Vanishing cohomology on a double cover

Abstract

In this paper, we prove the irreducibility of the monodromy action on the anti-invariant part of the vanishing cohomology on a double cover of a very general element in an ample hypersurface of a complex smooth projective variety branched at an ample divisor. As an application, we study dominant rational maps from a double cover of a very general surfaceSof degree > 7inP3branched at a very general quadric surface to smooth projective surfacesZ. Our method combines the classification theory of algebraic surfaces, deformation theory, and Hodge theory.