Mathematical Research Letters

Volume 18 (2011)

Number 3

Tau function and moduli of differentials

Pages: 447 – 458



D. Korotkin (Concordia University, 1455 de Maisonneuve West, Montreal, H3G 1M8 Quebec, Canada)

P. Zograf (Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023 Russia)


The tau function on the moduli space of generic holomorphic 1-differentials on complex algebraic curves is interpreted as a section of a line bundle on the projectivized Hodge bundle over the moduli space of stable curves. The asymptotics of the tau function near the boundary of the moduli space of generic 1-differentials is computed, and an explicit expression for the pullback of the Hodge class on the projectivized Hodge bundle in terms of the tautological class and the classes of boundary divisors is derived. This expression is used to clarify the geometric meaning of the Kontsevich-Zorich formula for the sum of the Lyapunov exponents associated with the Teichmüller flow on the Hodge bundle.

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