On the -cohomology and the geometry of metrics on moduli spaces of curves
Lizhen JiUniversity of Michigan, Ann Arbor, USA
Steven ZuckerThe Johns Hopkins University, Baltimore, USA
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Let be the moduli space of algebraic curves of genus with punctures, which is a noncompact orbifold. Let denote its Deligne–Mumford compactification. Then admits a class of canonical Riemannian and Finsler metrics. We probe the analogy between (resp. Teichmüller spaces) with these metrics and certain noncompact locally symmetric spaces (resp. symmetric spaces of noncompact type) with their natural metrics. In this chapter, we observe that for all , the -cohomology of with respect to these Riemannian metrics that are complete can be identified with the (ordinary) cohomology of , and hence the -cohomology is the same for different values of . This suggests a “rank-one nature” of the moduli space from the point of view of -cohomology. On the other hand, the -cohomology of with respect to the incomplete Weil Petersson metric is either the cohomology of or that of itself, depending on whether or not. At the end of the chapter, we pose several natural problems on the geometry and analysis of these complete Riemannian metrics.