Generalized holomorphic analytic torsion

19-22-26 Novembre 2010 - Gerard Freixas i Montplet


Gerard Freixas i Montplet (CNRS, Jussieu) terra' un ciclo di 3 seminari dal titolo "Generalized holomorphic analytic torsion".

Venerdi 19 novembre, 14:30 1BC50
Lunedi 22 novembre, 14:30 1BC50
Venerdi 26 novembre, 14:00 1A150

The aim of this series of talks is to present joint work with J.I. Burgos and R. Litcanu on holomorphic analytic torsion. Analytic torsion forms are differential forms that transgress the Grothendieck-Riemann-Roch theorem to the level of differential forms. The existing constructions impose a number of restrictions. For instance, one can only define analytic torsion forms of hermitian vector bundles with respect to proper submersions of complex manifolds.
We develop a formalism to extend the theory of analytic torsion forms to complexes of coherent sheaves with suitable hermitian structures and to arbitrary projective morphisms. We will divide the exposition in three talks. The rough contents will be as follows:

Lecture 1: The first talk will review the theory of Bott-Chern secondary classes and introduce the notion and basic properties of holomorphic analytic torsion forms. This includes work of Bismut and coworkers. We will try to insist on the motivating points.

Lecture 2: We will present the formalism of hermitian structures on objects of the bounded derived category of coherent sheaves. In particular we will introduce the category of complex algebraic varieties and projective morphisms with metrics. All this will reveal useful to extend the notion of analytic torsion forms to the case of coherent sheaves and arbitrary projective morphisms.

Lecture 3: In the last talk we will state an existence theorem and a classification of all possible theories of analytic torsion forms. Some applications will be presented, in relation to Grothendieck duality and Quillen metrics of degenerating families of curves.

Rif. int. A. Bertapelle