Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections
Abstract
Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville-Bogomolov pairing. A divisor E on X is called a prime exceptional divisor, if E is reduced and irreducible and of negative Beauville-Bogomolov degree. Let E be a prime exceptional divisor on X. We first observe that associated to E is a monodromy involution of the integral cohomology of X, which acts on the second cohomology lattice as the reflection by the cohomology class of E (Theorem 1.1). We then specialize to the case that X is deformation equivalent to the Hilbert scheme of length n zero-dimensional subschemes of a K3 surface. We determine the set of classes of exceptional divisors on X (Theorem 1.11). This leads to a determination of the closure of the movable cone of X.
Keywords
Cite
@article{arxiv.0912.4981,
title = {Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections},
author = {Eyal Markman},
journal= {arXiv preprint arXiv:0912.4981},
year = {2015}
}
Comments
v2: 53 pages, Latex. The main Conjecture 1.11 is now Theorem 1.11. Final version. To appear in KJM, Maruyama memorial volume