Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces

2010/11/04 by David Harvey, Brendan Hassett, Harvey, David +3 · 1 citation
Mathematics · #11G05 #14C34 #14J32 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1011.1285

openalex publication_date 2010/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic-symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation-equivalent to the Hilbert scheme of length-three subschemes of a K3 surface. The class of the projective space in the cohomology ring has prescribed intersection properties, which translate into Diophantine equations. Possible homology classes correspond to integral points on an explicit elliptic curve; our proof entails showing that the only such point is two-torsion.

Citations

• Hodge theory and Lagrangian planes on generalized Kummer fourfolds

Cited by

• Lagrangian 4-planes in holomorphic symplectic varieties of K3^[4] type

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