Galois closure and Lagrangian varieties

Let X be a complex projective variety and consider the morphismψk:⋀kH0(X,ΩX1)→H0(X,ΩXk). We use Galois closures of finite rational maps to introduce a new method for producing varieties such that ψk 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 in Barja et al. (2007) [3].