Vanishing cohomology on a double cover

TL;DR

This paper proves the irreducibility of monodromy action on certain vanishing cohomology groups of double covers in algebraic geometry and explores rational maps from these covers to other surfaces.

Contribution

It establishes the irreducibility of monodromy on anti-invariant vanishing cohomology for double covers of hypersurfaces, combining classification, deformation, and Hodge theories.

Findings

Monodromy action is irreducible on anti-invariant vanishing cohomology.

Rational maps from double covers of surfaces of degree ≥7 in P^3 to other surfaces are studied.

Method integrates classification theory, deformation theory, and Hodge theory.

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 surface $S$ of degree$\geq 7$ in ${\mathbb P}^3$ branched at a very general quadric surface to smooth projective surfaces $Z$. Our method combines the classification theory of algebraic surfaces, deformation theory, and Hodge theory.