Tate tame symbol and the joint torsion of commuting operators

Abstract

We investigate determinants of Koszul complexes of holomorphic functions of a commuting tuple of bounded operators acting on a Hilbert space. Our main result shows that the analytic joint torsion, which compares two such determinants, can be computed by a local formula which involves a tame symbol of the involved holomorphic functions. As an application we are able to extend the classical tame symbol of meromorphic functions on a Riemann surface to the more involved setting of transversal functions on a complex analytic curve. This follows by spelling out our main result in the case of Toeplitz operators acting on the Hardy space over the polydisc. Contents1. Introduction 1 2. Determinants, torsion and joint torsion 6 2.1. Determinants of vector spaces 6 2.2. Fredholm complexes 9 2.3. Joint torsion 10 3. Joint torsion of commuting operators 11 3.1. The Koszul complex 11 3.2. Joint torsion transition numbers 13 3.3. Local indices 14 4. Multiplicative Lefschetz numbers 15 4.1. Lefschetz numbers of Koszul complexes 15 5. Localization of the joint torsion 17 6. Application: Tame symbols of complex analytic curves 21 6.1. Preliminaries on complex analytic spaces 21 6.2. The Tate tame symbol 22 References 25

Functional Calculus and Joint Torsion of Pairs of Almost Commuting Operators

“…Recently in [17], J. Kaad and R. Nest investigate the local behavior of joint torsion transition numbers associated to commuting tuples of operators. They generalize the above Carey-Pincus formula and extend the notion of tame symbol to the setting of transversal functions on a complex analytic curve.…”