Local structure of closed symmetric 2-differentials

In the authors's previous work on symmetric differentials and their connection to the topological properties of the ambient manifold, a class of symmetric differentials was introduced: closed symmetric differentials ([BoDeO11] and [BoDeO13]). In this article we give a description of the local structure of closed symmetric 2-differentials on complex surfaces, with an emphasis towards the local decompositions as products of 1-differentials. We show that a closed symmetric 2-differential $w$ of rank 2 (i.e. defines two distinct foliations at the general point) has a subvariety $B_w\subset X$ outside of which $w$ is locally the product of closed holomorphic 1-differentials. The main result, theorem 2.6, gives a complete description of a (locally split) closed symmetric 2-differential in a neighborhood of a general point of $B_w$. A key feature of theorem 2.6 is that closed symmetric 2-differentials still have a decomposition as a product of 2 closed 1-differentials (in a generalized sense) even at the points of $B_w$. The (possibly multi-valued) closed 1-differentials can have essential singularities along $B_w$, but one still has a control on these essential singularities. The essential singularities come from exponentials of meromorphic functions acquiring poles along the irreducible components of $B_w$ of order bounded by the order of contact of the 2 foliations defined by the symmetric 2-differential along that irreducible component.