Rational Curves on Hyperkähler Manifolds

Amerik, Ekaterina; Verbitsky, Misha

doi:10.1093/imrn/rnv133

Cited by 35 publications

(82 citation statements)

References 36 publications

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“…In [2] we have seen that this implies some a priori stronger statements on the Hodge monodromy action.…”

Section: Theorem 213 ([1]) Suppose That the Picard Number ρ(M) > 3 mentioning

confidence: 74%

“…Geometrically, the MBM classes are characterized among negative integral (1, 1)-classes as those which are, up to a scalar multiple, represented by minimal rational curves on deformations of M under the identification of H 2 (M, Q) with H 2 (M, Q) given by the BBF form [2,3,13].…”

Section: Mbm Classesmentioning

confidence: 99%

“…The following theorems summarize the main results about MBM classes from [2]. In what follows, we shall also consider the positive cone in the algebraic part NS(M) ⊗ R of H One of the ample chambers is, obviously, the ample cone of M, hence the name.…”

Section: Mbm Classesmentioning

confidence: 99%

“…For reader's convenience, let us briefly sketch the proof (for details, see sections 3 and 6 of [2]). It consists in remarking that a face of a chamber is given by a flag P s ⊃ P s−1 ⊃ · · · ⊃ P 1 where P s is the supporting hyperplane (of dimension s = h 1,1 − 1), P s−1 supports a face of our face, etc., and for each P i an orientation ("pointing inwards the chamber") is fixed.…”

Section: Corollary 214 the Hodge Monodromy Group Has Only Finitely Mmentioning

confidence: 99%

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Hyperbolic geometry of the ample cone of a hyperkähler manifold

Amerik

Verbitsky

2016

Res Math Sci

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Let M be a compact hyperkähler manifold with maximal holonomy (IHS). The group H 2 (M, R) is equipped with a quadratic form of signature (3, b 2 − 3), called Bogomolov-Beauville-Fujiki form. This form restricted to the rational Hodge lattice H 1,1 (M, Q) has signature (1, k). This gives a hyperbolic Riemannian metric on the projectivization H of the positive cone in H 1,1 (M, Q). Torelli theorem implies that the Hodge monodromy group Γ acts on H with finite covolume, giving a hyperbolic orbifold X = H/Γ . We show that there are finitely many geodesic hypersurfaces, which cut X into finitely many polyhedral pieces in such a way that each of these pieces is isometric to a quotient P(M )/ Aut(M ), where P(M ) is the projectivization of the ample cone of a birational model M of M, and Aut(M ) the group of its holomorphic automorphisms. This is used to prove the existence of nef isotropic line bundles on a hyperkähler birational model of a simple hyperkähler manifold of Picard number at least 5 and also illustrates the fact that an IHS manifold has only finitely many birational models up to isomorphism (cf.

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“…In [2] we have seen that this implies some a priori stronger statements on the Hodge monodromy action.…”

Section: Theorem 213 ([1]) Suppose That the Picard Number ρ(M) > 3 mentioning

confidence: 74%

Section: Mbm Classesmentioning

confidence: 99%

Section: Mbm Classesmentioning

confidence: 99%

Section: Corollary 214 the Hodge Monodromy Group Has Only Finitely Mmentioning

confidence: 99%

See 2 more Smart Citations

Hyperbolic geometry of the ample cone of a hyperkähler manifold

Amerik

Verbitsky

2016

Res Math Sci

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“…This is the expected dimension of any family of rational curves on a (2k − 2)-dimensional hyperkähler manifold [Ra], whence (cf. [AV,Pf. of Cor.…”

Section: Introductionmentioning

confidence: 99%

Severi varieties and Brill–Noether theory of curves on abelian surfaces

Knutsen

Lelli-Chiesa

Mongardi

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Severi varieties and Brill-Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface S with polarization L of type (1, n), we prove nonemptiness and regularity of the Severi variety parametrizing δ-nodal curves in the linear system |L| for 0 ≤ δ ≤ n−1 = p −2 (here p is the arithmetic genus of any curve in |L|). We also show that a general genus g curve having as nodal model a hyperplane section of some (1, n)-polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many (1, n)-polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus g curve in S equigenerically to a nodal curve. The rest of the paper deals with the BrillNoether theory of curves in |L|. It turns out that a general curve in |L| is Brill-Noether general. However, as soon as the Brill-Noether number is negative and some other inequalities are satisfied, the locus |L|

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Rational Curves and MBM Classes on Hyperkähler Manifolds: A Survey

Amerik

Verbitsky

2021

Rationality of Varieties

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Rational Curves on Hyperkähler Manifolds

Cited by 35 publications

References 36 publications

Hyperbolic geometry of the ample cone of a hyperkähler manifold

Hyperbolic geometry of the ample cone of a hyperkähler manifold

Severi varieties and Brill–Noether theory of curves on abelian surfaces

Rational Curves and MBM Classes on Hyperkähler Manifolds: A Survey

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