Résume | **Attention, la soutenance aura lieu dans la salle 115 du bâtiment Olympe de Gouges**
Title: On some problems of holomorphic analytic torsion
Abstract:
The goal of this thesis is to study the analytic torsion in two different contexts.
In the first context, we study the asymptotics of the analytic torsion, when a Hermitian holomorphic vector bundle is twisted by an increasing power of a positive line bundle.
In the second context, we generalise the theory of analytic torsion for surfaces with hyperbolic cusps. Motivated by singularities appearing in complete metrics of constant scalar curvature -1 on stable Riemann surfaces, we suppose that the metric on the surface is smooth outside a finite number points in the neighborhood of which it can have singularities of Poincaré type. We fix a Hermitian holomorphic vector bundle which has at worst logarithmic singularities in the neighborhood of the marked points. For these data, by renormalising the trace of the heat operator, we construct the analytic torsion and study its properties.
Then we study the analytic torsion for families of Riemann surfaces. We prove the curvature theorem, which refines Riemann-Roch-Grothendieck theorem on the level of differential forms. We study the behavior of the analytic torsion when the cusps are created by degeneration and we give some applications to the moduli spaces of pointed curves.
Lieu du pot: Espace commun de l’étage 6, bâtiment Sophie Germain |