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