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Canonical holomorphic sections of determinant line bundles

  • Jens Kaad EMAIL logo and Ryszard Nest

Abstract

We investigate the analytic properties of torsion isomorphisms (determinants) of mapping cone triangles of Fredholm complexes. Our main tool is a generalization to Fredholm complexes of the perturbation isomorphisms constructed by R. Carey and J. Pincus for Fredholm operators. A perturbation isomorphism is a canonical isomorphism of determinants of homology groups associated to a finite rank perturbation of Fredholm complexes. The perturbation isomorphisms allow us to establish the invariance properties of the torsion isomorphisms under finite rank perturbations. We then show that the perturbation isomorphisms provide a holomorphic structure on the determinant lines over the space of Fredholm complexes. Finally, we establish that the torsion isomorphisms and the perturbation isomorphisms provide holomorphic sections of certain determinant line bundles.

Funding statement: The second author is partially supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation.

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Received: 2015-04-23
Revised: 2015-12-20
Published Online: 2016-04-07
Published in Print: 2019-01-01

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