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23
Equivalences of derived categories and K3 surfaces
, 1996
"... 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. ..."
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Cited by 91 (6 self)
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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.
Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional 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 ..."
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Cited by 68 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional 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
Affine structures and nonarchimedean analytic spaces
"... In this paper we propose a way to construct an analytic space over a nonarchimedean 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 StromingerYauZaslow conjectu ..."
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Cited by 35 (3 self)
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In this paper we propose a way to construct an analytic space over a nonarchimedean 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 StromingerYauZaslow 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 piecewiselinear transformations on the 2dimensional sphere S 2 equipped with naturally defined singular affine structure.
Yu.: Rational curves on holomorphic symplectic fourfolds
 Geom. Funct. Anal
, 2001
"... 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 ..."
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Cited by 18 (7 self)
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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
Instanton strings and HyperKahler geometry
 Nucl. Phys
, 1999
"... hepth/9810210 utfa98/26 spin98/4 ..."
Abelian fibrations and rational points on symmetric products
"... Given a variety over a number field, are its rational points potentially dense, i.e., does there exist a finite extension over which rational points are Zariski dense? We study the question of potential density for symmetric products of surfaces. Contrary to the situation for curves, rational points ..."
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Cited by 17 (5 self)
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Given a variety over a number field, are its rational points potentially dense, i.e., does there exist a finite extension over which rational points are Zariski dense? We study the question of potential density for symmetric products of surfaces. Contrary to the situation for curves, rational points are not necessarily potentially dense on a sufficiently high symmetric product. Our main result is that rational points are potentially dense for the Nth symmetric product of a K3 surface, where N is explicitly determined by the geometry of the surface. The basic construction is that for some N, the Nth symmetric power of a K3 surface is birational to an abelian fibration over P N. It is an interesting geometric problem to find the smallest N with this property. 1
Duality Orbits, Dyon Spectrum and Gauge Theory Limit of Heterotic String Theory
 on T 6 ,” JHEP 0803 (2008) 022 arXiv:0712.0043 [hepth
"... For heterotic string theory compactified on T 6, we derive the complete set of Tduality 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 degener ..."
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Cited by 16 (9 self)
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For heterotic string theory compactified on T 6, we derive the complete set of Tduality 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
Moving and ample cones of holomorphic symplectic fourfolds
, 2007
"... Abstract. We analyze the ample and moving cones of holomorphic symplectic manifolds, in light of recent advances in the minimal model program. As an application, we establish a numerical criterion for ampleness of divisors on fourfolds deformationequivalent to punctual Hilbert schemes of K3 surfaces ..."
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Cited by 9 (5 self)
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Abstract. We analyze the ample and moving cones of holomorphic symplectic manifolds, in light of recent advances in the minimal model program. As an application, we establish a numerical criterion for ampleness of divisors on fourfolds deformationequivalent to punctual Hilbert schemes of K3 surfaces. 1.
INTERSECTION NUMBERS OF EXTREMAL RAYS ON HOLOMORPHIC SYMPLECTIC VARIETIES
, 909
"... Suppose X is a smooth projective complex variety. Let N1(X, Z) ⊂ H2(X, Z) and N 1 (X, Z) ⊂ H 2 (X, Z) denote the group of curve classes modulo homological equivalence and the NéronSeveri group respectively. The monoids of effective classes in each group generate cones ..."
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Cited by 9 (3 self)
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Suppose X is a smooth projective complex variety. Let N1(X, Z) ⊂ H2(X, Z) and N 1 (X, Z) ⊂ H 2 (X, Z) denote the group of curve classes modulo homological equivalence and the NéronSeveri group respectively. The monoids of effective classes in each group generate cones
Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces
 Manuscripta Math
"... 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 ..."
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Cited by 7 (1 self)
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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 [Beau1], [Beau2], the dth 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 2ddimensional irreducible