Moduli Spaces Of Meromorphic Differentials Research Articles

Moduli space of meromorphic differentials with marked horizontal separatrices

We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat geometry surfaces "near" the Deligne-Mumford boundary.We compute the number of connected components of the corresponding strata, and give a simple topological invariant that distinguishes them. In particular we see that for $g>0$, there are at most two such components, except in the hyperelliptic case.

Connected components of the strata of the moduli space of meromorphic differentials

We study the translation surfaces corresponding to meromorphic di fferentials on compact Riemann surfaces. Such geometric structures naturally appear when studying compactifications of the strata of the moduli space of Abelian di fferentials. We compute the number of connected components of the strata of the moduli space of meromorphic di fferentials. We show that in genus greater than or equal to two, one has up to three components with a similar description as the one of Kontsevich–Zorich for the moduli space of Abelian di fferentials. In genus one, one can obtain an arbitrarily large number of connected components that are distinguished by a simple topological invariant.

Baker–Akhiezer Spinor Kernel and Tau-functions on Moduli Spaces of Meromorphic Differentials

In this paper we study Baker-Akhiezer spinor kernel on moduli spaces of meromorphic differentials on Riemann surfaces. We introduce the Baker-Akhiezer tau-function which is related to both Bergman tau-function (which was studied before in the context of Hurwitz spaces and spaces of holomorphic and quadratic differentials) and KP tau-function on such spaces. In particular, we derive variational formulas of Rauch-Ahlfors type on moduli spaces of meromorphic differentials with prescribed singularities: we use the system of homological coordinates, consisting of absolute and relative periods of the meromorphic differential, and show how to vary the fundamental objects associated to a Riemann surface (the matrix of $b$-periods, normalized Abelian differentials, the Bergman bidifferential, the Szeg\"o kernel and the Baker-Akhiezer spinor kernel) with respect to these coordinates. The variational formulas encode dependence both on the moduli of the Riemann surface and on the choice of meromorphic differential (variation of the meromorphic differential while keeping the Riemann surface fixed corresponds to flows of KP type). Analyzing the global properties of the Bergman and Baker-Akhiezer tau-functions we establish relationships between various divisor classes on the moduli spaces.