The holomorphic torsion forms of the de Rham complex
type: Item
instance of: scholarly article
title: The holomorphic torsion forms of the de Rham complex
zbMATH Open document ID: 1019.32016
DOI: 10.1016/S1631-073X(02)02469-X
author: Jean-Michel Bismut
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-09-15T00:00:00Z
review text: Let \(\pi: M \to S\) be a holomorphic submersion of complex varieties with compact fiber \(X\), and let \(E\) be a holomorphic fiber bundle on \(M\). Let \(h^E\) be a Hermitian metric on \(E\), and let \(\omega^M\) be a (1,1) closed form on \(M\) that induces a Hermitian metric \(h^{TX}\) on the holomorphic tangent fiber bundle \(TX=TM/S\). Let \(G\) be a compact Lie group acting holomorphically on \(M\) along the fibers \(X\). For \(g\in G\), consider the forms of equivariant holomorphic torsion \(T_g(\omega^M, h^E)\), which are the sums of forms of type \((p, p)\) on \(S\) such that \[ \frac{\overline{\partial}\partial}{2i\pi}T_g\left(\omega^M, h^E\right)= \text{ch}_g\left(R^\cdot\pi_\ast E,h^{R^\cdot\pi_\ast E}\right)-\int_{X_g} \text{Td}_g\left(TX,h^{TX}\right)\text{ch}_g\left(E,h^E\right). \tag{1} \] The characteristic classes on the right of (1) are the classes appearing in the Lefschetz-Atiyah-Bott formula. Apply the previous construction to a \(\mathbb Z\)-graduated fiber bundle \(\Lambda^\cdot(T^\ast X)\). Let \(T_g(\omega^M)\) be the forms of the corresponding holomorphic torsion. It is easily verified that in this case (1) becomes \[ \frac{\overline{\partial}\partial} {2i\pi} T_g(\omega^M)=0.\tag{2} \] If \(S\) is compact and Kählerian, it is deduced from (2) that \(T_g(\omega^M)\) defines a class of cohomology on \(S\). The objective of this note is to announce that in general, the forms \(T_g(\omega^M)\) are not only closed but exact.
Mathematics Subject Classification ID: 32L20
Mathematics Subject Classification ID: 14F40
zbMATH DE Number: 1854346
zbMATH Keywords: vanishing of the holomorphic torsion forms
zbMATH Keywords: forms of analytic torsion of de Rham complex
zbMATH Keywords: exterior algebra
zbMATH Keywords: Clifford algebra
zbMATH Keywords: Kähler fiber bundle
zbMATH Keywords: Hermitian metric
zbMATH Keywords: Levi-Civita superconnections
zbMATH Keywords: superconnection forms
zbMATH Keywords: characteristic classes
zbMATH Keywords: Lefschetz-Atiyah-Bott formula
MaRDI profile type: Publication
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full work available at URL: s1631-073x(02)02469-x
OpenAlex ID: W2039612489
description: scientific article; zbMATH DE number 1854346