Integral constraints on the monodromy group of the hyperkahler resolution of a symmetric product of a K3 surface

Details Integral constraints on the monodromy group of the hyperkahler resolution of a symmetric product of a K3 surface Integral constraints on the monodromy group of the hyperkahler resolution of a symmetric product of a K3 surface

2006-01-12

arxiv.org/abs/math/0601304v3

International Journal of Mathematics, volume 21, issue 2, 169-223 (2010)

Mathematics

Algebraic Geometry

Version 3: Latex, 54 pages. Expository changes

Scientific paper

10.1142/S0129167X10005957

Let M be a 2n-dimensional Kahler manifold deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface S. Let Mon be the group of automorphisms of the cohomology ring of M, which are induced by monodromy operators. The second integral cohomology of M is endowed with the Beauville-Bogomolov bilinear form. We prove that the restriction homomorphism from Mon to the isometry group O[H^2(M)] is injective, for infinitely many n, and its kernel has order at most 2, in the remaining cases. For all n, the image of Mon in O[H^2(M)] is the subgroup generated by reflections with respect to +2 and -2 classes. As a consequence, we get counter examples to a version of the weight 2 Torelli question, when n-1 is not a prime power.

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