Stable Sheaf Research Articles

The cohomology of the general stable sheaf on a K3 surface

Let X be a K3 surface with Picard group Pic(X)≅ZH such that H2=2n. Let MH(v) be the moduli space of Gieseker semistable sheaves on X with Mukai vector v. We say that v satisfies weak Brill-Noether if the general sheaf in MH(v) has at most one nonzero cohomology group. We show that given any rank r≥2, there are only finitely many Mukai vectors of rank r on K3 surfaces of Picard rank one where weak Brill-Noether fails. We give an algorithm for finding the potential counterexamples and classify all such counterexamples up to rank 20 explicitly. Moreover, in each of these cases we calculate the cohomology of the general sheaf. Given r, we give sharp bounds on n, d, and a that guarantee that v satisfies weak Brill-Noether. As a corollary, we obtain another proof of the classification of Ulrich bundles on K3 surfaces of Picard rank one. In addition, we discuss the question of when the general sheaf in MH(v) is globally generated. Our techniques make crucial use of Bridgeland stability conditions.

Virtual counts on $\operatorname{Quot}$ schemes and the higher rank local DT/PT correspondence

We show that the Quot scheme $\text{Quot}_{\mathbf{A}^3}(\mathcal{O}^r,n)$ admits a symmetric obstruction theory, and we compute its virtual Euler characteristic. We extend the calculation to locally free sheaves on smooth $3$-folds, thus refining a special case of a recent Euler characteristic calculation of Gholampour-Kool. We then extend Toda's higher rank DT/PT correspondence on Calabi-Yau $3$-folds to a local version centered at a fixed slope stable sheaf. This generalises (and refines) the local DT/PT correspondence around the cycle of a Cohen-Macaulay curve. Our approach clarifies the relation between Gholampour-Kool's functional equation for Quot schemes, and Toda's higher rank DT/PT correspondence.

Tilt-stability, vanishing theorems and Bogomolov-Gieseker type inequalities

We investigate the tilt-stability of stable sheaves on projective varieties with respect to certain tilt-stability conditions depends on two parameters constructed by Bridgeland [11] (see also [1,6,5]). For a stable sheaf, we give effective bounds of these parameters such that the stable sheaf is tilt-stable. These allow us to prove new vanishing theorems for stable sheaves and an effective Serre vanishing theorem for torsion free sheaves. Using these results, we also prove Bogomolov-Gieseker type inequalities for the third Chern character of a stable sheaf on P3.

The ample cone of moduli spaces of sheaves on the plane

Fix a Chern character of a stable sheaf on the plane. Assume either the rank is at most 6 or the rank and first Chern class are coprime and the discriminant is sufficiently large. We use recent results of Bayer and Macri on Bridgeland stability to compute the ample cone of the moduli space of Gieseker semistable sheaves with the given Chern character. We recover earlier results, such as those by Stromme and Yoshioka, as special cases.

Spectral bundles and the DRY-Conjecture

Supersymmetric heterotic string models, built from a Calabi–Yau threefold X endowed with a stable vector bundle V, usually start from a phenomenologically motivated choice of a bundle Vv in the visible sector, the spectral cover construction on an elliptically fibered X being a prominent example. The ensuing anomaly mismatch between c2(Vv) and c2(X), or rather the corresponding differential forms, is often ‘solved’, on the cohomological level, by including a fivebrane. This leads to the question whether the difference can be alternatively realized by a further stable bundle. The ‘DRY’-conjecture of Douglas, Reinbacher and Yau in math.AG/0604597 gives a sufficient condition on cohomology classes on X to be realized as the Chern classes of a stable sheaf. In 1010.1644 [hep-th], we showed that infinitely many classes on X exist for which the conjecture is true. In this note, we give the sufficient condition for the mentioned fivebrane classes to be realized by a further stable bundle in the hidden sector. Using a result obtained in 1011.6246 [hep-th], we show that corresponding bundles exist, thereby confirming this version of the DRY-Conjecture.

Higher rank stable pairs on $K3$ surfaces

We define and compute higher rank analogs of Pandharipande– Thomas stable pair invariants in primitive classes for K3 surfaces. Higher rank stable pair invariants for Calabi–Yau threefolds have been defined by Sheshmani [26, 27] using moduli of pairs of the form O^n →F for F purely one-dimensional and computed via wall-crossing techniques. These invariants may be thought of as virtually counting embedded curves decorated with a (n − 1)- dimensional linear system. We treat invariants counting pairs O^n → E on a K3 surface for E an arbitrary stable sheaf of a fixed numerical type (“coherent systems” in the language of [16]) whose first Chern class is primitive, and fully compute them geometrically. The ordinary stable pair theory of K3 surfaces is treated by [22]; there they prove the KKV conjecture in primitive classes by showing the resulting partition functions are governed by quasimodular forms. We prove a “higher” KKV conjecture by showing that our higher rank partition functions are modular forms.

Moduli Spaces of Reflexive Sheaves of Rank 2

AbstractLet ℱ be a coherent rank 2 sheaf on a scheme Y ⊂ ℙn of dimension at least two and let X ⊂ Y be the zero set of a section σ ∈ H0(ℱ). In this paper, we study the relationship between the functor that deforms the pair (ℱ, σ) and the two functors that deform F on Y, and X in Y, respectively. By imposing some conditions on two forgetful maps between the functors, we prove that the scheme structure of e.g., the moduli scheme MY(P) of stable sheaves on a threefold Y at (ℱ), and the scheme structure at (X) of the Hilbert scheme of curves on Y become closely related. Using this relationship, we get criteria for the dimension and smoothness of MY(P) at (ℱ), without assuming Ext2(ℱ,ℱ) = 0. For reflexive sheaves on Y = ℙ3 whose deficiencymodule M = H*1 (ℱ) satisfies 0Ext2(M,M) = 0 (e.g., of diameter at most 2), we get necessary and sufficient conditions of unobstructedness that coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of H*0 (ℱ). Moreover, we show that every irreducible component of Mℙ3 (P) containing a reflexive sheaf of diameter one is reduced (generically smooth) and we compute its dimension. We also determine a good lower bound for the dimension of any component of Mℙ3 (P) that contains a reflexive stable sheaf with “small” deficiency module M.

Dominance of a rational map to the Coble quartic

We show the dominance of the restriction map from a moduli space of stable sheaves on the projective plane to the Coble sixfold quartic. With the dominance and the interpretation of a stable sheaf on the plane in terms of hyperplane arrangements, we expect these tools to reveal the geometry of the Coble quartic.

On the symplectic structures on moduli space of stable sheaves over a K3 or abelian surface and on Hilbert scheme of points

Fix a smooth very ample curve C on a K3 or abelian surface X. Let $ \mathcal{M} $ denote the moduli space of pairs of the form (F, s), where F is a stable sheaf over X whose Hilbert polynomial coincides with that of the direct image, by the inclusion map of C in X, of a line bundle of degree d over C, and s is a nonzero section of F. Assume d to be sufficiently large such that F has a nonzero section. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over X is a holomorphic 2-form on $ \mathcal{M} $. On the other hand, $ \mathcal{M} $ has a map to a Hilbert scheme parametrizing 0-dimensional subschemes of X that sends (F, s) to the divisor, defined by s, on the curve defined by the support of F. We prove that the above 2-form on $ \mathcal{M} $ coincides with the pullback of the symplectic form on the Hilbert scheme.

Sheaves on toric varieties for physics

In this paper we give an inherently toric description of a special class of sheaves (known as equivariant sheaves) over toric varieties, due in part to A. A. Klyachko. We apply this technology to heterotic compactifications, in particular to the (0,2) models of Distler, Kachru, and also discuss how knowledge of equivariant sheaves can be used to reconstruct information about an entire moduli space of sheaves. Many results relevant to heterotic compactifications previously known only to mathematicians are collected here -- for example, results concerning whether the restriction of a stable sheaf to a Calabi-Yau hypersurface remains stable are stated. We also describe substructure in the Kahler cone, in which moduli spaces of sheaves are independent of Kahler class only within any one subcone. We study F theory compactifications in light of this fact, and also discuss how it can be seen in the context of equivariant sheaves on toric varieties. Finally we briefly speculate on the application of these results to (0,2) mirror symmetry.