A local families index formula for \({\overline{\partial}}\)-operators on punctured Riemann surfaces. (English) Zbl 1171.30017
The paper under review is a sequel to the paper by the same authors [Int. Math. Res. Not. 2009, No. 4, 625–697 (2009; Zbl 1184.58008)]. Using an index formula obtained in the paper cited, along with heat kernel methods obtained by B. Vaillant in his PhD dissertation (Bonn, 2001), the authors obtain a local index formula for families of \(\overline{\partial}\)-operators on the Teichmüller universal curve of Riemann surfaces of genus \(g\) with \(n\) punctures. The formula also holds on the orbifold moduli space of such surfaces, where it can be written in terms of Mumford-Morita-Miller classes. The degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the Quillen connection, a result originally obtained by Takhtajan and Zograf. The authors also note an interpretation of their local index formula as a local version of the Grothendieck-Riemann-Roch theorem.
Reviewer: Athanase Papadopoulos (Strasbourg)
MSC:
| 30F60 | Teichmüller theory for Riemann surfaces |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 58J20 | Index theory and related fixed-point theorems on manifolds |
Keywords:
universal Teichmüller curve; canonical connection; moduli space; index formula; determinant line bundle; Quillen connectionCitations:
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