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Canonical holomorphic sections of determinant line bundles

  • Jens Kaad EMAIL logo and Ryszard Nest


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.


[1] M. Breuning, Determinant functors on triangulated categories, J. K-Theory 8 (2011), no. 2, 251–291. 10.1017/is010006009jkt120Search in Google Scholar

[2] L. G. Brown, Operator algebras and algebraic K-theory, Bull. Amer. Math. Soc. 81 (1975), no. 6, 1119–1121. 10.1090/S0002-9904-1975-13943-7Search in Google Scholar

[3] R. Carey and J. Pincus, Joint torsion of Toeplitz operators with H symbols, Integral Equations Operator Theory 33 (1999), no. 3, 273–304. 10.1007/BF01230735Search in Google Scholar

[4] R. Carey and J. Pincus, Perturbation vectors, Integral Equations Operator Theory 35 (1999), no. 3, 271–365. 10.1007/BF01193903Search in Google Scholar

[5] R. W. Carey and J. D. Pincus, Steinberg symbols modulo the trace class, holonomy, and limit theorems for Toeplitz determinants, Trans. Amer. Math. Soc. 358 (2006), no. 2, 509–551. 10.1090/S0002-9947-05-03858-4Search in Google Scholar

[6] A. Connes and M. Karoubi, Caractère multiplicatif d’un module de Fredholm, K-Theory 2 (1988), no. 3, 431–463. 10.1007/BF00533391Search in Google Scholar

[7] R. E. Curto, Fredholm and invertible n-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1981), no. 1, 129–159. Search in Google Scholar

[8] D. S. Freed, On determinant line bundles, Mathematical aspects of string theory (San Diego 1986), Adv. Ser. Math. Phys. 1, World Scientific Publishing, Singapore (1987), 189–238. 10.1142/9789812798411_0011Search in Google Scholar

[9] I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monogr. 18, American Mathematical Society, Providence 1969. Search in Google Scholar

[10] J. Kaad, Joint torsion of several commuting operators, Adv. Math. 229 (2012), no. 1, 442–486. 10.1016/j.aim.2011.08.012Search in Google Scholar

[11] F. F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves. I: Preliminaries on “det” and “Div”, Math. Scand. 39 (1976), no. 1, 19–55. 10.7146/math.scand.a-11642Search in Google Scholar

[12] D. Kvillen, Determinants of Cauchy–Riemann operators on Riemann surfaces, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 37–41. Search in Google Scholar

[13] D. Quillen, Higher algebraic K-theory. I, Algebraic K-theory. Vol. I: Higher K-theories (Seattle 1972), Lecture Notes in Math. 341, Springer, Berlin (1973), 85–147. 10.1007/BFb0067053Search in Google Scholar

[14] H. Widom, Eigenvalue distribution of nonselfadjoint Toeplitz matrices and the asymptotics of Toeplitz determinants in the case of nonvanishing index, Topics in operator theory: Ernst D. Hellinger memorial volume, Oper. Theory Adv. Appl. 48, Birkhäuser, Basel (1990), 387–421. Search in Google Scholar

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

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