Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants

Abstract

A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We 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 geodesics. We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geodesics) which might be found on a generic flat surface S. We count the number of saddle connections of length less than L on a generic flat surface S; we also count the number of admissible configurations of pairs (triples,...) of saddle connections; we count the analogous numbers of configurations of families of closed geodesics. By the previous result of [EMa] these numbers have quadratic asymptotics <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>·</mml:mo> <mml:mo>(</mml:mo> <mml:mi>π</mml:mi> <mml:msup> <mml:mi mathvariant="normal">L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Here we explicitly compute the constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> for a configuration of every type. The constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> is found from a Siegel-Veech formula. To perform this computation we elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms and we find the numerical value of the normalized volume of the tubular neighborhood of the boundary. We use this for evaluation of integrals over the moduli space.