Kavli IPMU Komaba Seminar

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Date, time & place Monday 16:30 - 18:00 002Room #002 (Graduate School of Math. Sci. Bldg.)

2008/05/19

17:00-18:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Jean-Michel Bismut (Univ. Paris-Sud, Orsay)
A survey of Quillen metrics

[ Abstract ]
In this lecture, I will survey basic results
on Quillen metrics.

Indeed let $X$ be a complex K\\"ahler manifold, and let $E$ be a
holomorphic Hermitian vector bundle on $X$. Let $\\lambda$ be the complex line
which is the determinant of the cohomology of $E$. The Quillen metric
is a metric on the line $\\lambda$, which one obtains using a spectral
invariant of the Hodge Laplacian, the Ray-Singer analytic torsion.

The Quillen metrics have a number of remarkable properties. Among them
the curvature theorem says that when one considers a family of such
$X$, the curvature of the holomorphic Hermitian connection on
$\\lambda$ is given by a formula of Riemann-Roch-Grothendieck type.

I will explain some of the ideas which go into the proof of these
properties, which includes Quillen's superconnections.