Geometric families of 4-dimensional Galois representations with generically large images

We study compatible families of four-dimensional Galois representations constructed in the étale cohomology of a smooth projective variety. We prove a theorem asserting that the images will be generically large if certain conditions are satisfied. We only consider representations with coefficients in an imaginary quadratic field. We apply our result to an example constructed by Jasper Scholten (A non-selfdual 4-dimensional Galois representation, , 1999), obtaining a family of linear groups and one of unitary groups as Galois groups over . Research partially supported by MEC grant MTM2006-04895.