Stability conditions for generic K3 categories

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.

We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasimodular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by Gottsche-Shende. A formula for hyperelliptic curve counting in terms of Jacobi forms is proven (modulo a transversality statement). For abelian threefolds, complete conjectures in terms of Jacobi forms for the generating series of curve counts in primitive classes are presented. The base cases make connections to classical lattice counts of Debarre, Gottsche, and Lange-Sernesi. Further evidence is provided by Donaldson-Thomas partition function computations for abelian threefolds. A multiple cover structure is presented. The abelian threefold conjectures open a new direction in the subject.

We relate the Brauer group of a Kummer surface to the Brauer group of the corresponding abelian surface. For many pairs of elliptic curves over the rational numbers we prove that the Kummer surface attached to their product has trivial Brauer group.

We determine all complex K3 surfaces with Picard rank 20 over ℚ. Here the Neron–Severi group has rank 20 and is generated by divisors which are defined over ℚ. Our proof uses modularity, the Artin–Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell–Weil rank 18 over ℚ is impossible for an elliptic K3 surface. We apply our methods to general singular K3 surfaces, that is, those with Neron–Severi group of rank 20, but not necessarily generated by divisors over ℚ.

These notes are based on a series of talks given by the authors at the CIMPA Summer School on Algebraic Geometry and Hypergeometric Functions held in Istanbul in Summer of 2005. They provide an introduction to recent work on the complex ball uniformization of the moduli spaces of del Pezzo surfaces, K3 surfaces and algebraic curves of lower genus. We discuss the relationship of these constructions with the Deligne-Mostow theory of periods of hypergeometric differential forms. For convenience to a non-expert reader we include an introduction to the theory of periods of integrals on algebraic varieties with emphasis on abelian varieties and K3 surfaces.

Three new examples of 4-dimensional irreducible symplectic V-manifolds are constructed. Two of them are relative compactified Prymians of a family of genus-3 curves with involution, and the third one is obtained from a Prymian by Mukai's flop. They have the same singularities as two of Fujiki's examples, namely, 28 isolated singular points analytically equivalent to the Veronese cone of degree 8, but a different Euler number. The family of curves used in this construction forms a linear system on a K3 surface with involution. The structure morphism of both Prymians to the base of the family is a Lagrangian fibration in abelian surfaces with polarization of type (1,2). No example of such fibration is known on nonsingular irreducible symplectic varieties.

Let S be a smooth surface over the complex number field r An a u t o m o r p h i s m q~ of S is cohomologically trivial (resp. numerically trivial) if the induced a u t o m o r p h i s m q~* of the cohomology ring H * (S, 7/) (resp. H * (S, I1~)) is trivial. We denote by A o (S) (resp. A (S)) the quot ient o f the group o fcohomolog ica l ly (resp. numerically) trivial au tomorph i sms of S by its connected component . I t is known tha t A (S) is a finite g roup i f S is Kiihler, [4, 3]. A (S) is even trivial, for example, in the case of ra t ional surfaces, abelian surfaces and K3 surfaces, [9, 5]. In this article, we shall s tudy A (S) for an Enriques surface S. In this case, it is no more true that Ao (S) is always trivial. So far two examples are known.

A remarkable result of Peter O’Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville’s splitting principle, we formulate a conjectural Section Property which predicts that for smooth projective holomorphic symplectic varieties there exists such a section of algebra whose image contains all the Chern classes of the variety. In this paper, we investigate this property for (not necessarily symplectic) varieties with a Chow motive of abelian type. We introduce the notion of a symmetrically distinguished abelian motive and use it to provide a sufficient condition for a smooth projective variety to admit such a section. We then give a series of examples of varieties for which our theory works. For instance, we prove the existence of such a section for arbitrary products of varieties with Chow groups of finite rank, abelian varieties, hyperelliptic curves, Fermat cubic hypersurfaces, Hilbert schemes of points on an abelian surface or a Kummer surface or a K3 surface with Picard number at least 19, and generalized Kummer varieties. The latter cases provide evidence for the conjectural Section Property and exemplify the mantra that the motives of holomorphic symplectic varieties should behave as the motives of abelian varieties, as algebra objects.

In this article, we show that some semi-rigid μ-stable sheaves on a projective K3 surface X with Picard number 1 are stable under Bridgeland's stability condition. As a consequence of our work, we show that the special set U(X)⊂Stab(X) introduced by Bridgeland reconstructs X itself. This gives a sharp contrast to the case of an abelian surface.

The Picard group of S is generated by two effective divisors C and F with C = −2, F 2 = 0 and C · F = 1. It can be realized as an elliptic fibration over P with a unique section C, fibers F and λ = 2. Here λ = c1(π∗ω) is the first Chern class of the Hodge bundle π∗ω of the fibration π : S → P (see [H-M]). It is a standard result that the number of nodal fibers of an elliptic fibration are given by 12λ [H-M, p. 158]. So there are exactly 24 rational nodal curves in the linear series |F | for S general. This is the same special K3 surface used by Bryan and Leung in their counting of curves on K3 surfaces [B-L]. It is actually the attempt to understand their method that leads us to our proof. We will call a K3 surface with Picard lattice (1.1) a BL K3 surface. A BL K3 surface S lies on the boundary of the moduli space of K3 surfaces of genus g with C+gF as the corresponding primitive divisor. Every curve in the linear series |OS(C + gF )| is “totally reducible”, i.e., it consists of the −2 curve C and g elliptic “tails” attached to C. A curve D ∈ |OS(C + gF )| is the image of a stable rational map only if D = C ∪m1F1∪m2F2∪ ...∪m24F24, where F1, F2, ..., F24 are 24

These are notes of a course given at the 'school on moduli spaces' at the Newton Institute in January 2011. The abstract theory of stability conditions (due to Bridgeland and Douglas) on abelian and triangulated categories is developed via tilting and t-structures. Special emphasis is put on the bounded derived category of coherent sheaves on smooth projective varieties (in particular for curves and K3 surfaces). The lectures were targeted at an audience with little prior knowledge of triangulated categories and stability conditions but with an interest in vector bundles on curves.

We give a one-parameter family of Bridgeland stability conditions on the derived category of a smooth projective complex surface S and describe "wall-crossing behavior'' for objects with the same invariants as \mathcal O_C(H) when H generates Pic (S) and C \in |H| . If, in addition, S is a K3 or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgeland-stable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover a natural generalization of Thaddeus' stable pairs for curves embedded in the moduli spaces.

An object in the bounded derived category Db(X) of coherent sheaves on a complex projective K3 surface X is spherical if it is rigid and simple. Although spherical objects form only a discrete set in the moduli stack of complexes, they determine much of the structure of X and Db(X). Here we show that a stability condition on Db(X) is determined by the stability of spherical objects.

This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

We develop a mixed-characteristic version of the Mori-Mukai technique for producing rational curves on K3 surfaces. We reduce modulo p, produce rational curves on the resulting K3 surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K3 surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.