The heat kernel Lefschetz fixed point formula for the spin-c dirac operator
type: Item
instance of: scholarly article
title: The heat kernel Lefschetz fixed point formula for the spin-c dirac operator
zbMATH Open document ID: 0858.58045
author: Hans Duistermaat
published in: Progress in Nonlinear Differential Equations and Their Applications
publication date: 1996-01-14T00:00:00Z
review text: Let \(M\) be a compact \(2n\)-dimensional \(C^\infty\) manifold provided with an almost complex structure \(J\), and let \(\pi_L : L\to M\) be a smooth complex vector bundle over \(M\) of (complex) fiber dimension \(l\). In the complex analytic case (i.e., when \(M\) and \(L\) are holomorphic and \(n= \dim_\mathbb{C} M)\), one defines the Riemann-Roch number \(RR (M,L): = \sum^n_{g=0} (-1)^q \dim_\mathbb{C} H^q (M,{\mathcal O} (L))\), where \({\mathcal O}(L)\) is the sheaf of holomorphic sections of \(L\) over \(M\), and, more generally, one defines the holomorphic Lefschetz number of \(\gamma\), a complex analytic automorphism of \(L\) (so it acts on \({\mathcal O} (L))\), by \(\chi_{M,L} (\gamma): = \sum^n_{q=0} (-1)^q \text{trace}_\mathbb{C} (\gamma |H^q (M, {\mathcal O} (L)))\). Hence, \(\chi_{M,L} \text{(id)} = RR(M,L)\). In general, we suppose \(M\) endowed with a Hermitian metric (one has also to fix a Hermitian connection on \(K^*\), the dual of the canonical bundle of \(M\), and on the bundle \(L)\). Denote by \(E^q: = \Lambda^{(0,q)} (\mathbb{C} \otimes T^*M)\) the bundle of \((0,q)\)-forms on \(M\), and by \(\Gamma (M,E^q \otimes L)\) the smooth sections of the latter bundle tensored with \(L\). One constructs an operator (the Dirac operator) \(D: \Gamma (M,E^q \otimes L)\to \Gamma (M,(E^{q+1} \oplus E^{q-1}) \otimes L)\). Write \(E^+\) (resp. \(E^-)\) for the sum of the \(E^q\) over even \(q\)'s (resp. over odd \(q\)'s), and define \(D^\pm: =D |_{\Gamma (M,E^\pm \otimes L)}: \Gamma (M,E^\pm \otimes L) \to\Gamma (M,E^\pm \otimes L)\). Define the virtual character by \(\chi (\gamma) = \text{trace}_\mathbb{C} \gamma |\text{Ker} D^+-\text{trace}_\mathbb{C} \gamma |\text{Ker} D^-\). Notice that \(\chi (\text{id}) = \text{index} D^+ = \dim_\mathbb{C} \text{Ker} D^+ - \dim_\mathbb{C} \text{Ker} (D^+)^*\). Moreover, define \(Q^\pm: = D^\mp \circ D^\pm : \Gamma (M,E^\pm \otimes L) \to \Gamma (M,E^\pm \otimes L)\), and \(Q=Q^+ \oplus Q^-\). Consider (it is the Ber…
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58-02
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 58J37
Mathematics Subject Classification ID: 32L10
zbMATH DE Number: 833389
zbMATH Keywords: symplectic geometry
zbMATH Keywords: almost complex structure
zbMATH Keywords: Riemann-Roch number
zbMATH Keywords: Lefschetz number
zbMATH Keywords: virtual character
zbMATH Keywords: Hirzebruch-Riemann-Roch formula
zbMATH Keywords: Dirac operator
zbMATH Keywords: Clifford bundles
zbMATH Keywords: asymptotics
zbMATH Keywords: heat kernels
zbMATH Keywords: Lefschetz formulas
zbMATH Keywords: orbifolds
reviewed by: Alberto Parmeggiani
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 833389