Galois closure and Lagrangian varieties,

Let X be a complex projective variety and consider the cup product morphism \psi_k from the k-th exterior product of H^{1,0} to H^{k,0}. We use Galois closures of finite rational maps to introduce a new method for producing varieties such that \psi_k 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 g ≥ 2. The topological index of these surfaces is negative and this provides a counterexample to a conjecture on Lagrangian surfaces formulated by Barja-Naranjo-Pirola.