Symplectic Complex Spaces

Kirschner, Tim

doi:10.1007/978-3-319-17521-8_3

Cited by 1 publication

(2 citation statements)

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“…3.4.2] applies and first order locally trivial deformations are parametrised by H 1 (X − Sing X, Ω 1 X ) ≃ H 1,1 (X). Now [Ki,Thm. 3.4.4] provides the period map as stated above.…”

Section: Generalities On Ihs Manifoldsmentioning

confidence: 99%

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Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds

Knutsen

Lelli-Chiesa

Mongardi

2018

Trans. Amer. Math. Soc.

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Abstract. Rational curves on Hilbert schemes of points on K3 surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up to isometry, as dual divisors to such rational curves. The locus covered by the rational curves is then described, thus exhibiting algebraically coisotropic subvarieties. This provides strong evidence for a conjecture by Voisin concerning the Chow ring of irreducible holomorphic symplectic manifolds. Some general results concerning the birational geometry of irreducible holomorphic symplectic manifolds are also proved, such as a nonprojective contractibility criterion for wall divisors. IntroductionRational curves play a pivotal role in the study of the birational geometry and the Chow ring of algebraic varieties. The present paper concerns a specific class of varieties, namely, irreducible holomorphic symplectic (IHS) manifolds and, more precisely, Hilbert schemes of points on K3 surfaces and generalised Kummer manifolds (cf. §1), and is focused on some special rational curves arising from the Brill-Noether theory of normalisations of curves lying on K3 and abelian surfaces. In order to treat the two cases simultaneously, we introduce the following notation: we set ε = 0 (respectively, ε = 1) when S is a K3 (resp., abelian) surface, and we denote by S [k] ε the Hilbert scheme of k points on S when ε = 0 and the 2k-dimensional generalised Kummer variety on S when ε = 1.In the last few years, some classical results concerning (−2)-curves on K3 surfaces have been generalised to higher dimension and in particular it was shown that rational curves fully control the birational geometry of IHS manifolds. More precisely, Ran [Ra] proved that extremal rational curves can be deformed together with the ambient IHS manifold, and this was exploited by Bayer, Hassett and Tschinkel [BHT] in order to determine the structure of the ample cone. The same result was independently obtained by the third named author [Mo1] using intrinsic properties of IHS manifolds and a deformation invariant class of divisors, the so-called wall divisors (cf. Definition 2.2), which contains all divisors dual to extremal rays. and the space of stability conditions can be used towards computing their ample cones.In this paper we use Brill-Noether theory of nodal curves on abelian and K3 surfaces in order to exhibit rational curves in S [k] ε and describe, in many cases, the locus they cover. Our construction proceeds as follows. Let (S, L) be a general primitively polarized K3 or abelian surface of genus p := p a (L) and let C ∈ |L| be a δ-nodal curve whose normalization C has a linear series of type g 1 k+ε . Existence of a family of such curves having the expected dimension (and satisfying certain additional properties) has been proved in [CK, KLM] under suitable conditions on the triple (p, k, δ), cf. Theorem 3.1. Any pencil of degree k + ε on C defines a rational curve in S [k] ε , whose cla...

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“…3.4.2] applies and first order locally trivial deformations are parametrised by H 1 (X − Sing X, Ω 1 X ) ≃ H 1,1 (X). Now [Ki,Thm. 3.4.4] provides the period map as stated above.…”

Section: Generalities On Ihs Manifoldsmentioning

confidence: 99%

“…The latter is smooth by Theorem 1.1. After replacing X with a small locally trivial deformation, we can suppose that Def(X) lt is smooth, therefore [Ki,Cor. 3.4.2] applies and first order locally trivial deformations are parametrised by H 1 (X − Sing X, Ω 1 X ) ≃ H 1,1 (X).…”

Section: Generalities On Ihs Manifoldsmentioning

confidence: 99%