Abstract. We consider derived categories of coherent sheaves on smooth projective varieties. We prove that any equivalence between them can be represented by an object on the product. Using this, we give a necessary and sufficient condition for equivalence of derived categories of two K3 surfaces.

(i) Topology of embedded surfaces. Let X be a smooth, simply-connected 4-manifold, and ξ a 2-dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly

In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is motivated by the junction of Homological Mirror conjecture and geometric Strominger-Yau-Zaslow conjecture. In particular, we glue from “flat pieces ” an analytic K3 surface. As a byproduct of our approach we obtain an action of an arithmetic subgroup of the group SO(1,18) by piecewise-linear transformations on the 2-dimensional sphere S 2 equipped with naturally defined singular affine structure.

hep-th/9810210 utfa-98/26 spin-98/4

One main problem in the theory of irreducible holomorphic symplectic manifolds is the description of the ample cone in the Picard group. The goal of

For heterotic string theory compactified on T 6, we derive the complete set of T-duality invariants which characterize a pair of charge vectors (Q, P) labelling the electric and magnetic charges of the dyon. Using this we can identify the complete set of dyons to which the previously derived degeneracy formula can be extended. By going near special points in the moduli space of the theory we derive the spectrum of quarter BPS dyons in N = 4 supersymmetric gauge theory with simply laced gauge groups. The results are in agreement with those derived from

One of the dualities in string theory, the F-theory/heterotic string duality in eight dimensions [32], predicts an interesting correspondence between two seemingly disparate geometrical objects. On one side of the

Abstract. We give examples of non-isotrivial K3 surfaces over complex function fields with Zariski-dense rational points and Néron-Severi rank one. 1.

Let C denote a smooth projective curve of genus q over C, and S ′ ⊂ C a finite set of points, and f: X0 → C \ S ′ a smooth family of algebraic K3 surfaces, which extends to a family f: X → C with semi-stable singular fibres over S ′. Let S ⊂ S ′ denote the subset where the local monodromies of R2f∗ZX0 have infinite orders. Let ωX/C denote the dualizing sheaf. It is known that f∗ωX/C is ample on C by Fujita if f is not isotrivial [6] (See [9] [29] when the base of higher dimension). Jost and Zuo showed the following inequality [8]: (0.0.1) deg f∗ωX/C ≤ deg Ω 1 C(log S). If the iterated Kodaira-Spencer map of this family is zero, one shows then a stronger inequality deg f∗ωX/C ≤ 1

Abstract. A rational Lagrangian fibration f on an irreducible symplecitc variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov–Beauville square 0. This conjecture is proved in the case when V is the punctual Hilbert scheme of a generic algebraic K3 surface S. The construction of f uses a twisted Fourier–Mukai transform which induces an isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier–Mukai partner. According to Beauville [Beau-1], [Beau-2], the d-th symmetric power S (d) of a K3 surface S has a natural resolution of singularities, the punctual Hilbert scheme S [d] = Hilb d S, which is a 2d-dimensional irreducible