地点：海纳苑2幢204室

时间：4月20日，16:15 - 17:35

报告人：Gerard Freixas i Montplet（巴黎综合理工学院）

摘要:  

Pretalk:  Determinants of Laplacians and the geometry they encode

Abstract: Given a well-behaved Laplace type operator, say on a compact Riemannian manifold, it is possible to define its determinant as a regularized product of its eigenvalues. The regularization is obtained via zeta functions and heat kernel theory. Determinants of Laplacians encode some geometric features of the manifold. In this talk, I will explain this construction and provide some simple examples of the geometric properties they reflect.

Research talk:  Analytic torsion and degnerations of Calabi-Yau varieties

Abstract: Given a compact Kähler manifold and a Hermitian vector bundle E on it, the holomorphic analytic torsion is defined as a suitable combination of determinants of the Laplacians acting on (0,p)-forms with coefficients on E. This is a deep invariant involved in a differential form version of the Grothendieck-Riemann-Roch formula. For a Calabi-Yau manifold, one can consider another suitable combination of analytic torsions of sheaves of holomorphic differentials. This quantity, called the BCOV torsion, can then be normalized in order to produce a real invariant which does not depend on any choice of Kähler metric. It is called the BCOV invariant. In joint work with Dennis Eriksson, we obtain general expressions for the degeneration of the BCOV invariant for some degenerations of Calabi-Yau varieties. As an application, we obtain a necessary numerical criterion for the existence of a smooth filling of the degeneration. For some simple degenerations, such as those acquiring ADE singularities, this criterion entails a non-smooth filling result. This generalizes results by Voisin and others for A_1 and A_2 singularities. In the talk I will give an overview of these facts.