Prime exceptional divisors on holomorphic symplectic varieties and monodromy reflections

Let X be a K3 surface and E an irreducible curve on X. Then the following are well known to be equivalent: 1) E is a smooth rational curve. 2) E has self-intersection -2. 3) E is the exceptional divisor of a birational morphism from X to a normal projective surface Y with an isolated singular point. Furthermore, if e is a cohomology class of Hodge type (1,1) and self-intersection -2, then e=[E] or e=-[E] for an effective divisor E, and E becomes irreducible, under a generic small deformation of the pair (X,e). Thus, the condition (e,e)=-2 is a numerical characterization of irreducible exceptional divisors. We show that the above classical numerical characterization has an analogue for irreducible exceptional divisors on higher dimensional simply connected projective holomorphic-symplectic varieties. One of the key ingredients is the classification of Picard-Lefschetz monodromy involutions (the reflection with respect to the class [E] in the K3 case).