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Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants. (English) Zbl 1037.32013

A holomorphic 1-form on a compact Riemann surface \(S\) naturally defines a flat metric on \(S\) with cone-type singularities. The authors present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on \(S\) having the same direction and the same length as the initial one. A similar phenomenon is valid for the families of parallel closed geodesies.
The authors give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geodesies) which might be found on a generic flat surface \(S\). They compute the number of saddle connections of length less than \(L\) on a generic flat surface \(S\), the number of admissible configurations of pairs (triples, …) of saddle connections, and the analogous numbers of configurations of families of closed geodesies.
These numbers have quadratic asymptotics \(c\cdot(\pi\text{L}^2\)). Here the constant \(c\) is explicitly computed from a Siegel-Veech formula for a configuration of every type. To perform this computation the authors elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms, and find the numerical value of the normalized volume of the tubular neighborhood of the boundary. This is used for evaluation of integrals over the moduli space.

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F10 Compact Riemann surfaces and uniformization
30-06 Proceedings, conferences, collections, etc. pertaining to functions of a complex variable
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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