A global Torelli theorem for singular symplectic varieties

Abstract We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized K3[2]-type varieties, the polarization is base point free.

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FUJITA DECOMPOSITION AND HODGE LOCI

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748018000452 ◽

2018 ◽

Vol 19 (4) ◽

pp. 1389-1408 ◽

Cited By ~ 3

Author(s):

Paola Frediani ◽

Alessandro Ghigi ◽

Gian Pietro Pirola

Keyword(s):

Totally Geodesic ◽

Exterior Power ◽

Period Map ◽

Flat Part ◽

Torelli Theorem

This paper contains two results on Hodge loci in $\mathsf{M}_{g}$. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibers and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fiber is contained in a proper Hodge locus. The second result deals with divisors in $\mathsf{M}_{g}$. It is proved that the image under the period map of a divisor in $\mathsf{M}_{g}$ is not contained in a proper totally geodesic subvariety of $\mathsf{A}_{g}$. It follows that a Hodge locus in $\mathsf{M}_{g}$ has codimension at least 2.

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PROJECTIVITY CRITERION OF MOISHEZON SPACES AND DENSITY OF PROJECTIVE SYMPLECTIC VARIETIES

International Journal of Mathematics ◽

10.1142/s0129167x02001277 ◽

2002 ◽

Vol 13 (02) ◽

pp. 125-135 ◽

Cited By ~ 5

Author(s):

YOSHINORI NAMIKAWA

Keyword(s):

Kähler Manifold ◽

Projective Manifold ◽

Singular Case ◽

Isolated Singularities ◽

Rational Singularities ◽

Kahler Manifold ◽

Moishezon Space ◽

Symplectic Varieties

A Moishezon manifold is a projective manifold if and only if it is a Kähler manifold [13]. However, a singular Moishezon space is not generally projective even if it is a Kähler space [14]. Vuono [19] has given a projectivity criterion for Moishezon spaces with isolated singularities. In this paper we shall prove that a Moishezon space with 1-rational singularities is projective when it is a Kähler space (Theorem 1.6). We shall use Theorem 1.6 to show the density of projective symplectic varieties in the Kuranishi family of a (singular) symplectic variety (Theorem 2.4), which is a generalization of the result by Fujiki [4, Theorem 4.8] to the singular case. In the Appendix we give a supplement and a correction to the previous paper [15] where singular symplectic varieties are dealt with.

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The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces

Compositio Mathematica ◽

10.1112/s0010437x12000607 ◽

2013 ◽

Vol 149 (3) ◽

pp. 481-494 ◽

Cited By ~ 21

Author(s):

François Charles ◽

Eyal Markman

Keyword(s):

Hilbert Scheme ◽

K3 Surfaces ◽

Cohomology Algebra ◽

Hilbert Schemes ◽

Projective Varieties ◽

Hilbert Scheme Of Points ◽

Complex Projective ◽

Symplectic Varieties

AbstractWe prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky’s theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.

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