Gepner point and strong Bogomolov-Gieseker inequality for quintic 3-folds
aa r X i v : . [ m a t h . AG ] M a y GEPNER POINT AND STRONG BOGOMOLOV-GIESEKERINEQUALITY FOR QUINTIC 3-FOLDS
YUKINOBU TODA
Abstract.
We propose a conjectural stronger version of Bogomolov-Gieseker inequality for stable sheaves on quintic 3-folds. Our conjectureis derived from an attempt to construct a Bridgeland stability conditionon graded matrix factorizations, which should correspond to the Gep-ner point via mirror symmetry and Orlov equivalence. We prove ourconjecture in the rank two case.
Contents
1. Introduction 12. Background 33. Stronger BG inequality for quintic 3-folds 64. Clifford type bound for quintic surfaces 15References 181.
Introduction
Bogomolov-Gieseker (BG) inequality.
First of all, let us recall thefollowing classical result by Bogomolov and Gieseker:
Theorem 1.1. ([Bog78], [Gie79])
Let X be a smooth projective complexvariety and H an ample divisor in X . For any torsion free H -slope stablesheaf E on X , we have ∆( E ) · H dim X − ≥ . Here ∆( E ) is the discriminant ∆( E ) : = ch ( E ) − ( E ) ch ( E ) . It has been an interesting problem to improve the BG inequality for higherrank stable sheaves (cf. [Jar07], [Nak07]). So far such an improvement isonly known for some particular surfaces, e.g. K3 surfaces or Del Pezzosurfaces, which easily follows from Riemann-Roch theorem and Serre duality(cf. Lemma 16, [DRY, Appendix A]). In the 3-fold case, such an improvementis only known for rank two stable sheaves on P by Hartshorne [Har78].In a case of other 3-fold, even a conjectural improvement is not known.The purpose of this note is to propose a conjectural improvement of BGinequality for stable sheaves on quintic 3-folds, motivated by an idea frommirror symmetry and matrix factorizations. We first state the resultingconjecture: Conjecture 1.2.
Let X ⊂ P C be a smooth quintic 3-fold and H : = c ( O X (1)) .Then for any torsion free H -slope stable sheaf on X with c ( E ) / rank( E ) = − H/ , we have the following inequality: ∆( E ) · H rank( E ) > . · · · . (1)The RHS of (1) is a certain irrational real number contained in Q ( e π √− / ),and the detail will be discussed in Conjecture 3.3. Our conjecture is derivedfrom an attempt to construct a Bridgeland stability condition on D b Coh( X )corresponding to the Gepner point in the stringy K¨ahler moduli space of X .The RHS of (1) is related to the coefficient of the corresponding centralcharge. It seems that Conjecture 1.2 does not appear in literatures even inthe rank two case, which we will give a proof in this note: Proposition 1.3.
Conjecture 1.2 is true if rank( E ) = 2 . The above result will be proved in Subsection 3.7. Based on a similaridea, we also propose a conjectural Clifford type bound for stable coherentsystems on quintic surfaces (cf. Section 4). Below we discuss background ofthe derivation of the above conjecture.1.2.
Background.
The notion of stability conditions on triangulated cate-gories introduced by Bridgeland [Bri07] has turned out to be an importantmathematical object to study. However it has been a problem for more thanten years to construct Bridgeland stability conditions on the derived cate-gories of coherent sheaves on quintic 3-folds. From a picture of the mirrorsymmetry, the space of stability conditions on a quintic 3-fold is expected tobe related to its stringy K¨ahler moduli space, which is described in Figure 1.In Figure 1, we see three special points, large volume limit, conifold pointand Gepner point. A conjectural construction of a Bridgeland stability con-dition near the large volume limit was proposed by Bayer, Macri and theauthor [BMT], and we reduced the problem to showing a BG type inequal-ity evaluating ch ( ∗ ) for certain two term complexes. The main conjecturein [BMT] is not yet proved except in the P case [Mac], and we face our lackof knowledge on the set of Chern characters of stable objects.In this note, we focus on the Gepner point. A corresponding stability con-dition is presumably constructed as a Gepner type stability condition [Toda]with respect to the pair (cid:18) ST O X ◦ ⊗O X (1) , (cid:19) where ST O X is the Seidel-Thomas twist [ST01] associated to O X . Combinedwith Orlov’s result [Orl09], as discussed in [Wal], such a stability conditionis expected to give a natural stability condition on graded matrix factor-izations of the defining polynomial of the quintic 3-fold. One may expectthat constructing a Gepner point also requires such a conjectural inequality.It seems worth formulating a conjectural BG type inequality which arisesin an attempt to construct a Gepner point, so that making it clear whatwe should know on Chern characters of stable sheaves. Our Conjecture 1.2is the resulting output. The inequality (1) itself is interesting since there EPNER POINT 3 have been several attempts to improve the classical BG inequality. Assum-ing Conjecture 1, we construct data which presumably give a Bridgelandstability condition corresponding to the Gepner point.Compared to the lower degree cases studied in [Toda], constructing Gep-ner type stability conditions is much harder in quintic cases, and most ofthe attempts are still conjectural. This is the reason we have separated thearguments for the quintic case from the previous paper [Toda]. We hopethat the arguments in this note lead to future developments of the study ofChern characters of stable objects on 3-folds. <---
Large Volume Limit<--- Conifold point<--- Gepner point
Figure 1.
Stringy K¨ahler moduli space of a quintic 3-fold1.3.
Acknowledgment.
The author would like to thank Kentaro Hori, Ky-oji Saito and Atsushi Takahashi for valuable discussions. The author alsowould like to thank Johannes Walcher for pointing out the reference [Wal].This work is supported by World Premier International Research CenterInitiative (WPI initiative), MEXT, Japan. This work is also supported byGrant-in Aid for Scientific Research grant (22684002) from the Ministry ofEducation, Culture, Sports, Science and Technology, Japan.1.4.
Notation and convention.
All the varieties or polynomials are de-fined over complex numbers. For a smooth projective variety X of dimension n and E ∈ Coh( X ), we write its Chern character as a vectorch( E ) = (ch ( E ) , ch ( E ) , · · · , ch n ( E ))for ch i ( E ) ∈ H i ( X ). For a triangulated category D and a set of objects S in D , we denote by hSi ex the smallest extension closed subcategory in D which contains S . 2. Background
Bridgeland stability condition.
Let D be a triangulated categoryand K ( D ) its Grothendieck group. We first recall Bridgeland’s definition ofstability conditions on it. Definition 2.1. ([Bri07])
A stability condition σ on D consists of a pair ( Z, {P ( φ ) } φ ∈ R ) Z : K ( D ) → C , P ( φ ) ⊂ D (2) where Z is a group homomorphism (called central charge) and P ( φ ) is a fullsubcategory (called σ -semistable objects with phase φ ) satisfying the followingconditions: YUKINOBU TODA • For = E ∈ P ( φ ) , we have Z ( E ) ∈ R > exp( √− πφ ) . • For all φ ∈ R , we have P ( φ + 1) = P ( φ )[1] . • For φ > φ and E i ∈ P ( φ i ) , we have Hom( E , E ) = 0 . • For each = E ∈ D , there is a collection of distinguished triangles E i − → E i → F i → E i − [1] , E N = E, E = 0 with F i ∈ P ( φ i ) and φ > φ > · · · > φ N . The full subcategory P ( φ ) ⊂ D is shown to be an abelian category, and itssimple objects are called σ -stable. In [Bri07], Bridgeland shows that thereis a natural topology on the set of ‘good’ stability conditions Stab( D ), andits each connected component has a structure of a complex manifold. LetAut( D ) be the group of autoequivalences on D . There is a left Aut( D )-action on the set of stability conditions on D . For Φ ∈ Aut( D ), it acts onthe pair (2) as follows:Φ ∗ ( Z, {P ( φ ) } φ ∈ R ) = ( Z ◦ Φ − , { Φ( P ( φ )) } φ ∈ R ) . There is also a right C -action on the set of stability conditions on D . For λ ∈ C , it acts on the pair (2) as follows:( Z, {P ( φ ) } φ ∈ R ) · ( λ ) = ( e −√− πλ Z, {P ( φ + Re λ ) } φ ∈ R ) . The notion of Gepner type stability conditions is defined as follows:
Definition 2.2. ([Toda])
A stability condition σ on D is called Gepner typewith respect to (Φ , λ ) ∈ Aut( D ) × C if the following condition holds: Φ ∗ σ = σ · ( λ ) . Gepner type stability conditions on graded matrix factoriza-tions.
Let W be a homogeneous element W ∈ A := C [ x , x , · · · , x n ](3)of degree d such that ( W = 0) ⊂ C n has an isolated singularity at the origin.For a graded A -module P , we denote by P i its degree i -part, and P ( k ) thegraded A -module whose grade is shifted by k , i.e. P ( k ) i = P i + k . Definition 2.3.
A graded matrix factorization of W is data P p → P p → P ( d )(4) where P i are graded free A -modules of finite rank, p i are homomorphisms ofgraded A -modules, satisfying the following conditions: p ◦ p = · W, p ( d ) ◦ p = · W. The category HMF gr ( W ) is defined to be the triangulated category whoseobjects consist of graded matrix factorizations of W (cf. [Orl09]). The gradeshift functor P • P • (1) induces the autoequivalence τ of HMF gr ( W ),which satisfies the following identity: τ × d = [2] . (5)The following is the main conjecture in [Toda]: EPNER POINT 5
Conjecture 2.4.
There is a Gepner type stability condition σ G = ( Z G , {P G ( φ ) } φ ∈ R ) ∈ Stab(HMF gr ( W )) with respect to ( τ, /d ) , whose central charge Z G is given by Z G ( P • ) = str( e π √− /d : P • → P • ) . (6)The definition of the central charge Z G first appeared in [Wal]. It is moreprecisely written as follows: since P i are free A -modules of finite rank, theyare written as P i ∼ = m M j =1 A ( n ij ) , n ij ∈ Z . Then (6) is written as Z G ( P • ) = m X j =1 (cid:16) e n j π √− /d − e n j π √− /d (cid:17) . So far Conjecture 2.4 is proved when n = 1 [Tak], d < n = 3 [KST07], and n ≤ d ≤ n = d = 5,in which the variety X is a quintic Calabi-Yau 3-fold.2.3. Orlov’s theorem.
We recall Orlov’s theorem [Orl09] relating the tri-angulated category HMF gr ( W ) with the derived category of coherent sheaveson the smooth projective variety X := ( W = 0) ⊂ P n − . (7)We only use the results for d = n case, i.e. X is a Calabi-Yau manifold, and d = n + 1 case, i.e. X is general type. Theorem 2.5. ([Orl09, Theorem 2.5], [BFK12, Proposition 5.8]) If d = n ,there is an equivalence of triangulated categories Ψ : D b Coh( X ) ∼ → HMF gr ( W ) such that the following diagram commutes: D b Coh( X ) Ψ F HMF gr ( W ) τ D b Coh( X ) Ψ HMF gr ( W ) . Here F is the autoequivalence given by F = ST O X ◦ ⊗O X (1) . Recall that ST O X is the Seidel-Thomas twist [ST01], given byST O X ( ∗ ) = Cone( R Hom( O X , ∗ ) ⊗ O X → ∗ ) . Theorem 2.6. ([Orl09, Theorem 2.5], [Toda, Proposition 3.22]) If d = n +1 ,then there is a fully faithful functor Ψ : D b Coh( X ) ֒ → HMF gr ( W ) such that we have the semiorthogonal decomposition HMF gr ( W ) = h C (0) , Ψ D b Coh( X ) i YUKINOBU TODA where C (0) is a certain exceptional object. Moreover the subcategory A W : = h C (0) , Ψ Coh( X ) i ex is the heart of a bounded t-structure on HMF gr ( W ) , and there is an equiva-lence of abelian categories Θ : Syst( X ) ∼ → A W . Here
Syst( X ) is the abelian category of coherent systems on X . Recall that a coherent system on X consists of data V ⊗ O X s → F where V is a finite dimensional C -vector space, F ∈ Coh( X ) and s is amorphism in Coh( X ). The set of morphisms in Syst( X ) is given by thecommutative diagrams in Coh( X ) V ⊗ O X s FV ′ ⊗ O X s ′ F ′ . The equivalence Θ sends ( O X →
0) to C (0) and (0 → F ) for F ∈ Coh( X )to Ψ( F ) ∈ A W .3. Stronger BG inequality for quintic 3-folds
In this section, we take W to be a quintic homogeneous polynomial withfive variables W ∈ C [ x , x , x , x , x ] , deg( W ) = 5 . (8)The variety X := ( W = 0) ⊂ P is a smooth quintic Calabi-Yau 3-fold. This is the most interesting case inthe study of Conjecture 2.4. We have an equivalence by Theorem 2.5Ψ : D b Coh( X ) ∼ → HMF gr ( W ) . (9)The goal of this section is to translate Conjecture 2.4 in terms of D b Coh( X ),and relate it to a stronger version of BG inequality for stable sheaves on X .3.1. Stringy K¨ahler moduli space of a quintic 3-fold.
Let us firstrecall a mirror family of a quintic 3-fold X and its stringy K¨ahler modulispace. The mirror family of X is a simultaneous crepant resolution b Y ψ → Y ψ of the following one parameter family of quotient varieties [CdlOGP91]: Y ψ := ( X i =0 y i − ψ Y i =0 y i = 0 ) /G. Here [ y : y : y : y : y ] is the homogeneous coordinate of P , and G =( Z / Z ) acts on P by ξ · [ y : y : y : y : y ] = [ ξ y : ξ y : ξ y : ξ − ξ − ξ − y : 1] EPNER POINT 7 for ξ = ( ξ i ) ≤ i ≤ ∈ G . Let α be the root of unity α := e π √− / . Note that we have the isomorphism b Y ψ ∼ = → b Y αψ (10)by y i y i for 1 ≤ i ≤ y αy . Also b Y ψ is a non-singular Calabi-Yau3-fold if and only if ψ = 1. Hence the mirror family b Y ψ is parametrized bythe following quotient stack (see Figure 1) M K := (cid:20) { ψ ∈ C : ψ = 1 } µ (cid:21) where the generator of µ acts on C by the multiplication of α . The stack M K is called the stringy K¨ahler moduli space of X . We see that there are3-special points in Figure 1: • The point ψ = ∞ , called Large volume limit . • The point ψ = 1, called Conifold point . • The point ψ = 0, called Gepner point .The mirror variety b Y ψ is non-singular except at the first two special points.It is also non-singular at the Gepner point, but there admits a non-trivial Z / Z -action by the isomorphism (10).3.2. Relation to Bridgeland stability.
We discuss a relationship betweenthe space M K and the Bridgeland’s spaceStab( X ) := Stab( D b Coh( X ))based on the papers [Asp], [Bri09]. Let Auteq( X ) be the group of autoe-quivalences of D b Coh( X ). It is expected that there is an embedding I : M K ֒ → [Auteq( X ) \ Stab( X ) / C ](11)such that, if we write I ( ψ ) = ( Z ψ , {P ψ ( φ ) } φ ∈ R )then the central charge Z ψ ( E ) for E ∈ D b Coh( X ) is a solution of thePicard-Fuchs (PF) equation which the period integrals of the mirror family b Y ψ should satisfy. Using the following notation z := 5 − ψ − , θ z := z ddz the PF equation is given by θ z Φ − z (5 θ z + 1)(5 θ z + 2)(5 θ z + 3)(5 θ z + 4)Φ = 0 . (12)The solution space of the above PF equation is known to be four dimensional.In the ψ -variable, the basis is given by (cf. [CdlOGP91]) ̟ j ( ψ ) := − ∞ X m =1 Γ( m/ m )Γ(1 − m/ (5 α j ψ ) m YUKINOBU TODA for 0 ≤ j ≤
3. For an object E ∈ D b Coh( X ), the central charge Z ψ ( E )should satisfy the above PF equation, hence is written as Z ψ ( E ) = X i =0 Φ i ( ψ ) · H − i ch i ( E )where H := c ( O X (1)) and Φ i ( ψ ) is a linear combination of the basis { ̟ j ( ψ ) } ≤ j ≤ which is independent of E . Here we have identified H ( X, Q )with Q via the integration map. On the other hand, around the large vol-ume limit and the conifold point, the monodromy transformations inducelinear isomorphisms M L , M C on the solution space of the PF equation (12).Hence that monodromy transformations act on the central charge Z ψ ( E ),which are expected to coincide with the actions of autoequivalences ⊗O X (1),ST O X respectively. Namely we should have the following identities: Z ψ ( E ⊗ O X (1)) = X i =0 M L Φ i ( ψ ) · H − i ch i ( E ) Z ψ (ST O X ( E )) = X i =0 M C Φ i ( ψ ) · H − i ch i ( E ) . The coefficients of Φ i ( ψ ) are uniquely determined by the above matchingproperty of the monodromy transformations on both sides of (11).Indeed, the above idea is used to give an embedding similar to (11) when X is the local projective plane in [BM11]. In the quintic 3-fold case, basedon a similar idea as above, the central charges Z ψ ( E ) for line bundles E = O X ( m ) are computed by Aspinwall [Asp, Equation (217)]: Z ψ ( O X ( m )) = 16 (5 m + 3 m + 16 m + 6) ̟ ( ψ ) −
12 (3 m + 3 m + 2) ̟ ( ψ ) − m ̟ ( ψ ) − m ( m − ̟ ( ψ ) . Since e mH for m ∈ Z span H even ( X, Q ), the above formula uniquely deter-mines Φ i ( ψ ). A direct computation shows thatΦ ( ψ ) = 15 ( ̟ ( ψ ) − ̟ ( ψ ))Φ ( ψ ) = 130 (16 ̟ ( ψ ) − ̟ ( ψ ) + 3 ̟ ( ψ ))Φ ( ψ ) = 15 ( ̟ ( ψ ) − ̟ ( ψ ) − ̟ ( ψ ) − ̟ ( ψ ))Φ ( ψ ) = ̟ ( ψ ) . As a result, Z ψ ( E ) is written as( ̟ ( ψ ) − ̟ ( ψ )) ch ( E ) + 130 (16 ̟ ( ψ ) − ̟ ( ψ ) + 3 ̟ ( ψ )) H ch ( E )+ 15 ( ̟ ( ψ ) − ̟ ( ψ ) − ̟ ( ψ ) − ̟ ( ψ )) H ch ( E ) + ̟ ( ψ ) ch ( E ) . EPNER POINT 9
Gepner point and Gepner type stability conditions.
Let us con-sider a conjectural stability condition σ G ∈ Stab( X ) satisfying[ σ G ] = I ( ψ = 0) ∈ [Auteq( X ) \ Stab( X ) / C ]where I is an expected embedding (11). Since the point ψ = 0 (Gepnerpoint) in M K is an orbifold point with stabilizer group Z / Z , the stabilitycondition σ G should also have the stabilizer group Z / Z with respect tothe Auteq( X ) × C action on Stab( X ). Under a suitable choice of σ G , thegenerator of the above stabilizer group should be given by (cid:18) ST O X ◦ ⊗O X (1) , − (cid:19) ∈ Auteq( X ) × C (13)since the action of ST O X ◦ ⊗O X (1) on H even ( X, Q ) corresponds to the com-position of monodromy transformations at the large volume limit and theconifold point under the embedding (11), and the five times composition ofST O X ◦ ⊗O X (1) coincides with [2]. (This is a consequence of Theorem 2.5and the identity (5).) The property of σ G fixed by (13) is nothing butthe Gepner type property with respect to (ST O X ◦ ⊗O X (1) , / Z † G so that Z † G ( O x ) = − x ∈ X . Under thisnormalization, Z † G is given by Z † G ( E ) := lim ψ → − Z ψ ( E ) /̟ ( ψ )= − ch ( E ) + 15 ( α + 2 α + 3 α − H ch ( E )+ 130 ( − α + 9 α − H ch ( E ) + ( α −
1) ch ( E ) . Indeed, the coefficients α † j ∈ C H − j of Z † G ( E ) at ch j ( E ) are checked to formthe unique solution of the linear equation( α † , · · · , α † ) · M = α · ( α † , · · · , α † ) , α † = − M is given by the composition of matrices (cf. [Toda, Subsection 4.1]) M := − (td X ) −
10 1 0 00 0 1 00 0 0 1 H H / H H / H / H . Here (td X ) = 5 H / H , ( X )-component of td X . The above matrix M induces the isomorphism on H even ( X ), which is identified with the actionof ST O X ◦ ⊗O X (1) on it. By [Toda, Proposition 4.4], the central charge Z † G is related to the central charge Z G on HMF gr ( W ) given by (6) as Z G (Ψ( E )) = − (1 − α ) · Z † G ( E ) for any E ∈ D b Coh( X ). Here Ψ is the equivalence (9). By applying C -actionon Stab( X ), Conjecture 2.4 for the polynomial (8) leads to the followingconjecture: Conjecture 3.1.
Let X ⊂ P C be a smooth quintic 3-fold, H : = c ( O X (1)) and α : = e π √− / . Then there is a Gepner type stability condition ( Z † G , {P † G ( φ ) } φ ∈ R ) ∈ Stab( X )(14) with respect to (ST O X ◦ ⊗O X (1) , / , whose central charge Z † G is given by Z † G ( E ) = − ch ( E ) + 15 ( α + 2 α + 3 α − H ch ( E )+ 130 ( − α + 9 α − H ch ( E ) + ( α −
1) ch ( E ) . Some observations.
Let us try to construct a desired stability con-dition in Conjecture 3.1. By [Bri07, Proposition 5.3], giving data (14) isequivalent to giving the heart of a bounded t-structure A G ⊂ D b Coh( X )satisfying Z † G ( A G \ { } ) ⊂ { r exp( √− πφ ) : r > , φ ∈ (0 , } (15)and any object E ∈ A G admits a Harder-Narasimhan filtration with respectto the Z † G -stability. We propose that a desired heart A G is constructed asa double tilting of Coh( X ), similar to the one in [BMT]. This is motivatedby the following observations:Firstly in [Toda], we constructed a Gepner type stability condition for aquartic K3 surface S via a tilting of Coh( S ). The construction is similarto the one near the large volume limit in [Bri08], [AB]. A different point isthat, although we only need a classical BG inequality to construct a stabilitycondition near the large volume limit, a construction at the Gepner pointrequires a stronger version of BG inequality given as follows: Lemma 3.2.
Let S be a K3 surface and E a torsion free stable sheaf E on S with rank( E ) ≥ . Then we have the following inequality ∆( E )rank( E ) ≥ − E ) ≥ . (16)The above lemma is an easy consequence of the Riemann-Roch theoremand Serre duality (cf. [Muk87, Corollary 2.5]) and a similar improvementis not known for other surfaces except Del Pezzo surfaces. By the aboveobservation, we expect that a desired Gepner type stability condition on aquintic 3-fold is also constructed in a way similar to the one near the largevolume limit, after an an improvement of BG inequality.Secondly we can rewrite the central charge Z † G ( E ) in the following way: − ch B ( E ) + aH ch B ( E ) + √− (cid:0) bH ch B ( E ) + c ch B ( E ) (cid:1) . (17)Here B = − H/ B ( E ) is the twisted Chern characterch B ( E ) := e − B ch( E ) . EPNER POINT 11
In (17), a, b, c are some real numbers in Q ( α, √− a = − α − α − b √− α + 25 α + 35 α + 310 c √− α + 14 α + 58 α + 516 . They are approximated by a = − . · · · , b = 0 . · · · , c = 0 . · · · . The expression (17) is very similar to the central charge near the largevolume limit, given by Z B,tH ( E ) := − Z X e −√− tH ch B ( E )for t ∈ R > . The above integration is expanded as − ch B ( E ) + t H ch B ( E ) + √− (cid:18) tH ch B ( E ) − t B ( E ) (cid:19) . (18)By comparing (17) with (18), although they are in a similar form, we seethat some signs of the coefficients are different. In [BMT], we constructeda double tilting of Coh( X ) which, together with the central charge (18),conjecturally gives a Bridgeland stability condition near the large volumelimit. We propose to construct the heart A G via a double tilting of Coh( X )in a way similar to [BMT], by taking the difference of the signs of thecoefficients into consideration.3.5. Conjectural stronger Bogomolov-Gieseker inequality.
We im-itate the argument in [BMT] to construct A G . In what follows, we fix B = − H/
2. Let µ B,H be the twisted slope function on Coh( X ) defined by µ B,H ( E ) := H ch B ( E )rank( E ) . Here we set µ B,H ( E ) = ∞ if E is a torsion sheaf. The above slope functiondefines the classical slope stability on Coh( X ). We define the pair of fullsubcategories ( T B,H , F B,H ) of Coh( X ) to be T B,H := h E : µ B,H -semistable with µ B,H ( E ) > i ex F B,H := h E : µ B,H -semistable with µ B,H ( E ) ≤ i ex . The above subcategories form a torsion pair in Coh( X ). The associatedtilting B B,H is defined to be B B,H := hF B,H [1] , T B,H i ex . The category B B,H is the heart of a bounded t-structure on D b Coh( X ).In [BMT, Lemma 3.2.1], it is observed that the central charge (18) sat-isfies the following condition: an object E ∈ B B,H with H ch B ( E ) = 0satisfies Im Z B,tH ( E ) ≥
0. The classical BG inequality is used to showthe above property. We propose that a similar property also holds for the central charge Z † G , i.e. an object E ∈ B B,H with H ch B ( E ) = 0 satisfiesIm Z † G ( E ) ≥
0. Note that such an object E is contained in the category h F [1] , Coh ≤ ( X ) : F is µ B,H -stable with H ch B ( F ) = 0 i ex where Coh ≤ ( X ) is the category of coherent sheaves T ∈ Coh( X ) withdim Supp( T ) ≤
1. Also noting the equality∆( E ) = ch B ( E ) − B ( E ) ch B ( E )the above requirement leads to the following conjecture: Conjecture 3.3.
Let X ⊂ P be a smooth quintic 3-fold and E a torsionfree slope stable sheaf on X with c ( E ) / rank( E ) = − H/ . Then we havethe following inequality: ∆( E ) · H rank( E ) > cb = 1 . · · · . (19)The RHS of (19) is irrational, hence the equality is not achieved. Notethat the RHS in (19) is very close to the RHS in (16) for the K3 surfacecase. Remark 3.4.
A stronger BG inequality similar to (19) is predicted by [DRY] without the condition c ( E ) / rank( E ) = − H/ . The prediction in [DRY] isshown to be false in [Jar07] , [Nak07] . Conjecture 3.3 does not contradictto the results in [Jar07] , [Nak07] since we restrict to the sheaves with fixedslope c ( E ) / rank( E ) = − H/ . There are few examples of stable sheaves on quintic 3-folds in literatures.The following example is taken in [Jar07]:
Example 3.5.
Let E be the kernel of the morphism O ⊕ X → O X (1) ⊕ givenby the matrix (cid:18) x x x x x x x x (cid:19) . Here [ x : x : x : x : x ] is the homogeneous coordinates in P . By [Jar07] , E is a stable vector bundle on X with ch( E ) = (4 , − H, − H , − H / . Then we have ∆( E ) · H rank( E ) = 154 > . · · · . The rank two case will be treated in Subsection 3.7.3.6.
Conjectural construction of the Gepner point.
We now give aconjectural construction of a desired A G assuming Conjecture 3.3. Similarlyto [BMT, Lemma 3.2.1], we have the following lemma: Lemma 3.6.
Suppose that Conjecture 3.3 is true. Then for any non-zero E ∈ B B,H , we have the following: • We have H ch B ( E ) ≥ . • If H ch B ( E ) = 0 , then we have Im Z † G ( E ) ≥ . EPNER POINT 13 • If H ch B ( E ) = Im Z † G ( E ) = 0 , then − Re Z † G ( E ) > .Proof. The same argument of [BMT, Lemma 3.2.1] is applied by using Con-jecture 3.3 instead of the classical BG inequality. (cid:3)
The above lemma shows that the triple( H ch B ( E ) , Im Z † G ( E ) , − Re Z † G ( E ))should behave like (rank , c , ch ) on coherent sheaves on algebraic surfaces.Similarly to the slope function on coherent sheaves, we consider the followingslope function on B B,H ν G ( E ) := Im Z † G ( E ) H ch B ( E ) . Here we set µ G ( E ) = ∞ if H ch B ( E ) = 0. If we assume Conjecture 3.3,then Lemma 3.6 shows that the slope function ν G satisfies the weak see-sawproperty. Definition 3.7.
An object E ∈ B B,H is ν G -(semi)stable if, for any non-zeroproper subobject F ⊂ E in B B,H , we have the inequality ν B,H ( F ) < ( ≤ ) ν B,H ( E/F ) . We have the following lemma:
Lemma 3.8.
Suppose that Conjecture 3.3 is true. Then the ν G -stability on B B,H satisfies the Harder-Narasimhan property.Proof.
Although the central charge Z † G ( ∗ ) is irrational, the values H ch B ( ∗ )are contained in + Z , hence they are discrete. This is enough to apply thesame argument of [BMT, Lemma 3.2.4], [Bri08, Proposition 7.1] to show theexistence of Harder-Narasimhan filtrations with respect to ν G -stability. (cid:3) Assuming Conjecture 3.3, we define the full subcategories in B B,H T G := h E : ν G -semistable with ν G ( E ) > i ex F G := h E : ν G -semistable with ν G ( E ) ≤ i ex . As before, the pair ( T G , F G ) forms a torsion pair on B B,H . By taking thetilting, we obtain the heart of a bounded t-structure A G := hF G [1] , T G i ex . We propose the following conjecture:
Conjecture 3.9.
Let X ⊂ P be a smooth quintic 3-fold and assume thatConjecture 3.3 is true. Then the pair ( Z † G , A G )(20) determines a Gepner type stability condition on D b Coh( X ) with respect to (ST O X ◦ ⊗O X (1) , / . Remark 3.10.
By the construction and the irrationality of Z † G , the pair(20) satisfies the condition (15). On the other hand, the irrationality of Z † G makes it hard to prove the Harder-Narasimhan property of the pair (20). Conjecture 3.3 for the rank two case.
We show that Conjecture 3.3is true in the rank two case.
Proposition 3.11.
Conjecture 3.3 is true when rank( E ) = 2 .Proof. Since we have the inequality∆( E ∨∨ ) · H ≥ ∆( E ) · H we may assume that E is reflexive. Since rank( E ) = 2, we have c ( E ) = − H and ∆( E ) · H = − H + 4 c ( E ) · H. The classical BG inequality implies that ∆( E ) · H ≥
0, i.e. c ( E ) · H ≥ / c ( E ) · H > . · · · .It is enough to exclude the case c ( E ) · H = 2, or equivalently ch ( E ) · H =1 / ( E ) · H = 1 /
2. Let us set F := E ∨ ,which is also a torsion free slope stable sheaf. Since F is reflexive, we have E xt i ( F, O X ) = 0 , i ≥ Q := E xt ( F, O X ) is a zero dimensional sheaf by [HL97, Proposi-tion 1.1.10]. This implies that there is a distinguished triangle F ∨ → D ( F ) → Q [ − D ( ∗ ) is the derived dual R H om ( ∗ , O X ). Therefore if we writech( F ) = (2 , H, ch ( F ) , ch ( F ))(21)then we have ch ( F ∨ ) = ch ( F ) = ch ( E ) andch( F ∨ ) = (2 , − H, ch ( E ) , − ch ( F ) + | Q | ) . (22)Here | Q | is the length of the zero dimensional sheaf Q . On the other hand,since F is a rank two reflexive sheaf, we have the isomorphism (cf. [Har80,Proposition 1.10]) F ∼ = F ∨ ⊗ det( F ) . Noting that det( F ) = O X ( H ), and (21), (22), we have(2 , H, ch ( E ) , ch ( F )) = e H · (2 , − H, ch ( E ) , − ch ( F ) + | Q | ) . The above equality and the assumption ch ( E ) · H = 1 / ( F ) = −
16 + | Q | . (23)Noting that c ( X ) = 10 H , the Riemann-Roch theorem and (23) imply that χ ( F ) := X i =0 ( − i dim H i ( X, F )= 4 + | Q | . (24)We divide into two cases: Case 1. H ( X, F ) = 0 . EPNER POINT 15
By the Serre duality and stability, we have H ( X, F ) ∼ = H ( F, O X ) ∼ = 0 . Therefore, by the assumption H ( X, F ) = 0 and (24), we havedim Ext ( F, O X ) = dim H ( X, F ) ≥ . (25)Let us take the universal extension0 → O X ⊗ Ext ( F, O X ) ∨ → U → F → . Then by [Todb, Lemma 2.1], the sheaf U is a torsion free slope stable sheaf.Applying the BG inequality to U , we obtain the inequality( H − ( E )(2 + dim Ext ( F, O X ))) · H ≥ . The above inequality implies that dim Ext ( F, O X ) ≤
3, which contradictsto (25).
Case 2. H ( X, F ) = 0 . Let us take a non-zero element s ∈ H ( X, F ), and an exact sequence0 → O
X s → F → M → . (26)By [Todb, Lemma 2.2], the sheaf M is a torsion free slope stable sheaf.Therefore it is written as M ∼ = O X ( H ) ⊗ I Z for some subscheme Z ⊂ X with dim Z ≤
1. We have the equalities ofChern characters ch ( F ) = 12 H − [ Z ]ch ( F ) = 16 H − H · [ Z ] − χ ( O Z ) . Because ch ( F ) · H = ch ( E ) · H = 1 /
2, we have H · [ Z ] = 2. Hence weobtain ch ( F ) = − − χ ( O Z ) . On the other hand, (23) implies that ch ( F ) ≥ − /
6, hence we have χ ( O Z ) ≤−
1. By taking the generic projection of the one dimensional subscheme Z ⊂ P to P , the Castelnuovo inequality implies g ( Z ) := h ( O Z ) ≤
12 ( H · [ Z ] − H · [ Z ] − . Since H · [ Z ] = 2, we have h ( O Z ) = 0, which contradicts to χ ( O Z ) ≤ − (cid:3) Clifford type bound for quintic surfaces
In this section, we take W ′ to be a quintic homogeneous polynomial withfour variables W ′ ∈ C [ x , x , x , x ] , deg( W ′ ) = 5 . We consider Conjecture 2.4 in this case. We relate it with some Cliffordtype bound for stable coherent systems on the smooth quintic surface S := ( W ′ = 0) ⊂ P . Computation of the central charge.
The surface S is a hyperplanesection ( x = 0) of a quintic 3-fold X := ( W = 0) ⊂ P , where W is definedby W := W ′ + x ∈ C [ x , x , x , x , x ] . By Theorem 2.6, there is the heart of a bounded t-structure A W ′ ⊂ HMF gr ( W ′ ),and an equivalence Θ : Syst( S ) ∼ → A W ′ . Below we abbreviate Θ and regard a coherent system ( O ⊕ RS → F ) as anobject in A W ′ . There is a natural push-forward functor (cf. [Ued]) i ∗ : HMF gr ( W ′ ) → HMF gr ( W )such that by [Toda, Lemma 3.12] and [Toda, Lemma 4.5], we have i ∗ ( O ⊕ RS → F ) ∼ = Ψ( O ⊕ RX → i ∗ F ) . Here i ∗ F is the usual sheaf push-forward for the embedding i : S ֒ → X ,Ψ : D b Coh( X ) ∼ → HMF gr ( W ) an equivalence in Theorem 2.5 and( O ⊕ RX → i ∗ F ) ∈ D b Coh( X )is an object in the derived category with i ∗ F located in degree zero. Let usconsider the central charge Z ′ † G on HMF gr ( W ′ ) defined by Z ′ † G ( P ) := Z † G (Ψ − i ∗ P ) , P ∈ HMF gr ( W ′ )where Z † G is the central charge (17) on D b Coh( X ) considered in the previoussection. By the argument in [Toda, Section 4], the central charge Z ′ † G onHMF gr ( W ′ ) differs from (6) only up to a scalar multiplication. For F ∈ Coh( S ), let us writech( F ) = ( r, l, n ) ∈ H ( S ) ⊕ H ( S ) ⊕ H ( S )with r ∈ Z and n ∈ + Z . By setting H = c ( O X (1)) and B = − H/
2, wehave ch B (Ψ − i ∗ ( O ⊕ RS → F ))= ch B ( O ⊕ RX → i ∗ F )= (cid:18) − R, (cid:18) r − R (cid:19) H, i ∗ l − R H , n + 524 r − R (cid:19) . Applying the computation of Z † G in the previous section, we have Z ′ † G ( O ⊕ RS → F ) = − n − r + 548 R + 5 a (cid:18) r − R (cid:19) + √− (cid:18) b (cid:18) h · l − R (cid:19) − cR (cid:19) . Here h := H | S and a, b, c are irrational numbers given in (17). EPNER POINT 17
Conjectural Clifford type bound.
We expect that a desired Gepnertype stability condition in this case is constructed via double tilting of A W ′ ,similarly to the previous section. Let µ ′ be the slope function on A W ′ , givenby (using the notation in the previous subsection) µ ′ ( O ⊕ RS → F ) := − ch B ( i ∗ F ) · H R = 5 (cid:18) − rank( F ) R (cid:19) . Here we set µ ′ ( ∗ ) = −∞ if R = 0. (Also see [Toda, Subsection 5.4].)The above slope function defines the µ ′ -stability on A W ′ , which satisfiesthe Harder-Narasimhan property (cf. [Toda, Lemma 5.14]). Following thesame argument in the previous section, we expect that any µ ′ -stable object E ∈ A W ′ with µ ′ ( E ) = 0 satisfies Im Z ′ † G ( E ) ≥
0. It leads to the followingconjecture:
Conjecture 4.1.
Let S ⊂ P be a smooth quintic surface and h = c ( O S (1)) .For a µ ′ -stable coherent system ( O ⊕ RS → F ) on S with R = 2 rank( F ) > ,we have the following inequality c ( F ) · hR >
58 + cb = 1 . · · · . If we assume the above conjecture, we are able to construct a doubletilting A ′ G of A W ′ , such that the pair ( Z ′ † G , A ′ G ) satisfies Z ′ † G ( A ′ G \ { } ) ⊂ { r exp( √− πφ ) : r > , φ ∈ (0 , } . We conjecture that the pair ( Z ′ † G , A ′ G ) gives a Gepner type stability conditionon HMF gr ( W ′ ) with respect to ( τ, / A ′ G is similarto A G in the previous section, and we leave the readers to give its explicitconstruction. We just check the easiest case of Conjecture 4.1: Lemma 4.2.
Conjecture 4.1 is true if R = 2 rank( F ) = 2 .Proof. Let ( O ⊕ S s → F ) be a µ ′ -stable coherent system on S with rank( F ) =1. The inequality in Conjecture 4.1 is equivalent to c ( F ) · h > . · · · . Itis enough to show that c ( F ) · h ≥
3. Let F ։ F ′ be a torsion free quotient.There is a surjection in A W ′ ( O ⊕ S → F ) ։ ( O ⊕ S → F ′ )whose kernel is of the form (0 → F ′′ ) for a torsion sheaf F ′′ on S . Obviously( O ⊕ S → F ′ ) is also µ ′ -stable, and c ( F ′ ) · h ≤ c ( F ) · h . Hence we mayassume that F is torsion free. Also note that h ( F ) ≥
2, since otherwisethere is an injection in A W ′ ( O S → ֒ → ( O ⊕ S → F )satisfying µ ′ ( O S →
0) = 5 / > µ ′ ( O ⊕ S → F )which contradicts to the µ ′ -stability of ( O ⊕ S → F ). Let us set L := F ∨∨ ,and take a smooth member C ∈ | h | . Note that L is a line bundle satisfying h ( L ) ≥
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