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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

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
Full Text: DOI
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