Counter-example to global Torelli problem for irreducible symplectic manifolds

Abstract

A simply connected compact Kaehler manifoldX is an irreducible symplectic manifold if there is an everywhere non-degenerate holomorphic 2-form Ω on X with H0(X,Ω2X) = C[Ω]. By definition, X has even complex dimension. There is a canonical symmetric form qX on H (X,Z), which is called the BeauvilleBogomolov form (cf. [Be]). On the other hand, since X is Kaehler, H(X,Z) has a natural Hodge structure of weight 2. If two irreducible symplectic manifolds X and Y are bimeromorphically equivalent, then there is a natural Hodge isometry between (H(X,Z), qX) and (H (Y,Z), qY ) (cf. [O, Proposition (1.6.2)], [Huy, Lemma 2.6]). Debarre [De] has constructed bimeromorphically equivalent irreducible symplectic manifolds X and Y such that X and Y are not isomorphic. This is a counter-example to the following problem.