Characteristic numbers of compact hyper-Kähler manifolds are expressed in graphtheoretical form, considering them as a special case of the curvature invariants introduced by L. Rozansky and E. Witten. The appropriate graphs are generated by "wheels," and the recently proved Wheeling theorem is used to give a formula for the L 2 -norm of the curvature of an irreducible hyper-Kähler manifold in terms of the volume and Pontryagin numbers. The formula involves the multiplicative sequence that is the square root of theÂ-polynomial.

Let Hilb g S be the Hilbert scheme of g points on a K3 surface S. Suppose that PicS ∼ = ZC where C is a smooth curve with C 2 = 2(g − 1)n 2 . We prove that Hilb g S is a Lagrangian fibration.

We apply the methods of Cȃldȃraru to construct a twisted Fourier-Mukai transform between a pair of holomorphic symplectic four-folds. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a certain four-fold and the derived category of twisted sheaves on its 'mirror' partner. As corollaries, we show that the two spaces are connected by a one-parameter family of deformations through Lagrangian fibrations, and we extend the original Fourier-Mukai transform to degenerations of abelian surfaces. * 2000 Mathematics Subject Classification. 14J60; 14D06; 18E30; 53C26. and Thaddeus' examples via the construction of Donagi, Ein, and Lazarsfeld [15] (namely, the compactified Hitchin system is a degeneration of the Beauville-Mukai system).The paper is organized as follows. In Section 2 we review results of Mukai, Bridgeland, Maciocia, and Cȃldȃraru on Fourier-Mukai and twisted Fourier-Mukai transforms. In Section 3 we introduce a pair of holomorphic symplectic four-folds which are fibred by abelian varieties and collect together some results about them. In Section 4 we construct a twisted Fourier-Mukai transform relating the derived category and twisted derived category of the pair of four-folds from Section 3. This is followed by our applications. This paper was begun during a visit to the Institut des HautesÉtudes Scientifiques and finished at the University of Kyoto; the author is grateful for the hospitality he received at both those places. The author has benefited from conversations with many people on the topics presented here: he thanks them all, particularly

For a prime p, we study subgroups of order p of the Brauer group Br(S) of a general complex polarized K3 surface of degree 2d, generalizing earlier work of van Geemen. These groups correspond to sublattices of index p of the transcendental lattice TS of S; we classify these lattices up to isomorphism using Nikulin's discriminant form technique. We then study geometric realizations of p-torsion Brauer elements as Brauer-Severi varieties in a few cases via projective duality. We use one of these constructions for an arithmetic application, giving new kinds of counter-examples to weak approximation on K3 surfaces of degree two, accounted for by transcendental Brauer-Manin obstructions.