Connected components of strata of residueless meromorphic differentials (Q6623099)

The aim of this paper is to investigate the flat surfaces that arise from meromorphic differentials on compact Riemann surfaces with vanishing residues at each pole. Generalized strata of meromorphic differentials are loci in the usual strata of differentials, where certain sets of residues sum up to zero. They appear naturally in the boundary of the multi-scale compactification of the usual strata. Enumerating the connected components of generalized strata is necessary to understand the boundary complex of the multi-scale compactification. \N\NIn this paper, the author classifies connected components of the strata of residueless meromorphic differentials, which are the strata with the maximum possible number of conditions imposed on the residues of the poles. He proves that the connected components of strata of residueless meromorphic differentials are classified by the well-known topological invariants called hyperellipticity, spin parity, and rotation number. He proves that hyperellipticity, spin parity and rotation number still suffice to distinguish all connected components of the generalized strata, with few exceptional cases. \N\NThis work consists of the following parts : 1) Introduction. This section is an introduction to the subject and statement of results. 2) Flat surfaces and their deformations. In this section, the author gives basic definitions related to flat surfaces, and recalls the \(GL^+(2,\mathbb{R})\)-action on the strata and related concepts such as cores and polar domains. He also classifies zero-dimensional projectivized generalized strata. 3) The multi-scale compactification and degenerations of flat surfaces. Here, the author recalls the definition and the properties of the multi-scale compactification of the generalized strata, and describes how he can shrink a collection of parallel saddle connections using the contraction flow. 4) The principal boundary of residueless strata. In this section, the author recalls the definition of the principal boundary of strata and how a flat surface degenerates to the principal boundary. He explains that two surgeries introduced in [\textit{M. Kontsevich} and \textit{A. Zorich}, Invent. Math. 153, No. 3, 631--678 (2003; Zbl 1087.32010)] can be considered as smoothing processes from certain multi-scale differentials, and they can be reversed by degeneration into the principal boundary under certain conditions. \N\N5) Hyperelliptic components. Here, the author describes hyperelliptic components of the stratum and their principal boundary. 6) Multiplicity one saddle connections. In this section, the author proves the existence of a flat surface with a multiplicity one saddle connection for all connected component but hyperelliptic components with \(2g+2\) fixed marked points. 7) Genus one single-zero strata. Here, the author classifies the non-hyperelliptic components of genus one single-zero strata. 8) Classification of hyperelliptic components. In this section, the author classifies the hyperelliptic components of strata. 9) Genus one multiple-zero strata. Here, the author classifies the connected components of genus one multiple-zero strata. 10) Higher genus strata. In this section, the author classifies the non-hyperelliptic components of strata of higher genus.