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 (Formula presented.) of degree (Formula presented.) in (Formula presented.) branched at a very general quadric surface to smooth projective surfaces (Formula presented.). Our method combines the classification theory of algebraic surfaces, deformation theory, and Hodge theory.
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