Surveys in Differential Geometry

Volume 14 (2009)

The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces

Pages: 111 – 130

DOI: http://dx.doi.org/10.4310/SDG.2009.v14.n1.a4

Authors

Samuel Grushevsky (Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, U.S.A.)

Igor Krichever (Department of Mathematics, Columbia University, Broadway, New York, NY 10027, U.S.A.; Landau Institute for Theoretical Physics, Moscow, Russia.)

Abstract

We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define local coordinates and construct foliations on the moduli spaces. We use these constructions to give a new direct proof of the Diaz’ bound on the dimension of complete subvarieties of the moduli spaces. Geometrically, we study the properties of meromorphic differentials with real periods and their degenerations.

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