INTEGRAL CONSTRAINTS ON THE MONODROMY GROUP OF THE HYPERKÄHLER RESOLUTION OF A SYMMETRIC PRODUCT OF A K3 SURFACE

Source abstract

Let S [n] be the Hilbert scheme of length n subschemes of a K3 surface S. H 2 (S [n] ,ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL [H*(S [n] ,ℤ)] generated by monodromy operators, and let Mon 2 be its image in OH 2 (S [n] ,ℤ). We prove that Mon 2 is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon 2 does not surject onto OH 2 (S [n] ,ℤ)/(±1), when n - 1 is not a prime power. As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKähler varieties X deformation equivalent to S [n] . The weight 2 Hodge structure on H 2 (X,ℤ) does not determine the bimeromorphic class of X, whenever n - 1 is not a prime power (the first case being n = 7). There are at least 2 ρ(n - 1) - 1 distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n - 1) is the Euler number of n - 1. The second main result states, that if a monodromy operator acts as the identity on H 2 (S [n] ,ℤ), then it acts as the identity on H k (S [n] ,ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon 2 , if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6).