The space of stability conditions on abelian threefolds, and on some CalabiYau threefolds
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Abstract
 1.
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 BogomolovGieseker type inequality involving the third Chern character of “tiltstable” twoterm complexes on smooth projective threefolds; we extend it from complexes of tiltslope zero to arbitrary tiltslope.
 2.
We show that this stronger conjecture implies the socalled support property of Bridgeland stability conditions, and the existence of an explicit open subset of the space of stability conditions.
 3.
We prove our conjecture for abelian threefolds, thereby reproving and generalizing a result by Maciocia and Piyaratne.
Mathematics Subject Classification
Primary 14F05 Secondary 14J30 18E301 Introduction
In this paper, we determine the space of Bridgeland stability conditions on abelian threefolds and on CalabiYau 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 \(H^{3i} \mathop {\mathrm {ch}}\nolimits _i(\underline{\,\,})\), \(i = 0, 1, 2, 3\), of the Chern character^{1} with respect to a given polarization H, and that satisfy the support property.
1.1 Stability conditions on threefolds via a conjectural BogomolovGieseker type inequality
The existence of stability conditions on threedimensional varieties in general, and more specifically on CalabiYau 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 DonaldsonThomas invariants [43, 45] to Reidertype 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 tiltstability for twoterm complexes, and a conjectural BogomolovGieseker type inequality for the third Chern character of tiltstable 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 \(\omega , B \in \mathop {\mathrm {NS}}\nolimits (X)_\mathbb {R}\) with \(\omega \) 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 BogomolovGieseker type inequality for tiltstable objects, Conjecture 2.4, holds when X is an abelian threefold, and \(\omega \) is a real multiple of an integral ample divisor class.
There are CalabiYau threefolds that admit an abelian variety as a finite étale cover; we call them CalabiYau threefolds of abelian type. Our result applies similarly in these cases:
Theorem 1.2
Conjecture 2.4 holds when X is a CalabiYau threefold of abelian type, and \(\omega \) 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 CalabiYau 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 Gequivariant derived category of X and the BKRequivalence [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 CalabiYau threefold of abelian type, or a Kummer threefold.
1.2 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 wellbehaved wallcrossing.
In order to prove the support property, we first need a quadratic inequality for all tiltstable complexes, whereas Conjecture 2.4 only treats complexes E with tiltslope zero. We state such an inequality in Conjecture 4.1 for the case where \(\omega , B\) are proportional to a given ample class H:
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.
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.
1.3 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 \(\mathop {\mathrm {Hom}}\nolimits (\mathbb {Q}^4, \mathbb {C})\), consider the open subset \(\mathfrak V\) of linear maps Z whose kernel does not intersect the (real) twisted cubic \(\mathfrak C\subset \mathbb {P}^3(\mathbb {R})\) parametrized by \((x^3, x^2y, \frac{1}{2} xy^2, \frac{1}{6} y^3)\); it is the complement of a real hypersurface. Such a linear map Z induces a morphism \(\mathbb {P}^1(\mathbb {R}) \cong \mathfrak C\rightarrow \mathbb {C}^*/\mathbb {R}^* = \mathbb {P}^1(\mathbb {R})\); we define \(\mathfrak P\) be the component of \(\mathfrak V\) for which this map is an unramified cover of topological degree \(+3\) with respect to the natural orientations. Let \(\widetilde{\mathfrak P}\) be its universal cover.
We let \(\mathop {\mathrm {Stab}}\nolimits _H(X)\) 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 CalabiYau threefold of abelian type, or a Kummer threefold. Then \(\mathop {\mathrm {Stab}}\nolimits _H(X)\) has a connected component isomorphic to \(\widetilde{\mathfrak P}\).
1.4 Approach
We will now explain some of the key steps of our approach.
1.4.1 Reduction to a limit case
1.4.2 Abelian threefolds
In the case of an abelian threefold, we make extensive use of the multiplication by m map \(\underline{m}:X \rightarrow X\) in order to establish inequality (2). The key fact is that if E is tiltstable, then so is \(\underline{m}^*E\).
1.4.3 Support property
 (a)
The kernel of the central charge (as a subspace of \(\mathbb {R}^4\)) is negative definite with respect to Q, and
 (b)
Every semistable object E satisfies \(Q(v_H(E)) \ge 0\).
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 wallcrossing. The essential arguments involve elementary linear algebra of quadratic forms.
Tiltstability 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 wallcrossings between tiltstability and any of our stability conditions, thereby establishing the desired support property.
1.4.4 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 \(\widetilde{\mathfrak P}\) described above.
In Sect. 9, we show that this set is in fact an entire component. The proof is based on the observation that semihomogeneous vector bundles E with \(c_1(E)\) proportional to H are stable everywhere on \(\mathfrak P\); their Chern classes (up to rescaling) are dense in \(\mathfrak C\).
This fact is very unique to varieties admitting étale covers by abelian threefolds. In particular, while Conjecture 4.1 implies that \(\widetilde{\mathfrak P}\) is a subset of the space of stability conditions, one should in general expect the space to be much larger than this open subset.
1.5 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

\(L^3 > 49\alpha \),

\(L^2 D \ge 7\alpha \) for every integral divisor class D with \(L^2 D > 0\) and \(L D^2 < \alpha \), and

\(L.C \ge 3\alpha \) for every curve \(C \subset X\).
In addition, if \(L = A^{\otimes 5}\) 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]. \(\square \)
Setting \(\alpha = 2\) we obtain a Reidertype criterion for L to be very ample. The statement for \(A^{\otimes 5}\) confirms (the very ampleness case of) Fujita’s conjecture for such X. The best known bounds for CalabiYau threefolds say that \(A^{\otimes 8}\) is very ample if \(L^3 > 1\) [18, Corollary 1], \(A^{\otimes 10}\) is very ample in general, and that \(A^{\otimes 5}\) induces a birational map [33, Theorem I]. For abelian varieties, much stronger statements are known, see [37, 38].
Corollary 1.6
The assumptions hold when \(\mathop {\mathrm {NS}}\nolimits (X)\) 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 \(\mathbb {P}^3\), 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 \(E = I_C \otimes L\) is the twist of an ideal sheaf of a curve C, this inequality is not known, see [49].
1.6 Open questions
1.6.1 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 \(\underline{m}\) with ramified coverings. This would immediately yield the set \(\widetilde{\mathfrak P}\) as an open subset of the space of stability conditions.
1.6.2 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 BogomolovGieseker type inequality, depending on \(\mathop {\mathrm {ch}}\nolimits _1\) and \(\mathop {\mathrm {ch}}\nolimits _2\) directly, not just on \(H^2 \mathop {\mathrm {ch}}\nolimits _1\) and \(H \mathop {\mathrm {ch}}\nolimits _2\). We point out that the obvious guess, namely to replace \(\left( H^2 \mathop {\mathrm {ch}}\nolimits _1\right) ^2\) by \(H \mathop {\mathrm {ch}}\nolimits _1^2 \cdot H^3\), and \(\left( H \mathop {\mathrm {ch}}\nolimits _2\right) ^2\) by an appropriate quadratic form on \(H^4(X)\), does not work in general: for \(\alpha \rightarrow +\infty \), such an inequality fails for torsion sheaves supported on a divisor D with \(H D^2 < 0\).
1.6.3 Higher dimension
Our work also clarifies the expectations for higher dimensions. The definition of \(\mathfrak P\) directly generalizes to dimension n in an obvious way, by replacing the twisted cubic with the rational normal curve \(\left( x^n, x^{n1}y, \frac{1}{2} x^{n2}y^2, \dots , \frac{1}{n!} y^n\right) \). Let \(\widetilde{\mathfrak P}_n \rightarrow \mathfrak P_n\) denote the corresponding universal covering.
Conjecture 1.7
Let (X, H) be a smooth polarized ndimensional variety. Its space \(\mathop {\mathrm {Stab}}\nolimits _H(X)\) of stability conditions contains an open subset \(\widetilde{\mathfrak P}_n\), for which skyscraper sheaves of points are stable. In the case of abelian varieties, \(\widetilde{\mathfrak P}_n \subset \mathop {\mathrm {Stab}}\nolimits _H(X)\) 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 \(H^n \mathop {\mathrm {ch}}\nolimits _0, H^{n1} \mathop {\mathrm {ch}}\nolimits _1, \dots , H^{ni} \mathop {\mathrm {ch}}\nolimits _i\). Semistable objects would have to satisfy a quadratic inequality \(Q_i\) involving \(\mathop {\mathrm {ch}}\nolimits _{i+1}\). 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 seminegative 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 \(\mathop {\mathrm {ch}}\nolimits _3\) for stable objects on a fourfold would follow from a MehtaRamanathan 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 tiltstable object on a threefold restricts to a Bridgelandstable object on a divisor of sufficiently high degree.
1.7 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 tiltstability with respect to FourierMukai transforms; in addition to constructing stability conditions, they show their invariance under FourierMukai 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 FourierMukai 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 pullbacks 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 \(\mathbb {P}^3\), and in [39] for the case of the quadric in \(\mathbb {P}^4\); 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 \(\mathbb {P}^4\) via graded matrix factorizations, proposed by Toda [46, 47]. It is more specific, but would yield a stability condition that is invariant under certain autoequivalences; it would also lie outside of our set \(\widetilde{\mathfrak P}\). His approach would require a stronger BogomolovGieseker inequality already for slopestable vector bundles, and likely lead to very interesting consequences for generating functions of DonaldsonThomas invariants.
Conjecture 2.4 can be specialized to certain slopestable sheaves, similar to Corollary 1.6; see [11, Conjecture 7.2.3]. This statement was proved by Toda for certain CalabiYau threefolds, including the quintic hypersurface, in [48]. Another case of that conjecture implies a certain Castelnuovotype 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, semihomogeneous bundles are examples of the Lagrangianinvariant objects considered by Polishchuk, are semistable for our stability conditions, and their phases behave as predicted.
1.8 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 wallcrossing.
Sections 2 and 3 and Appendix 2 review basic properties of tiltstabilty, its deformation properties (fixing a small inaccuracy in [11]), the conjectural inequality proposed in [11] and variants of the classical BogomolovGieseker inequality satisfies by tiltstable 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 \(\widetilde{\mathfrak P}\) of stabilty conditions described above. Section 9 shows that in the case of abelian threefolds, \(\widetilde{\mathfrak P}\) is in fact a connected component, and Sect. 10 extends these results to (crepant resolutions) of quotients of abelian threefolds.
1.9 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].
2 Review: tiltstability and the conjectural BG inequality
In this section, we review the notion of tiltstability for threefolds introduced in [11]. We then recall the conjectural BogomolovGieseker type inequality for tiltstable complexes proposed there; see Conjecture 2.4 below.
2.1 Slopestability
Let X be a smooth projective complex variety and let \(n\ge 1\) be its dimension. Let \(\omega \in \mathrm {NS}(X)_\mathbb {R}\) be a real ample divisor class.
Definition 2.1
2.2 The tilted category
Definition 2.2
By the general theory of torsion pairs and tilting [20], \(\mathop {\mathrm {Coh}}\nolimits ^{\omega , B}(X)\) is the heart of a bounded tstructure on \(\mathrm {D}^{b}(X)\); in particular, it is an abelian category.
2.3 Tiltstability and the main conjecture
Definition 2.3
Tiltstability gives a notion of stability, in the sense that HarderNarasimhan filtrations exist.
The following conjecture is the main topic of [11]:
Conjecture 2.4
2.4 Properties of tiltstability
To prove that tiltstability is a wellbehaved property, one needs to use variants of the classical BogomolovGieseker inequality for slopesemistable sheaves; in particular, this leads to the following statements:
Remark 2.5
 (a)
Tiltstability is an open property. More precisely, assume that \(E\in \mathrm {D}^{b}(X)\) is \(\nu _{\omega , B}\)stable with \(\omega = \sqrt{3}\alpha H\). Then the set of pairs \((\alpha ', B') \in \mathbb {R}_{>0} \times \mathop {\mathrm {NS}}\nolimits (X)_\mathbb {R}\) such that E is \(\nu _{\sqrt{3}\alpha 'H, B'}\)stable is open.
 (b)
The boundary of the above subset of \(\mathbb {R}_{>0} \times \mathop {\mathrm {NS}}\nolimits (X)_\mathbb {R}\) where \(E \in \mathrm {D}^{b}(X)\) is tiltstable 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
 (a)
\(\overline{Z}_{\omega , B}(F)\) and \(\overline{Z}_{\omega , B}(E)\) are linearly dependent, or that
 (b)
\(\nu _{\omega , B}(E) = +\infty \).
In the limit \(\omega \rightarrow +\infty \cdot H\), tiltstability becomes closely related to slopestability:
Lemma 2.7
 (a)
The category \(\mathop {\mathrm {Coh}}\nolimits ^{\omega , B}(X)\) is independent of \(\alpha \).
 (b)
Moreover, its subcategory of objects E with \(\nu _{\omega , B}(E) = +\infty \) is independent of \(\alpha \).
 (c)If \(E \in \mathop {\mathrm {Coh}}\nolimits ^{H, B}(X)\) is \(\nu _{\omega , B}\)semistable for \(\alpha \gg 0\), then it satisfies one of the following conditions:Conversely, assume \(E \in \mathop {\mathrm {Coh}}\nolimits (X)\) is a \(\mu _{\omega ,B}\)stable torsionfree sheaf.
 (i)
\(H^{1}(E) = 0\) and \(H^0(E)\) is a \(\mu _{\omega ,B}\)semistable torsionfree sheaf.
 (ii)
\(H^{1}(E) = 0\) and \(H^0(E)\) is a torsion sheaf.
 (iii)
\(H^{1}(E)\) is a \(\mu _{\omega ,B}\)semistable sheaf and \(H^0(E)\) is either 0, or supported in dimension \(\le 1\).
 (i)
If \(H^2\mathop {\mathrm {ch}}\nolimits _1^B(E)>0\), then \(E\in \mathop {\mathrm {Coh}}\nolimits ^{H,B}(X)\) and it is \(\nu _{\omega , B}\)stable for \(\alpha \gg 0\).
 (ii)
If \(H^2\mathop {\mathrm {ch}}\nolimits _1^B(E)\le 0\), then \(E[1]\in \mathop {\mathrm {Coh}}\nolimits ^{H,B}(X)\); if moreover E is a vector bundle, then it is \(\nu _{\omega , B}\)stable for \(\alpha \gg 0\).
 (i)
3 Classical BogomolovGieseker type inequalities
In this section, we review a result from [11] that shows that tiltstable objects on X satisfy variants of the classical BogomolovGieseker inequality.
We continue to assume that X is a smooth projective threefold. Throughout this section, let \(H \in \mathop {\mathrm {NS}}\nolimits (X)\) be a polarization, \(\omega = \sqrt{3} \alpha H\) for \(\alpha > 0\), and \(B \in \mathop {\mathrm {NS}}\nolimits (X)_\mathbb {R}\) arbitrary.
First we recall the classical BogomolovGieseker inequality:
Definition 3.1
Theorem 3.2
(Bogomolov, Gieseker) Assume that E is a \(\mu _{H}\)semistable torsionfree sheaf on X. Then \(\Delta _H(E) \ge 0\).
However, a sheaf F supported on a divisor \(D \subset X\) does not necessarily satisfy \(\Delta _H(F) \ge 0\) (even if it is the pushforward of a slopestable 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 tiltstable objects. We first need the following easy observation (see, for example, the proof of [11, Corollary 7.3.3]):
Lemma 3.3
(Note that for abelian threefolds, we may take \(C_H = 0\).)
Definition 3.4
Theorem 3.5
This was proved for rational B in [11]; we will give a selfcontained proof of the rational case with a slightly different presentation below, and extend it to arbitrary B in Appendix 2.
Notice that \(\overline{Z}_{\omega , B}\) as defined in equation (5) factors via \(v_H^B\). Its relation to \(q_H^B\) is controlled by the following immediate consequences of the Hodge index theorem:
Lemma 3.6
The quadratic form \(q_H^B\) has signature \((2, \rho (X))\).
The kernel of \(\overline{Z}_{\omega , B}\) is negative definite with respect to \(q_H^B\).
This makes our situation analogous to the one in Appendix 1; in particular, Theorem 3.5 implies a version of the support property for tiltstable objects.
Lemma 3.7
 (a)For any object \(E \in \mathop {\mathrm {Coh}}\nolimits ^{\omega , B}(X)\) with \(\nu _{\omega , B}(E) = \nu \), we have$$\begin{aligned} v_H^B(E) \in \mathbb {H}_{\omega , B, \nu }. \end{aligned}$$
 (b)
The intersection of \(\mathbb {H}_{\omega , B, \nu }\) with the set defined by \(q_H^B(\underline{\,\,}) \ge 0\) is a real convex cone.
Proof
We define \(\mathbb {H}_{\omega , B, \nu }\) as the preimage under \(\overline{Z}_{\omega , B}\) of the ray in the complex plane that has slope \(\nu \), starting at the origin; this ensures the first claim. The second claim is a general fact about quadratic forms, see Lemma 11.7. \(\square \)
Note that by definition, a halfspace is closed; indeed, we may have \(v_H^B(E) = 0\) iff \(\nu = +\infty \).
Remark 3.8
Proof of Theorem 3.5, case \(H^2B \in \mathbb {Q}\) We prove the statement for \(\Delta _{H, B}^C \) under the assumption that \(H^2B\) is rational. The proof for \(\overline{\Delta }_H^B\) follows similarly due to Remark 3.8, and the nonrational case will be treated in Appendix 2.
We proceed by induction on \(H^2 \mathop {\mathrm {ch}}\nolimits _1^B(E)\), which by our assumption is a nonnegative function with discrete values on objects of \(\mathop {\mathrm {Coh}}\nolimits ^{H, B}(X)\).
We start increasing \(\alpha \). If E remains stable as \(\alpha \rightarrow +\infty \), we apply Lemma 2.7, (c); by Theorem 3.2 (for torsionfree slopesemistable 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 \(\alpha \) increases, all possible destabilizing subobjects and quotients have strictly smaller \(H^2 \mathop {\mathrm {ch}}\nolimits _1^B\), which satisfy the desired inequality by our induction assumption. This is enough to ensure that E satisfies wellbehaved wallcrossing: following the argument of [14, Proposition 9.3] it is enough to know a support property type statement for all potentially destabilizing classes.
We now turn to some consequences of Theorem 3.5.
Lemma 3.9
Proof
This follows immediately from the easy fact that if \(x, y \in \mathcal C^+  \{0\}\), then the bilinear form associated to Q satisfies \((x,y) \ge 0\), with equality if and only if x, y are proportional with \(Q(x)= Q(y) = 0\). \(\square \)
Corollary 3.10
The same statements hold with \(\overline{\Delta }_H^B\) and \(\overline{v}_H^B\) replaced by \(\Delta _{H, B}^C \) and \(v_H^B\), respectively.
The case \(\nu = +\infty \) is excluded as in that case we may have \(\overline{v}_H^B(E_i) = 0\) or \(\overline{v}_H^B(E_i) = \overline{v}_H^B(E)\).
Proof
Let \(x_i := \overline{v}_H^B(E_i)\) and \(x := \overline{v}_H^B(E)\). By Lemmas 3.6 and 3.7, they satisfy the assumptions of Lemma 3.9, which then implies our claim. \(\square \)
As another application, one obtains the tiltstability of certain slopestable sheaves (see also [11, Proposition 7.4.1]):
Corollary 3.11
 (a)
Let F be a \(\mu _{H,B}\)stable vector bundle with \(\Delta _{H, B}^C (F) = 0\) or \(\overline{\Delta }_H^B(F) = 0\). Then F or F[1] is a \(\nu _{\omega , B}\)stable object of \(\mathop {\mathrm {Coh}}\nolimits ^{H,B}(X)\).
 (b)
In particular, 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 \(\nu _{\omega , B}\)stable.
 (c)
Conversely, consider an object \(E \in \mathop {\mathrm {Coh}}\nolimits ^{H, B}(X)\) that is \(\nu _{\omega , B}\)stable with \(\Delta _{H, B}^C (E) = 0\) or \(\overline{\Delta }_H^B(E) = 0\). Then either \(E = H^0(E)\) is a \(\mu _H\)semistable sheaf, or \(E = H^0(E)\) is supported in dimension \(\le 2\), or \(H^{1}(E) \ne 0\) is a \(\mu _H\)semistable sheaf and \(H^0(E)\) has zerodimensional support. In addition, E is \(\nu _{\omega ', B}\)stable for all \(\omega '\) proportional to H.
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 \(\nu _{\omega , B}\)stable with \(\overline{\Delta }_H^B(E) = 0\) or \(\Delta _{H, B}^C (E) = 0\). By Corollary 3.10, E can never become strictly semistable with respect to \(\nu _{\omega ', B'}\) as long as \(\omega '\) is proportional to \(\omega \). Combined with Lemma 2.7, (c) this implies all our claims. \(\square \)
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.
Proposition 3.12
Assume that B is rational, and let \(E \in \mathop {\mathrm {Coh}}\nolimits ^{H, B}(X)\) be a \(\nu _{\omega , B}\)stable object with \(\overline{\Delta }_H^B(E) = 0\) and \(\nu _{\omega ,B}(E)=0\). Then E satisfies Conjecture 2.4.
Proof
If F is a \(\mu _{\omega ,B}\)semistable reflexive sheaf on X with \(\overline{\Delta }_H^B(F) = 0\), then F is a vector bundle by [24, Proposition 3.12], Further, if E is \(\nu _{\omega ,B}\)semistable with \(\nu _{\omega ,B}(E)<+\infty \), then \(H^{1}(E)\) is reflexive by [24, Proposition 3.1]. Hence, the case \(H^{1}(E) \ne 0\) of part (c) in Corollary 3.11 can actually be made much more precise: in this case, \(H^0(E) = 0\) and \(H^{1}(E)\) is a vector bundle. In the other case, if \(\nu _{\omega ,B}(E)=0\), \(\overline{\Delta }_H^B(E)=0\), and \(H^{1}(E)=0\), then \(H^0(E)\) is a torsionfree sheaf and its doubledual is again locallyfree with \(\overline{\Delta }_H^B=0\). In either case, a classical result of Simpson (see [41, Theorem 2] and [22, Theorem 4.1]) implies that E satisfies Conjecture 2.4; see [11, Proposition 7.4.2]. \(\square \)
4 Generalizing the main conjecture
The goal of this section is to generalize Conjecture 2.4 to arbitrary tiltsemistable objects, not just those satisfying \(\nu _{\alpha , \beta } = 0\). This generalization relies on the structure of walls for tiltstability in \(\mathbb {R}_{>0}\times \mathbb {R}\); it is completely analogous to the case of walls for Bridgeland stability on surfaces, treated most systematically in [25].
Conjecture 4.1
Theorem 4.2
Let X be a smooth projective threefold, and \(H \in \mathop {\mathrm {NS}}\nolimits (X)\) an ample class. Then Conjecture 4.1 holds if and only if Conjecture 2.4 holds for all \(\omega , B\) proportional to H.
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 \(\nu _{\alpha , \beta }(E) \ne +\infty \). Then the object E is \(\nu _{\alpha , \beta }\)semistable along the semicircle \(\mathcal C_{\alpha , \beta }(E)\) in the \((\alpha , \beta )\)plane \(\mathbb {R}_{>0}\times \mathbb {R}\) with center \((0, \beta + \nu _{\alpha , \beta }(E))\) and radius \(\sqrt{\alpha ^2 + \nu _{\alpha , \beta }(E)^2}\).
Proof
In addition, a simple computation shows \(H^2 \mathop {\mathrm {ch}}\nolimits _1^{\beta '}(E) > 0\) for \((\alpha ', \beta ') \in \mathcal C_{\alpha , \beta }(E)\); therefore, the semicircle cannot intersect a wall given by \(\nu _{\alpha ', \beta '}(E) = +\infty \) either. \(\square \)
Proof of Theorem 4.2
We first note that due to Theorem 3.5, Conjecture 4.1 holds for all objects E with \(H^2 \mathop {\mathrm {ch}}\nolimits _1^\beta (E) = 0\). We may therefore assume \(\nu _{\alpha , \beta }(E) \ne +\infty \) throughout the proof.
 (*)Assume that E is \(\nu _{\alpha , \beta }\)stable with \(\nu _{\alpha , \beta }(E) \ne +\infty \). Let \(\beta ':= \beta + \nu _{\alpha , \beta }(E)\). Then$$\begin{aligned} \mathop {\mathrm {ch}}\nolimits ^{\beta '}_3(E) \le \frac{1}{6} \left( \alpha ^2 + \nu _{\alpha , \beta }(E)^2\right) H^2\mathop {\mathrm {ch}}\nolimits _1^{\beta '}(E). \end{aligned}$$(12)
Example 4.4
Remark 4.5
The inequality (13) holds when X is an abelian threefold, or a CalabiYau threefold of abelian type. Moreover, since Conjecture 4.1 is equivalent to Conjecture 2.4, and since the latter has been verified for \(\mathbb {P}^3\) 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 \(\mathbb {P}^3\): 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.
5 Reduction to small \(\alpha \)
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 \(\alpha \rightarrow 0\) and \(\nu _{\alpha , \beta } \rightarrow 0\).
The other motivation for the definition of \(\bar{\beta }\) lies in the following observations, extending Lemma 3.6.
Lemma 5.1
The kernel of \(\overline{Z}_{0, \overline{\beta }(E)}\) (as a subspace of \(\mathbb {R}^3\)) is contained in the quadric \(\overline{q} =0\), and the map \((\alpha , \beta ) \rightarrow \mathop {\mathrm {Ker}}\nolimits \overline{Z}_{\alpha , \beta }\) extends to a continuous map from of \(\mathbb {R}_{\ge 0} \times \mathbb {R}\) to the projectivization \(\mathcal C^/\mathbb {R}^*\) of the cone \(\mathcal C^ \subset \mathbb {R}^3\) given by \(\overline{q} \le 0\).
Moreover, if \(\overline{\Delta }_H(E) > 0\), then the quadratic form \(\overline{q}\) is positive semidefinite on the 2plane spanned by \(\overline{v}_H(E)\) and the kernel of \(\overline{Z}_{0, \overline{\beta }(E)}\).
In other words, the vector \(\overline{v}_H(E)\) is contained in the tangent plane to the quadric \(\overline{q} = 0\) at the kernel of \(\overline{Z}_{0, \overline{\beta }(E)}\); see Fig. 1.
Remark 5.2
The map \((\alpha , \beta ) \mapsto \mathop {\mathrm {Ker}}\nolimits \overline{Z}_{\alpha , \beta }\) gives a homeomorphism from \(\mathbb {R}_{\ge 0} \times \mathbb {R}\) onto its image in the closed unit disc \(\mathcal C^/\mathbb {R}^*\). This can be a helpful visualization, as a central charge is, up to the action of \(\mathop {\mathrm {GL}}\nolimits _2(\mathbb {R})\), determined by its kernel.
Proof of Lemma 5.1
The kernel of \(\overline{Z}_{\alpha , \beta }\) is spanned by the vector \(\big (1, \beta , \frac{1}{2} (\alpha ^2 + \beta ^2)\big )\), which has Hdiscriminant \(q_H(1, \beta , \frac{1}{2} (\alpha ^2 + \beta ^2))= \alpha ^2\). This proves the first claim.
For the second claim, we just observe that \(\left( 1, \overline{\beta }(E), \frac{1}{2} \overline{\beta }(E)^2\right) \) and \(\overline{v}_H(E)\) are orthogonal with respect to the bilinear form on \(\mathbb {R}^3\) associated to \(\overline{q}\). \(\square \)
The following is a limit case of Conjecture 4.1.
Conjecture 5.3
Unless \(\overline{\Delta }_H(E) = 0\), we can always make U small enough such that \(H^2\mathop {\mathrm {ch}}\nolimits _1^\beta (E) > 0\) for \((\alpha , \beta ) \in U\); then E itself is an object of \(\mathop {\mathrm {Coh}}\nolimits ^\beta (X)\).
A strengthening of the methods of [26] leads to the main result of this section:
Lemma 5.5
Let \(E \in \mathrm {D}^{b}(X)\) be an object with \(\overline{\Delta }_H(E) > 0\) that is \(\nu _{\alpha ,\beta }\)stable for some \((\alpha , \beta ) \in \mathbb {R}_{>0}\times \mathbb {R}\). The point \((0, \overline{\beta }(E))\) cannot be an endpoint of a wall of tiltstability for E. Moreover, each of the semicircles of Lemma 4.3 (along which E has to remain stable) contains \((0, \overline{\beta }(E))\) in its interior.
Proof
Recall the description of walls in Remark 2.6. As \(\overline{\Delta }_H(E) > 0\) implies \(H^2\mathop {\mathrm {ch}}\nolimits _1^{\overline{\beta }(E)}(E) > 0\), we can exclude the possibility of a wall given by \(\nu _{\omega , B}(E) = +\infty \). The other type of walls can equivalently be defined by the property that the kernel of \(\overline{Z}_{\alpha , \beta }(E)\) is contained in the 2plane \(\Pi \subset \mathbb {R}^3\) spanned by \(\overline{v}_H(F)\) and \(\overline{v}_H(E)\), for some destabilizing subobject \(F \hookrightarrow E\). The signature of \(\overline{q}\) restricted to \(\Pi \) has to be (1, 1) (as it contains \(\overline{v}_H(E)\) and the kernel of \(\overline{Z}_{\alpha , \beta }\) for some \(\alpha > 0\)). If \((0, \overline{\beta }(E))\) was an endpoint of this wall, then by Lemma 5.1 the kernel of \(Z_{0, \overline{\beta }(E)}\) would also be contained in \(\Pi \); 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 \(\mathop {\mathrm {Ker}}\nolimits \overline{Z}_{\alpha , \beta }\) is contained in a given plane through \(\overline{v}_H(E)\).) As we shrink the radius of the circles, their center has to converge to the point with \(\alpha = 0\) and \(\nu _{\alpha , \beta }(E) = 0\).
\(\square \)
Proof
Proposition 3.12 combined with Theorem 4.2 ensures that such an object satisfies Conjecture 4.1. If E in addition satisfies the assumptions of Conjecture 5.3, we consider inequality (10) nearby \((0, \overline{\beta }(E))\). The first term vanishes identically, the second vanishes to second order at \((0, \overline{\beta }(E))\). Therefore, we must have \(\mathop {\mathrm {ch}}\nolimits _3^{\overline{\beta }(E)} (E)= 0\); otherwise the third term would only have a simple zero, in contradiction to Conjecture 4.1. \(\square \)
Proof of Theorem 5.4
By the previous lemma, we can restrict to the case \(\overline{\Delta }_H(E) > 0\) 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 \((\alpha , \beta ) \rightarrow (0, \overline{\beta })\). Evidently the first term \(\alpha ^2 \overline{\Delta }_H(E)\) goes to zero; by equation (17), the same holds for the second term \((H \mathop {\mathrm {ch}}\nolimits _2^\beta (E))^2\). Since \(H^2 \mathop {\mathrm {ch}}\nolimits _1^{\overline{\beta }(E)}> 0\), the limit yields exactly (18).
 (a)
Consider a semicircle given by Lemma 4.3. By the proof of Theorem 4.2, inequality (10) either holds for all points on the semicircle, or it is violated for all such points; indeed, it is equivalent to inequality (12), which is just the original Conjecture 2.4 applied at the point where this semicircle intersects the curve given by \(\nu _{\alpha , \beta }(E) = 0\).
 (b)
Once we fix \(\beta \), it is clear from Theorem 3.5 that if (10) holds for a given \(\alpha _0\), then it holds for all \(\alpha \ge \alpha _0\).
 (c)
Finally, if we consider the semicircles of Lemma 4.3 at all points \((\alpha , \beta )\) with \(\alpha > 0, \beta = \overline{\beta }(E)\), then by Lemma 5.5 they fill up all points of \(\mathbb {R}_{>0} \times \mathbb {R}\) with \(H^2 \mathop {\mathrm {ch}}\nolimits _1^\beta (E) > 0\).
For contradiction, let E be an object that is \(\nu _{\alpha , \beta }\)stable, with \(\overline{\Delta }_H(E) > 0\), and that violates conjecture (10) at this point. By Lemma 5.5 and observation (a) above, we may assume \(\beta = \overline{\beta }(E)\).
Now fix \(\beta = \overline{\beta }(E)\) and start decreasing \(\alpha \). Since we assume (10) to be violated, we must have \(\mathop {\mathrm {ch}}\nolimits _3^{\overline{\beta }(E)}(E) > 0\). If E were to remain stable as \(\alpha \rightarrow 0\), then by Lemma 5.5 it would be stable in a neighborhood of \((0, \overline{\beta }(E))\) as in the conditions of Conjecture 5.3; this is a contradiction.
Therefore there must be a point \(\alpha _0\) where E is strictly \(\nu _{\alpha _0,\overline{\beta }(E)}\)semistable; let \(E_i\) be the list of its JordanHölder factors. By observation (b), E still violates Conjecture (10) at \((\alpha _0, \overline{\beta }(E))\). On the other hand, by Corollary 3.10, \(\overline{\Delta }_H(E_i) < \overline{\Delta }_H(E)\) 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 lefthandside of (10) as a quadratic form on \(\mathbb {R}^4\) with coordinates \((H^3 \mathop {\mathrm {ch}}\nolimits _0^\beta , H^2 \mathop {\mathrm {ch}}\nolimits _1^\beta , H \mathop {\mathrm {ch}}\nolimits _2^\beta , \mathop {\mathrm {ch}}\nolimits _3^\beta )\). The kernel of \(\overline{Z}_{\alpha , \beta }\), considered as a subspace of \(\mathbb {R}^4\), is negative semidefinite with respect to the quadratic form. Therefore, the claim follows from Lemma 11.7. \(\square \)
6 Tilt stability and étale Galois covers
Consider an étale Galois cover \(f :Y \rightarrow X\) with covering group G; in other words, G acts freely on Y with quotient \(X = Y/G\). In this section, we will show that tiltstability is preserved under pullback by f.
For this section, we again let \(\omega , B \in \mathop {\mathrm {NS}}\nolimits (X)_\mathbb {R}\) be arbitrary classes with \(\omega \) a positive real multiple of an ample.
Proposition 6.1
 (a)
\(E\in \mathop {\mathrm {Coh}}\nolimits ^{\omega , B}(X)\) if and only if \(f^*E\in \mathop {\mathrm {Coh}}\nolimits ^{f^*\omega , f^*B}(Y)\), and
 (b)
E is \(\nu _{\omega ,B}\)semistable if and only if \(f^*E\) is \(\nu _{f^*\omega ,f^*B}\)semistable.
Proof
Now consider \(E \in \mathop {\mathrm {Coh}}\nolimits ^{\omega , B}(X)\). Part (a) and the above computation shows that if E is tiltunstable, then so is \(f^*E\). Conversely, assume that \(f^*E\) is tiltunstable. Let \(F \hookrightarrow f^*E\) be the first step in its HarderNarasimhan filtration with respect to \(\nu _{f^*\omega , f^*B}\). Since \(f^*E\) is Gequivariant, and since the HN filtration is unique and functorial, the object F must also be Gequivariant. Hence it is the pullback of an object \(F'\) in \(\mathrm {D}^{b}(X)\). Using part (a) again, we see that \(F'\) must be an object of \(\mathop {\mathrm {Coh}}\nolimits ^{\omega , B}(X)\). Applying the same arguments to the quotient \(f^*E/F\), we see that \(F'\) is a destabilizing subobject of E in \(\mathop {\mathrm {Coh}}\nolimits ^{\omega , B}(X)\). \(\square \)
Example 6.2
Let \(n\in \mathbb {Z}_{>0}\). Let \(X=Y\) be an abelian threefold and let \(\underline{n}:X\rightarrow X\) be the multiplication by n map. Then \(\underline{n}\) has degree \(n^6\), and \(\underline{n}^*H=n^2 H\) for any class \(H\in \mathrm {NS}(X)\); 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 tiltstability with respect to \(\nu _{f^*\omega , f^*B}\) on Y, then it also holds for tiltstability with respect to \(\nu _{\omega , B}\) on X.
7 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 \(\omega \) and B are proportional to H. For \((\alpha ,\beta )\in \mathbb {R}_{>0}\times \mathbb {R}\), we let \(\omega =\sqrt{3} \alpha H\) and \(B = \beta H\). 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.
7.1 Idea of the proof
The proof naturally divides into two cases: if \(\overline{\beta }(E)\) is rational, then \(\underline{n}^* \left( E(\overline{\beta }(E)H)\right) \) exists when n is sufficiently divisible, and the above approach works verbatim; otherwise, we need to use Diophantine approximation of \(\overline{\beta }(E)\).
7.2 Proof of Conjecture 5.3, rational case
We assume that \(\overline{\beta }(E)\) is a rational number.
7.2.1 Reduction to \(\overline{\beta }(E)=0\)

\(\overline{\beta }(E)=0\), and so \(H.\mathop {\mathrm {ch}}\nolimits _2(E)=0\), and

\(\mathop {\mathrm {ch}}\nolimits _3(E)>0\), and so \(\mathop {\mathrm {ch}}\nolimits _3(E)\ge 1\).
7.2.2 Asymptotic Euler characteristic
7.2.3 First bound
7.2.4 Homvanishing from stability
7.2.5 Restriction to divisors
We will use this Homvanishing to restrict sections to divisors; we will repeatedly apply the following immediate observation.
Lemma 7.1
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 \(F_j(D) \rightarrow F_j\) is injective. \(\square \)
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.
7.2.6 Bound on \(\hom (\mathcal O_X,\underline{n}^*E)\)
Theorem 7.2
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.
Lemma 7.3
Proof
7.2.7 Bound on \(\mathop {\mathrm {ext}}\nolimits ^2(\mathcal O_X,\underline{n}^*E)\)
7.2.8 Conclusion
7.3 Proof of Conjecture 5.3, irrational case
Now assume that \(\overline{\beta }(E)\in \mathbb {R}{\setminus }\mathbb {Q}\) is an irrational number. As a consequence \(\mathop {\mathrm {ch}}\nolimits _0(E)\ne 0\) and, for all \(\beta \in \mathbb {Q}\), \(H\mathop {\mathrm {ch}}\nolimits _2^\beta (E)\ne 0\).
7.3.1 The Euler characteristic
7.3.2 Homvanishing
7.3.3 Bound on \(\hom (\mathcal O_X,\underline{m}^*F_n)\) and conclusion
Lemma 7.4
Proof
7.4 Proof of Theorem 1.1
Let now \(B\in \mathrm {NS}(X)_\mathbb {R}\) be an arbitrary divisor class and \(\omega \) 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 \(E\in \mathop {\mathrm {Coh}}\nolimits ^{\omega ,B}(X)\) be as in Conjecture 2.4. We first assume that \(B\in \mathrm {NS}(X)_\mathbb {Q}\) is rational. Then, by Proposition 6.1, we can assume B integral. By taking the tensor product with \(\mathcal O_X(B)\), we can then assume E is \(\nu _{\omega ,0}\)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 JordanHölder factors we can assume E is \(\nu _{\omega ,B}\)stable. By using Theorem 3.5 and Remark 2.5, we can deform \((\omega , B)\) to \((\omega ', B')\) with \(B'\) rational (and \(\omega '\) still proportional to H), such that E is still \(\nu _{\omega ',B'}\)stable with \(\nu _{\omega ', B'}(E) = 0\). But, if (6) does not hold for \((\omega , B)\), then it does not hold for \((\omega ', B')\) sufficiently close, giving a contradiction to what we just proved.
8 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 wellbehaved wallcrossing.
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.
8.1 Statement of results
Definition 8.1
The goal of this section is the following precise version of Theorem 1.3:
Theorem 8.2
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.
8.2 Alternative description of \(\mathfrak P\)
We will need a more explicit description of the set \(\mathfrak P\) before proceeding to prove our main result.
The group \(\mathop {\mathrm {GL}}\nolimits _2^+(\mathbb {R})\) of \(2 \times 2\)matrices with positive determinant acts on \(\mathfrak P\) on the left by postcomposing a central charge with the induced \(\mathbb {R}\)linear map of \(\mathbb {R}^2 \cong \mathbb {C}\). There is also an action of \(\mathbb {R}\) on \(\mathfrak P\) on the right: for \(\beta \in \mathbb {R}\), the multiplication by \(e^{\beta H}\) in \(K(\mathrm {D}^{b}(X))\) corresponds to a linear selfmap of \(\Lambda _H \otimes \mathbb {R}\) which leaves \(\mathfrak C\) invariant; therefore we can act on \(\mathfrak P\) by precomposing with this linear map.
Lemma 8.3
It follows that it is simultaneously a slice of the \(\widetilde{\mathop {\mathrm {GL}}\nolimits }_2^+(\mathbb {R})\)action on \(\widetilde{\mathfrak P}\).
Proof
We now use the action of \(\mathbb {R}\) on \(\mathfrak P\) 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 \(\mathbb {R}^2\) to normalize it to be \(+\frac{1}{2}\). Since the sign of \(i(0) = \mathfrak {I}Z(\mathcal O_X)\) is constant within this slice, it has to be negative; hence there exists a unique \(\alpha \in \mathbb {R}_{>0}\) such that \(i(0) =  \frac{1}{2} \alpha ^2\).
Conversely, given a central charge \(Z_{\alpha , \beta }^{a, b}\) as described in the lemma, we can first use the action of \(\mathbb {R}\) to reduce to the case \(\beta = 0\). The coefficients of the linear functions \(\mathfrak {I}Z, \mathfrak {R}Z\) are in onetoone 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 \(Z_H^{\mathrm {basic}}\). \(\square \)
Remark 8.4
From the proof of the lemma one can also deduce the following more intrinsic description of the set \(\mathfrak P\). Consider the twisted cubic \(\overline{\mathfrak C}\) in projective space \(\mathbb {P}^3(\mathbb {R})\). There is an open subset of central charges Z with the following properties: the hyperplanes \(\mathfrak {I}Z = 0\) and \(\mathfrak {R}Z = 0\) both intersect \(\overline{\mathfrak C}\) in three distinct points; moreover, their configuration on \(\overline{\mathfrak C} \cong S^1\) are such that the zeros of the two functions alternate. This open set has two components: one of them is \(\mathfrak P\), the other is obtained from \(\mathfrak P\) by composing central charges with complex conjugation.
Moreover, one can also deduce the description given in the introduction.
Lemma 8.5
Proof
8.3 Review: construction of stability conditions
 (a)For any \(0 \ne E\in \mathcal A\) the central charge \(Z(v_H(E))\) lies in the following semiclosed upper halfplane:We can use \(\mathfrak {R}Z\) and \(\mathfrak {I}Z\) to define a notion of slopestability on the abelian category \(\mathcal A\) via the slope function \(\lambda _\sigma (E)= \frac{\mathfrak {R}Z(v_H(E))}{\mathfrak {I}Z(v_H(E))}\)$$\begin{aligned} Z(v_H(E)) \in \mathbb {R}_{>0} \cdot e^{(0,1]\cdot i\pi } \end{aligned}$$(30)
 (b)
With this notion of slopestability, every object in \(E \in \mathcal A\) has a HarderNarasimhan filtration \(0 = E_0 \hookrightarrow E_1 \hookrightarrow \dots \hookrightarrow E_n = E\) such that each \(E_i/E_{i1}\) is \(\lambda _\sigma \)semistable, with \(\lambda _\sigma (E_1/E_0)> \lambda _\sigma (E_2/E_1)> \dots > \lambda _\sigma (E_n/E_{n1})\).
 (c)(support property) There is a constant \(C>0\) such that, for all \(\lambda _\sigma \)semistable object \(E\in \mathcal A\), we havewhere \(\underline{\,\,}\) is a fixed norm on \(\Lambda _H\otimes \mathbb {R}\cong \mathbb {R}^4\).$$\begin{aligned} v_H(E) \le C Z(v_H(E)) , \end{aligned}$$
8.4 Explicit construction of stability conditions
We start by reviewing (a slightly generalized version of) the construction of stability conditions in [11].
Theorem 8.6
[11] Let (X, H) be a polarized threefold for which Conjecture 4.1 holds. Assume that \(\alpha , \beta \in \mathbb {Q}\), and that \(\alpha , \beta , a, b\) satisfy (29). Then the pair \(\sigma = \left( Z_{\alpha , \beta }^{a, b}, \mathcal A^{\alpha , \beta }(X)\right) \) satisfy conditions (a) and (b) above.
Proof
The case \(b = 0\) is [11, Corollary 5.2.4], and the same arguments apply here; let us review them briefly.

T is a zerodimensional torsion sheaf, and

\(F\in \mathop {\mathrm {Coh}}\nolimits ^\beta (X)\) is \(\nu _{\alpha ,\beta }\)semistable with \(\nu _{\alpha ,\beta }(E)=0\) (in particular, \(H^2\mathop {\mathrm {ch}}\nolimits _1^{\beta }(F)> 0\)).
8.5 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 \(\sigma = (Z, \mathcal A) \in \widetilde{\mathfrak P} \subset \mathop {\mathrm {Stab}}\nolimits _H(X)\) be a stability condition in the open subset given in Theorem 8.6. We may assume that \(Z = Z_{\alpha , \beta }^{a, b}\) is of the form given in Lemma 8.3. We also choose a constant \(K \in I_{\alpha }^{a, b}\) in accordance with Lemma 8.5.
Theorem 8.7
We will treat only the case \(b = 0\); then \(I_{\alpha }^{a, b} = (\alpha ^2, 6a)\). We will also shorten notation and write \(Z_{\alpha , \beta }^a\) instead of \(Z_{\alpha , \beta }^{a, 0}\), and \(I_{\alpha }^a\) instead of \(I_{\alpha }^{a,0}\). The case \(b\ne 0\) 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:
Lemma 8.8
Let (X, H) be a polarized threefold and \((\alpha ,\beta )\in \mathbb {Q}_{>0}\times \mathbb {Q}\). Assume that Conjecture 4.1 holds for this pair \((\alpha ,\beta )\). Then for any \(a > \frac{1}{6} \alpha ^2\), the pair \(\sigma _{\alpha ,\beta }^a=(Z_{\alpha ,\beta }^a,\mathcal A^{\alpha ,\beta }(X))\) satisfies the support property; more precisely, the inequality (31) holds for all \(\sigma _{\alpha ,\beta }^a\)semistable objects E and all \(K \in I_{\alpha }^a\).
We first need an analogue of Lemma 2.7.
Let us denote by \(H_\beta ^i\) the ith cohomology object with respect to the tstructure \(\mathop {\mathrm {Coh}}\nolimits ^{\beta }(X)\).
Lemma 8.9
 (a)
\(H^{1}_\beta (E)=0\) and \(H^0_\beta (E)\) is \(\nu _{\alpha ,\beta }\)semistable;
 (b)
\(H^{1}_\beta (E)\) is \(\nu _{\alpha ,\beta }\)semistable and \(H^0_\beta (E)\) is either 0 or supported in dimension 0.
Proof
In the limit \(a \rightarrow +\infty \), we have \(Z_{\alpha , \beta }^a \rightarrow \overline{Z}_{\alpha , \beta }\) up to rescaling of the real part; this implies the \(\nu _{\alpha , \beta }\)semistability of the cohomology objects in both cases.
\(\square \)
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 \(\alpha \) and \(\beta \).
If E is strictly \(\sigma _{\alpha ,\beta }^a\)semistable, and if (31) holds for all of the JordanHölder 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 \(F\in \mathop {\mathrm {Coh}}\nolimits ^{\beta }(X)\) is \(\nu _{\alpha ,\beta }\)semistable, then Conjecture 4.1 and Theorem 3.5 show that in particular, it satisfies \(Q_K^\beta (F) \ge 0\) for every \(K > \alpha ^2\).
We proceed by induction on \(f(E):=H\mathop {\mathrm {ch}}\nolimits _2^\beta (E)  \frac{\alpha ^2 \, H^3}{2} \mathop {\mathrm {ch}}\nolimits _0^\beta (E) = \mathfrak {I}Z_{\alpha , \beta }^a(E)\), which is a nonnegative function on \(\mathcal A^{\alpha , \beta }(X)\) with discrete values.
We fix \(a_0 > \frac{1}{6} \alpha ^2\) and \(K \in (\alpha ^2, 6a_0)\). Let E be a \(\sigma _{\alpha ,\beta }^{a_0}\)stable object in \(\mathcal A^{\alpha ,\beta }(X)\).
Otherwise, E will be unstable for a sufficiently big. Every possibly destabilizing subobject or quotient F has \(f(F) < f(E)\) (since f is nonnegative, and since the subcategory of objects \(F \in \mathcal A^{\alpha , \beta }(X)\) with \(f(F) = 0\) has maximum possible slope with respect to \(Z_{\alpha , \beta }^a\) for all a).
Therefore they obey the induction assumption; since \(K \in (\alpha ^2, 6a_0) \subset (\alpha ^2, 6a)\), this means that all these possible subobject or quotients satisfy (31) with respect to our choice of K. Since \(Z_{\alpha , \beta }^a\) has negative definite kernel with respect to \(Q_K^\beta \) for all \(a \ge a_0\), this is equivalent to a support property type statement, see Appendix 1. It follows that E satisfies wellbehaved wallcrossing along our path. Hence, there will exist \(a_1> a_0\) such that E is strictly \(\sigma _{\alpha ,\beta }^{a_1}\)semistable. But all the JordanHölder factors \(E_i\) of E have strictly smaller f. Using the induction assumption again, we see that they satisfy \(Q_K^\beta (E_i) \ge 0\); therefore, we can again apply Lemma 11.6 to deduce the same claim for E. \(\square \)
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 \(\alpha , \beta , a, b\) as in Theorem 8.6 (in particular \(\alpha , \beta \in \mathbb {Q}\)), we obtain an open subset \(U(\alpha , \beta , a, b) \subset \mathop {\mathrm {Stab}}\nolimits _H(X)\) of stability conditions by deforming the pair \((Z_{\alpha , \beta }^{a, b}, \mathcal A^{\alpha , \beta }(X))\). The associated open subsets \(\mathcal Z(U(\alpha , \beta , a, b))\) of central charges combine to cover the set \(\mathfrak P\). To conclude the proof of Theorems 8.2 and 8.7, we need to show that the sets \(U(\alpha , \beta , a, b)\) glue to form a continuous family covering \(\widetilde{\mathfrak P}\).
This is done by the following analogue of Proposition 12.2.
Proposition 8.10
Indeed, deformations of the central charge \(Z_{\alpha , \beta }^{a, b}\) for \(b \ne 0\) (while keeping \(\alpha , \beta , a\) fixed) do not change the heart, as modifying b only affects the real part of the central charge. Acting on these stability conditions by \(\mathop {\mathrm {GL}}\nolimits _2^+(\mathbb {R})\) produces the entire set \(\widetilde{\mathfrak P}\).
To prove Proposition 8.10, we need a few preliminary results. We will use the notion of a prestability 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
 (a)
Their central charges agree.
 (b)There exists a heart \(\mathcal B\) of a bounded tstructure such that each \(\mathcal A_i\) can be obtained as a tilt of \(\mathcal B\):$$\begin{aligned} \mathcal A_1, \mathcal A_2 \subset \langle \mathcal B, \mathcal B[1] \rangle . \end{aligned}$$
Proof
Lemma 8.12
Proof
Lemma 8.13
We keep the notation as in the previous lemma. If \(E \in \mathop {\mathrm {Coh}}\nolimits ^\beta (X)\) is \(\nu _{\alpha , \beta }\)stable with \(\left\nu _{\alpha , \beta }(E)\right < \epsilon \), then \(E \in \mathcal P_{\alpha , \beta }^{a}((\frac{1}{2}, \frac{1}{2}))\).
Proof
We consider just the case \(0 < \nu _{\alpha , \beta }(E)\); the opposite case follows from dual arguments.
By construction we know \(E \in \mathcal A^{\alpha , \beta } = \mathcal P_{\alpha , \beta }^{a}((0,1])\). Let A be the HNfiltration factor of E with respect to \(\sigma _{\alpha , \beta }^a\) and with the largest phase, and consider the associated short exact sequence \(A \hookrightarrow E \twoheadrightarrow B\) in \(\mathcal A^{\alpha , \beta }\). The associated long exact cohomology with respect to \(\mathop {\mathrm {Coh}}\nolimits ^\beta (X)\) shows that \(A \in \mathop {\mathrm {Coh}}\nolimits ^\beta (X) \cap \mathcal A^{\alpha , \beta } = \mathcal T'_{\alpha , \beta }\); moreover, there is a sequence \(H^{1}(B) \hookrightarrow A \rightarrow E\) exact on the left with \(H^{1}(B) \in \mathcal F'_{\alpha , \beta }\).
Now consider the slopes appearing in the HarderNarasimhan filtration of A for tiltstability with respect to \(\nu _{\alpha , \beta }\). By standard arguments using the observations in the previous paragraph, all these slopes lie in the interval \((0, \epsilon )\). Lemma 8.12 then implies \(\mathfrak {R}Z_{\alpha , \beta }^a(A) > 0\), and therefore \(E \in \mathcal P_{\alpha , \beta }^a((0, \frac{1}{2}))\) as we claimed. \(\square \)
Proof of Proposition 8.10
Consider a stability condition \(\sigma _0 = (Z_0, \mathcal P_0) := \sigma _{\alpha _0, \beta _0}^{a_0}\). Let \(\epsilon := \epsilon (\alpha _0, \beta _0, a_0)\) be as in Lemmas 8.12 and 8.13. Consider \((\alpha , \beta , a)\) sufficiently close to \((\alpha _0, \beta _0, a_0)\) (which we will make precise shortly). Let \(\sigma _1 := \sigma _{\alpha , \beta }^{a}\), and let \(\sigma _2 = (Z_{\alpha , \beta }^a, \mathcal P_2)\) be the stability condition with central charge \(Z_{\alpha , \beta }^{a}\) obtained by deforming \(\sigma _0\). We want to apply Lemma 8.11 with \(\mathcal B= \mathcal P_0((\frac{1}{2}, \frac{1}{2}])\).

If E is \(\sigma _2\)stable of phase \(\phi \), then \(E \in \mathcal P_0((\phi \epsilon , \phi +\epsilon ))\), and

The analogous statement for tiltstability with respect to \(\nu _{\alpha , \beta }\) and \(\nu _{\alpha _0, \beta _0}\), respectively. This means that if \(E \in \mathop {\mathrm {Coh}}\nolimits ^{\beta }(X)\) is \(\nu _{\alpha , \beta }\)semistable, and if \(A_1, \dots , A_m\) are the HarderNarasimhan filtration factors of E for tiltstability with respect to \(\nu _{\alpha _0, \beta _0}\), then the phases of \(\overline{Z}_{\alpha _0, \beta _0}(A_i)\) differs by at most \(\epsilon \) from the phase of \(\overline{Z}_{\alpha , \beta }(E)\),
We have verified all the assumptions of Lemma 8.11, which implies \(\sigma _1 = \sigma _2\).
\(\square \)
Let us also mention the following property:
Proposition 8.14
[29, Proposition 2.1] Skyscraper sheaves are stable for all \(\sigma \in \widetilde{\mathfrak P}\).
Proof (sketch)
Using the long exact cohomology sequence with respect to the heart \(\mathop {\mathrm {Coh}}\nolimits (X)\), one sees that k(x) is a minimal object of \(\mathop {\mathrm {Coh}}\nolimits ^\beta (X)\): otherwise, there would be a short exact sequence \(E \hookrightarrow k(x) \twoheadrightarrow F[1]\) in \(\mathop {\mathrm {Coh}}\nolimits ^\beta (X)\) coming from a short exact sequence \(F \hookrightarrow E \twoheadrightarrow k(x)\) of sheaves; this is a contradiction to \(\mu _{H, \beta }(F) < 0\) and \(\mu _{H, \beta }(E) \ge 0\). Similarly, taking the long exact cohomology sequence with respect to \(\mathop {\mathrm {Coh}}\nolimits ^\beta (X)\) of short exact sequences in \(\mathcal A^{\alpha , \beta }(X)\), we see that k(x) is a minimal object of \(\mathcal A^{\alpha , \beta }(X)\). \(\square \)
9 The space of stability conditions on abelian threefolds
In this section we prove the following:
Theorem 9.1
Let (X, H) be a polarized abelian threefold. Then \(\widetilde{\mathfrak P} \hookrightarrow \mathop {\mathrm {Stab}}\nolimits _H(X)\) is a connected component of the space of stability conditions.
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 \(\mathfrak C\).
The above theorem is essentially based on the following result:
Proposition 9.2
The semihomogeneous vector bundle \(E_{p/q}\) is \(\sigma \)stable for every \(\sigma \in \widetilde{\mathfrak P}\).
Proof
As mentioned above, \(E_{p/q}\) is slopestable. By Corollary 3.11, either \(E_{p/q}\) or \(E_{p/q}[1]\) is a \(\nu _{\alpha , \beta }\)stable object of \(\mathop {\mathrm {Coh}}\nolimits ^\beta (X)\) for all \(\alpha > 0, \beta \in \mathbb {R}\).
One can prove in general that \(\nu _{\alpha , \beta }\)stable vector bundles are \(\sigma _{\alpha , \beta }^{a, b}\)stable for \(a \gg 0\); but in our situation one can argue more easily as follows. Choose \(\alpha , \beta \) with \(\beta < \frac{p}{q}\) (and therefore \(E_{p/q} \in \mathop {\mathrm {Coh}}\nolimits ^\beta (X)\)) and \(\nu _{\alpha , \beta }(E) = 0\). Then \(E[1] \in \mathcal A^{\alpha , \beta }(X)\) with \(\mathfrak {I}Z_{\alpha , \beta }^{a, b} = 0\) for all a, b, i.e. it has maximal possible slope; therefore it is \(\sigma _{\alpha , \beta }^{a, b}\)semistable. By Lemma 11.7, it must actually be strictly stable. \(\square \)
Proof of Theorem 9.1
Assume for a contradiction that there is a stability condition \(\sigma = (Z, \mathcal P) \in \partial \widetilde{\mathfrak P}\) in the boundary of \(\widetilde{\mathfrak P}\) inside \(\mathop {\mathrm {Stab}}\nolimits _H(X)\). Since \(\widetilde{\mathfrak P} \rightarrow \mathfrak P\) is a covering map, the central charge Z must be in the boundary \(\partial \mathfrak P\) of \(\mathfrak P\subset \mathop {\mathrm {Hom}}\nolimits (\Lambda _H, \mathbb {C})\); by definition, this means that there is a point \((x^3, x^2y, \frac{1}{2} xy^2, \frac{1}{6} y^3)\) on the twisted cubic \(\mathfrak C\) that is contained in the kernel of Z.
If \(\mu := \frac{y}{x} = \frac{p}{q}\) is rational, then we observe that every semihomogeneous bundle \(E_{p/q}\) is \(\sigma \)semistable, because being \(\sigma \)semistable is a closed condition on \(\mathop {\mathrm {Stab}}\nolimits _H(X)\). This is an immediate contradiction, as \(Z\left( E_{p/q}\right) = 0\). Similarly, if \(x = 0\), we get \(Z(\mathcal O_x) = 0\); yet skyscraper sheaves of points are \(\sigma \)semistable by 8.14.
\(\square \)
10 The space of stability conditions on some CalabiYau 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 Gequivariant derived category from Ginvariant stability conditions on X; similar results are due to Polishchuk, see [35, Sect. 2.2]. We use it to construct stability conditions on CalabiYau threefolds that are (crepant resolutions of) quotients of abelian threefolds, thus proving Theorems 1.2, 1.3 and 1.4.
10.1 The equivariant derived category
We let \(\mathop {\mathrm {Coh}}\nolimits ([X/G])\) be the abelian category of Gequivariant coherent sheaves on X, and \(\mathrm {D}^{b}([X/G]):=\mathrm {D}^{b}(\mathop {\mathrm {Coh}}\nolimits ([X/G]))\). As explained in [17], the category \(\mathrm {D}^{b}([X/G])\) is equivalent to the category of the Gequivariant objects in \(\mathrm {D}^{b}(X)\).
10.2 Inducing stability conditions
Theorem 10.1
[28] Let (X, H) be a polarized threefold with an action by a finite group G fixing the polarization. Then \(\mathop {\mathrm {Stab}}\nolimits _H^G(X) \subset \mathop {\mathrm {Stab}}\nolimits _H(X)\) is 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 \(\Lambda _H\). 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 prestability conditions as in Definition 11.1 with respect to the lattice \(\Lambda _H\). Thus we will freely quote the results there.
We now observe that if \(\sigma \) is a Ginvariant prestability condition on \(\mathrm {D}^{b}(X)\), then \(\sigma \) satisfies the support property with respect to \(v_H\) if and only if \((f^*)^{1}(\sigma )\) satisfies the support property with respect to \(v_H^G\). This is rather obvious, given the definition of \((f^*)^{1}(\sigma )\) above, the fact that \(\Lambda _H\) is invariant under the action of G and that the semistable objects in \((f^*)^{1}(\sigma )\) are the image under \(f^*\) of the semistable objects in \(\sigma \) (see [28, Theorem 1.1]). Hence [28, Proposition 2.17] applies and \((f^*)^{1}\) yields a welldefined and closed embedding.
It remains to point out that \(\mathop {\mathrm {Stab}}\nolimits _H^G(X)\) is a union of connected components of \(\mathop {\mathrm {Stab}}\nolimits _H(X)\). This is clear in view of the arguments in [28, Lemma 2.15] and, again, of the fact that \(\Lambda _H\) is invariant under the action of G. Thus the image of \((f^*)^{1}\) is a union of connected components as well. \(\square \)
An immediate consequence of the results of Sect. 8 and Theorem 10.1 is the following, which completes the proof of Theorem 1.3 (see also Examples 10.4, 10.5 below):
Proposition 10.2
Let (X, H) be a smooth polarized threefold with an action of a finite group G fixing the polarization. Assume that Conjecture 4.1 holds for (X, H). Then, given \(\alpha , \beta \in \mathbb {R}\) and \(\alpha , \beta , a, b\) satisfying (29), the stability condition \((Z_{\alpha , \beta }^{a, b}, \mathcal A^{\alpha , \beta }(X))\) is in \(\mathop {\mathrm {Stab}}\nolimits _H^G(X)\), and \((f^*)^{1}(\mathop {\mathrm {Stab}}\nolimits _H^G(X))\) is a nonempty union of connected components of \(\mathop {\mathrm {Stab}}\nolimits _H([X/G])\).
Proof
As an immediate consequence we get the following.
Corollary 10.3
Let (X, H) be a polarized abelian threefold with an action of a finite group G fixing the polarization. Then \((f^*)^{1}(\widetilde{\mathfrak P})\) is a connected component of \(\mathop {\mathrm {Stab}}\nolimits _H([X/G])\).
Proof
By Theorem 9.1, the open subset \(\widetilde{\mathfrak P}\) is a connected component of \(\mathop {\mathrm {Stab}}\nolimits _H(X)\). By Proposition 10.2, we have that \(\widetilde{\mathfrak P}\cap \mathop {\mathrm {Stab}}\nolimits _H^G(X)\) is not empty. Since \(\mathop {\mathrm {Stab}}\nolimits _H^G(X)\) is a union of connected components of \(\mathop {\mathrm {Stab}}\nolimits _H(X)\) (see Theorem 10.1), we get that \(\widetilde{\mathfrak P}\) is a connected component of \(\mathop {\mathrm {Stab}}\nolimits _H^G(X)\). Again by Theorem 10.1, we conclude that \((f^*)^{1}(\widetilde{\mathfrak P})\) is a connected component of \(\mathop {\mathrm {Stab}}\nolimits _H([X/G])\). \(\square \)
10.3 Applications
When the action of the finite group G is free, the quotient \(Y=X/G\) is smooth and \(\mathrm {D}^{b}(Y)\cong \mathrm {D}^{b}([X/G])\). In this case, an ample class H on X induces an ample class \(H_Y\) on Y. If we take B on X to be Ginvariant as well, and write \(B_Y\) for the induced class on \(\mathop {\mathrm {NS}}\nolimits (Y)_\mathbb {R}\), we then have, by Proposition 6.3, that Conjecture 2.4 holds for \(\nu _{\sqrt{3}\alpha H_Y, B_Y}\)stability on Y if it holds for \(\nu _{\sqrt{3}\alpha H, B}\)stability on X.
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
(ii) Let A be an abelian surface and let E be an elliptic curve. We write \(X:= A \times E\). 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 nontrivial (torsion) canonical bundle. The easiest example is by taking A as the product \(E_1\times E_2\) of two elliptic curve, and the action of G only on the second factor so that \(E_2/G\cong \mathbb {P}^1\). Then \(Y=E_1 \times S\), 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 Gequivariant sheaf. By [8], the quotient X / G admits a crepant resolution Y with an equivalence \(\Phi _{\mathrm {BKR}} :\mathrm {D}^{b}(Y)\rightarrow \mathrm {D}^{b}([X/G])\). By a slightly more serious abuse of notation, we will continue to write \(\mathop {\mathrm {Stab}}\nolimits _H(Y)\) for the space of stability conditions with respect to the lattice \(\Lambda _H\) and the map \(v_H^G \circ (\Phi _{\mathrm {BKR}})_* :K(Y) \rightarrow \Lambda _H\). By Corollary 10.3, we obtain a connected component as \(\left( \Phi _{\mathrm {BKR}}\right) ^* (f^*)^{1} \left( \widetilde{\mathfrak P}\right) \subset \mathop {\mathrm {Stab}}\nolimits _H(Y)\).
Example 10.5
 (i)
Let E be an elliptic curve, and let \(X=E\times E\times E\). We consider a finite subgroup \(G\subset \mathrm {SL}(3,\mathbb {Z})\) and let it act on X via the identification \(X=\mathbb {Z}^3\otimes _\mathbb {Z}E\). 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 \(X=E\times E\times E\). We can take \(G=\mathbb {Z}/3\mathbb {Z}\) acting on X via the diagonal action. Then the crepant resolution Y of X / G is a simply connected rigid CalabiYau threefold containing 27 planes, see [7, Sect. 2].
One can also take \(G \subset (\mathbb {Z}/3\mathbb {Z})^3\) 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 \(G=\mathbb {Z}/7\mathbb {Z}\) acts on X, and again the crepant resolution Y of X / G is a simply connected rigid CalabiYau threefold.
 (iv)
We can also provide easy examples involving three nonisomorphic elliptic curves \(E_1\), \(E_2\) and \(E_3\). Indeed, take the involutions \(\iota _i :E_i\rightarrow E_i\) such that \(\iota _i(e)=e\), for \(i=1,2,3\), and set \(G:=\langle \iota _1\times \iota _2\times \mathop {\mathrm {id}}\nolimits _{E_3}, \iota _1\times \mathop {\mathrm {id}}\nolimits _{E_2}\times \iota _3\rangle \). The quotient \((E_1\times E_2\times E_3)/G\) admits a crepant resolution Y which is a CalabiYau threefold. This is a very simple instance of the so called BorceaVoisin construction (see [12, 50]). This yields smooth projective CalabiYau threefolds as crepant resolutions of the quotient \((S\times E)/G\), where S is a K3 surface, E is an elliptic curve and G is the group generated by the automorphism \(f\times \iota \) of \(S\times E\), with f an antisymplectic involution on S and \(\iota \) the natural involution on E above. Example 2.32 in [34] is yet another instance of this circle of ideas.
Footnotes
Notes
Acknowledgments
The paper benefitted from many useful discussions with Aaron Bertram, Izzet Coskun, Alice Garbagnati, Bert van Geemen, Daniel Huybrechts, Martí Lahoz, Antony Maciocia, Eric Miles, Rahul Pandharipande, Dulip Piyaratne, Benjamin Schmidt, Yukinobu Toda, and we would like to thank all of them. The first author is particularly grateful to Dulip Piyaratne for many hours explaining the details of [29, 30], including a long session under the disguise of a PhD defense. We are grateful to the referee for a very careful reading of the manuscript. We also would like to thank for their hospitality the Ohio State University, the University of Bonn, and the University of Edinburgh, where parts of this paper were written. AB is supported by ERC starting Grant No. 337039 “WallXBirGeom”. EM is partially supported by the NSF Grants DMS1160466 and DMS1302730/DMS1523496. PS is partially supported by the Grant FIRB 2012 “Moduli Spaces and Their Applications” and by the National Research Project “Geometria delle Varietà Proiettive” (PRIN 201011).
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