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Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves
 

Summary: Combinatorial and algebro-geometric cohomology classes
on the moduli spaces of curves
Enrico Arbarello1
and Maurizio Cornalba1,2
Dedicated to the memory of Claude Itzykson
Introduction
Given a compact Riemann surface C of genus g, n points on it, and n positive real
numbers (2g-2+n > 0), Strebel's theory of quadratic differentials [15] provides a canonical
way of dissecting C into n polygons and assigning lengths to their sides. As Mumford first
noticed, this can be used to give a combinatorial description of the moduli space Mg,n of
n-pointed smooth curves of given genus g. If one looks at moduli spaces from this point
of view, one can construct combinatorial cycles in them (cf. [7], for instance). It is then
natural to ask how these may be related to the algebraic geometry of moduli space. It
was first conjectured by Witten that the combinatorial cycles can be expressed in terms
of Mumford-Morita-Miller classes. The first result in this direction is due to Penner [13];
we will comment on his work at the end of section 2. As we shall briefly explain now, and
more extensively in section 3, our approach to the question has its origin in the papers [16],
[17] and [7] by Witten and Kontsevich. The combinatorial cycles we are talking about will
be denoted by the symbols Wm,n, where m = (m0, m1, m2, . . . ) is an infinite sequence of
non-negative integers, almost all zero, and n a positive integer. On the moduli space Mg,n

  

Source: Arbarello, Enrico - Dipartimento di Matematica, UniversitÓ di Roma "La Sapienza"

 

Collections: Mathematics