Mukai Vector Research Articles

The second integral cohomology of moduli spaces of sheaves on K3 and Abelian surfaces

In this paper we study the second integral cohomology of moduli spaces of semistable sheaves on projective K3 surfaces. If S is a projective K3 surface, v a Mukai vector and H a polarization on S that is general with respect to v, we show that H2(Mv,Z) is a free Z-module of rank 23 carrying a pure weight-two Hodge structure and a lattice structure, with respect to which H2(Mv,Z) is Hodge isometric to the Hodge sublattice v⊥ of the Mukai lattice of S. Similar results are proved for Abelian surfaces.

Counting special Lagrangian classes and semistable mukai vectors for K3 surfaces

Counting special Lagrangian classes and semistable mukai vectors for K3 surfaces

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.

Derived categories of hyper-Kähler manifolds: extended Mukai vector and integral structure

We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-Kähler manifolds as well as an integral lattice associated to the derived category of hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.

Elliptic quintics on cubic fourfolds, O'Grady 10, and Lagrangian fibrations

For a smooth cubic fourfold Y, we study the moduli space M of semistable objects of Mukai vector 2λ1+2λ2 in the Kuznetsov component of Y. We show that with a certain choice of stability conditions, M admits a symplectic resolution M˜, which is a smooth projective hyperkähler manifold, deformation equivalent to the 10-dimensional examples constructed by O'Grady. As applications, we show that a birational model of M˜ provides a hyperkähler compactification of the twisted family of intermediate Jacobians associated to Y. This generalizes the previous result of Voisin [58] in the very general case. We also prove that M˜ is the MRC quotient of the main component of the Hilbert scheme of quintic elliptic curves in Y, confirming a conjecture of Castravet.

A note on a question of Markman

Let F be a vector bundle on a complex projective algebraic variety X. If F deforms along a first order deformation of X, its Mukai vector remains of Hodge type along this deformation. We prove an analogous statement for all polyvector fields, not only for those in H1(X,TX) corresponding to deformations of the complex structure. This answers a question of Markman. We also explore a Lie theoretic analogue of the statement above.

Two polarised K3 surfaces associated to the same cubic fourfold

AbstractFor infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarised K3 surfaces of degree d via their Hodge structures. For half of the d, each associated K3 surface (S, L) canonically yields another one, (Sτ, Lτ). We prove that Sτ is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3, L, d/6). We also explain for which d the Hilbert schemes Hilbn (S) and Hilbn (Sτ) are birational.

Birational geometry of moduli spaces of stable objects on Enriques surfaces

Using wall-crossing for K3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic Enriques surfaces for different stability conditions. As an application, we prove in the case of a Mukai vector of odd rank that they are birational to Hilbert schemes. The argument makes use of a new Chow-theoretic result, showing that moduli spaces on an Enriques surface give rise to constant cycle subvarieties of the moduli spaces of the covering K3.

Moduli spaces of bundles over nonprojective K3 surfaces

We study moduli spaces of sheaves over non-projective K3 surfaces. More precisely, if $v=(r,\xi,a)$ is a Mukai vector on a K3 surface $S$ with $r$ prime to $\xi$ and $\omega$ is a "generic" K\"ahler class on $S$, we show that the moduli space $M$ of $\mu_{\omega}-$stable sheaves on $S$ with associated Mukai vector $v$ is an irreducible holomorphic symplectic manifold which is deformation equivalent to a Hilbert scheme of points on a K3 surface. If $M$ parametrizes only locally free sheaves, it is moreover hyperk\"ahler. Finally, we show that there is an isometry between $v^{\perp}$ and $H^{2}(M,\mathbb{Z})$ and that $M$ is projective if and only if $S$ is projective.

Birational geometry of singular moduli spaces of O'Grady type

Following Bayer and Macrì, we study the birational geometry of singular moduli spaces M of sheaves on a K3 surface X which admit symplectic resolutions. More precisely, we use the Bayer–Macrì map from the space of Bridgeland stability conditions Stab(X) to the cone of movable divisors on M to relate wall-crossing in Stab(X) to birational transformations of M. We give a complete classification of walls in Stab(X) and show that every minimal birational model of M in the sense of the log minimal model program appears as a moduli space of Bridgeland semistable objects on X. An essential ingredient of our proof is an isometry between the orthogonal complement of a Mukai vector inside the algebraic Mukai lattice of X and the Néron–Severi lattice of M which generalises results of Yoshioka, as well as Perego and Rapagnetta. Moreover, this allows us to conclude that the symplectic resolution of M is deformation equivalent to the 10-dimensional irreducible holomorphic symplectic manifold found by O'Grady.

A note on the existence of stable vector bundles on Enriques surfaces

We prove the non-emptiness of $M_{H,Y}(v)$, the moduli space of Gieseker-semistable sheaves on an unnodal Enriques surface $Y$ with Mukai vector $v$ of positive rank with respect to a generic polarization $H$. This completes the chain of progress initiated by H. Kim in \cite{Kim98}. We also show that the stable locus $M^s_{H,Y}(v)\neq\varnothing$ for $v^2>0$. Finally, we prove irreducibility of $M_{H,Y}(v)$ in case $v^2=0$ and $v$ primitive.

The wall-crossing behavior for Bridgeland’s stability conditions on abelian and K3 surfaces

Abstract The wall-crossing behavior for Bridgeland’s stability conditions on the derived category of coherent sheaves on K3 or abelian surface is studied. We introduce two types of walls. One is called the wall for categories, where the t-structure encoded by stability condition is changed. The other is the wall for stabilities, where stable objects with prescribed Mukai vector may get destabilized. Some fundamental properties of walls and chambers are studied, including the behavior under Fourier–Mukai transforms. A wall-crossing formula of the counting of stable objects will also be derived. As an application, we will explain previous results on the birational maps induced by Fourier–Mukai transforms on abelian surfaces. These transformations turns out to coincide with crossing walls of certain property.

A note on singular moduli spaces of sheaves on K3 surfaces

This paper studies deformations and birational maps between singular moduli spaces of torsion free semistable sheaves with 2-divisible Mukai vectors on K3 surfaces. It is showed that when the greatest common divisor of the rank and the first Chern class is 2, two such moduli spaces of the same dimension can be connected by deformations and birational maps.

Factoriality Properties of Moduli Spaces of Sheaves on Abelian and K3 Surfaces

In this paper we study factoriality properties of some moduli spaces of semistable sheaves on abelian or projective K3 surface S. If v = 2w is a Mukai vector, w is primitive, w 2 = 2 and H is a generic polarization, let Mv(S,H) be the moduli space of H semistable sheaves on S with Mukai vector v. If S is abelian, we show that the fiber Kv(S,H) of the Albanese map of Mv(S,H) is 2 factorial. If S is K3, we show that Mv(S,H) is either locally factorial or 2 factorial, and we give an example of both cases.

Vector bundles on Fano varieties of genus ten

In this note we describe a unique linear embedding of a prime Fano 4-fold $$F$$ of genus 10 into the Grassmannian $$G(3,6)$$ . We use this to construct some moduli spaces of bundles on linear sections of $$F$$ . In particular the moduli space of bundles with Mukai vector $$(3,L,3)$$ on a generic polarized K3 surface $$(S,L)$$ of genus 10 is constructed as a double cover of $$\mathbf P ^2$$ branched over a smooth sextic.

Deformation of the O'Grady moduli spaces

In this paper we study moduli spaces of sheaves on an abelian or projective K3 surface. If S is a K3, v = 2w is a Mukai vector on S, where w is primitive and w(2) = 2, and H is a v-generic polarization on S, then the moduli space M-v of H-semistable sheaves on S whose Mukai vector is v admits a symplectic resolution (M) over tilde (v). A particular case is the 10-dimensional O'Grady example (M) over tilde (10) of an irreducible symplectic manifold. We show that (M) over tilde (v) is an irreducible symplectic manifold which is deformation equivalent to (M) over tilde (10) and that H-2 (M-v, Z) is Hodge isometric to the sublattice v(perpendicular to) of the Mukai lattice of S. Similar results are shown when S is an abelian surface.

Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups

In a series of our papers with Carlo Madonna (2002–2008), we described self-correspondences of a K3 surface over ℂ via moduli of sheaves with primitive isotropic Mukai vectors for the Picard number 1 or 2 of the K3 surfaces. Here we give a natural and functorial answer to the same problem for an arbitrary Picard number. As an application, we characterize, in terms of self-correspondences via moduli of sheaves, K3 surfaces with reflective Picard lattice, that is, when the automorphism group of the lattice is generated by reflections up to finite index. It is known since 1981 that the number of reflective hyperbolic lattices is finite. We also formulate some natural unsolved related problems.

The 2-factoriality of the O’Grady moduli spaces

The aim of this work is to show that the moduli space M10 introduced by O’Grady is a 2-factorial variety. Namely, M10 is the moduli space of semistable sheaves with Mukai vector v: = (2, 0, −2) in \({H^{ev}(X,\mathbb{Z})}\) on a projective K3 surface X. As a corollary to our construction, we show that the Donaldson morphism gives a Hodge isometry between \({v^{\perp}}\) (sublattice of the Mukai lattice of X) and its image in \({H^{2} (\widetilde{M}_{10}, \mathbb{Z})}\), lattice with respect to the Beauville form of the 10-dimensional irreducible symplectic manifold \({\widetilde{M}_{10}}\), obtained as symplectic resolution of M10. Similar results are shown for the moduli space M6 introduced by O’Grady to produce its 6-dimensional example of irreducible symplectic variety.

Fourier–Mukai transform on abelian surfaces

We study moduli spaces of stable sheaves on abelian surfaces whose Mukai vectors are related by a cohomological Fourier-Mukai transform. We show that there is a Fourier-Mukai transform inducing a birational map between them.

Explicit correspondences of a K3 surface with itself

Let be a K3-surface with a polarization of degree , . We consider the moduli space of sheaves over with a primitive isotropic Mukai vector . This space is again a K3-surface. In earlier papers, we gave necessary and sufficient conditions (in terms of the Picard lattice ) for and to be isomorphic. Here we show that these conditions imply the existence of an isomorphism between and which is a composite of certain universal geometric isomorphisms between moduli of sheaves over and Tyurin's geometric isomorphism between moduli of sheaves over and itself. It follows that a general K3-surface with is isomorphic to if and only if there is an isomorphism which is a composite of universal isomorphisms and Tyurin's isomorphism.