Canonical holomorphic sections of determinant line bundles

Jens Kaad, Ryszard Nest

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Resumé

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.

OriginalsprogEngelsk
TidsskriftJournal für die reine und angewandte Mathematik
Vol/bind746
Sider (fra-til)67-116
Antal sider50
ISSN0075-4102
DOI
StatusUdgivet - 2019

Fingeraftryk

Line Bundle
Torsional stress
Isomorphism
Determinant
Perturbation
Invariance
Torsion
Cones
Finite Rank
Fredholm Operator
Homology Groups
Triangle
Cone
Line

Citer dette

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Canonical holomorphic sections of determinant line bundles. / Kaad, Jens; Nest, Ryszard.

I: Journal für die reine und angewandte Mathematik, Bind 746, 2019, s. 67-116.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - Canonical holomorphic sections of determinant line bundles

AU - Kaad, Jens

AU - Nest, Ryszard

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

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DO - 10.1515/crelle-2015-0114

M3 - Journal article

VL - 746

SP - 67

EP - 116

JO - Journal für die reine und angewandte Mathematik

JF - Journal für die reine und angewandte Mathematik

SN - 0075-4102

ER -