Local structure of closed symmetric 2-differentials

TL;DR

This paper characterizes the local structure of closed symmetric 2-differentials on complex surfaces, showing they decompose into products of 1-differentials with controlled singularities outside a specific subvariety.

Contribution

It provides a complete local description of closed symmetric 2-differentials, including their decomposition into 1-differentials even at singular points, extending previous work on their topological properties.

Findings

Closed symmetric 2-differentials decompose into products of 1-differentials outside a subvariety.

At general points, these differentials have a local product structure.

Essential singularities of the 1-differentials are controlled and related to the geometry of the subvariety.

Abstract

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.