Antisymplectic involutions of holomorphic symplectic manifolds

Let X be a holomorphic symplectic manifold, of dimension divisible by 4, and s an antisymplectic involution of X . The fixed locus F of s is a Lagrangian submanifold of X ; we show that its \^A-genus is 1. As an application, we determine all possibilities for the Chern numbers of F when X is a deformation of the Hilbert square of a K3 surface.Comment: Final version, accepted for publication by the Journal of Topolog

Counting rational curves on K3 surfaces

The aim of these notes is to explain the remarkable formula found by Yau and Zaslow to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families F(g) (g>0); a surface in F(g) admits a g-dimensional linear system of curves of genus g. Such a system contains a positive number, say n(g), of rational (highly singular) curves. The formula is \sum n(g) q^g = q/D((q), where D(q) = q \prod (1-q^n)^{24} is the well-known modular form of weight 12.Comment: Plain TeX, 11 page

Some surfaces with maximal Picard number

For a smooth complex projective variety, the rank of the N\'eron-Severi group is bounded by the Hodge number h^{1,1}. Varieties with rk NS = h^{1,1} have interesting properties, but are rather sparse, particularly in dimension 2. We discuss in this note a number of examples, in particular those constructed from curves with special Jacobians.Comment: Some comments and references adde