Foliations on the moduli space of curves, vanishing in cohomology, and Calogero-Moser curves

Details Foliations on the moduli space of curves, vanishing in cohomology, and Calogero-Moser curves Foliations on the moduli space of curves, vanishing in cohomology, and Calogero-Moser curves

2011-08-21

arxiv.org/abs/1108.4211v1

Mathematics

Algebraic Geometry

Scientific paper

Using meromorphic differentials with real periods, we show that a certain tautological homology class on the moduli space of smooth algebraic curves of genus g vanishes. The vanishing of the entire tautological ring for degree g-1 and higher, part of Faber's conjecture, is known in both homology and Chow --- it was proven by Looijenga, Ionel, and Graber-Vakil, and the class that we show vanishes is just one such tautological class. However, our approach, motivated by the Whitham perturbation theory of soliton equations, is completely new, elementary in the sense that no techniques beyond elementary complex analysis are used, and also leads to a natural non-speciality conjecture, which would imply many more vanishing results and relations among tautological classes. In the course of the proof we define and study foliations of the moduli space of curves constructed using periods of meromorphic differentials, in a way providing for meromorphic differentials a theory similar to that developed for abelian differentials by Kontsevich and Zorich. In our setting we can construct local coordinates near any point of the moduli space, while for abelian differentials only coordinates along the strata with a fixed configuration of zeroes are known. The results we obtain are of independent interest for the study of singularities of solutions of the Whitham equations.

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