On the homology of the moduli space of stable curves. (English) Zbl 0575.14024

Wolpert, Scott

Ann. Math. (2) 118, 491-523 (1983).

Stable curves were introduced by P. Deligne and D. Mumford in Publ. Math., Inst. Haut. Étud. Sci. 36 (1969), 75-109 (1970; Zbl 0181.488). Their moduli space \(\bar {\mathcal M}_ g\) is a compactification of the classical moduli space \({\mathcal M}_ g\) of Riemann surfaces. Both spaces have dimension 3g-3 and the locus \({\mathcal D}=\bar {\mathcal M}_ g- {\mathcal M}_ g\) is the sum of \(1+[g/2]\) divisors of \(\bar {\mathcal M}_ g\). These define homology classes in \(H_{6g-8}(\bar {\mathcal M}_ g;{\mathbb{Q}})\); the Weil-Peterson Kähler form \(\omega\) on \({\mathcal M}_ g\) extends to a closed form on \(\bar {\mathcal M}_ g\) and so defines, by Poincaré duality, another class in \(H_{6g-8}(\bar {\mathcal M}_ g;{\mathbb{Q}})\). It is proved in this paper that these \(2+[g/2]\) cycles essentially form a basis of \(H_{6g-8}(\bar {\mathcal M}_ g;{\mathbb{Q}})\), if \(g>2\) (theorem 5.1). From this result, the author deduces interesting consequences about the form \(\omega\). The main technique is the construction of \(2+[g/2]\) analytic 2-cycles in \(\bar {\mathcal M}_ g\) and the computation of their intersection pairing with the above \(2+[g/2]\) classes in \(H_{6g-8}({\mathcal M}_ g;{\mathbb{R}})\), which turns out to be a non-singular pairing. This technique for constructing analytic cycles can also be used to construct higher dimensional homology classes.

Reviewer: J.A.Seade

MSC:

14H10	Families, moduli of curves (algebraic)
14D20	Algebraic moduli problems, moduli of vector bundles
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14C99	Cycles and subschemes

Keywords:

moduli space of stable curves; Weil-Peterson Kähler form; analytic cycles