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172
Numerical characterization of the Kähler cone of a compact Kähler manifold
, 2001
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On the cobordism class of the Hilbert scheme of a surface
- J. Algebraic Geometry
"... Let S be a smooth projective surface and S [n] the Hilbert scheme of zerodimensional subschemes of S of length n. We proof that the class of S [n] in the complex cobordism ring depends only on the class of the surface itself. Moreover, we compute the cohomology and holomorphic Euler characteristics ..."
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Cited by 70 (4 self)
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Let S be a smooth projective surface and S [n] the Hilbert scheme of zerodimensional subschemes of S of length n. We proof that the class of S [n] in the complex cobordism ring depends only on the class of the surface itself. Moreover, we compute the cohomology and holomorphic Euler characteristics of certain tautological sheaves on S [n] and prove results on the general structure of certain integrals over polynomials in Chern classes of tautological sheaves. Let S be a smooth projective surface over the field of complex numbers. For a nonnegative integer n let S [n] denote the Hilbert scheme parameterizing zerodimensional subschemes of length n. By a well-known result of Fogarty [10] the scheme S [n] is smooth and projective of dimension 2n, and is irreducible if S is irreducible. Let Ω = Ω U ⊗ Q be the complex cobordism ring with rational coefficients. Milnor [20] showed that Ω is a polynomial ring freely generated by the cobordism classes [CPi] for i ∈ N. For a smooth and projective complex surface we define
Counting rational curves on K3 surfaces
"... The aim of these notes is to explain the remarkable formula found by Yau and Zaslow [Y-Z] to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families (Fg)g≥1; a surface in Fg admits a g-dimensional linear system of curves of genus g. A naïve cou ..."
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Cited by 65 (0 self)
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The aim of these notes is to explain the remarkable formula found by Yau and Zaslow [Y-Z] to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families (Fg)g≥1; a surface in Fg admits a g-dimensional linear system of curves of genus g. A naïve count of constants suggests
A survey of Torelli and monodromy results for holomorphic-symplectic varieties
- In Complex and differential geometry. Conference held at Leibniz Universität
, 2009
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A global Torelli theorem for hyperkähler manifolds, arXiv:0908.4121. version 6.5
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MMP FOR MODULI OF SHEAVES ON K3S VIA WALL-CROSSING: NEF AND MOVABLE CONES, LAGRANGIAN FIBRATIONS
"... ABSTRACT. We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: (a) We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. (b) ..."
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Cited by 35 (2 self)
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ABSTRACT. We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: (a) We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. (b) We establish a long-standing conjecture that predicts the existence of a birational Lagrangian fibration on M whenever M admits an integral divisor class D of square zero (with respect to the Beauville-Bogomolov form). These results are proved using a natural map from the space of Bridgeland stability conditions Stab(X) to the cone Mov(X) of movable divisors on M; this map relates wall-crossing in Stab(X) to birational transformations of M. In particular, every minimal model of M appears as a moduli space of Bridgeland-stable objects on X. CONTENTS
The Kähler cone of a compact hyperkähler manifold
- Math. Ann
"... This paper is a sequel to [12]. We study a number of questions only touched upon in [12] in more detail. In particular: What is the relation between two birational compact hyperkähler manifolds? What is the shape of the cone of all Kähler classes on such a manifold? How can the birational Kähler con ..."
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Cited by 35 (2 self)
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This paper is a sequel to [12]. We study a number of questions only touched upon in [12] in more detail. In particular: What is the relation between two birational compact hyperkähler manifolds? What is the shape of the cone of all Kähler classes on such a manifold? How can the birational Kähler cone be described? Most of the results are motivated by either the well-established two-dimensional theory, i.e. the theory of K3 surfaces, or the theory of Calabi-Yau threefolds and string theory. As was pointed out by D. Kaledin, the proof of the projectivity criterion in [12] contains a serious mistake (from dimension four on). This criterion asserts that a compact hyperkähler manifold is projective if and only if its positive cone contains an intergral class. It is used in a crucial way in many of the arguments in [12] and of this paper. As I have been unable to fix the proof of the projectivity criterion, almost all the results formulated here and many of those in [12] are, for the moment, not fully proved. Warning: All the results of Sect. 1-5 depend on the projectivity criterion [12, Thm. 3.11] and are, therefore, conjectural!!! I have decided to post this prepint despite its conjectural character, because, first of all, it
Projectivity and birational geometry of Bridgeland moduli spaces
, 2012
"... ABSTRACT. We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby gene ..."
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Cited by 34 (2 self)
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ABSTRACT. We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wall-crossing for Bridgeland-stability and the minimal model program for the moduli space. We give three applications of our method for classical moduli spaces of sheaves on a K3 surface: 1. We obtain a region in the ample cone in the moduli space of Gieseker-stable sheaves only depending on the lattice of the K3. 2. We determine the nef cone of the Hilbert scheme of n points on a K3 surface of Picard rank one when n is large compared to the genus. 3. We verify the “Hassett-Tschinkel/Huybrechts/Sawon ” conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.
Moduli spaces of irreducible symplectic manifolds
, 2008
"... We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3 [2] manifolds with polarisation of degree 2d and split type is of general type if d ..."
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Cited by 31 (7 self)
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We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3 [2] manifolds with polarisation of degree 2d and split type is of general type if d ≥ 12. MSC 2000: 14J15, 14J35, 32J27, 11E25, 11F55 0
