Consider the moduli space of pairs (C, ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.

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 c · (πL 2 ). Here we explicitly compute the constant c for a configuration of every type. The constant c 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.

1 2 ALEX ESKIN, MAXIM KONTSEVICH, AND ANTON ZORICH 4.1. Comparison of flat and hyperbolic geometry (after K. Rafi) 4.2. Flat pairs of pants 4.3. Geometric Compactification Theorem 4.4. The (δ, η)-thick-thin decomposition 4.5. Uniform bounds for the conformal factor 5. Analytic Riemann-Roch Theorem 5.1. Proof based on the results of J. Fay 5.2. Alternative proof based on results of A. Kokotov, D. Korotkin and P. Zograf 6. Relating flat and hyperbolic Laplacians by means of Polyakov formula 6.1. Polyakov formula revisited 6.2. Polyakov Formula applied to smoothed flat and hyperbolic metrics 6.3. Integration over a neighborhood of a cusp 7. Comparison of relative determinants of Laplace operators near the boundary of the moduli space 7.1. Admissible pairs of subsurfaces 7.2. Estimate for the thick part 7.3. Estimate for the thin part. 7.4. Proof of Theorem 11 8. Determinant of Laplacian near the boundary of the moduli space 8.1. Determinant of hyperbolic Laplacian near the boundary of the moduli space 8.2. Proof of Theorem 8 9. Cutoff near the boundary of the moduli space 9.1. Green's Formula and cutoff near the boundary 9.2. Restriction to cylinders of large modulus sharing parallel core curves 9.3. The Determinant of the Laplacian and the Siegel-Veech constant 10. Evaluation of Siegel-Veech constants 10.1. Arithmetic Teichmüller discs 10.2. Siegel-Veech constants for square-tiled surfaces Appendix A. Conjectural approximate values of individual Lyapunov exponents in small genera Appendix B. Square-tiled surfaces and permutations B.1. Alternative interpretation of Siegel-Veech constant for arithmetic Teichmüller discs B.2. Non varying phenomenon B.3. Global average Acknowledgments References