2005/05/07 by Dmitry Gerenrot, Gerenrot, Dmitry · 1 citation
Mathematics · #57R20(primary) 57R22 #58J20 (secondary) #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AT #math.DG #msc:57R22 #msc:58J20
paper · pdf · doi:10.48550/arxiv.math/0505121
38 pages with 4 figures
arxiv created 2005/05/07 · openalex publication_date 2005/05/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Chern character of a complex vector bundle is most conveniently defined as the exponential of a curvature of a connection. It is well known that its cohomology class does not depend on the particular connection chosen. It has been shown by Quillen that a connection may be perturbed by an endomorphism of the vector bundle, such as a symbol of some elliptic differential operator. This point of view, as we intend to show, allows one to relate Chern character to a non-commutative sibling formulated by Connes and Moscovici. The general setup for our problem is purely geometric. Let σbe the symbol of a Dirac-type operator acting on sections of a \Z2-graded vector bundle E. Let ∇ be a connection on E, pulled back to T^*M. Suppose also that ∇ respects the Z2-grading. The object ∇+σis a superconnection on T^*M in the sense of Quillen. We obtain a formula for the H_*(M)-valued Poincare dual of Quillen's Chern character ch(D)=trace(exp(∇+σ)2) in terms of residues of Γ(z)trace(∇+σ)-2z. We also compute two examples.