Diff\'erentielles \`a singularit\'es prescrites

Abstract

We study the local invariants that a meromorphic $k$-differential on a Riemann surface of genus $g\geq0$ can have. These local invariants are the orders of zeros and poles, and the $k$-residues at the poles. We show that for a given pattern of orders of zeroes, there exists, up to a few exceptions, a primitive $k$-differential having these orders of zero. The same is true for meromorphic $k$-differentials and in this case, we describe the tuples of complex numbers that can appear as $k$-residues at their poles. For genus $g\geq2$, it turns out that every expected tuple appears as $k$-residues. On the other hand, some expected tuples are not the $k$-residues of a $k$-differential in some remaining strata. This happens in the quadratic case in genus $1$ and in genus zero for every $k$. We also give consequences of these results in algebraic and flat geometry.