The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

We describe a connected component of the space of stability conditions on abelian threefolds, and on Calabi-Yau threefolds obtained as (the crepant resolution of) a finite quotient of an abelian threefold. Our proof includes the following essential steps: We simultaneously strengthen a conjecture by the first two authors and Toda, and prove that it follows from a more natural and seemingly weaker statement. This conjecture is a Bogomolov-Gieseker type inequality involving the third Chern character of “tilt-stable” two-term complexes on smooth projective threefolds; we extend it from complexes of tilt-slope zero to arbitrary tilt-slope. We show that this stronger conjecture implies the so-called support property of Bridgeland stability conditions, and the existence of an explicit open subset of the space of stability conditions. We prove our conjecture for abelian threefolds, thereby reproving and generalizing a result by Maciocia and Piyaratne. Important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall-crossing, closely related to the support property. We simultaneously strengthen a conjecture by the first two authors and Toda, and prove that it follows from a more natural and seemingly weaker statement. This conjecture is a Bogomolov-Gieseker type inequality involving the third Chern character of “tilt-stable” two-term complexes on smooth projective threefolds; we extend it from complexes of tilt-slope zero to arbitrary tilt-slope. We show that this stronger conjecture implies the so-called support property of Bridgeland stability conditions, and the existence of an explicit open subset of the space of stability conditions. We prove our conjecture for abelian threefolds, thereby reproving and generalizing a result by Maciocia and Piyaratne.


Introduction
In this paper, we determine the space of Bridgeland stability conditions on abelian threefolds and on Calabi-Yau threefolds obtained either as a finite quotient of an abelian threefold, or as the crepant resolution of such a quotient. More precisely, we describe a connected component of the space of stability conditions for which the central charge only depends on the degrees H 3−i ch i ( ), i = 0, 1, 2, 3, of the Chern character 1 with respect to a given polarization H , and that satisfy the support property.

Stability conditions on threefolds via a conjectural Bogomolov-Gieseker type inequality
The existence of stability conditions on three-dimensional varieties in general, and more specifically on Calabi-Yau threefolds, is often considered the biggest open problem in the theory of Bridgeland stability conditions. Until recent work by Maciocia and Piyaratne [29,30], they were only known to exist on threefolds whose derived category admits a full exceptional collection. Possible applications of stability conditions range from modularity properties of generating functions of Donaldson-Thomas invariants [43,45] to Reider-type theorems for adjoint linear series [6].
In [11], the first two authors and Yukinobu Toda, also based on discussions with Aaron Bertram, proposed a general approach towards the construction of stability conditions on a smooth projective threefold X . The construction is based on the auxiliary notion of tilt-stability for twoterm complexes, and a conjectural Bogomolov-Gieseker type inequality for the third Chern character of tilt-stable objects; we review these notions in Sect. 2 and the precise inequality in Conjecture 2.4. It depends on the choice of two divisor classes ω, B ∈ NS(X ) R with ω ample. It was shown that this conjecture would imply the existence of Bridgeland stability conditions, 2 and, in the companion paper [6], a version of an open case of Fujita's conjecture, on the very ampleness of adjoint line bundles on threefolds.
Our first main result is the following, generalizing the result of [29,30] for the case when X has Picard rank one: Theorem 1. 1 The Bogomolov-Gieseker type inequality for tilt-stable objects, Conjecture 2.4, holds when X is an abelian threefold, and ω is a real multiple of an integral ample divisor class.
There are Calabi-Yau threefolds that admit an abelian variety as a finite étale cover; we call them Calabi-Yau threefolds of abelian type. Our result applies similarly in these cases:

Theorem 1.2 Conjecture 2.4 holds when X is a Calabi-Yau threefold of abelian type, and ω is a real multiple of an integral ample divisor class.
Combined with the results of [11], these theorems imply the existence of Bridgeland stability conditions in either case. There is one more type of Calabi-Yau threefolds whose derived category is closely related to those of abelian threefolds: namely Kummer threefolds, that are obtained as the crepant resolution of the quotient of an abelian threefold X by the action of a finite group G. Using the method of "inducing" stability conditions on the G-equivariant derived category of X and the BKR-equivalence [8], we can also treat this case. Overall this leads to the following result (which we will make more precise in Theorem 1.4). Theorem 1. 3 Bridgeland stability conditions on X exist when X is an abelian threefold, or a Calabi-Yau threefold of abelian type, or a Kummer threefold.

Support property
The notion of support property of a Bridgeland stability condition is crucial in order to apply the main result of [13], namely that the stability condition can be deformed; moreover, it ensures that the space of such stability conditions satisfies well-behaved wall-crossing.
In order to prove the support property, we first need a quadratic inequality for all tilt-stable complexes, whereas Conjecture 2.4 only treats complexes E with tilt-slope zero. We state such an inequality in Conjecture 4.1 for the case where ω, B are proportional to a given ample class H : Conjecture 4.1 Let (X, H ) be a smooth polarized threefold, and ω = √ 3α H , B = β H , for α > 0, β ∈ R. If E ∈ D b (X ) is tilt-semistable with respect to ω, B, then where ch B := e −B ch. In Theorem 4.2, we prove that this generalized conjecture is in fact equivalent to the original Conjecture 2.4. Moreover, in Theorem 8.7 we prove that it implies a similar quadratic inequality for objects that are stable with respect to the Bridgeland stability conditions constructed in Theorem 1.3, thereby obtaining a version of the support property.
To be precise, we consider stability conditions whose central charge Z : K (X ) → C factors via (1) (In the case of Kummer threefolds, we apply the BKR-equivalence before taking the Chern character.) We prove the support property with respect to v H ; this shows that a stability condition deforms along a small deformation of its central charge, if that deformation still factors via v H .
We discuss the relation between support property, quadratic inequalities for semistable objects and deformations of stability conditions systematically in Appendix 1. In particular, we obtain an explicit open subset of stability conditions whenever Conjecture 4.1 is satisfied, see Theorem 8.2.

The space of stability conditions
In each of the cases of Theorem 1.3, we show moreover that this open subset is a connected component of the space of stability conditions. We now give a description of this component.
Inside the space Hom(Q 4 , C), consider the open subset V of linear maps Z whose kernel does not intersect the (real) twisted cubic C ⊂ P 3 (R) parametrized by (x 3 , x 2 y, 1 2 x y 2 , 1 6 y 3 ); it is the complement of a real hypersurface. Such a linear map Z induces a morphism P 1 (R) ∼ = C → C * /R * = P 1 (R); we define P be the component of V for which this map is an unramified cover of topological degree +3 with respect to the natural orientations. Let P be its universal cover.
We let Stab H (X ) be the space of stability conditions for which the central charge factors via the map v H as in equation (1) (and satisfying the support property). Theorem 1.4 Let X be an abelian threefold, or a Calabi-Yau threefold of abelian type, or a Kummer threefold. Then Stab H (X ) has a connected component isomorphic to P.

Approach
We will now explain some of the key steps of our approach.

Reduction to a limit case
The first step applies to any smooth projective threefold. Assume that ω, B are proportional to a given ample polarization H of X . We reduce Conjecture 4.1 to a statement for objects E that are stable in the limit as ω(t) → 0 and ν ω(t),B(t) (E) → 0; if B := lim B(t), the claim is that X e −B ch(E) ≤ 0. ( The reduction is based on the methods of [26]: as we approach this limit, either E remains stable, in which case the above inequality is enough to ensure that E satisfies our conjecture everywhere. Otherwise, E will be strictly semistable at some point; we then show that all its Jordan-Hölder factors have strictly smaller "H -discriminant" (which is a variant of the discriminant appearing in the classical Bogomolov-Gieseker inequality). This allows us to proceed by induction.

Abelian threefolds
In the case of an abelian threefold, we make extensive use of the multiplication by m map m : X → X in order to establish inequality (2). The key fact is that if E is tilt-stable, then so is m * E.
To illustrate these arguments, assume that B is rational. Via pull-back we can then assume that B is integral; by tensoring with O X (B) we reduce to the case of B = 0. We then have to prove that ch 3 (E) ≤ 0; in other words, we have to prove an inequality of the Euler characteristic of E. To obtain a contradiction, assume that ch 3 (E) > 0, and consider further pull-backs: Similar bounds for h 2 lead to a contradiction to (3).

Support property
As pointed out by Kontsevich and Soibelman in [21,Sect. 2.1], the support property is equivalent to the existence of a real quadratic form Q : Q 4 → R such that (a) The kernel of the central charge (as a subspace of R 4 ) is negative definite with respect to Q, and (b) Every semistable object E satisfies Q(v H (E)) ≥ 0.
The inequality in Conjecture 4.1 precisely gives such a quadratic form. We therefore need to show that this inequality is preserved when we move from tilt-stability to actual Bridgeland stability conditions.
We establish a more basic phenomenon of this principle in Appendix 1, which may be of independent interest: if a stability condition satisfies the support property with respect to Q, and if we deform along a path for which the central charges all satisfy condition (a), then condition (b) remains preserved under this deformation, i.e., it is preserved under wall-crossing. The essential arguments involve elementary linear algebra of quadratic forms.
Tilt-stability can be thought of as a limiting case of a path in the set of stability conditions we construct. In Sect. 8 we show that the principle described in the previous paragraph similarly holds in this case: we show that a small perturbation of the quadratic form in Conjecture 4.1 is preserved under the wall-crossings between tilt-stability and any of our stability conditions, thereby establishing the desired support property.

Connected component
In Appendix 1, we also provide a more effective version of Bridgeland's deformation result. In particular, the proof of the support property yields large open sets of stability conditions, which combine to cover the manifold P described above.
In Sect. 9, we show that this set is in fact an entire component. The proof is based on the observation that semi-homogeneous vector bundles E with c 1 (E) proportional to H are stable everywhere on P; their Chern classes (up to rescaling) are dense in C.
This fact is very unique to varieties admitting étale covers by abelian threefolds. In particular, while Conjecture 4.1 implies that P is a subset of the space of stability conditions, one should in general expect the space to be much larger than this open subset.

Applications
Our work has a few immediate consequences unrelated to derived categories. Although these are fairly specific, they still serve to illustrate the power of Conjecture 4.1. Corollary 1.5 Let X be a Calabi-Yau threefold of abelian type. Given α ∈ Z >0 , let L be an ample line bundle on X satisfying • L 2 D ≥ 7α for every integral divisor class D with L 2 D > 0 and L D 2 < α, and • L .C ≥ 3α for every curve C ⊂ X.
In addition, if L = A ⊗5 for an ample line bundle A, then L is very ample.
Proof Since Conjecture 2.4 holds for X by our Theorem 1.2, we can apply Theorem 4.1 and Remark 4.3 of [6].
Setting α = 2 we obtain a Reider-type criterion for L to be very ample. The statement for A ⊗5 confirms (the very ampleness case of) Fujita's conjecture for such X . The best known bounds for Calabi-Yau threefolds say that A ⊗8 is very ample if L 3 > 1 [18, Corollary 1], A ⊗10 is very ample in general, and that A ⊗5 induces a birational map [33,Theorem I]. For abelian varieties, much stronger statements are known, see [37,38]. Corollary 1.6 Let X be one of the following threefolds: projective space, the quadric in P 4 , an abelian threefold, or a Calabi-Yau threefold of abelian type. Let H be a polarization, and let c ∈ Z >0 be the minimum positive value of H 2 D for integral divisor classes D. If E is a sheaf that is slope-stable with respect to H , and with H 2 c 1 (E) = c, then The assumptions hold when NS(X ) is generated by H , and c 1 (E) = H . We refer to Example 4.4 and Remark 4.5 for a proof and more discussion. Even for vector bundles on P 3 , this statement was not previously known for rank bigger than three.
It is a special case of Conjecture 4.1. Even when X is a complete intersection threefold and E = I C ⊗ L is the twist of an ideal sheaf of a curve C, this inequality is not known, see [49].

Open questions
General proof of Conjecture 4.1 While Conjecture 4.1 for arbitrary threefolds remains elusive, our approach seems to get a bit closer: in our proof of Theorem 1.1 (in Sects. 2, 3, 4, 5, 6, 7), only Sect. 7 is specific to abelian threefolds. One could hope to generalize our construction by replacing the multiplication map m with ramified coverings. This would immediately yield the set P as an open subset of the space of stability conditions.

Strengthening of Conjecture 4.1
In order to construct a set of stability conditions of dimension equal to the rank of the algebraic cohomology of X , we would need a stronger Bogomolov-Gieseker type inequality, depending on ch 1 and ch 2 directly, not just on H 2 ch 1 and H ch 2 . We point out that the obvious guess, namely to replace H 2 ch 1 2 by H ch 2 1 ·H 3 , and (H ch 2 ) 2 by an appropriate quadratic form on H 4 (X ), does not work in general: for α → +∞, such an inequality fails for torsion sheaves supported on a divisor D with H D 2 < 0.

Higher dimension
Our work also clarifies the expectations for higher dimensions. The definition of P directly generalizes to dimension n in an obvious way, by replacing the twisted cubic with the rational normal curve x n , x n−1 y, 1 2 x n−2 y 2 , . . . , 1 n! y n . Let P n → P n denote the corresponding universal covering. Conjecture 1.7 Let (X, H ) be a smooth polarized n-dimensional variety. Its space Stab H (X ) of stability conditions contains an open subset P n , for which skyscraper sheaves of points are stable. In the case of abelian varieties, P n ⊂ Stab H (X ) is a connected component.
Such stability conditions could be constructed by an inductive procedure; the ith induction step would be an auxiliary notion of stability with respect to a weak notion of central charge Z i depending on H n ch 0 , H n−1 ch 1 , . . . , H n−i ch i . Semistable objects would have to satisfy a quadratic inequality Q i involving ch i+1 . The precise form of Q i would depend on the parameters of the stability condition; it would always be contained in the defining ideal of the rational normal curve, and the kernel of Z i would be semi-negative definite with respect to Q i .
One could hope to prove such inequalities for i < n using a second induction by dimension: for example, an inequality for ch 3 for stable objects on a fourfold would follow from a Mehta-Ramanathan type restriction theorem, showing that such objects restrict to semistable objects on threefolds. As a first test case, one should try to prove that a given tilt-stable object on a threefold restricts to a Bridgeland-stable object on a divisor of sufficiently high degree.

Related work
As indicated above, the first breakthrough towards constructing stability conditions on threefolds (without using exceptional collections) is due to Maciocia and Piyaratne, who proved Theorem 1.1 in the case of principally polarized abelian varieties of Picard rank one in [29,30]. Their method is based on an extensive analysis of the behavior of tilt-stability with respect to Fourier-Mukai transforms; in addition to constructing stability conditions, they show their invariance under Fourier-Mukai transforms.
Our approach is very different, as it only uses the existence of the étale selfmaps given by multiplication with m. Nevertheless, there are some similarities. For example, a crucial step in their arguments uses restriction to divisors and curves to control a certain cohomology sheaf of the Fourier-Mukai transform of E, see the proof of [29,Proposition 4.15]; in Sect. 7 we use restriction of divisors explicitly and to curves implicitly (when we use Theorem 7.2) to control global sections of pull-backs of E.
As mentioned earlier, it is easy to construct stability conditions on any variety admitting a complete exceptional collection; however, it is still a delicate problem to relate them to the construction proposed in [11]. This was done in [11,26] for the case of P 3 , and in [39] for the case of the quadric in P 4 ; these are the only other cases in which Conjecture 2.4 is known.
There is an alternative conjectural approach towards stability conditions on the quintic hypersurface in P 4 via graded matrix factorizations, proposed by Toda [46,47]. It is more specific, but would yield a stability condition that is invariant under certain auto-equivalences; it would also lie outside of our set P. His approach would require a stronger Bogomolov-Gieseker inequality already for slope-stable vector bundles, and likely lead to very interesting consequences for generating functions of Donaldson-Thomas invariants.
Conjecture 2.4 can be specialized to certain slope-stable sheaves, similar to Corollary 1.6; see [11,Conjecture 7.2.3]. This statement was proved by Toda for certain Calabi-Yau threefolds, including the quintic hypersurface, in [48]. Another case of that conjecture implies a certain Castelnuovo-type inequality between the genus and degree of curves lying on a given threefold; see [49] for its relation to bounds obtained via classical methods.
Our results are at least partially consistent with the expectations formulated in [36]; in particular, semi-homogeneous bundles are examples of the Lagrangian-invariant objects considered by Polishchuk, are semistable for our stability conditions, and their phases behave as predicted.

Plan of the paper
Appendix 1 may be of independent interest. We review systematically the relation between support property, quadratic inequalities for semistable objects and deformations of stability conditions, and their behaviour under wall-crossing. Sections 2 and 3 and Appendix 2 review basic properties of tilt-stabilty, its deformation properties (fixing a small inaccuracy in [11]), the conjectural inequality proposed in [11] and variants of the classical Bogomolov-Gieseker inequality satisfies by tilt-stable objects.
In Sect. 4 we show that a more general form of Conjecture 2.4 is equivalent to the original conjecture, whereas Sect. 5 shows that both conjectures follows from a special limiting case.
This limiting case is proved for abelian threefolds in Sect. 7; in the following Sect. 8 we show that this implies the existence of the open subset P of stabilty conditions described above. Section 9 shows that in the case of abelian threefolds, P is in fact a connected component, and Sect. 10 extends these results to (crepant resolutions) of quotients of abelian threefolds.

Update (March 2016)
Counterexamples due to Schmidt [40] and Martinez [27] indicate that Conjectures 2.4 and 4.1 need to be modified in the case of a threefold obtained as the blowup at a point of another threefold; on the other hand, they have been verified for all Fano threefolds of Picard rank one [23].

Review: tilt-stability and the conjectural BG inequality
In this section, we review the notion of tilt-stability for threefolds introduced in [11]. We then recall the conjectural Bogomolov-Gieseker type inequality for tilt-stable complexes proposed there; see Conjecture 2.4 below.

Slope-stability
Let X be a smooth projective complex variety and let n ≥ 1 be its dimension.
Let ω ∈ NS(X ) R be a real ample divisor class.
For an arbitrary divisor class B ∈ NS(X ) R , we will always consider the twisted Chern character ch B (E) = e −B ch(E); more explicitly, we have We define the slope μ ω,B of a coherent sheaf E on X by otherwise.
Observe that if a sheaf is slope-semistable, then it is either torsion-free or torsion. Harder-Narasimhan filtrations (HN-filtrations, for short) with respect to slope-stability exist in Coh(X ): given a non-zero sheaf E ∈ Coh(X ), there is a filtration

The tilted category
Let X be a smooth projective threefold. As above, let ω, B be real divisor classes with ω ample. There exists a torsion pair (T ω,B , F ω,B ) in Coh(X ) defined as follows: Equivalently, T ω,B and F ω,B are the extension-closed subcategories of Coh(X ) generated by slope-stable sheaves of positive and non-positive slope, respectively.
By the general theory of torsion pairs and tilting [20], Coh ω,B (X ) is the heart of a bounded t-structure on D b (X ); in particular, it is an abelian category.

Tilt-stability and the main conjecture
We now define the following slope function, called tilt, on the abelian category Coh ω,B (X ): for an object E ∈ Coh ω,B (X ), its tilt ν ω,B (E) is defined by otherwise.
We think of this as induced by the "reduced" central charge

Properties of tilt-stability
We will often fix B and vary ω along a ray in the ample cone via for some given integral ample class H ∈ NS(X ). 3 To prove that tilt-stability is a well-behaved property, one needs to use variants of the classical Bogomolov-Gieseker inequality for slope-semistable sheaves; in particular, this leads to the following statements: Remark 2.5 (a) Tilt-stability is an open property. More precisely, assume that tilt-stable is given by a locally finite collection of walls, i.e., submanifolds of real codimension one.
Unfortunately, a slightly stronger statement was claimed in [11, Corollary 3.3.3], but (as noted first by Yukinobu Toda) the proof there only yields the above claims. We will therefore review these statements in more detail in Sect. 3 and Appendix 2; one can also deduce them with the same arguments as in the surface case, treated in detail in [44,Sect. 3].
Remark 2.6 It can be helpful to distinguish between two types of walls for tiltstability, see Proposition 12.5. Locally, a wall for tilt-stability of E is described by the condition ν ω,B (F) = ν ω,B (E) for a destabilizing subobject F. This translates into the condition that either (a) Z ω,B (F) and Z ω,B (E) are linearly dependent, or that (b) ν ω,B (E) = +∞.

Lemma 2.7 Let H, B be fixed divisor classes with H ample, and let
then it satisfies one of the following conditions:

Classical Bogomolov-Gieseker type inequalities
In this section, we review a result from [11] that shows that tilt-stable objects on X satisfy variants of the classical Bogomolov-Gieseker inequality.
We continue to assume that X is a smooth projective threefold. Throughout this section, let H ∈ NS(X ) be a polarization, ω = √ 3α H for α > 0, and B ∈ NS(X ) R arbitrary.
First we recall the classical Bogomolov-Gieseker inequality: However, a sheaf F supported on a divisor D ⊂ X does not necessarily satisfy H (F) ≥ 0 (even if it is the push-forward of a slope-stable sheaf); indeed, we may have H D 2 < 0. This leads us to modify the inequality to a form that also holds for torsion sheaves, and in consequence for tilt-stable objects. We first need the following easy observation (see, for example, the proof of [ (Note that for abelian threefolds, we may take C H = 0.) Definition 3. 4 We define the H -discriminant as the following quadratic form: For the second definition, choose a rational non-negative constant C H satisfying the conclusion of Lemma 3.3. Then  This was proved for rational B in [11]; we will give a self-contained proof of the rational case with a slightly different presentation below, and extend it to arbitrary B in Appendix 2.
We think of C H,B as the composition where the first map is given by This makes our situation analogous to the one in Appendix 1; in particular, Theorem 3.5 implies a version of the support property for tilt-stable objects.

Lemma 3.7 Let ν ∈ R ∪ {+∞}. Then there exists a half-space
of codimension one with the following properties: Proof We define H ω,B,ν as the preimage under Z ω,B of the ray in the complex plane that has slope ν, starting at the origin; this ensures the first claim. The second claim is a general fact about quadratic forms, see Lemma 11.7. Note that by definition, a half-space is closed; indeed, we may have v B H (E) = 0 iff ν = +∞.
and q B H with the obvious quadratic form q B H on R 3 , then H and the analogues of Lemmas 3.6 and 3.7 hold.
Proof of Theorem 3.5, case H 2 B ∈ Q We prove the statement for C H,B under the assumption that H 2 B is rational. The proof for B H follows similarly due to Remark 3.8, and the non-rational case will be treated in Appendix 2.
We proceed by induction on H 2 ch B 1 (E), which by our assumption is a non-negative function with discrete values on objects of Coh H,B (X ).
We start increasing α. If E remains stable as α → +∞, we apply Lemma 2.7, (c); by Theorem 3.2 (for torsion-free slope-semistable sheaves) and Lemma 3.3 (for torsion sheaves) one easily verifies that E satisfies the conclusion in any of the possible cases.
Otherwise, E will get destabilized. Note that as α increases, all possible destabilizing subobjects and quotients have strictly smaller H 2 ch B 1 , which satisfy the desired inequality by our induction assumption. This is enough to ensure that E satisfies well-behaved wall-crossing: following the argument of [14,Proposition 9.3] it is enough to know a support property type statement for all potentially destabilizing classes.
Hence there will be a wall be a short exact sequence where both E 1 and E 2 have the same tilt as E. Then both E 1 and E 2 have strictly smaller H 2 ch B 1 ; so they satisfy the inequality We now turn to some consequences of Theorem 3.5.

Lemma 3.9
Let Q be a quadratic form of signature (1, r ). Let C + be the closure of one of the two components of the positive cone given by Q(x) > 0.

with equality if and only if for all i, we have that x i is proportional to x and
Proof This follows immediately from the easy fact that if x, y ∈ C + − {0}, then the bilinear form associated to Q satisfies (x, y) ≥ 0, with equality if and only if x, y are proportional with Q(x) = Q(y) = 0.

Equality holds if and only if all
Proof Let By Lemmas 3.6 and 3.7, they satisfy the assumptions of Lemma 3.9, which then implies our claim.
As another application, one obtains the tilt-stability of certain slope-stable sheaves (see also [11,Proposition 7.4.1]):

if L is a line bundle, and if in addition either c 1 (L) − B is proportional to H , or we can choose the constant C H of Lemma 3.3 to be zero, then L or L[1] is
Note that the choice C H = 0 in particular applies to abelian threefolds (or more generally any threefold whose group of automorphisms acts transitively on closed points), or to any threefold of Picard rank one.
Proof Consider an object E that is ν ω,B -stable with B H (E) = 0 or C H,B (E) = 0. By Corollary 3.10, E can never become strictly semistable with respect to ν ω ,B as long as ω is proportional to ω. Combined with Lemma 2.7, (c) this implies all our claims.
The analogue to the case C H = 0 of part (b) for Bridgeland stability on surfaces is due to Arcara and Miles, see [2, Theorem 1.1], with a very different proof.

Generalizing the main conjecture
For this and the following section, we assume that ω and B are proportional to a given ample class H ∈ NS(X ): We will abuse notation and write ch We will also write H instead of B H , as it is independent of the choice of β. The goal of this section is to generalize Conjecture 2.4 to arbitrary tiltsemistable objects, not just those satisfying ν α,β = 0. This generalization relies on the structure of walls for tilt-stability in R >0 × R; it is completely analogous to the case of walls for Bridgeland stability on surfaces, treated most systematically in [25].

Conjecture 4.1 Let X be a smooth projective threefold, and H ∈ NS(X ) an ample class. Assume that E is ν H
α,β -semistable. Then We begin with the following aspect of "Bertram's Nested Wall Theorem" [25, Theorem 3.1]:

Lemma 4.3 Assume the situation and notation of Conjecture 4.1 with
Proof We have to show that C α,β (E) does not intersect any wall for tiltstability, which are described in Remark 2.6 or Proposition 12.5. In our situation, all reduced central charges Z α,β factor via the map The first type of wall, case (a) in Proposition 12.5, can thus equivalently be described as the set of (α , β ) for which v H (F) (for some destabilizing subobject F → E) is contained in the two-dimensional subspace of Q 3 spanned by v H (E) and the kernel of Z α ,β .
As an auxiliary step, consider the following statement: Evidently, Conjecture 2.4 (for the case of ω, B proportional to H ) is a special case of (*). Conversely, consider the assumptions of (*). By Lemma 4.3, E is ν α ,β -semistable, where β is as above, and Moreover, a simple computation shows ν α ,β (E) = 0. Therefore, Conjecture 2.4 implies the statement (*).
This generalizes [11,Conjecture 7.2.3]. In particular, let C ⊂ X be a curve of genus g and degree d = HC; then E = I C ⊗ O(H ) is supposed to satisfy (13). Let K ∈ Z such that the canonical divisor class K X = K H. By the Hirzebruch-Riemann-Roch Theorem, we have the inequality (13) specializes to the following Castelnuovo type inequality between genus and degree of the curve (where D = H 3 is the degree of the threefold): Even for complete intersection threefolds, this inequality does not follow from existing results; see [49,Sect. 3] for progress in that direction.

Remark 4.5
The inequality (13) holds when X is an abelian threefold, or a Calabi-Yau threefold of abelian type. Moreover, since Conjecture 4.1 is equivalent to Conjecture 2.4, and since the latter has been verified for P 3 in [11,26], and for the quadric threefold in [39], it also applies in these two cases.
The inequality is new even in the case of P 3 : for sheaves of rank three, it is slightly weaker than classically known results, see [16,Theorem 4.3] and [31, Theorem 1.2], but no such results are known for higher rank.

Reduction to small α
The goal of this section is to reduce Conjecture 4.1 to a more natural inequality, that can be interpreted as an Euler characteristic in the case of abelian threefolds, and which considers the limit as α → 0 and ν α,β → 0.
We continue to assume that X is a smooth projective threefold with an ample polarization H ∈ NS(X ). To give a slightly better control over the limit α → 0, we will modify the definition of the reduced central charge of (5) to the following form (which is equivalent for α = 0): It factors via the map v H of (11). Also, as observed in Remark 3.8, the Hdiscriminant can be written as the composition H = q • v H where q is the quadratic form on Q 3 given by Given any E ∈ Coh β (X ), we define β(E) as follows: The motivation behind this definition is that β(E) is the limit of a curve in other words, for which the right-hand-side of the inequality (12) goes to zero: this follows from We also point out that H 2 ch The other motivation for the definition ofβ lies in the following observations, extending Lemma 3.6. In other words, the vector v H (E) is contained in the tangent plane to the quadric q = 0 at the kernel of Z 0,β(E) ; see Fig. 1.

Remark 5.2
The map (α, β) → Ker Z α,β gives a homeomorphism from R ≥0 × R onto its image in the closed unit disc C − /R * . This can be a helpful visualization, as a central charge is, up to the action of GL 2 (R), determined by its kernel.
For the second claim, we just observe that 1, β(E), 1 2 β(E) 2 and v H (E) are orthogonal with respect to the bilinear form on R 3 associated to q. The following is a limit case of Conjecture 4.1.

Conjecture 5.3 Let E ∈ D b (X ) be an object with the following property: there exists an open neighborhood U
Unless H (E) = 0, we can always make U small enough such that H 2 ch β 1 (E) > 0 for (α, β) ∈ U ; then E itself is an object of Coh β (X ). A strengthening of the methods of [26] leads to the main result of this section:  The signature of q restricted to has to be (1, 1) (as it contains v H (E) and the kernel of Z α,β for some α > 0). If (0, β(E)) was an endpoint of this wall, then by Lemma 5.1 the kernel of Z 0,β(E) would also be contained in ; this is a contradiction to the second assertion of Lemma 5.1.
For the second claim, recall that the semicircles of Lemma 4.3 do not intersect. (For example, in Fig. 1, they are given by the condition that Ker Z α,β is contained in a given plane through v H (E).) As we shrink the radius of the circles, their center has to converge to the point with α = 0 and ν α,β (E) = 0. Proof of Theorem 5.4 By the previous lemma, we can restrict to the case H (E) > 0 throughout. First assume that Conjecture 4.1 holds. Let E be an object as in the assumptions of Conjecture 5.3 and consider the limit of (10) as (α, β) → (0, β). Evidently the first term α 2 H (E) goes to zero; by equation (17), the same holds for the second term (H ch β 2 (E)) 2 . Since H 2 ch β(E) 1 > 0, the limit yields exactly (18).
For the converse, we start with three observations on inequality (10). with α > 0, β = β(E), then by Lemma 5.5 they fill up all points of Now assume that Conjecture 5.3 holds. We proceed by induction on H (E) (recall that H only obtains non-negative integers for tilt-stable objects E).
For contradiction, let E be an object that is ν α,β -stable, with H (E) > 0, and that violates conjecture (10) at this point. By Lemma 5.5 and observation (a) above, we may assume β = β(E). Now fix β = β(E) and start decreasing α. Since we assume (10) to be violated, we must have ch β(E) 3 (E) > 0. If E were to remain stable as α → 0, then by Lemma 5.5 it would be stable in a neighborhood of (0, β(E)) as in the conditions of Conjecture 5.3; this is a contradiction.
Therefore there must be a point α 0 where E is strictly ν α 0 ,β(E) -semistable; let E i be the list of its Jordan-Hölder factors. By observation (b), E still violates Conjecture (10) at (α 0 , β(E)). On the other hand, by Corollary 3.10, H (E i ) < H (E) for each i; by the induction assumption, E i satisfies Conjecture 4.1.
Now the conclusion follows just as in Lemma 11.6: consider the left-handside of (10) as a quadratic form on R 4 with coordinates (H 3 ch . The kernel of Z α,β , considered as a subspace of R 4 , is negative semi-definite with respect to the quadratic form. Therefore, the claim follows from Lemma 11.7.

Tilt stability and étale Galois covers
Consider an étale Galois cover f : Y → X with covering group G; in other words, G acts freely on Y with quotient X = Y /G. In this section, we will show that tilt-stability is preserved under pull-back by f .
For this section, we again let ω, B ∈ NS(X ) R be arbitrary classes with ω a positive real multiple of an ample.
Proof The pull-back formula for Chern characters immediately gives Now consider E ∈ Coh ω,B (X ). Part (a) and the above computation shows that if E is tilt-unstable, then so is f * E. Conversely, assume that f * E is tiltunstable. Let F → f * E be the first step in its Harder-Narasimhan filtration with respect to ν f * ω, f * B . Since f * E is G-equivariant, and since the HN filtration is unique and functorial, the object F must also be G-equivariant. Hence it is the pull-back of an object F in D b (X ). Using part (a) again, we see that F must be an object of Coh ω,B (X ). Applying the same arguments to the quotient f * E/F, we see that F is a destabilizing subobject of E in Coh ω,B (X ). Example 6.2 Let n ∈ Z >0 . Let X = Y be an abelian threefold and let n : X → X be the multiplication by n map. Then n has degree n 6 , and n * H = n 2 H for any class H ∈ NS(X ); see e.g. [9, Corollary 2.3.6 and Chapter 16].
We also obtain directly the following consequence: Proposition 6.3 If Conjecture 2.4 holds for tilt-stability with respect to ν f * ω, f * B on Y , then it also holds for tilt-stability with respect to ν ω,B on X.

Abelian threefolds
Let (X, H ) be a polarized abelian threefold. In this section we prove Theorem 1.1.
Most of this section will be concerned with proving Conjecture 5.3, the case where ω and B are proportional to H . For (α, β) ∈ R >0 × R, we let ω = √ 3α H and B = β H . We can also assume that H is the class of a very ample divisor, which, by abuse of notation, will also be denoted by H .
We let E ∈ D b (X ) be an object satisfying the assumptions of Conjecture 5.3.
By Lemma 5.6, we can also assume H (E) > 0, and so H 2 ch (E) > 0. We proceed by contradiction, and assume that

Idea of the proof
Consider the Euler characteristic of the pull-backs via the multiplication by n map. If we pretend that E(−β(E)H ) exists, this Euler characteristic grows proportional to n 6 ; we will show a contradiction via restriction of sections to divisors.
The proof naturally divides into two cases: if β(E) is rational, then n * E(−β(E)H ) exists when n is sufficiently divisible, and the above approach works verbatim; otherwise, we need to use Diophantine approximation of β(E).

Proof of Conjecture 5.3, rational case
We assume that β(E) is a rational number.

Reduction to β(E) = 0
Let q ∈ Z >0 such that qβ(E) ∈ Z, and consider the multiplication map q : X → X . By Proposition 6.1, q * E still violates Conjecture 5.3. By definition, we have Replacing E with q * E, we may assume that β(E) is an integer. Replacing E again with E ⊗O X (−β(E)H ), we may assume that E satisfies the assumptions of Conjecture 5.3, as well as • β(E) = 0, and so H. ch 2 (E) = 0, and • ch 3 (E) > 0, and so ch 3 (E) ≥ 1.

Asymptotic Euler characteristic
We look at χ(O X , n * E), for n → ∞. By the Hirzebruch-Riemann-Roch Theorem, we have The goal is to bound χ(O X , n * E) from above with a lower order in n.

First bound
We claim that Indeed, both n * E and O X [1] are objects of Coh β=0 (X ). Hence, for all k ∈ Z >0 , we have

Hom-vanishing from stability
To bound the above cohomology groups, we use Hom-vanishing between line bundles and n * E. By Corollary 3.11, all objects of Coh β (X ) of the form O X (u H) and O X (−u H) [1] are ν α,β -stable, for all u > 0 and β close to 0. For (β, α) → (0, 0), we have and therefore Applying the standard Hom-vanishing between stable objects and Serre duality, we conclude

Restriction to divisors
We will use this Hom-vanishing to restrict sections to divisors; we will repeatedly apply the following immediate observation. Proof We choose D such that it does not contain any of the associated points of F j , i.e., such that the natural map F j (−D) → F j is injective.
In particular, for general D, a finite number of short exact sequences restrict to exact sequences on D, and taking cohomology sheaves of a complex E commutes with restriction to D.
We consider the exact triangle in D b (X ) where D is a general smooth linear section of H . By (22), we have We consider the cohomology sheaves of E and the exact triangle in D b (X ) Since D is general, Lemma 7.1 gives The bound (23) will then follow from Lemma 7.3 below. We first recall a general bound on global sections of sheaves restricted to hyperplane sections, which is due to Simpson and Le Potier, and can be deduced as a consequence of the Grauert-Mülich Theorem: Notice that in the actual statement of [19,Corollary 3.3.3] there is a factor H n ; this is already included in our definition of slope. Proof We assume first that Q is torsion-free. Notice that the multiplication map n preserves slope-stability and the rank. Therefore, by Theorem 7.2, we have The h 2 -estimate follows similarly, by using Serre Duality on D. Finally, the Hirzebruch-Riemann-Roch Theorem on D gives This finishes the proof in the torsion-free case. For a general sheaf Q, we take a resolution with N locally-free and M torsion-free. Since D is very general, Lemma 7.1 applies, giving Hence the result follows from the previous case.
This is similar to the previous case. We consider the exact triangle Again, we apply (22), Lemmas 7.1 and 7.3 and reach

Proof of Conjecture 5.3, irrational case
Now assume that β(E) ∈ R\Q is an irrational number. As a consequence ch 0 (E) = 0 and, for all β ∈ Q, H ch β 2 (E) = 0. By assumption, there exists > 0 such that E is ν α,β -stable for all (α, β) in By the Dirichlet approximation theorem, there exists a sequence β n = p n q n n∈N of rational numbers such that for all n, and with q n → +∞ as n → +∞.

The Euler characteristic
The function f (β) = ch Consider the multiplication map q n : X → X . We let F n := q n * E ⊗ O X (− p n q n H ).
By Lemma 6.1, F n is ν α,0 -stable, for all α > 0 sufficiently small. We have By (20), it is again enough to bound both hom(O X , F n ) and ext 2 (O X , F n ) from above.

Hom-vanishing
As α → 0, we have We can bound this term as follows: Here we used H 2 ch β(E) 2 (E) = 0 in the second equality, and H 2 ch By comparison with (21), it follows that for α → 0 and n sufficiently large; therefore Bound on hom(O X , m * F n ) and conclusion Proceeding as in the rational case, we consider the exact triangle where D is a general smooth surface in the linear system |3H |. By (27), we have The following is the analogue of Lemma 7.3:

Lemma 7.4 Let Q be a sheaf on X and let L be a line bundle. Then
for all i, and for D a general smooth surface in |3H |.
Proof By the same argument as in the proof of Lemma 7.3, we may assume that Q is torsion-free. Applying Theorem 7.2 in our case we obtain, for general D, The h 1 and h 2 bounds follow from Serre duality and the Riemann-Roch Theorem.
Applying Lemma 7.4 to the cohomology sheaves of E in combination with Lemma 7.1, we get The same argument gives a similar bound on ext 2 (O X , F n ) and a contradiction to (26). This completes the proof of Conjecture 5.3, and therefore Conjecture 4.1, for abelian threefolds.

Proof of Theorem 1.1
Let now B ∈ NS(X ) R be an arbitrary divisor class and ω a positive multiple of H . In the abelian threefold case, we can use Conjecture 5.3 to deduce Conjecture 2.4 in this more general case.
We let E ∈ Coh ω,B (X ) be as in Conjecture 2.4. We first assume that B ∈ NS(X ) Q is rational. Then, by Proposition 6.1, we can assume B integral. By taking the tensor product with O X (−B), we can then assume E is ν ω,0 -semistable. Conjecture 2.4 then follows directly from Conjecture 4.1 and Theorem 5.4.
Finally, we take B irrational. Since (6) is additive, by considering its Jordan-Hölder factors we can assume E is ν ω,B -stable. By using Theorem 3.5 and Remark 2.5, we can deform (ω, B) to (ω , B ) with B rational (and ω still proportional to H ), such that E is still ν ω ,B -stable with ν ω ,B (E) = 0. But, if (6) does not hold for (ω, B), then it does not hold for (ω , B ) sufficiently close, giving a contradiction to what we just proved.

Construction of Bridgeland stability conditions
It was already established in [11] that Conjecture 2.4 implies the existence of Bridgeland stability conditions on X , except that the notion of support property was ignored. This property ensures that stability conditions deform freely, and exhibit well-behaved wall-crossing.
In this section, we show that the equivalent Conjecture 4.1 is in fact strong enough to deduce the support property, and to construct an explicit open subset of the space of stability conditions. In the following section, we will show that in the case of abelian threefolds, this open set is in fact an entire component of the space of stability conditions.

Statement of results
Fix a threefold X with polarization H ; we assume throughout this section that Conjecture 4.1 is satisfied for the pair (X, H ). We consider the lattice H ∼ = Z 4 generated by vectors of the form together with the obvious map v H : K (X ) → H . We refer to Appendix 1 for the definition of stability conditions on D b (X ) with respect to ( H , v H ); it is given by a pair σ = (Z , P), where P is a slicing, and the central charge Z is a linear map Z : H → C. The main result of [13] shows that the space Stab H (X ) of such stability conditions is a four-dimensional complex manifold such that is a local isomorphism. In Proposition 11.5 we make this deformation result more effective. This result will be essential in the following, where we will construct an explicit open subset of this manifold. We let C ⊂ H ⊗ R ∼ = R 4 be the cone over the twisted cubic which contains v H (O X (u H)) for all u ∈ Z.
Let P be its universal covering.
The goal of this section is the following precise version of Theorem We will prove this theorem by constructing an explicit family of stability conditions following the construction of [11], and then applying the deformation arguments of Proposition 11.5.

Alternative description of P
We will need a more explicit description of the set P before proceeding to prove our main result. The group GL + 2 (R) of 2 × 2-matrices with positive determinant acts on P on the left by post-composing a central charge with the induced R-linear map of R 2 ∼ = C. There is also an action of R on P on the right: for β ∈ R, the multiplication by e −β H in K (D b (X )) corresponds to a linear selfmap of H ⊗R which leaves C invariant; therefore we can act on P by pre-composing with this linear map. for all α, β, a, b ∈ R satisfying α > 0 and a > 1 6 This slice is simply-connected.
It follows that it is simultaneously a slice of the GL + 2 (R)-action on P. Proof Consider a central charge Z ∈ P. Since Z (0, 0, 0, 1) = 0 by definition of P, we may use the action of rotations and dilations to normalize to the assumption Z (0, 0, 0, 1) = −1. Now consider the functions for Z ∈ P normalized as above; their coefficients vary continuously with Z . They can never vanish simultaneously, by definition of P. In the case of Z basic H , the function r (x) = − 1 6 x 3 + 1 2 x has zeros as x = − whereas i(x) = 1 2 x 2 − 1 6 has zeros at x = ± 1 3 . This configuration of zeros on the real line will remain unchanged as Z varies: r (x) will always have three zeros, and i(x) will have two zeros lying between the first and second, and the second and third zero of r (x), respectively.
We now use the action of R on P from the right to ensure that x = 0 is always the midpoint of the two zeros of i(x). The sign of the leading coefficient of i(x) must remain constant as Z varies; therefore, we can use vertical rescaling of R 2 to normalize it to be + 1 2 . Since the sign of i(0) = Z (O X ) is constant within this slice, it has to be negative; hence there exists a unique α ∈ R >0 such that i(0) = − 1 2 α 2 . On the slice we have constructed thus far, we still have the action of R given by sheerings of R 2 ∼ = C that leave the real line fixed. In the case of Z basic H , we have α = 1 3 , b = 0 and a = 1 2 , which is bigger than the right-hand-side. It follows that the inequality (29) holds in the whole connected component of our slice.
Conversely, given a central charge Z a,b α,β as described in the lemma, we can first use the action of R to reduce to the case β = 0. The coefficients of the linear functions Z , Z are in one-to-one correspondence with the coefficients of r (x) and i(x), respectively; these are, up to scaling, uniquely determined by the configurations of zeros of r (x) and i(x) on the real line. But our conditions ensure that we can continuously deform the configuration of zeros into the one corresponding to Z basic H .

Remark 8.4
From the proof of the lemma one can also deduce the following more intrinsic description of the set P. Consider the twisted cubic C in projective space P 3 (R).
There is an open subset of central charges Z with the following properties: the hyperplanes Z = 0 and Z = 0 both intersect C in three distinct points; moreover, their configuration on C ∼ = S 1 are such that the zeros of the two functions alternate. This open set has two components: one of them is P, the other is obtained from P by composing central charges with complex conjugation. Moreover, one can also deduce the description given in the introduction.
Recall the H -discriminant defined in (7), Let us also introduce a notation of the remainder term of (10):  1, 0, a). The intersection matrix for the symmetric pairing associated to The diagonal entries are negative for K ∈ (α 2 , 6a) (which is non-empty by the assumptions on a). In case b = 0, we additionally need to ensure that the determinant is positive. Solving the quadratic equation, one obtains a subinterval of (α 2 , 6a) symmetric around the midpoint K = 1 2 α 2 + 6a with the properties as claimed.

Review: construction of stability conditions
We will use [13,Proposition 5.3] to construct stability conditions. It says that a stability condition is equivalently determined by a pair σ = (Z , A), where Z : H → C is a group homomorphism (called central charge) and A ⊂ D b (X ) is the heart of a bounded t-structure, which have to satisfy the following three properties: (a) For any 0 = E ∈ A the central charge Z (v H (E)) lies in the following semi-closed upper half-plane: We can use Z and Z to define a notion of slope-stability on the abelian category A via the slope function With this notion of slope-stability, every object in E ∈ A has a Harder- where is a fixed norm on H ⊗ R ∼ = R 4 . For brevity, we will write Z (E) instead of Z (v H (E)). Shifts of λ σ -(semi)stable objects are called σ -(semi)stable.

Explicit construction of stability conditions
We start by reviewing (a slightly generalized version of) the construction of stability conditions in [11].

Support property
The next step towards proving Theorem 8.2 is to establish the support property for the stability conditions constructed in Theorem 8.6. Our overall goal is the following analogue of Theorem 3.5.
Let σ = (Z , A) ∈ P ⊂ Stab H (X ) be a stability condition in the open subset given in Theorem 8.6. We may assume that Z = Z a,b α,β is of the form given in Lemma 8.3. We also choose a constant K ∈ I a,b α in accordance with Lemma 8.5.

Moreover, up to shift the heart A is of the form A = A α,β (X ).
We will treat only the case b = 0; then I a,b α = (α 2 , 6a). We will also shorten notation and write Z a α,β instead of Z a,0 α,β , and I a α instead of I a,0 α . The case b = 0 will then follow directly by Proposition 11.5.
The analogy between Theorems 3.5 and 8.7 is reflected also in their proof. We first treat the rational case:  for this pair (α, β). Then for any a > 1 6 α 2 , the pair σ a α,β = (Z a α,β , A α,β (X )) satisfies the support property; more precisely, the inequality (31) holds for all σ a α,β -semistable objects E and all K ∈ I a α .
We first need an analogue of Lemma 2.7. Let us denote by H i β the i-th cohomology object with respect to the tstructure Coh β (X ).

Lemma 8.9
Let E ∈ A α,β (X ) be a σ a α,β -semistable object, for all a 1 sufficiently big. Then it satisfies one of the following conditions: Proof Consider the exact sequence Their imaginary parts are constant, with Z a α,β H 0 In the limit a → +∞, we have Z a α,β → Z α,β up to rescaling of the real part; this implies the ν α,β -semistability of the cohomology objects in both cases.
We have already proved the analogue of Lemma 3.6, as part of Lemma 8.5. This also enables us to use the result from Appendix 1.
Proof of Lemma 8.8 Throughout the proof, we fix α and β.
If E is strictly σ a α,β -semistable, and if (31) holds for all of the Jordan-Hölder factors E i of E, then by Lemma 11.6, it also holds for E. We may therefore assume that E is stable.
We also notice that if F ∈ Coh β (X ) is ν α,β -semistable, then Conjecture 4.1 and Theorem 3.5 show that in particular, it satisfies Q β K (F) ≥ 0 for every K > α 2 .
We proceed by induction on f (E) which is a non-negative function on A α,β (X ) with discrete values.
If E remains σ a α,β -semistable, for all a > a 0 , then by Lemma 8.9 either E = H 0 β (E) is ν α,β -semistable, or H −1 β (E) is ν α,β -semistable and H 0 (E) is either 0 or supported in dimension 0. In the first case, we already pointed out above that E satisfies (31). In the second case, H 2 ch Since (31) holds for H −1 β (E), it holds also for E. Otherwise, E will be unstable for a sufficiently big. Every possibly destabilizing subobject or quotient F has f (F) < f (E) (since f is non-negative, and since the subcategory of objects F ∈ A α,β (X ) with f (F) = 0 has maximum possible slope with respect to Z a α,β for all a). Therefore they obey the induction assumption; since K ∈ (α 2 , 6a 0 ) ⊂ (α 2 , 6a), this means that all these possible subobject or quotients satisfy (31) with respect to our choice of K . Since Z a α,β has negative definite kernel with respect to Q β K for all a ≥ a 0 , this is equivalent to a support property type statement, see Appendix 1. It follows that E satisfies well-behaved wall-crossing along our path. Hence, there will exist a 1 > a 0 such that E is strictly σ a 1 α,βsemistable. But all the Jordan-Hölder factors E i of E have strictly smaller f . Using the induction assumption again, we see that they satisfy Q β K (E i ) ≥ 0; therefore, we can again apply Lemma 11.6 to deduce the same claim for E.
The combination of Lemma 8.3, Theorem 8.6 and Lemma 8.8 together with Proposition 11.5 leads to the following result: for each tuple α, β, a, b as in Theorem 8.6 (in particular α, β ∈ Q), we obtain an open subset U (α, β, a, b) ⊂ Stab H (X ) of stability conditions by deforming the pair (Z a,b α,β , A α,β (X )). The associated open subsets Z (U (α, β, a, b)) of central charges combine to cover the set P. To conclude the proof of Theorems 8.2 and 8.7, we need to show that the sets U (α, β, a, b) glue to form a continuous family covering P. This is done by the following analogue of Proposition 12.2.

Proposition 8.10 There is a continuous family of Bridgeland stability conditions in Stab H (X ), parameterized by the set
Indeed, deformations of the central charge Z a,b α,β for b = 0 (while keeping α, β, a fixed) do not change the heart, as modifying b only affects the real part of the central charge. Acting on these stability conditions by GL + 2 (R) produces the entire set P.
To prove Proposition 8.10, we need a few preliminary results. We will use the notion of a pre-stability condition, which is a stability condition that does not necessarily satisfy the support property; see Appendix 1. The first result already appears implicitly in [14,Sect. 10].

Lemma 8.11
Assume that σ 1 = (Z , A 1 ) and σ 2 = (Z , A 2 ) are two prestability conditions with the following properties: (a) Their central charges agree. (b) There exists a heart B of a bounded t-structure such that each A i can be obtained as a tilt of B: .
Proof By [35, Lemma 1.1.2], for i = 1, 2, A i is a tilt of B with respect to the torsion pair We need to show that T 1 = T 2 and F 1 = F 2 ; in fact, since F i = T ⊥ i , it is enough to show T 1 = T 2 . Observe that, since the central charges agree, we have We let T ∈ T 2 . Consider the exact sequence in B with T 1 ∈ T 1 and F 1 ∈ F 1 . Since the torsion part of any torsion pair is closed under quotients, F 1 ∈ T 2 , contradicting the observation above. Hence, T ∈ T 1 , and so T 2 ⊆ T 1 . The reverse inclusion follows similarly.

Lemma 8.12
There exists a continuous positive function (α, β, a) > 0 with the following property: if E ∈ Coh β (X ) is ν α,β -stable with then Z a α,β (E) > 0. Proof We first apply Conjecture 4.1, rewriting (10) as Now we apply Theorem 3.5. First of all, we can rewrite H (E) ≥ 0 as By assumption, and therefore Summing up the last two equations we obtain Plugging this into (32), we obtain the desired claim.
By construction we know E ∈ A α,β = P a α,β ((0, 1]). Let A be the HNfiltration factor of E with respect to σ a α,β and with the largest phase, and consider the associated short exact sequence A → E B in A α,β . The associated long exact cohomology with respect to Coh β (X ) shows that A ∈ Coh β (X ) ∩ A α,β = T α,β ; moreover, there is a sequence H −1 (B) → A → E exact on the left with H −1 (B) ∈ F α,β . Now consider the slopes appearing in the Harder-Narasimhan filtration of A for tilt-stability with respect to ν α,β . By standard arguments using the observations in the previous paragraph, all these slopes lie in the interval (0, ). Lemma 8.12 then implies Z a α,β (A) > 0, and therefore E ∈ P a α,β ((0, 1 2 )) as we claimed.
We have verified all the assumptions of Lemma 8.11, which implies σ 1 = σ 2 .
Let us also mention the following property: Proposition 8.14 [29, Proposition 2.1] Skyscraper sheaves are stable for all σ ∈ P.
Proof (sketch) Using the long exact cohomology sequence with respect to the heart Coh(X ), one sees that k(x) is a minimal object of Coh β (X ): otherwise, there would be a short exact sequence E → k(x) F [1] in Coh β (X ) coming from a short exact sequence F → E k(x) of sheaves; this is a contradiction to μ H,β (F) < 0 and μ H,β (E) ≥ 0. Similarly, taking the long exact cohomology sequence with respect to Coh β (X ) of short exact sequences in A α,β (X ), we see that k(x) is a minimal object of A α,β (X ).

The space of stability conditions on abelian threefolds
In this section we prove the following: The fundamental reason behind Theorem 9.1 is the abundance of projectively flat vector bundles on abelian threefolds; their Chern classes are dense in the projectivization of the twisted cubic C.
Consider a slope μ = p q ∈ Q with p, q coprime and q > 0. Then there exists a family of simple vector bundles E p/q that are semi-homogeneous in the sense of Mukai, have slope p q and Chern character see [32,Theorem 7.11]. They can be constructed as the push-forward of line bundles via an isogeny Y → X [32, Theorem 5.8], and are slope-stable [32, Proposition 6.16]. The above theorem is essentially based on the following result: The semi-homogeneous vector bundle E p/q is σ -stable for every σ ∈ P.
Proof As mentioned above, E p/q is slope-stable. By Corollary 3.11, either Also observe that for all K , β ∈ R, we have The open subsets of P where the central charges are negative definite with respect to Q β K = K H + ∇ β H for some K , β form a covering of P; by Proposition 11.8, it is therefore enough to find a single stability condition σ ∈ P for which E p/q is σ -stable.
One can prove in general that ν α,β -stable vector bundles are σ a,b α,β -stable for a 0; but in our situation one can argue more easily as follows. Choose α, β with β < p q (and therefore E p/q ∈ Coh β (X )) and ν α,β (E) = 0. Then α,β = 0 for all a, b, i.e. it has maximal possible slope; therefore it is σ a,b α,β -semistable. By Lemma 11.7, it must actually be strictly stable.
Proof of Theorem 9.1 Assume for a contradiction that there is a stability condition σ = (Z , P) ∈ ∂ P in the boundary of P inside Stab H (X ). Since P → P is a covering map, the central charge Z must be in the boundary ∂P of P ⊂ Hom( H , C); by definition, this means that there is a point (x 3 , x 2 y, 1 2 x y 2 , 1 6 y 3 ) on the twisted cubic C that is contained in the kernel of Z . If μ := y x = p q is rational, then we observe that every semi-homogeneous bundle E p/q is σ -semistable, because being σ -semistable is a closed condition on Stab H (X ). This is an immediate contradiction, as Z E p/q = 0. Similarly, if x = 0, we get Z (O x ) = 0; yet skyscraper sheaves of points are σ -semistable by 8.14.
Otherwise, if μ ∈ R\Q, consider a sequence ( p n , q n ) with lim n→∞ p n q n = μ, let E n := E p n /q n , and let r n = rk E n . Then This is a contradiction to the condition that σ satisfies the support property.

The space of stability conditions on some Calabi-Yau threefolds
Let X be a projective threefold with an action of a finite group G. In this section, we recall the main result of [28], which induces stability conditions on the G-equivariant derived category from G-invariant stability conditions on X ; similar results are due to Polishchuk, see [35,Sect. 2.2]. We use it to construct stability conditions on Calabi-Yau threefolds that are (crepant resolutions of) quotients of abelian threefolds, thus proving Theorems 1.2, 1.3 and 1.4.

The equivariant derived category
We

Inducing stability conditions
Following [28], we consider Here the action of g on Stab H (X ) is given by Theorem 10.1 [28] Let (X, H ) be a polarized threefold with an action by a finite group G fixing the polarization. Then Stab G H (X ) ⊂ Stab H (X ) is a union of connected components.
Moreover, the pull-back f * induces an embedding whose image is again a union of connected components.
Proof The theorem is essentially a reformulation of Theorem 1.1 in [28] but some subtle issues have to be clarified. First of all, Theorem 1.1 in [28] deals with stability conditions whose central charge is defined on the Grothendieck group K (X ) rather than on the lattice H . On the other hand, the same argument as in [28,Remark 2.18] shows that all the results in [28,Sect. 2.2], with the obvious changes in the statements and in the proofs, hold true if we consider pre-stability conditions as in Definition 11.1 with respect to the lattice H . Thus we will freely quote the results there.

Applications
When the action of the finite group G is free, the quotient Y = X/G is smooth Here is a list of examples where X is an abelian threefold and this discussion can be implemented, concluding the proof of Theorems 1.2 and 1.4.

Example 10.4 (i) A
Calabi-Yau threefold of abelian type is an étale quotient Y = X/G of an abelian threefold X by a finite group G acting freely on X such that the canonical line bundle of Y is trivial and H 1 (Y, C) = 0. In [34, Theorem 0.1], those Calabi-Yau manifolds are classified; the group G can be chosen to be (Z/2) ⊕2 or D 8 , and the Picard rank of Y is 3 or 2, respectively. The following concrete example is usually referred to as Igusa's example (see Example 2.17 in [34]). Take three elliptic curves E 1 , E 2 and E 3 and set X = E 1 × E 2 × E 3 . Pick three non-trivial elements τ 1 , τ 2 and τ 3 in the 2-torsion subgroups of E 1 , E 2 and E 3 , respectively. Then we define two automorphisms a and b of X by setting By taking G := a, b , the quotient Y = X/G is a Calabi-Yau threefold of abelian type.
(ii) Let A be an abelian surface and let E be an elliptic curve. We write X := A × E. Consider a finite group G acting on A and E, where the action on E is given by translations. Then the diagonal action on X is free, but it may have non-trivial (torsion) canonical bundle. The easiest example is by taking A as the product E 1 × E 2 of two elliptic curve, and the action of G only on the second factor so that E 2 /G ∼ = P 1 . Then Y = E 1 × S, where S is a bielliptic surface.
Let us now assume that X is an abelian threefold, that G acts faithfully, and that the dualizing sheaf is locally trivial as a G-equivariant sheaf. By [8], the quotient X/G admits a crepant resolution Y with an equivalence . By a slightly more serious abuse of notation, we will continue to write Stab H (Y ) for the space of stability conditions with respect to the lattice H and the map v G H • ( BKR ) * : K (Y ) → H . By Corollary 10.3, we obtain a connected component as Example 10. 5 We say that a Calabi-Yau threefold is of Kummer type if it is obtained as a crepant resolution of a quotient X/G of an abelian threefold X . Skyscraper sheaves will be semistable but not stable with respect to the stability conditions induced from X . We mention a few examples.
(i) Let E be an elliptic curve, and let X = E × E × E. We consider a finite subgroup G ⊂ SL(3, Z) and let it act on X via the identification These examples were studied in [3] and classified in [15]; there are 16 examples, and G has size at most 24. The singularities of the quotient X/G are not isolated. (ii) Let E be the elliptic curve with an automorphism of order 3, and let X = E × E × E. We can take G = Z/3Z acting on X via the diagonal action. Then the crepant resolution Y of X/G is a simply connected rigid Calabi-Yau threefold containing 27 planes, see [7,Sect. 2]. One can also take G ⊂ (Z/3Z) 3 to be the subgroup of order 9 preserving the volume form. These examples were influential at the beginning of mirror symmetry, see [5] and references therein. (iii) Let X be the Jacobian of the Klein quartic curve. The group G = Z/7Z acts on X , and again the crepant resolution Y of X/G is a simply connected rigid Calabi-Yau threefold. (iv) We can also provide easy examples involving three non-isomorphic elliptic curves E 1 , E 2 and E 3 . Indeed, take the involutions ι i : E i → E i such that ι i (e) = −e, for i = 1, 2, 3, and set G := ι 1 ×ι 2 ×id E 3 , ι 1 ×id E 2 ×ι 3 . The quotient (E 1 × E 2 × E 3 )/G admits a crepant resolution Y which is a Calabi-Yau threefold. This is a very simple instance of the so called Borcea-Voisin construction (see [12,50]). This yields smooth projective Calabi-Yau threefolds as crepant resolutions of the quotient (S × E)/G, where S is a K3 surface, E is an elliptic curve and G is the group generated by the automorphism f × ι of S × E, with f an antisymplectic involution on S and ι the natural involution on E above. Example 2.32 in [34] is yet another instance of this circle of ideas.
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Appendix 1: Support property via quadratic forms
In this appendix, we clarify the relation between support property, quadratic inequalities for Chern classes of semistable objects, and effective deformations of Bridgeland stability conditions.

Equivalent definitions of the support property
Let D be a triangulated category, for which we fix a finite rank lattice with a surjective map v : K (D) . We recall the main definition of [13] with a slight change of terminology: a stability condition not necessarily satisfying the support property will be called a pre-stability condition: Definition 11.1 A pre-stability condition on D is a pair (Z , P) where • The central charge Z is a linear map Z : → C, and • P is a collection of full subcategories P(φ) ⊂ D for all φ ∈ R, such that (a) P(φ + 1) = P(φ) [1]; (b) For φ 1 > φ 2 , we have Hom(P(φ 1 ), P(φ 2 )) = 0; (c) For 0 = E ∈ P(φ), we the complex number Z (v(E)) is contained in the ray R >0 · e iπφ ; and (d) Every E admits an HN-filtration. We This notion is equivalent to σ being "full" in the sense of [14], see [10,Proposition 12.4]. The definition is quite natural: it implies that if W is in an -neighborhood of Z with respect to the operator norm on Hom( R , C) induced by and the standard norm on C, then W (E) is in a disc of radius C |Z (E)| around Z (E) for all semistable objects E; in particular, we can bound the difference of the arguments of the complex numbers Z (E) and W (E).
Moreover, it is equivalent to the following notion; we follow Kontsevich-Soibelman and also call it "support property": Definition 11.3 The pre-stability condition σ = (Z , P) satisfies the support property if there exists a quadratic form Q on the vector space R such that • The kernel of Z is negative definite with respect to Q, and • For any σ -semistable object E ∈ D b (X ), we have evidently satisfies both properties of Definition 11.3. Conversely, assume we are given a quadratic form Q as in Definition 11.3. The non-negative quadratic form |Z (w)| 2 is strictly positive on the set where −Q(w) ≤ 0; by compactness of the unit ball, there exists a constant C such that is a positive definite quadratic form. Then Z clearly satisfies (33) with respect to the induced norm on R .

Statement of deformation properties
By Stab (D) we denote the space of stability conditions satisfying the support property with respect to ( , v). By the main result of [13], the forgetful map In other words, this proposition gives an effective version of Bridgeland's deformation result [13,Theorem 1.2], and shows that Chern classes of semistable objects for varieties continue to satisfy the same inequalities within this class of deformations.

The quadratic form and wall-crossing
We start with the observation that the quadratic form is preserved by wallcrossing: Proof Let H E ⊂ R be the half-space of codimension one given as the preimage of the ray R ≥0 · Z (E), and let C + ⊂ H E be the subset defined by Q ≥ 0. By the following Lemma, C + is a convex cone, implying the claim.
Lemma 11.7 Let Q be a quadratic form an a real vector space V , and let Z : V → C be a linear map such that the kernel of Z is semi-negative definite with respect to Q. Let ρ be a ray in the complex plane starting at the origin.
Then the intersection is a convex cone. Moreover, if we assume that Q has signature (2, dim V − 2), and that the kernel of Z is negative definite, then any vector w ∈ C + with Q(w) = 0 generates an extremal ray of C + .
Proof To prove convexity we just need to show that if w 1 , w 2 ∈ C + , then Q(w 1 + w 2 ) ≥ 0. According to the taste of the reader, this can either be seen by drawing a picture of 2-plane spanned by w 1 , w 2 -the only interesting case being where Q| has signature (1, 1)-, or by the following algebraic argument. Assume that Q(w 1 +w 2 ) < 0. Since w 1 , w 2 ∈ Z −1 (ρ), there exists λ > 0 such that w 1 − λw 2 is in the kernel of Z . We therefore have This configuration is impossible, since the quadratic function f (x) := Q(w 1 + xw 2 ) would have too many sign changes.
To prove the second statement, observe that under these stronger assumptions and for w 1 , w 2 , λ as above, we have Q(w 1 − λw 2 ) < 0. This implies Q(w 1 + w 2 ) > 0, from which the claim follows.
Before returning to the proof of Proposition 11.5, let us add one additional consequence: Proposition 11.8 Assume that the quadratic form Q has signature (2, rk H − 2). Let U ⊂ Stab (D) be a path-connected set of stability conditions that satisfy the support property with respect to Q. Let E ∈ D b (X ) be an object with Q(E) = 0 that is σ -stable for some σ ∈ U . Then E is σ -stable for all σ ∈ U .
Proof Otherwise there would be a wall at which E becomes strictly semistable. However, by the previous Lemma, v H (E) is an extremal ray of the cone C + . Therefore, all the Jordan-Hölder factors E i must have v H (E i ) proportional to v H (E), in contradiction to E being strictly stable for some nearby central charges.

Proof of the deformation property
In a sense, Lemma 11.6 is the key observation in the proof of Proposition 11.5; the remainder boils down to a careful application of local finiteness of wall-crossing, and of the precise version of the deformation result proved by Bridgeland. To this end, we need to recall the definition of the metric on Stab (D).
Definition 11.9 [13, Proposition 8.1] The following is a generalized metric on Stab (D): Bridgeland's proof of the deformation result in fact proves the following stronger statement: Theorem 11.10 [13,Sects. 6,7] Assume that σ = (Z , P) is a stability condition on D, and let C > 0 be a constant with respect to which σ satisfies the support property condition (33). Let < 1 8 , and consider the neighborhood B C (Z ) of Z taken with respect to the operator norm on Hom( , C). Then there exists an open neighborhood U ⊂ Stab (D) containing σ , such that Z restricts to a homeomorphism . Therefore, Stab (D) is a complex manifold; moreover, the generalized metric of Definition 11.9 is finite on every connected component of Stab (D).
Proof of Proposition 11.5 Consider the subset V ⊂ U of stability conditions that do not satisfy the second claim; we want to prove that V is empty, thereby establishing the second claim.
Given σ ∈ V, there exists a σ -semistable object E with with Q(v(E)) < 0; by Lemma 11.6, we may assume that E is stable. By openness of stability of E, there exists a neighborhood of σ contained in V; therefore, V ⊂ U is open.
We claim that V ⊂ U is also a closed subset; since U is a manifold and V ⊂ U is open, it is enough to show that if σ : [0, 1] → U is a piece-wise linear path with σ (t) ∈ V for 0 ≤ t < 1, then σ (1) ∈ V. By the definition of V and Lemma 11.6 there exists an object E 0 that is σ (0)-stable with Q(v(E 0 )) < 0. Since σ (1) / ∈ V, there must be 0 < t 1 < 1 such that E 0 is strictly semistable; applying Lemma 11.6 again, it must have a Jordan-Hölder factor σ (t 1 )-stable factor E 1 with Q(v(E 1 )) < 0. Proceeding by induction, we obtain an infinite sequence 0 = t 0 < t 1 < t 2 < t 3 < · · · < 1 of real numbers and objects E i such that E i is σ (t)-stable for t i ≤ t < t i+1 , strictly semistable with respect to σ (t i+1 ) (having E i+1 as a Jordan-Hölder factor), and satisfies Q(v(E i )) < 0. This is a contradiction by Lemma 11.11 below.
Therefore, since V ⊂ U is both open and closed, and does not contain σ , it must be empty.
It remains to prove the first claim. By Theorem 11.10, it is enough to show that there is a continuous function C : U → R >0 such that every σ ∈ U satisfies the support property with respect to C(Z(σ )). This is evident from the second claim and the proof of Lemma 11.4.

Lemma 11.11
Let σ : [0, 1] → Stab (D) be a piece-wise linear path in the space of stability condition satisfying the support property. Assume there is a sequence 0 = t 0 < t 1 < t 2 < · · · < 1 of real numbers and a sequence of objects E 0 , E 1 , E 2 , . . . with the following properties: • E i is σ (t i+1 )-semistable, and E i+1 is one of its Jordan-Hölder factors.
Such a sequence always terminates.
Proof Assume we are given an infinite such sequence. Let d i := d(σ (t i ), σ (t i+1 )); the assumptions imply that σ is a path of bounded length, and hence that On the other hand, if we write Z i for the central charge of σ (t i ), then using induction we deduce that the mass of all objects E i is bounded: By [14,Sect. 9], this implies that there is a locally finite collection of walls of semistability for all E i . Since our path is compact, it intersects only finitely many walls; since it is piece-wise linear, it intersects every wall only finitely many times.

Appendix 2: Deforming tilt-stability
The purpose of this appendix is to establish rigorously the deformation and wall-crossing properties of tilt-stability, in particular correcting [11,Corollary 3.3.3]. This will lead to variants of the results of Appendix 1 in this context. We assume that the reader of this appendix is familiar with the notion of tilt-stability as reviewed in Sects. 2 and 3, as well as with the proof of Bridgeland's deformation result for stability conditions in [13,Sects. 6,7].
Let X be a smooth projective threefold with polarization H ; the role of and v in the previous appendix will be played by , ch 1 (E), H ch 2 (E)) .
We will use a variant of the notion of "weak stability" of [42], adapted to our situation: Definition 12.1 A very weak stability condition on X is a pair σ = (Z , A), where A is the heart of a bounded t-structure on D b (X ), and Z : → C is a group homomorphism such that • Z satisfies the following weak positivity criterion for every E ∈ A: • If we let ν Z ,A : A → R ∪ +∞ be the induced slope function, then HN filtrations exist in A with respect to ν Z ,A -stability.
By induced slope function we mean that ν Z ,A (E) is the usual slope of the complex number Z (v H (E)) if its real part is positive, and ν Z ,A (E) = +∞ if Z (v H (E)) is purely imaginary or zero. The crucial difference to a Bridgeland stability condition is that Z (v H (E)) = 0 is allowed for non-zero objects E ∈ A. Given a very weak stability condition, one can define a slicing P = {P(φ) ⊂ D b (X )} φ∈R just as in the case of a proper stability condition constructed from a heart of a t-structure: for − 1 2 < φ ≤ 1 2 , we let P(φ) ⊂ A be the subcategory of ν Z ,A -semistable objects with slope corresponding to the ray R >0 · e iπφ ; this gets extended to all φ ∈ R via P(φ + n) = P(φ)[n] for n ∈ Z.
This allows one to define a topology on the set of very weak stability conditions; it is the coarsest topology such that the maps σ → Z and σ → φ ± σ (E) are continuous, for all E ∈ D b (X ). Our first goal is to show tilt-stability conditions vary continuously; note that we use a slightly different normalization of the central charge than in Sect. 2: For rational B, this stability condition can be constructed by proving directly that the pair Z α,B , Coh H,B (X ) admits Harder-Narasimhan filtrations, see [11,Lemma 3.2.4]. We will extend this to arbitrary B by deformations, and show simultaneously that these deformations glue to give a single family of very weak stability conditions. Let us first indicate the key difficulty that prevents us from applying the methods of [13,Sects. 6,7] directly. Let I be a small interval containing We now explain how to circumvent this problem. Fix α, B with B rational; we will use Z := Z α,B for the corresponding central charge. By the rational case of Theorem 3.5, proved in Sect. 3, the central charge Z satisfies the support property. 4 Let C > 0 be the constant appearing in the support property; we also write P for the associated slicing. Now consider a central charge W := Z α ,B , where α , B are sufficiently close to α, B such that W satisfies W − Z < C , for some sufficiently small > 0; recall that this implies that the phases of σ -semistable object change by at most . We choose < 1 8 and small enough such that H 2 (B − B) < αH 3 is automatically satisfied. For simplicity we also assume that H 2 (B − B) < 0; the other case can be dealt with analogously.
Let I = (a, b) be a small interval with a + < 1 2 < b − ; the key problem is to construct Harder-Narasimhan filtrations of objects in P(I ) with respect to W . Our first observation is that due to our assumption H 2 (B − B) < 0, central charges of objects in P(I ) can only "move to the left"; this is again based on the Bogomolov-Gieseker inequality for σ -stability: ((a, b)) is σ -semistable with Z (E) < 0, then also W (E) < 0.
By using Theorem 3.5, applied to the rational class B, we also have Therefore, we deduce Using H 2 (B − B) < αH 3 , this implies the claim.
As in [13,Sect. 7], we define the set of semistable objects Q(φ) to be objects of P((φ − , φ + )) that are W -semistable in a slightly larger category, e.g. in P ((φ − 2 , φ + 2 )). The key lemma overcoming the indicated difficulty above is the following: Lemma 12.4 Given E ∈ P ((a, b)), there exists a filtration 0 = E 0 → E 1 → E 2 → E 3 such that • E 1 ∈ Coh H,B (X ) [1] and E 1 has no quotients E 1 N in P ((a, b)) with W (N ) ≥ 0; Proof The t-structure associated to Coh H,B (X ) gives a short exact sequence E → E E in P ((a, b)) with E ∈ Coh H,B (X ) [1] and E ∈ Coh H,B (X ). Any quotient E N would necessarily satisfy N ∈ P(( 1 2 , b)); by Lemma 12.3, this implies W (N ) < 0. Thus, given a filtration as in the claim for E , its preimage in E will still satisfy all the claims.
We may therefore assume E ∈ Coh H,B (X ). Note that Let E 1 → E F be the short exact sequence associated to E via this torsion pair. Since T is closed under quotients, E 1 satisfies all the claims in the lemma; similarly, B only has subobjects with W ( ) ≥ 0.
The existence of E 2 now follows from the fact that Coh H,B (X ) admits a torsion pair whose torsion part is given by objects with W ( ) = 0; this is shown in the first paragraph of the proof of [11, Lemma 3.2.4], which does not use any rationality assumptions.
The existence of Harder-Narasimhan filtrations of E 1 and E 3 /E 2 can now be proved with the same methods as in [13,Sect. 7]; the same goes for any advantage is that the slicing associated to this polynomial stability condition is locally finite.
(b) Let us also explain precisely the problem with the statement of [11, Corollary 3.3.3]: if we allow arbitrary deformations of ω ∈ NS(X ) R , rather than just those proportional to a given polarization H , we would need to prove the support property for tilt-stable objects with respect to a non-degenerate quadratic form on the lattice = H 0 (X ) ⊕ NS(X ) ⊕ 1 2 N 1 (X ), v(E) = (ch 0 (E), ch 1 (E), ch 2 (E)) .
However, none of the variants of the classical Bogomolov-Gieseker inequality discussed in Sect. 3 give such a quadratic form, as they only depend on H ch 2 rather than ch 2 directly.