On Riemann-Roch-Grothendieck theorem for punctured curves with hyperbolic singularities

Abstract: We will present a refinement of Riemann-Roch-Grothendieck theorem on the level of differential forms for families of curves with hyperbolic cusps. The study of spectral properties of the Kodaira Laplacian on those surfaces, and more precisely of its determinant, lies in the heart of our approach.

When our result is applied directly to the moduli space of punctured stable curves, it expresses the extension of the Weil-Petersson form (as a current) to the boundary of the moduli space in terms of the first Chern form of a Hermitian line bundle. This provides a generalisation of a result of Takhtajan-Zograf.

We will also explain how our results imply some bounds on the growth of Weil-Petersson form near the compactifying divisor of the moduli space of punctured stable curves. This would permit us to give a new approach to some well-known results of Wolpert on the Weil-Petersson geometry.

number theoryrepresentation theory

Audience: researchers in the topic

Geometry, Number Theory and Representation Theory Seminar