We show that the moduli space M X (v) of Gieseker stable sheaves on a smooth cubic threefold X with Chern character v = (3, −H, −H 2 /2, H 3 /6) is smooth and of dimension four. Moreover, the Abel-Jacobi map to the intermediate Jacobian of X maps it birationally onto the theta divisor Θ, contracting only a copy of X ⊂ M X (v) to the singular point 0 ∈ Θ.We use this result to give a new proof of a categorical version of the Torelli theorem for cubic threefolds, which says that X can be recovered from its Kuznetsov component Ku(X) ⊂ D b (X). Similarly, this leads to a new proof of the description of the singularity of the theta divisor, and thus of the classical Torelli theorem for cubic threefolds, i.e., that X can be recovered from its intermediate Jacobian.
Let C be a curve of genus g = 11 or g ≥ 13 on a K3 surface whose Picard group is generated by the curve class [C]. We use wall-crossing with respect to Bridgeland stability conditions to generalise Mukai's program to this situation: we show how to reconstruct the K3 surface containing the curve C as a Fourier-Mukai transform of a Brill-Noether locus of vector bundles on C.
Fix a Calabi-Yau 3-fold X satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the quintic 3-fold. We express Joyce's generalised DT invariants counting Gieseker semistable sheaves of any rank r ≥ 1 on X in terms of those counting sheaves of rank 0 and pure dimension 2.The basic technique is to reduce the ranks of sheaves by replacing them by the cokernels of their Mochizuki/Joyce-Song pairs and then use wall crossing to handle their stability.It has long been speculated that the higher rank or "nonabelian" DT theory of a Calabi-Yau 3-fold might be governed by its rank one "abelian" theory. 1 This would be a 6dimensional analogue of the correspondence between nonabelian Donaldson theory and abelian Seiberg-Witten theory for smooth 4-manifolds. Combined with the MNOP conjecture [MNOP], now proved for most Calabi-Yau 3-folds [PP], it would express DT theory in any rank entirely in terms of Gromov-Witten invariants. Contents 1. Weak stability conditions 4 2. Semistable objects are sheaves 6 3. Destabilising objects are sheaves 13 4. Enumerative invariants and wall crossing formulae 21 5. Crossing the walls 28 Appendix A. Joyce-Song pairs 31 Appendix B. Rank 2 simplifications 33 Appendix C. Moduli stacks 36 References 39Plan. We work on a smooth complex projective threefold X. In Section 1 we review the weak stability conditions and Bogomolov-Gieseker inequality of [BMT, BMS]. Section 2 proves that for such stability conditions, semistable objects E ∈ D(X) of class v n are always sheaves. Moreover in Section 3 we show that their destabilising objects are also sheaves, with one exception -on the Joyce-Song wall, E can be destabilised by an exact triangleexpressing it in terms of a Joyce-Song pair (2) up to tensoring by some T ∈ Pic 0 (X).From Section 4 we assume X is Calabi-Yau. We review the invariants counting its (weak) semistable sheaves and their wall crossing formulae in Section 4. Section 5 gives the proof of Theorem 1. It hides the role of Joyce-Song stable pairs so we explain their relevance in Appendix A, and stronger results in rank 2 in Appendix B. Neither is necessary for the rest of the paper. Appendix C uses work of Piyaratne-Toda to check that moduli stacks of weak semistable objects are algebraic of finite type.Outlook. Up until Section 4 all of our results apply to any 3-fold satisfying the Bogomolov-Gieseker inequality. From Section 4 we restrict attention to Calabi-Yau 3-folds only to apply the wall crossing formula. Joyce is currently developing a a wall crossing formula that should apply to Fano 3-folds [GJT, Conjecture 4.2]; we would then expect our results to prove a version of Theorem 1 with insertions.On Calabi-Yau 3-folds with h 1 (O X ) > 0 the locally free action of Jac(X) on moduli spaces of rank r > 0 sheaves forces J(v) = 0. So from Section 4 we assume h 1 (O X ) = 0 without loss of generality. But it would be interesting to extend our results to counts of sheaves of fixed determinant when h 1 (O X ) > 0; this should not be too hard because in this case the wall cro...
We prove a general criterion which ensures that a fractional Calabi--Yau category of dimension $\leq 2$ admits a unique Serre-invariant stability condition, up to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. We apply this result to the Kuznetsov component $\text{Ku}(X)$ of a cubic threefold $X$. In particular, we show that all the known stability conditions on $\text{Ku}(X)$ are invariant with respect to the action of the Serre functor and thus lie in the same orbit with respect to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As an application, we show that the moduli space of Ulrich bundles of rank $\geq 2$ on $X$ is irreducible, answering a question asked by Lahoz, Macr\`i and Stellari.
Let (X, O(1)) be a smooth polarised complex projective threefold. Gieseker and slope (semi)stability of sheaves will always be defined by H := c 1 (O(1)). We denote the bounded derived category of coherent sheaves on X by D(X).Fix β in the image of H 4 (X, Z) → H 4 (X, Q) and m ∈ Z.
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