Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections

Details Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections

2009-12-29

arxiv.org/abs/0912.4981v1

Mathematics

Algebraic Geometry

62 pages. Latex

Scientific paper

Let X be a projective irreducible holomorphic symplectic variety. 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 [E] of E (Theorem 1.1). We then specialize to the case of X, which is deformation equivalent to the Hilbert scheme of length n zero-dimensional subschemes of a K3 surface, n>1. Let E be a prime exceptional divisor on X. We show that [E]=e or [E]=2e, for a class e satisfying the following numerical condition: e is an integral and primitive class of Beauville-Bogomolov degree -2 or 2-2n, and the class (e,-) in the dual lattice H^2(X,Z)^* is divisible by (e,e)/2 (Theorem 1.2). A class e satisfying this numerical condition is called monodromy-reflective. There are many examples of monodromy-reflective classes e of Hodge type (1,1), no multiple of which is effective. Instead, the reflection by e is induced by a birational involution of X. We introduce monodromy-invariants of monodromy-reflective classes e, and use them to formulate a conjectural numerical criterion for the effectiveness of e (Conjecture 1.11). We show that the Conjecture follows from a version of Torelli (Theorem 1.12).

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