Extension de la torsion analytique holomorphe aux fibr\'es en droites int\'egrables

TL;DR

This paper extends the concept of holomorphic analytic torsion and Quillen metrics to integrable line bundles on compact Kähler manifolds, broadening the scope of these invariants beyond smooth cases.

Contribution

It introduces a new extension of holomorphic analytic torsion to integrable line bundles on compact Kähler manifolds, under specific cohomological conditions.

Findings

Holomorphic analytic torsion extends to integrable line bundles with vanishing higher cohomology.

Quillen metric is generalized to a broader class of line bundles.

The extension maintains key properties of the classical torsion in the smooth setting.

Abstract

Let $X$ be a compact K\"ahler manifold. We extend the notion of Quillen metric to the set of integrable line bundles on $X$. In particular, we prove that the notion of holomorphic analytic torsion extends to integrable line bundles $\bar{L}$, such that $H^q(X,L)=0$ for $q\geq 1$.