Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections

TL;DR

This paper studies prime exceptional divisors on holomorphic symplectic varieties, revealing their connection to monodromy involutions and classifying their types on certain deformation classes, which helps understand the movable cone.

Contribution

It establishes a link between prime exceptional divisors and monodromy reflections, and classifies these divisors on deformation equivalent Hilbert schemes of K3 surfaces.

Findings

Prime exceptional divisors induce monodromy involutions acting as reflections.

Classified exceptional divisors on deformation equivalent Hilbert schemes.

Determined the closure of the movable cone for these varieties.

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.