Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces

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 deformationequivalent to the Hilbert scheme of length-3 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.

Extremal Rays and Automorphisms of Holomorphic Symplectic Varieties

“…We take (v, v) = 4. The case of Lagrangian P 3 's, where (a, a) = −2 and (a, v) = 2, was examined in [HHT12].…”

Mori cones of holomorphic symplectic varieties of K3 type

“…Further work on moving cones was presented in [HT09,Mar13], which built on Markman's analysis of monodromy groups. The characterization of extremal rays arising from Lagrangian projective spaces P n ֒→ X has been addressed in [HT09,HHT12] and [BJ14]. The paper [HT10] proposed a general framework describing all types of extremal rays; however, Markman found counterexamples in dimensions ≥ 10, presented in [BMT14].…”

Tautological classes on moduli spaces of hyper-Kähler manifolds

“…As G X is a connected component of the Zariski closure of Mon 2 (X) and Mon(X) → Mon 2 (X) has finite kernel, the monodromy action of Mon(X) also gives rise to an action of G X on H * (X) of X via automorphisms (cf. [24,Proposition 4.1]).…”