Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.

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The geometry of moduli spaces of sheaves

K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

Lectures on K3 Surfaces

In this paper, we consider basic problems on moduli spaces of stable sheaves on abelian surfaces. Our main assumption is the primitivity of the associated Mukai vector. We determine the deformation types, albanese maps, Bogomolov factors and their weight 2 Hodge structures. We also discuss the deformation types of moduli spaces of stable sheaves on K3 surfaces.

https://link.springer.com/content/pdf/10.1007/s002080100255.pdf

Moduli spaces of stable sheaves on abelian surfaces

A cubic fourfold is a smooth cubic hypersurface of dimension four; it is special if it contains a surface not homologous to a complete intersection. Special cubic fourfolds form a countably infinite union of irreducible families 
$$C_d $$

 , each a divisor in the moduli space 
$$C $$

 of cubic fourfolds. For an infinite number of these families, the Hodge structure on the nonspecial cohomology of the cubic fourfold is essentially the Hodge structure on the primitive cohomology of a K3 surface. We say that this K3 surface is associated to the special cubic fourfold. In these cases, 
$$C_d $$

 is related to the moduli space 
$$N_d $$

 of degree d K3 surfaces. In particular, 
$$C $$

 contains infinitely many moduli spaces of polarized K3 surfaces as closed subvarieties. We can often construct a correspondence of rational curves on the special cubic fourfold parametrized by the K3 surface which induces the isomorphism of Hodge structures. For infinitely many values of d, the Fano variety of lines on the generic cubic fourfold of 
$$C_d $$

 is isomorphic to the Hilbert scheme of length-two subschemes of an associated K3 surface.

/pdf/special-cubic-fourfolds-36ijp91sw8.pdf

Special Cubic Fourfolds

Algebra & Number Theory

There are three types of “building blocks” in the Bogomolov decomposition [B, Th.2] of compact Kahlerian manifolds with torsion c1, namely complex tori, CalabiYau varieties, and irreducible symplectic manifolds. We are interested in the last type, i.e. simply-connected compact Kahlerian manifolds carrying a holomorphic symplectic form which spans H. (The holonomy of a Ricci-flat Kahler metric is equal to Sp(r), hence these manifolds are hyperkahler [B].) The stock of available irreducible symplectic manifolds appears to be quite scarce, expecially if we think of the many examples of CalabiYau’s. Every known irreducible symplectic manifold is a deformation of one of the following varieties: the Hilbert scheme parametrizing zero-dimensional subschemes of a K3 of fixed length [B], the generalized Kummer variety parametrizing zero-dimensional subschemes of a complex torus of fixed length and whose associated 0-cycle sums up to 0 [B], the (10-dimensional) desingularization of the moduli space of rank-two semistable torsion-free sheaves on a K3 with c1 = 0, c2 = 4 constructed by the author [O1]. Briefly: all known examples are deformations of an irreducible factor in the Bogomolov decomposition of a moduli space of semistable sheaves on a surface with trivial canonical bundle or, as in the last case, of a symplectic desingularization of such a moduli space. This paper provides a new example in dimension 6: the manifold in question is an irreducible factor in the Bogomolov decomposition of a symplectic desingularization of a moduli space of sheaves on an abelian surface. To put our result in perspective we recall some results on moduli spaces of sheaves on a surface with trivial canonical bundle. Let X be such a surface and D an ample divisor on it: given a vector w ∈ H∗(X;Z), we let Mw(X,D) be the moduli space of D-semistable torsion-free sheaves F on X with Mukai vector

A new six-dimensional irreducible symplectic variety

Moduli spaces of semistable torsion-free sheaves on a K3 surface $X$ are often holomorphic symplectic varieties, deformation equivalent to a Hilbert scheme parametrizing zero-dimensional subschemes of $X$. In fact this should hold whenever semistability is equivalent to stability. In this paper we study a typical "opposite" case, i.e. the moduli space $M_c$ of semistable rank-two torsion-free sheaves on $X$ with trivial determinant and second Chern class equal to an even number $c$. The moduli space $M_c$ always contains points corresponding to strictly semistable sheaves. If $c$ is at least 4, then $M_c$ is singular along the locus parametrizing strictly semistable sheaves, and on the smooth locus of $M_c$ there is a symplectic holomorphic form. Thus it is natural to ask whether there is a symplectic desingularization of $M_c$. We construct such a desingularization for $c=4$; in another paper we show that this desingularization gives a new deformation class of (K\"ahler) holomorphic irreducible symplectic varieties (of dimension ten). We also study the case $c>4$. We describe what should be an interesting desingularization, however we are not able to produce a symplectic one. In fact we suspect there is no symplectic smooth model of $M_c$ if $c>4$ (and even, of course).

Desingularized moduli spaces of sheaves on a K3, I

We prove that the weight-two Hodge structure of mod- uli spaces of torsion-free sheaves on a K3 surface is as described by Mukai (the rank is arbitrary but we assume the first Chern class is primitive). We prove the moduli space is an irreducible symplectic variety (by Mukai's work it was known to be symplectic). By work of Beauville, this implies that its H 2 has a canonical integral non- degenerate quadratic form; Mukai's recepee for H 2 includes a descrip- tion of Beauville's quadratic form. As an application we compute higher-rank Donaldson polynomials of K3 surfaces. Recently Jun Li (Li1,Li2) determined the stable rational Hodge structure on the n-th cohomol- ogy, for n ≤ 2, of moduli spaces of rank-two torsion-free sheaves on an arbitrary projective surface (stable means: for large enough dimension of the moduli space). It seems worthwile to study in greater detail the integral Hodge structure of these moduli spaces: in this paper we consider the case of a K3 surface. Mukai (M1,M2,M3) studied extensively moduli of sheaves on a K3: in (M2) he determined the Hodge structure of two-dimensional moduli spaces, and in (M3, Th. (5.15)) there is a beautiful description of the weight-two Hodge structure when the rank is at most two, and the dimension of the moduli space is greater than two. In this paper we prove that Mukai's description of H 2 is valid in arbitrary rank. (But notice that we assume the first Chern class is primitive.) Statement of the results. Let S be a projective K3 surface. The Mukai lattice (M2, § 2) consists of H ∗ (S;Z) endowed with the symmetric bilinear form h α, βi := − Z S α ∗ ∧ β , (1) where for α = α 0 + α 1 + α 2 ∈ H ∗ (S), with α i ∈ H 2i (S), we let α ∗ := α 0 − α 1 + α 2 . Setting F 0 H ∗ (S) := H ∗ (S), F 1 H ∗ (S) := H 0 (S) ⊕ F 1 H 2 (S) ⊕ H 4 (S), F 2 H ∗ (S) := F 2 H 2 (S) , (H ∗ (S;Z), F • ) becomes a weight-two Hodge structure, with the additional datum of Mukai's quadratic form defined above. If F is a sheaf on S, following Mukai we set

The weight-two hodge structure of moduli spaces of sheaves on A K3 surface

Eisenbud Popescu and Walter have constructed certain special sextic hypersurfaces in P 5 as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3 surface (K3) [2] and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type (1,1) - thus we get an example similar to that (discovered by Beauville and Donagi) of the Fano variety of lines on a cubic 4-fold. Conversely suppose that X is a numerical (K3) [2] , that H is an ample divisor on X of square 2 for Beauville’s quadratic form and that the map X 99K |H| _ is the composition of the quotient X ! Y for an anti-symplectic involution on X followed by an immersion Y ,! |H| _ ; then Y is an EPW-sextic and X ! Y is the natural double cover.

Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics

We prove that moduli spaces of torsion-free sheaves on a projective smooth complex surface are irreducible, reduced and of the expected dimension, provided the expected dimension is large enough. Actually we prove more: given a line bundle on the surface, we show that the number of moduli of sheaves which have a non-zero twisted (by the chosen line-bundle) endomorphism grows slower than the expected dimension of the moduli space (for fixed rank and increasing discriminant). The bounds we get are effective: we concentrate on the case of rank two and we give a lower bound on the discriminant guaranteeing that the moduli space is reduced of the expected dimension. We also give an effective irreducibility result for complete intersections. All of the above follows from a theorem bounding the dimension of complete subsets of the moduli space which do not intersect the boundary.

Kieran G. O'Grady

Papers

A new six-dimensional irreducible symplectic variety

Desingularized moduli spaces of sheaves on a K3, I

The weight-two hodge structure of moduli spaces of sheaves on A K3 surface

Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics

Moduli of vector bundles on projective surfaces: some basic results