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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 wellknown 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 [YZ] 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 gdimensional 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 [YZ] 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 gdimensional linear system of curves of genus g. A naïve count of constants suggests
A survey of Torelli and monodromy results for holomorphicsymplectic 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 WALLCROSSING: NEF AND MOVABLE CONES, LAGRANGIAN FIBRATIONS
"... ABSTRACT. We use wallcrossing 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 wallcrossing 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 longstanding 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 BeauvilleBogomolov 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 wallcrossing in Stab(X) to birational transformations of M. In particular, every minimal model of M appears as a moduli space of Bridgelandstable 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 wellestablished twodimensional theory, i.e. the theory of K3 surfaces, or the theory of CalabiYau 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. 15 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 wallcrossing for Bridgelandstability 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 Giesekerstable 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 “HassettTschinkel/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