Holomorphic Quillen determinant line bundles on integral compact Kahler manifolds

TL;DR

This paper demonstrates a deep connection between integral Kähler manifolds and holomorphic families of Cauchy-Riemann operators, linking the Quillen determinant line bundle to tensor powers of the line bundle defined by the Kähler form.

Contribution

It establishes an isomorphism between the Quillen determinant line bundle and tensor powers of the line bundle from the Kähler form on integral compact Kähler manifolds, including a symplectic version.

Findings

The Quillen determinant line bundle corresponds to a high tensor power of the line bundle from the Kähler form.

A symplectic analogue of the main result is proven.

Conjecture for an equivariant version is proposed.

Abstract

We show that any compact Kahler manifold with integral Kahler form, parametrizes a natural holomorphic family of Cauchy-Riemann operators on the Riemann sphere such that the Quillen determinant line bundle of this family is isomorphic to a sufficiently high tensor power of the holomorphic line bundle determined by the integral Kahler form. We also establish a symplectic version of the result. We conjecture that an equivariant version of our result is true.