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Moduli spaces of curves and representation theory. (English) Zbl 0647.17010
Let $$d$$ be the Lie algebra of holomorphic vector fields on $${\mathbb C}^{\times}$$; it acts on the space $$V_ n$$ of holomorphic differentials of degree $$n$$ on $${\mathbb C}^{\times}$$. This gives a homomorphism $$d\to a_{\infty}$$, the Lie algebra of “restricted” endomorphisms of $$V_ n$$. The universal central extension of $$a_{\infty}$$ pulls back to $$(6n^ 2-6n+1)$$-times that of $$d$$.
On the other hand, if $$\pi: {\mathbb C}\to S$$ is a family of genus $$g$$ compact Riemann surfaces, $$\omega$$ the relative dualizing sheaf of $$\pi$$, $$\lambda_ n$$ the determinant line bundle of $$\omega^ n$$, then D. Mumford [Enseign. Math., II. Sér. 23, 39–110 (1977; Zbl 0363.14003)] has shown that $$c_ 1(\lambda_ n)=(6n^ 2-6n+1)c_ 1(\lambda_ 1).$$
The authors explain this coincidence by using Kodaira-Spencer deformation theory to produce a homomorphism $$d\to \text{Vect}(\widehat M_ g)$$, the vector fields on the moduli space of curves of genus $$g$$ with a basepoint $$p$$ and a local parameter near $$p$$. This implies that the tangent bundle $$T(\hat M_ g)$$ is a quotient of the trivial bundle with fiber $$d$$. This gives a homomorphism $$H^ 2(d)\to H^ 1(\Omega^ 1_{\hat M_ g})$$ and from this one deduces the coincidence of the two formulas.
The authors further deduce an isomorphism $$H^ 2(d)\to H^ 2(M_ g,{\mathbb C})$$ for $$g\geq 3$$, where $$M_ g$$ is the moduli space of curves of genus $$g$$. Applications are also given to the cohomology of Teichmüller spaces.
Reviewer: A.N.Pressley

MSC:
 17B66 Lie algebras of vector fields and related (super) algebras 58B25 Group structures and generalizations on infinite-dimensional manifolds 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 17B56 Cohomology of Lie (super)algebras
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