Abelian differentials with prescribed singularities (Q2050438)

Let \(X\) be a closed Riemann surface of genus \(g\) and let \(K_{X}\) be its canonical line bundle. An abelian differential on \(X\) is a meromorphic section of \(K_{X}\). It is well known that the local inviariants of an abelian differential \(\omega\) at a point \(p\) are the order of \(\omega\) at \(p\), and the residue at \(p\), if \(p\) is a pole. These local invariants are subject to some local and global constraints: \begin{itemize} \item the residue at a simple pole is always non-zero; \item the sum of all residues is equal to zero; \item the sum of the orders of the poles and the zeros is equal to \(2g-2\). \end{itemize} This paper answers the following natural question: can we find an abelian differential with prescribed orders of zeros and poles and prescribed residues at the poles, if all above constraints are satisfied? More formally, consider a partition of \(2g-2\) of the form \[ \mu=(a_{1}, \dots, a_{n}, -b_{1}, \dots, -b_{p}, -1, \dots, -1) \] where \(a_{i}\) are positive integers, \(b_{i}\) are greater than or equat to \(2\) and the number of \(-1\) is equal to \(s\). The stratum of abelian differentials of type \(\mu\) is denoted by \(\Omega\mathcal{M}_{g}(\mu)\) and consists of all abelian differentials over a closed Riemann surface of genus \(g\) having zeros of order \(a_{i}\), \(s\) simple poles and higher order poles of order \(b_{i}\). The possible residues of an abelian differential of order \(\mu\) belong to the set \[ \mathcal{R}_{g}(\mu)=\{ (r_{1}, \dots, r_{p+s}) \in \mathbb{C}^{p}\times (\mathbb{C}^{*})^{s} \ | \ r_{1}+\cdots +r_{p+s}=0\} \ . \] The main result of the paper shows what configuration of residues can be realized: Theorem 1.1. Let \(g\geq 1\). The map \(\Omega\mathcal{M}_{g}(\mu) \rightarrow \mathcal{R}_{g}(\mu)\) associating to each abelian differential its residues is surjective when restricted to each connected component of the stratum \(\Omega\mathcal{M}_{g}(\mu)\). In genus \(g=0\), the map \(\Omega\mathcal{M}_{0}(\mu) \rightarrow \mathcal{R}_{0}(\mu)\) is not surjective in general and two other cases may occur: 1. if \(s=0\) and there is an index \(i\) such that \[ a_{i} > \sum_{j=1}^{p} b_{j} - (p+1) \ , \] then the image of the map is \(\mathcal{R}_{0}(\mu) \setminus \{0\}\). 2. if \(s\geq 2\) and \(p=0\), then the image of the map is the complement of the union of the planes \(\mathbb{C}^{*}(x_{1}, \dots, x_{s_{1}}, -y_{1}, \dots, -y_{s_{2}})\), where \(x_{i}, y_{j} \in \mathbb{N}\) are coprime and \[ \sum_{i=1}^{s_{1}} a_{i} = \sum_{j=1}^{s_{2}} b_{j} \leq \max(a_{1}, \dots, a_{n}) \ . \]