On involutions with middle-dimensional fixed-point locus and holomorphic-symplectic manifolds

Abstract

Let $$g$$ be an involution of a compact closed manifold $$X$$ such that the fixed-point set $$X^{g}$$ is middle dimensional. Under the assumption that the normal bundle of the fixed-point set is either the tangent or co-tangent bundle conditions on the equivariant invariants of $$X$$ arise. In particular if $$X$$ is a holomorphic-symplectic manifold and $$g$$ an anti holomorphic-symplectic involution one arrives at a generalisation of Beauville’s result that for $$X$$ a hyper-Kahler manifold the $$\hat{A}$$ genus of $$X^{g}$$ is one.