptional objects agrees with the intersection mul-tiplicity of ∇ with C W [HLSh, Remark 4.7]. Our conjecture is a synthesis of theirswith the work of [APW].1.6. Acknowledgements.
E.S. would like to thank Paul Aspinwall for helpfulconversations.This project has received funding from the European Research Council (ERC)under the European Union Horizon 2020 research and innovation programme (grantagreement No.725010). A.K. was supported by the EPSRC [EP/L015234/1] via theLSGNT Centre for Doctoral Training.2.
Toric background
Notation and assumptions.
We are interested in toric varieties constructedas GIT quotients of a vector space V by a torus T . We specify the data of the torusaction as a complex of lattices0 −→ L Q ∨ −→ Z n A −→ N −→ −→ M A ∨ −→ Z n Q −→ L ∨ −→ • L is the lattice of 1-parameter subgroups of the torus T , so T = L C ∗ . • Z n is the lattice of Laurent monomials on V , i.e. V = Spec[ N n ] for thesubmonoid N n ⊂ Z n . • Q is the weight map . The images q i = Q ( e i ) of the standard basis vectorsare the weights of the action. • N is the cokernel of Q ∨ modulo torsion. • M is the kernel of Q and the dual of N . • A is the ray map . The images a i = A ( e i ) are the rays .By definition A is surjective and A ∨ is injective. We will always assume that Q ∨ is injective, so Q is surjective modulo torsion - this is the assumption that genericpoints of our GIT quotient stacks do not have infinite isotropy groups. It followsthat (2.1) and (2.2) are exact apart from a possible torsion group L ∨ / Im Q ∼ =Ker A/ Im Q ∨ .A stability condition is an element of L ∨ R . A choice of stability condition θ definesa semi-stable locus in V and hence a GIT quotient, which for us means the quotientstack: X θ = [ V ssθ / T ]We’ll generally only be interested in quotients with respect to generic θ , in whichcase X θ is (at worst) a DM stack. We’ll also refer to these generic GIT quotientsas the phases of the GIT problem. Each phase is a toric orbifold and has a corre-sponding fan in N . The ‘rays’ a i always span the rays of the fan (hence the name)but the higher dimensional cones change depending on the phase. Remark . If the weight map Q has some finite cokernel then the representation T → GL ( V ) has a finite kernel, so the GIT quotients X θ have finite isotropy groupsat all points. We need to allow this possibility, since even if it doesn’t apply to ourinitial toric variety X it can happen for the smaller-dimensional varieties Z thatappear in wall-crossing.Note that in this situation A does not determine Q . There is a theory of stackyfans but which solves this issue but we won’t need it because for us Q is thefundamental piece of data. ISCRIMINANTS AND SEMI-ORTHOGONAL DECOMPOSITIONS 7
The space of stability conditions has a wall-and-chamber structure whose cham-bers correspond to phases. If we consider all (non-empty) GIT quotients we geta fan in L ∨ called the secondary fan - the top-dimensional cones correspond tonon-empty phases and the lower-dimensional cones correspond to non-generic GITquotients. The rays of the secondary fan include those generated by the weights q i ,but in general there more rays than this. Corresponding to the secondary fan is atoric variety, the secondary toric variety F .2.2. The Calabi-Yau case.
An important special case is when the torus T actsthrough SL ( V ), which implies that each phase is Calabi-Yau.In terms of the toric data, the Calabi-Yau case is when the sum of the weights q i is zero. Equivalently, the rays a i are all contained in (and hence affinely span)an affine hyperplane of height 1. In this case is is helpful to consider the polytopeΠ ⊂ N given by the convex hull of the rays. Each phase corresponds to a fan in N , whichwhen intersected with the affine hyperplane determines a decomposition of Π. Thesedecompositions are exactly the coherent triangulations , i.e. triangulations inducedby a piece-wise linear function.2.3. Higgs and Coloumb GIT problems.
From our original GIT problem T (cid:121) V we will often extract a smaller GIT problem involving some subset of the toricdata, either by picking a subset of the weights, or a subset of the rays. The twomain ways this will happen are:(1) Suppose W ⊂ L ∨ R is a wall in the secondary fan, normal to some 1-parametersubgroup λ ∈ L . Then we can consider the subset of weights which areorthogonal to λ , i.e. which lie in the subspace (cid:104) W (cid:105) .(2) In the Calabi-Yau case we can choose a face Γ ⊂ Π of the toric polytope,and consider the set of rays lying in this face.Formally, suppose we pick a subset
S ⊂ { , ..., n } . We can view S as a subsetof the standard basis vectors { e , .., e n } in Z n so there is a corresponding set ofrays A ( S ) ⊂ N . We set N S ⊂ N to be the sublattice spanned by A ( S ), write A S : Z S → N S for the restriction of A , and set L S = Ker A S . Then we get a GITproblem: L S Q ∨S −→ Z S A S −→ N S We’ll refer to this as the
Coloumb GIT problem associated to the subset S .Alternatively we pick a subset T ⊂ { , ..., n } and consider the corresponding setof weights Q ( T ) ⊂ L ∨ . We define L ∨T as the primitive sublattice generated by theseweights L ∨T = L ∨ ∩ (cid:104) Q ( T ) (cid:105) R ⊂ L ∨ R and we get a GIT problem: M T A ∨T −→ Z T Q T −→ L ∨T We’ll call this the
Higgs GIT problem associated to T . Note that Q S is by definitionsurjective but Q T might not be ( c.f. Remark 2.3).Our ‘Higgs’ and ‘Coloumb’ terminology is based on the ‘Higgs GLSM’ and‘Coloumb GLSM’ from [APW], which are related to the Higgs and Coloumb branchesof the vacuum moduli space at singular values of the FI parameters.
ALEX KITE AND ED SEGAL Semi-orthogonal decompositions for toric varieties
Crossing a single wall.
Fix a toric GIT problem T (cid:121) V . Let C + and C − be two adjacent chambers of the secondary fan separated by a wall W , and labelledsuch that C + lies on the same side of W as the character det( V ). Let λ W ∈ L bethe primitive 1-parameter subgroup normal to this wall, oriented such that κ = (det V )( λ W ) ≥ i.e. C + lies on the λ W > X ± for the phases corresponding to thesetwo chambers.For this wall we have a Higgs GIT problem as described in Section 2.3. Let T bethe indexing set for the weights orthogonal to λ W , so Q ( T ) are all the weights lyingin the subspace (cid:104) W (cid:105) . The vector space corresponding to Z T is the fixed subspace V λ W ⊂ V . Also Q ( T ) necessarily span (cid:104) W (cid:105) , so L ∨T is exactly the orthogonal to λ W , i.e. it’s the character lattice of T /λ W . Hence this Higgs GIT problem is justdescribing the action of T /λ W on V λ W .The secondary fan for this Higgs GIT problem lives in the vector space (cid:104) W (cid:105) andthe cone W lies in some chamber of it. We write Z for the corresponding phase. Theorem 3.1. [BFK, Theorem 5.2.1]
We have a semi-orthogonal decomposition D b ( X + ) = (cid:10) D b ( X − ) , D b ( Z ) , ..., D b ( Z ) (cid:11) where κ copies of D b ( Z ) occur.Remark . This theorem is an application of the general theory of ‘windows’relating GIT and derived categories [BFK, HL, Seg], which applies to a generalGIT quotient of a variety by a reductive group. However, in the current state-of-the-art you cannot use this theory to compare two different
GIT quotients unlessyou assume that the wall-crossing is of a particularly simple form. Which theseones are.
Remark . If det( V ) lies on the wall then κ = 0 and the theorem states that D b ( X + ) and D b ( X − ) are equivalent. This is a toric flop. Example 3.4.
If we consider the standard action of C ∗ on C n +1 then X − = ∅ and we get D b ( P n ) = (cid:10) D b ( pt ) , ..., D b ( pt ) (cid:11) which recovers Beilinson’s result that P n has full exceptional collection of length n + 1. // Remark . If X + happens to be a blow-up of X − then Theorem 3.1 recoversOrlov’s blow-up formula for this toric situation. It’s possible to formulate thetheorem more generally in such a way that it directly generalizes Orlov’s result.3.2. The algorithm.
Theorem 3.1 immediately suggests the following recursivealgorithm for decomposing the derived category of a phase X :(1) Starting at the chamber for X we cross through a sequence of walls, alwaysmoving away from det( V ). At each wall we refine our decomposition.(2) We stop when we reach a minimal phase where no further such wall-crossings are possible.(3) Every factor occuring in this decomposition is the derived category of aphase of a smaller GIT problem, so we can apply this algorithm to eachfactor.Note that a phase is minimal if − (det V ) lies in the closure of that chamber, orequivalently if the canonical bundle of that phase is nef. ISCRIMINANTS AND SEMI-ORTHOGONAL DECOMPOSITIONS 9 (1) A (2) O ( − P (3) O ( − P × A (4) O ( − P Figure 2.
Remark . If X is projective then you can use this algorithm to recover Kawa-mata’s result [Kaw] that a projective toric variety has a full exceptional collection[BFK, Thm 5.2.3]. This is because the minimal phase will be empty (as in Example3.4), and moreover the minimal phase is empty in every Higgs GIT problem thatoccurs in the algorithm.In this paper we are more interested in quasi-projective examples. Example 3.7.
Take V = C and quotient by ( C ∗ ) using the following matrix ofweights: (cid:18) − − (cid:19) Observe that det( V ) = (1 , (cid:62) . This GIT problem has four phases and the sec-ondary fan is drawn in Figure 2. The phases are:(1) X = A . This is the unique minimal phase.(2) X = O ( − P , the total space of the tautological line bundle on P .(3) X = O ( − P × A .(4) X = O ( − P , the total space of the relative O ( −
1) line bundle over theprojective bundle P = P ( O ⊕ ⊕ O ( − → P .Firstly we decompose D b ( X ) by crossing the wall into chamber (1). The 1-parameter subgroup for this wall is (1 ,
1) so κ = 3. The Higgs GIT is C ∗ (cid:121) C withweight 1, and Z is the non-empty phase Z = pt . Hence Theorem 3.1 in this casesays D b ( X ) = (cid:10) D b ( X ) , D b ( pt ) , D b ( pt ) , D b ( pt ) (cid:11) which is an instance of Orlov’s blow-up formula (see Remark 3.5).To make the rest of this example more readable we’ll write this SOD and allfollowing ones in the compressed form: D b ( X ) = (cid:10) X , pt, pt, pt (cid:11) For this phase no futher refinements are possible, and the algorithm is finished.Next we apply the algorithm to phase 4. Let us choose to cross to phase (2)and then to phase (1). The wall-crossing between (2) and (4) is again a blow-up,it blows up the codimension 2 subvariety O ( − P . So crossing both walls gives: D b ( X ) = (cid:10) X , O ( − P (cid:11) = (cid:10) X , pt, pt, pt, O ( − P (cid:11) We are not yet finished, because we can still apply the algorithm to the factor D b ( O ( − P ). But this variety is just the blow-up of A at the origin, so the nextrefinement is: D b ( X ) = (cid:10) X , pt, pt, pt, A , pt (cid:11) (3.8)No further refinements are possible. What happens if make a different choice? We could instead have crossed to phase(3) before crossing to phase (1). The crossing (1) (cid:32) (3) blows up a plane, and thecrossing (3) (cid:32) (4) blows up a P , so crossing these walls gives the decomposition: D b ( X ) = (cid:10) X , P , P (cid:11) = (cid:10) X , A , P , P (cid:11) The factor D b ( P ) can be split into two exceptional objects (as in Example 3.4) sothe final step is: D b ( X ) = (cid:10) X , A , pt, pt, pt, pt (cid:11) (3.9)Note that in this example the quotienting torus is ( C ∗ ) , and for each phase weneeded to apply the recursive algorithm (at most) two times. For rank r it wouldneed r applications. //In the preceding example we noticed that when decomposing D b ( X ) we hadtwo choices, since there were two possible paths from chamber (4) to chamber (1). In the second step of the algorithm there was no such choice, since the Higgs GITproblems were all rank 1 and had only two chambers. In a higher rank examplethere will be many more choices because we need to choose a path at every stepexcept the last one.However, examining the decompositions (3.8) and (3.9) that resulted from ourtwo paths we can see evidence of our J¨ordan-Holder property - the decompositionsare different, but the multiplicities of the ‘irreducible factors’ agree. To state thisprecisely we need to think about what these ‘irreducible factors’ really are.3.3.
Relevant subspaces.
Recall that our initial GIT problem is given by a weightmatrix Q : Z n → L ∨ specifying a torus action T (cid:121) V . At any step in the algorithmthe Higgs GIT problem arises as the fixed subspace V T (cid:48) for some sub-torus T (cid:48) ⊂ T ,with a corresponding a sublattice L (cid:48) ⊂ L . The weights of the Higgs GIT problemare those weights which are orthogonal to L (cid:48) and they always span the subspace( L (cid:48) ) ⊥ R .The ‘irreducible factors’ of our decompositions are the derived categories of theminimal phases of each such Higgs GIT problem. However, some of these minimalphases will be empty. The condition for them to be non-empty is that the coneon the weights is the whole of ( L (cid:48) ) ⊥ R , so the secondary fan for that GIT problem iscomplete. Definition 3.10.
We call a subspace H ⊂ L ∨ R relevant if the cone spanned by theweights lying in H is the whole of H .Obviously there can only be finitely-many relevant subspaces. We allow H = 0(which is always relevant) and H = L ∨ R (which might not be). A 1-dimensionalrelevant subspace is a line which has weights on both its rays.The relevant subspaces index the ‘irreducible factors’ in our semi-orthogonaldecompositions. Each one defines a Higgs GIT problem with a non-empty minimalphase Z H , and the corresponding factor is D b ( Z H ). Example 3.11.
In Example 3.7 there are three relevant subspaces: the whole of R , the vertical axis, and the origin. They contribute the factors D b ( A ), D b ( A )and D b ( pt ) respectively. // Theorem 3.12.
Let X be a phase of a toric GIT problem and let H be a rele-vant subspace. The multiplicity of D b ( Z H ) in the semi-orthogonal decomposition of D b ( X ) is independent of all choices of paths. Presumably the different decompositions resulting from different choices of pathsare always related by mutations, but we haven’t checked this. By ‘path’ we really mean a sequence of adjacent chambers.
ISCRIMINANTS AND SEMI-ORTHOGONAL DECOMPOSITIONS 11
Proof of the main theorem.
We’ll prove Theorem 3.12 using the recursivestructure of the algorithm to reduce to the rank 2 case, i.e. when the GIT problemconsists of ( C ∗ ) (cid:121) V = C n . In the rank 1 case the theorem is vacuous since thereare no choices. Lemma 3.13.
Theorem 3.12 holds in the rank 2 case.Proof.
If det( V ) is the trivial character then all phases are derived equivalent andthe theorem is vacuously true, so we can assume det( V ) (cid:54) = 0. For simplicity weassume that neither det( V ) nor − det( V ) lie on a wall, so there is a unique minimalphase and a unique ‘maximal’ phase X max , whose chamber contains det( V ). Infact there could be up to two minimal or maximal phases, but crossing the wallsbetween them is a derived equivalence and we can ignore it. If we start at anynon-maximal phase then there are no choices to be made in the algorithm, but ifwe start at X max then we have exactly two choices of paths to reach X min . So theonly thing to check is that these two choices produce the same multiplicities.There are three classes of relevant subspace:(1) H = C . This is relevant iff X min is non-empty, in which case D b ( X min ) occursin D b ( X max ) with multiplicity one for either choice of path.(2) H a line, both rays of which are walls. The Higgs GIT for H has two non-emptyphases, let Z H be a minimal one and Z (cid:48) H be the other one.By assumption det( V ) doesn’t lie on H , so if λ H is a primitive normal1-parameter subgroup to H then κ = | λ H (det V ) | is strictly positive. The min-imal and maximal chambers lie on opposite sides of H so either choice of pathcrosses it; one choice contributes κ copies of D b ( Z H ) and the other contributes κ copies of D b ( Z (cid:48) H ). But the decomposition of D b ( Z (cid:48) H ) includes exactly onecopy of D b ( Z H ) so either way the multiplicity of D b ( Z H ) in D b ( X max ) is κ .(3) H = { } . This contributes the factor D b ( V T ), the subspace of V fixed by thewhole torus.Consider a line l ⊂ L ∨ R containing at least one weight, let q l be the sum ofthe weights on this line, and let µ l = | q l | be the lattice length of q l . There aretwo possibilities:(a) There are weights on both rays of l . Then l is a relevant subspace as incase (2), both rays are walls and the Higgs GIT has a non-empty minimalphase Z l . The derived category of the other phase Z (cid:48) l decomposes into onecopy of D b ( Z l ) and µ l copies of D b ( V T ).(b) There are only weights on one ray so only that ray is a wall. The HiggsGIT has an empty phase and the other phase decomposes into µ l copiesof D b ( V T ).In either case only one of our two paths will pick up any factors of D b ( V T )from this line l ; it’s the path that crosses l on the same side as q l , and thenumber of such factors it picks up is µ l κ l = µ l | λ l (det( V )) | where λ l is a primitive 1-parameter subgroup normal to l . So we may as wellassume that each such line contains only a single weight q i = q l , and hence onlythat ray of the the line is a wall.Now fix an orientation on our lattice L ∨ , i.e. a unit symplectic form ω . Thismeans that for the wall through q i we can produce a primitive normal subgroupby setting λ = ω ( ˆ q i , − ) where ˆ q i is a primitive vector in the direction of q i . Withthis choice one of our paths always crosses walls in the direction of increasing λ and the other path always crosses walls in the direction of decreasing λ . Soif the first path crosses the rays through q , ..., q r and the second path crosses the rays through q r +1 , ..., q s then the equality we want to show is: r (cid:88) i =1 µ i λ i (det( V )) = − s (cid:88) i = r +1 µ i λ i (det( V ))But this is true since s (cid:88) i =1 µ i λ i = s (cid:88) i =1 ω ( q i , − ) = ω (det( V ) , − )and ω (det( V ) , det( V )) = 0. (cid:3) Proof of Theorem 3.12.
Pick two paths from the chamber for X to the chamberfor a minimal phase, always moving away from det( V ). We can assume that ourtwo paths agree except at a single codimension-two wall where they travel oppositeways around; the general case follows from this since any two paths are related by asequence of such moves. For every Higgs GIT problem that our paths encounter wealso need to make choices, but those GIT problems have lower rank so by inductionwe can assume that those choices do not matter.Let U ⊂ V be the semi-stable locus for a character lying on our codimension-twowall. We have a non-linear GIT problem T (cid:121) U whose phases are exactly thosephases of T (cid:121) V whose chambers are adjacent to the wall. There is an importantsense in which this non-linear problem is rank two. If we let L (cid:48) ⊂ L be the rank 2sublattice normal to our codimension-two wall, and T (cid:48) ⊂ T be the correspondingsubtorus, then only subgroups lying in T (cid:48) can have fixed points in U . It followsthat the secondary fan for T (cid:121) U is just the secondary fan for T (cid:48) (cid:121) U , pulled-backvia the projection L ∨ R → ( L (cid:48) ) ∨ R .So in the region where our two paths differ we can think of them as paths in thesecondary fan for T (cid:48) (cid:121) U . And since they are different they both must start in amaximal chamber and end in a minimal chamber.Now consider the linear GIT problem T (cid:48) (cid:121) V . The secondary fan for T (cid:48) (cid:121) U isa coarsening of the one for T (cid:48) (cid:121) V ; every wall of the former is a wall of the latter,but not necessarily vice-versa since a subgroup λ ⊂ T (cid:48) could have fixed points in V but none in U . However, from the point-of-view of our algorithm there is no harmin regarding every wall for T (cid:48) (cid:121) V as corresponding to a wall for T (cid:121) U - it justhappens that some of them will be ‘fake walls’ where the semi-stable locus doesnot change. In the semi-orthogonal decomposition crossing a fake wall adds somenumber of copies of the zero category D b ( U λ //T ) = D b ( ∅ ). Note that if both raysof a line are a wall for T (cid:48) (cid:121) V then either both give genuine walls for T (cid:121) U orboth are fake. Also if there are any fake walls then U T (cid:48) is empty, which means thatthe codimension-two wall itself also contributes the zero category.If we include these zero categories then we have a bijection between the factors inthe decomposition algorithms for T (cid:48) (cid:121) V and for T (cid:121) U , and their multiplicitiesagree since these depend only on the restriction of the character det( V ) to thesubtorus T (cid:48) . Hence the result follows from Lemma 3.13. (cid:3) FI parameter spaces and discriminants
In this section we consider a
Calabi-Yau
GIT problem T (cid:121) V where T actsthrough the subgroup SL ( V ). This has a different flavour to the previous section,since all phases are Calabi-Yau and every wall-crossing is a derived equivalence, sono semi-orthogonal decompositions occur. Instead (as discussed in the introduction) ISCRIMINANTS AND SEMI-ORTHOGONAL DECOMPOSITIONS 13 we focus on autoequivalences of the phases and relate these to the fundamentalgroup of the FI parameter space.4.1.
Spherical functors.
Let T (cid:121) V be a Calabi-Yau toric GIT problem. Let X + and X − be two phases coming from two adjacent chambers C + and C − , separatedby a wall W . Let Z be the phase of the associated Higgs GIT problem for acharacter lying on W .Since det( V ) = 0, Theorem 3.1 tells us that D b ( X + ) and D b ( X − ) are equivalent.However, what the theory actually gives us is a countable set of equivalencesΨ i : D b ( X + ) ∼ −→ D b ( X − )indexed by the integers. They are related by the Picard groups of X + and X − . Theorem 4.1. [HLSh, Prop. 3.4]
There is a spherical functor F : D b ( Z ) → D b ( X + ) such that Φ − Φ is the twist around F . Recall that the twist around F is the endofunctor of D b ( X + ) defined by the coneon the counit T F = [ F R → R is the right adjoint to F , and that the key property of a spherical functoris that T F is an autoequivalence. See [AL] for more detail on spherical functors.Note that this cone of functors makes sense since we can interpret it as a cone ofFourier-Mukai kernels (or insert the prefix ‘dg’ where needed).The variety Z is toric - it’s a phase of the Higgs GIT problem - but it willnot usually be Calabi-Yau. So using the algorithm of Section 3 we can produce asemi-orthogonal decomposition: D b ( Z ) = (cid:10) C , ..., C r (cid:11) (4.2)Halpern-Leistner and Shipman observed that this implies:(1) The restriction of F to each piece gives a spherical functor F i : C i → D b ( X + ).(2) The twist T F factors as: T F = T F ◦ ... ◦ T F r (4.3)The formal result is [HLSh, Theorem 4.14] and it applies in this situation since thecotwist around F is (up to a shift) the Serre functor on D b ( Z ).The factors in the semi-orthogonal decomposition (4.2) are indexed by the rele-vant subspaces in the Higgs GIT problem for W , but these are simply the relevantsubspaces H ⊂ L ∨ R which are contained in the hyperplane (cid:104) W (cid:105) .4.2. Discriminants.
We now recall some of the theory of discriminant loci devel-oped by Gelfand–Kapranov–Zelevinsky [GKZ].Recall that our GIT problem is specified by a sequence of lattices, exact modulotorsion, or its dual: L Q ∨ −→ Z n A −→ NM A ∨ −→ Z n Q −→ L ∨ From this point on we need to make two mild additional assumptions:(1) We assume that the rays A ( e i ) are all distinct. We need this because for[GKZ] A is a subset of N . This excludes 1-parameter subgroups actingwith weights (0 , ..., , , − , , ...,
0) but these are very uninteresting from awall-crossing perspective. (2) We assume the weights Q ( e i ) are all non-zero. This is just for simplicity.A zero weight just contributes a factor of A to each phase.Tensoring our lattices by C ∗ gives two exact sequences of tori: L C ∗ Q ∨× −→ ( C ∗ ) n A × −→ N C ∗ M C ∗ A ∨× −→ ( C ∗ ) n Q × −→ L ∨ C ∗ The map A ∨× provides us with n characters of the torus M C ∗ . If we pick a vectorof coefficients a ∈ C n we can take a linear combination of these characters, thisgives us a Laurent monomial: W a : M C ∗ → C x (cid:55)→ (cid:104) a, A ∨× ( x ) (cid:105) In explicit coordinates this means W a = n (cid:88) i =1 a i m (cid:89) t =1 X A it t where X , ..., X m are coordinates on M C ∗ . This is the Hori-Vafa mirror to ourtoric GIT problem (or abelian GLSM), it’s a family of Landau-Ginzburg modelsparametrized by a .Since our GIT problem is Calabi-Yau we can choose co-ordinates such that thefirst column of A is entirely 1’s, hence W a = X (cid:102) W a where X doesn’t appear in (cid:102) W a .For a generic a the zero locus W a will be a smooth hypersurface in M C ∗ . Considerthe subset of non-generic a , i.e. D A = { a ∈ C n , ∃ x ∈ M C ∗ such that W a ( x ) = 0 and dW a ( x ) = 0 } This, or perhaps its closure, is the discriminant locus of the family W a . Thisdefinition is the correct one for general A ; since we’re in the Calabi-Yau case thefirst condition is redundant as ∂ X W a = 0 implies W a = 0.The closure of D A is an affine variety, which is always irreducible and usuallya hypersurface [GKZ, Ch. 9]. To understand why this is true observe that D a is a cone so there is an associated projective variety in P n − . It’s not hard tocompute that its projective dual is the closure of the image of M C ∗ in P n − , whichis evidently irreducible. But the projective dual to an irreducible variety is alwaysirreducible, and usually a hypersurface.[GKZ, Ch. 1] If D A is a hypersurface thenwe denote its defining polynomial by ∆ A .As well as being a cone D A is invariant under rescaling the X i variables, i.e. itis invariant under the action of the torus M C ∗ on C n . We can replace D A with theopen subset D A ∩ ( C ∗ ) n - if D A is a hypersurface this loses no information - andthen the quotient by M C ∗ is a subvariety: ∇ A ⊂ L ∨ C ∗ D A is a hypersurface iff ∇ A is, and in this case ∆ A is really a function on L ∨ C ∗ .4.2.1. Horn uniformization.
In the Calabi-Yau case there is a useful dominant ra-tional map P L C (cid:57)(cid:57)(cid:75) ∇ A called Horn uniformization, given by:[ λ ] (cid:55)→ Q × ◦ Q ∨ ( λ ) ISCRIMINANTS AND SEMI-ORTHOGONAL DECOMPOSITIONS 15
In explicit co-ordinates this says: λ : ... : λ r (cid:55)→ (cid:32) n (cid:89) i =1 (cid:16) r (cid:88) k =1 Q ik λ k (cid:17) Q i , ... , n (cid:89) i =1 (cid:16) r (cid:88) k =1 Q ik λ k (cid:17) Q ir (cid:33) Example 4.4.
Suppose L = Z has rank one, and write ( q , ..., q n ) for the vectorof weights. Then by the above ∇ A consists of the single point q q ...q q n n ∈ C ∗ (recall we are assuming that no weights are zero). In particular ∇ A is a hypersurfaceand non-empty. //Let’s explain why this works. We have: ∂ X s W a = 1 X s n (cid:88) i =1 a i A is m (cid:89) t =1 X A it t Invariantly, for a fixed x ∈ M C ∗ this says that dW a ( x ) is the linear map dW a ( x ) : M C −→ C given by composing: M C x − −→ M C A ∨ −→ C n A ∨× ( x ) −→ C n a −→ C Here the first map is the action of the element x − ∈ M C ∗ on M C , and similarly forthe third map. So dW a has a critical point at x iff a ◦ A ∨× ( x ) annihilates M C , i.e. iff a ◦ A ∨× ( x ) = Q ∨ ( λ )for some λ ∈ L C . So the image of the map M C ∗ × L C −→ C n ( x, λ ) (cid:55)→ a = (cid:0) A ∨× ( x ) (cid:1) − Q ∨ ( λ )is the subset where W a has a critical point, and in the Calabi-Yau case this isexactly D A .Next we compose this with the quotient map Q × : D A (cid:57)(cid:57)(cid:75) ∇ A and observe that Q × ( a ) = (cid:0) Q × A ∨× ( x ) (cid:1) − Q × Q ∨ ( λ ) = Q × Q ∨ ( λ )is independent of x , since Q × A ∨× ( x ) = 1. Hence Q × ◦ Q ∨ is a dominant rational mapfrom L C to ∇ A . Finally, the Calabi-Yau condition implies that this map descendsto P L C .If ∇ A is a hypersurface it has the same dimension as P L C , and in this caseHorn uniformization is a birational equivalence [GKZ, Thm 3.3]. The inverse is thelogarithmic Gauss map.4.2.2. Components of the discriminant.
Recall that the convex hull of the rays A ( e i ) is a polytope Π ⊂ N , which lies in an affine hyperplane of height 1.Choose a face Γ of Π. Associated to this face there is a Coulomb GIT problem,as described in Section 2.3. The face contains a certain subset of the rays, andwe’ll abuse notation and write Γ ⊂ { , ..., n } for the subset that indexes these rays.Then the Coulomb GIT problem is specified by an exact sequence of lattices L Γ Q ∨ −→ Z Γ A Γ −→ N Γ (4.5)where N Γ is the sublattice spanned by the face.We can define a discriminant locus associated to this face in the same way as wedid for the whole polytope. For any vector of coefficients a (cid:48) ∈ C Γ there is a Laurentmonomial W (cid:48) a (cid:48) on the torus M Γ C ∗ , where M Γ is the dual lattice to N Γ . To obtain W (cid:48) a (cid:48) from W a you just delete all the terms that don’t correspond to rays on Γ, thensince only some variables remain this function descends from M C ∗ to the quotient M Γ C ∗ . Proceeding as before, we obtain a discriminant subset D Γ ⊂ C Γ , a subvariety ∇ (cid:48) Γ ⊂ ( L ∨ Γ ) C ∗ and its preimage: ∇ Γ ⊂ ( L ∨ ) C ∗ Remark . What we’ve just done works for any subset of the rays, not just thesubsets corresponding to faces of Π. But the faces are the most important.Roughly, we are interested in the union of these subvarieties over all faces of Π.However, some faces don’t contribute anything. For example if Γ is a simplex then L Γ = 0 so ∇ Γ must be empty; indeed it’s easy to see that D Γ is just the origin inthis case.More generally suppose Γ contains a ray A ( e i ) which is linearly independent ofthe other rays in Γ. Then D Γ will be contained in the hyperplane a (cid:48) i = 0 andhence (cid:101) ∇ Γ is empty. If we want to access D Γ then we should try deleting this ray A ( e i ); this will give us a subface Σ ⊂ Γ with one less ray but with L Σ = L Γ . Then D Σ = D Γ under the inclusion C Σ (cid:44) → C Γ , but (cid:101) ∇ Σ might be a non-empty subvarietyof the torus ( L ∨ Γ ) C ∗ . This observation leads to us to the following: Definition 4.7.
A face Γ ⊂ Π is called minimal if every ray in Γ is linearlydependent on the other rays in Γ.Then we define:
Definition 4.8.
The discriminant locus ∇ ⊂ ( L ∨ ) C ∗ is the union of the subvarieties ∇ Γ , for each minimal face Γ ⊂ Π such that ∇ Γ is a hypersurface.The whole polytope Π is minimal since we’re assuming that there are no zeroweights. If ∇ Π = ∇ A is a hypersurface then we call it the principal component of ∇ . Remark . This definition comes from [GKZ]. It is not entirely clear to us whyone disregards the subvarieties ∇ Γ which are not hypersurfaces. In the exampleswe’ve calculated it makes no difference, i.e. each discriminant subvariety of highercodimension is contained in one which is a hypersurface. But we don’t know if thisis always true.If ∇ Γ is a hypersurface we write ∆ Γ for its defining polynomial, then the prod-uct of these cuts out the hypersurface Γ. GKZ modify this by introducing somemultiplicities µ Γ [GKZ, Ch. 10, 1.B] and then taking the product E A = (cid:89) Γ (∆ Γ ) µ Γ which they call the principal A -determinant . The µ Γ ’s are not relevant for us, butthere are two important theorems that they prove that are stated in terms of E A . Theorem 4.10. [GKZ, Ch. 10, Thm 1.4]
The Newton polytope of E A is dual tothe secondary fan. In fact they give a more precise definition of the secondary polytope ˇΠ - whichis in particular dual to the secondary fan - and their theorem is that the Newtonpolytope of E A is ˇΠ.Recall that the secondary fan is the fan of the secondary toric variety F . This isa compactification of L ∨ C ∗ so we can consider the closure: ∇ ⊂ F ISCRIMINANTS AND SEMI-ORTHOGONAL DECOMPOSITIONS 17
The theorem above suggests that this is a natural choice of compactification for ∇ .It implies that ∇ avoids all the toric fixed points in F and that the tropicalizationof ∇ is the secondary fan.Recall also that phases of our GIT problem correspond to coherent triangulationsof the polytope Π, meaning triangulations induced by a piece-wise linear function[GKZ]. More generally a non-generic stability condition induces a coherent sub-division of Π where not all the pieces are simplices. Such a stability conditioncorresponds to a face of the secondary polytope ˇΠ whose vertices are the phasesrefining this subdivision to a triangulation.Suppose we fix a coherent subdivision of Π, corresponding to a face ˇΓ ⊂ ˇΠ. Nowchoose one of the pieces of the subdivision, it is some polytope Σ i ⊂ Π. As usualwe abuse notation and also write Σ i ⊂ { , ..., n } for the indexing set of the raysappearing in this polytope. Associated to this subset Σ i we have a Coloumb GITproblem and a corresponding discriminant locus ∇ Σ i ⊂ L ∨ C ∗ (see Remark 4.6). IfΣ i is a simplex this discriminant locus is empty, so it’s only worth considering thenon-simplicial pieces of our subdivision.Going further we can consider the principal determinant E Σ i , which we mayview as a function on L ∨ C ∗ by pulling-back under the projection L ∨ → L ∨ Σ i . Thezero locus of E Σ i consists of the discriminant locus associated to Σ i as well as thediscriminant loci coming from all the faces of Σ i .On the other hand, Theorem 4.10 tells us that the face ˇΓ corresponds to somesubset of the monomials appearing in E A . Let us write ( E A ) ˇΓ for the sum of thisset of monomials. Theorem 4.11. [GKZ, Ch. 10, Thm 1.12]
For some positive integer multiplicities µ i and some non-zero constant ν we have ( E A ) ˇΓ = ν (cid:89) i ( E Σ i ) µ i where the product runs over the non-simplicial pieces of the subdivision. In fact we only care about one special case of this theorem: the case when ˇΓ isan edge of ˇΠ. Such an edge connects two phases, and corresponds to a wall W inthe secondary fan. In the secondary toric variety F the phases correspond to toricfixed points, and the wall W (or edge ˇΓ) corresponds to a toric rational curve C W ⊂ F connecting the two fixed points. We discussed this in Section 1.5. Corollary 4.12.
The discriminant locus ∇ intersects C W in exactly one point.Proof. The intersection of ∇ with C W is the zero locus of the restriction E A | C W and this restriction is the sum ( E A ) ˇΓ of the monomials appearing in the edge ˇΓ.This edge corresponds to a coherent subdivision of Π which has exactly one non-simplicial piece Σ, having two possible triangulations. By Theorem 4.11 the zerolocus of ( E A ) ˇΓ agrees with the zero locus of E Σ .But the zero locus of E Σ is the discriminant locus for Coloumb GIT problemassociated to Σ. This GIT problem has rank L Σ = 1 so by Example 4.4 its discrim-inant locus is a single point. (cid:3) In Lemma 4.16 below we will refine this result by identifying which componentsof ∇ can intersect with C W . Faces and subspaces.
In Section 3.3 we discussed relevant subspaces in L ∨ R ,these index the factors appearing in our SODs. In this section we show that rele-vant subspaces biject with minimal faces of the polytope Π; this is an elementaryobservation but crucial for formulating our conjecture.Let H ⊂ L ∨ R be a subspace and let q , .., q h ∈ H be the weights lying in H . Recallthat H is relevant if the cone spanned by these q i ’s is the whole of H . Lemma 4.13. H is relevant iff H is spanned by { q i } and there exist positiveintegers k , ..., k h with: k q + ... + k h q h = 0 Proof. If H is relevant then the vector − (cid:80) q i is a non-negative linear combinationof the q i ’s, hence the required strictly positive relation holds. Conversely if the q i ’s span H and there is such a relation then any vector in H can be written as anon-negative linear combination of the q i ’s. (cid:3) Now consider a subset
S ⊂ { , ..., n } and its complement S c . Let’s consider theColoumb GIT problem associated to S and the Higgs GIT problem associated to S c (Section 2.3). These are related by the following diagram: M S c Z S c L ∨S c M Z n L ∨ M S Z S L ∨S A ∨S c Q S c A ∨ QA ∨S Q S (4.14)The middle column is obviously exact, the other columns are exact modulotorsion. Let us also write H S c ⊂ L ∨ R for the subspace spanned by L ∨S c .As a special case we could consider a face of the polytope Π and let Γ be theindexing set for the rays on that face. Then we get an associated subspace H Γ c ⊂ L ∨ R . Proposition 4.15.
The map Γ (cid:55)→ H Γ c is a bijection between the minimal faces of Π and the relevant subspaces of L ∨ R .Proof. Generalizing Definition 4.7, let us call a subset
S ⊂ { , ..., n } minimal if theset of rays A ( S ) ⊂ N has the property that every ray in A ( S ) is linearly dependenton the remaining rays in A ( S ). This is the statement that no basis vectors map tozero under the map Z S → L ∨S , or equivalently that the only weights lying in H S c are Q ( S c ).Conversely, pick a subspace H ⊂ L ∨ R which is spanned by the weights it contains,and let S be the set of weights which do not lie in H . Then H = H S c and S isminimal. Hence the assignment S (cid:55)→ H S c is a bijection between the minimal subsetsof { , ..., n } and the subspaces of L ∨ R which are spanned by the weights they contain.Finally, by Lemma 4.13 the subspace H S c is relevant iff there is a vector k ∈ Z S c with strictly positive entries which maps to zero under Q . Such a vector is exactlyan element k ∈ N ∨ such that k ( A ( e i )) = 0 if i ∈ S and k ( A ( e i )) > i ∈ S c .Since the polytope Π lives in an affine hyperplane of height 1, the existence of sucha k is the statement that S is all the rays on a face of Π. (cid:3) The zero subspace H = 0 is always relevant, and since we assume there are nozero weights it corresponds to the whole polytope Π. The empty set is a face of Π ISCRIMINANTS AND SEMI-ORTHOGONAL DECOMPOSITIONS 19 (in the sense of the above proof), it corresponds to the subspace H = L ∨ R which istherefore relevant.4.4. The conjecture.
Let W be a wall separating two chambers in the secondaryfan. Recall that we have the following two objects associated to W :(1) A toric variety Z W . The wall W has an associated Higgs GIT problem, and Z W is the phase of this problem coming from a character on the relativeinterior of W .(2) A toric rational curve C W in the secondary stack. W is a codimension 1cone in the secondary fan and C W is the associated curve.We can decompose D b ( Z W ) using the algorithm of Section 3, and the factors thatappear are indexed by the relevant subspaces H ⊂ L ∨ R contained in the hyperplane (cid:104) W (cid:105) . Each such subspace defines a Higgs GIT with a non-empty minimal phase Z H , and by Theorem 3.12 the multiplicity of D b ( Z H ) in D b ( Z W ) is well-defined.Relevant subspaces correspond (by Proposition 4.15) to minimal faces Γ of thepolytope Π, and these in turn index the components of the discriminant locus. Asdiscussed in Section 1.5 we are interested in the intersection of ∇ Γ with the curve C W . Lemma 4.16.
Let Γ be a minimal face. If H Γ c is not contained in W then ∇ Γ doesn’t meet the curve C W .Proof. Consider the projection map π : L ∨ → L ∨ Γ or its real version L ∨ R → ( L ∨ Γ ) R , whose kernel is H Γ c . This map takes a stabilitycondition for the original GIT problem and restricts it to give one for the ColoumbGIT problem associated to Γ. If we take a chamber of stability conditions andrestrict them then they will all lie in a single chamber for the Coloumb GIT problem(if two stability conditions induce the same triangulation of Π then they evidentlyinduce the same triangulation of the face Γ). This says that π is a map of fans,between the secondary fan for the original problem and the secondary fan for theColoumb problem, hence it induces a toric morphism π : F → F Γ between the two secondary toric varieties.Recall that ∇ Γ is defined as the preimage of the discriminant locus ∇ (cid:48) Γ ⊂ ( L Γ ) ∨ C ∗ under the projection π : ( L ∨ ) C ∗ → ( L Γ ) ∨ C ∗ . Since π extends to the toric boundarywe can also say that ∇ Γ ⊂ F is the preimage of ∇ (cid:48) Γ ⊂ F Γ .The wall W is a codimension 1 cone in L ∨ . If it doesn’t contain H Γ c then π ( W )is a top-dimensional cone in L ∨ Γ so π ( C W ) is one of the toric fixed points in F Γ .Theorem 4.10 implies that ∇ (cid:48) Γ avoids all the toric fixed points hence ∇ Γ misses C W . (cid:3) Conjecture 4.17.
Let W ⊂ L ∨ R be a wall, let Γ ⊂ Π be a minimal face and let H = H Γ c the corresponding relevant subspace. Assume that H ⊆ (cid:104) W (cid:105) . Write n Γ ,W for the multiplicity of D b ( Z H ) in D b ( Z W ) , and m Γ ,W for the intersection multiplicity of ∇ Γ with C W .Then n Γ ,W = m Γ ,W .Remark . We could allow the case when H doesn’t lie in (cid:104) W (cid:105) : then D b ( Z H ) isnot a factor in D b ( Z W ) so we should set n Γ ,W = 0, and by Lemma 4.16 m Γ ,W = 0also. We will now prove various special cases of this conjecture. The most straight-forward case is when rank L Γ = 1 so H is a hyperplane, hence H = (cid:104) W (cid:105) . Proposition 4.19. If rank L Γ = 1 then n Γ ,W = m Γ ,W = 1 .Proof. In this case Z H is the minimal phase for the Higgs GIT problem that pro-duces Z W , so n Γ ,W = 1.As in Lemma 4.16 we consider the map π : F → F Γ . This map is an isomorphismfrom C W to F Γ , at least away from the fixed points, and ∇ Γ is the pre-image of ∇ (cid:48) Γ ⊂ F Γ . The latter is a single non-fixed point (Example 4.4) so m Γ ,W = 1. (cid:3) Remark . This proposition includes the case when L itself has rank 1 and hence H = W is the origin. This is a vacuous case of our conjecture: C W is the wholeof F , there is only the principal component of ∇ which is a single point, Z W is apoint, and the decomposition of D b ( Z W ) is trivial.We can get a less trivial special case by increasing the rank by one. Proposition 4.21. If rank L Γ = 2 then n Γ ,W = m Γ ,W .Proof. In this case H is a hyperplane in (cid:104) W (cid:105) . Since the projection π : L ∨ → L ∨ Γ is a map of fans π ( W ) must be a ray, so W must lie completely on one side of H .Pick a primitive one-parameter subgroup λ normal to W , then the GIT problemproducing Z W consists of the vector space V λ - these are weights that lie in (cid:104) W (cid:105) - acted on by the torus T /λ . Then H is normal to some primitive one-parametersubgroup µ ∈ L/λ and we orient µ so that it pairs positively with W . Recall that Z W is defined to be the phase associated to a generic character in W , so such acharacter pairs positively with µ . For the decomposition of D b ( Z W ) the importantquantity is κ = (det V λ )( µ ) = (cid:88) weights q i ∈(cid:104) W (cid:105) q i (˜ µ )for any ˜ µ ∈ L lifting µ . If κ > H and D b ( Z W )aquires κ copies of D b ( Z H ); if κ < H . Hence: n Γ ,W = max { κ, } Now we compute the intersection multiplicity m Γ ,W . To start with, let’s assumethat Γ = Π so L itself has rank 2 and H is the origin. Then we wish to compute theintersection multiplicity of the principal component ∇ A with the boundary curve C W . To do this we use the Horn uniformization map P ( L Γ ) C (cid:57)(cid:57)(cid:75) ∇ A from Section 4.2.1, which in this case is actually a morphism since P ( L C ) ∼ = P . Inexplicit co-ordinates, as a rational map to ( C ∗ ) , this is given by: λ : λ (cid:55)→ (cid:32) n (cid:89) i =1 (cid:16) Q i λ + Q i λ (cid:17) Q i , n (cid:89) i =1 (cid:16) Q i λ + Q i λ (cid:17) Q i (cid:33) Without loss of generality we may assume that W is the ray through (1 , C ∗ ) , the subset: C × C ∗ ⊂ F The subset where the first co-ordinate is zero is C W with its fixed points deleted.Since ∇ A avoids the fixed points the only way that λ : λ can map to C W is ifthere exists an i such that Q i λ + Q i λ = 0 and Q i = 0, hence λ : λ = 0 : 1.Then the intersection multiplicity is given by (cid:88) i | Q i =0 Q i if this sum is strictly positive, and zero otherwise. But these rows of Q are preciselythe weights q i that lie on (cid:104) W (cid:105) , and we may set ˜ µ = (1 , (cid:62) , so this sum is κ andhence m Γ ,W = n Γ ,W in this case.To finish we must compute m Γ ,W for rank L Γ = 2 but Γ (cid:32) Π. Once again we usethe projection π : F → F Γ . It maps W to a wall W (cid:48) for the rank 2 GIT problem,and hence it maps C W isomorphically (at least away from its fixed points) onto theboundary curve C W (cid:48) ⊂ F Γ . Since ∇ Γ is the preimage of ∇ (cid:48) Γ ⊂ F Γ it is enough tocompute the intersection multiplicity of ∇ Γ with C W (cid:48) inside the two-dimensionalspace F Γ . But this was the calculation we just performed, and the result is stillmax { κ, } since q i ∈ (cid:104) W (cid:105) iff π ( q i ) ∈ (cid:104) W (cid:48) (cid:105) . (cid:3) The above result and its proof are quite close to [HLSh, Prop. 4.4.].
Theorem 4.22. If rank L = 2 then Conjecture 4.17 holds.Proof. The wall W is a ray and there are only two possibilities for H : either H = (cid:104) W (cid:105) (if this is a relevant subspace) or H = 0. The first case is covered byProposition 4.19 and the second by Proposition 4.21. (cid:3) The main obstacle to extending our proofs to higher rank is the fact that Hornuniformization may no longer be a morphism so it becomes harder to compute theintersection multiplicity m Γ ,W . However in special cases it is still possible to verifythe conjecture - see [Kit, Sect. 10.2] for some more examples. References [AL] R. Anno, T. Logvinenko,
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