## Riemann-Roch-Grothendieck theorem for families of curves with hyperbolic cusps and its applications to the moduli space of curves |

## Siarhei Finski (Paris) |

## Tuesday, 2019-07-23 09:00 |

We’ll 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 a Riemann surface, 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, which provides a generalisation of a result of Takhtajan-Zograf. If time permits, we will explain how our result implies some bounds on the growth of the 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 of the moduli space of curves. |