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.

v2: 53 pages, Latex. The main Conjecture 1.11 is now Theorem 1.11. Final version. To appear in KJM, Maruyama memorial volume

References in corpus (5)

• A survey of Torelli and monodromy results for holomorphic-symplectic varieties
• Singular symplectic moduli spaces
• The Kodaira dimension of the moduli of K3 surfaces
• Integral constraints on the monodromy group of the hyperkahler resolution of a symmetric product of a K3 surface
• Poisson deformations of affine symplectic varieties II

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• Contraction centers in families of hyperkahler manifolds
• Monodromy Invariants and Polarization Types of Generalized Kummer Fibrations
• On polarization types of Lagrangian fibrations
• Rational Hodge isometries of hyper-Kahler varieties of K3[n]-type are algebraic
• Computing Riemann-Roch polynomials and classifying hyper-Kähler fourfolds
• On symplectic birational self-maps of projective hyperkähler manifolds of K3${}^{[n]}$-type
• The geometry of antisymplectic involutions, I