  We use Galois closures of finite rational maps between complex projective
varieties to introduce a new method for producing varieties such that the
holomorphic part of the cup product map has non-trivial kernel. We then apply
our result to the two-dimensional case and we construct a new family of
surfaces which are Lagrangian in their Albanese variety. Moreover, we analyze
these surfaces computing their Chern invariants, and proving that they are not
fibred over curves of genus greater than one.
