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Equivariant Chern character for coherent sheaves and Riemann-Roch-Grothendieck

Guangzhe Xu

TL;DR

The paper develops an equivariant Bott–Chern Chern character $ ext{ch}_{g, ext{BC}}$ for coherent sheaves on compact complex manifolds with a finite group action, taking values in Bott–Chern cohomology on the fixed-point set $X_g$ and satisfying functorial, multiplicative, and RR–Grothendieck properties. It extends antiholomorphic superconnection techniques to the $G$–equivariant setting, proves a category equivalence between equivariant antiholomorphic superconnections and equivariant derived category objects, and defines an equivariant Chern character that behaves well under pullbacks, tensor products, and cones. The main RR–Grothendieck theorems establish formulas for equivariant embeddings and projections, reducing to and generalizing classical RR results through deformation to the normal cone and Koszul resolutions, with a rigorous unicity theorem. The work also develops submersion, hypoelliptic, and exotic superconnection frameworks, showing convergence of hypoelliptic forms to elliptic counterparts and extending the RR–Grothendieck theory to more general geometric settings, including localization to fixed-point strata. Overall, this provides a robust, functorial, and unique equivariant RR machinery in Bott–Chern cohomology with potential applications to orbifolds and equivariant index theory in complex geometry.

Abstract

In this paper, we develope an equivariant theory of Chern characters for coherent sheaves on compact complex manifolds with finite group actions, taking values in Bott-Chern cohomology classes. Furthermore, we establish the corresponding Riemann-Roch-Grothendieck theorem in this context.

Equivariant Chern character for coherent sheaves and Riemann-Roch-Grothendieck

TL;DR

The paper develops an equivariant Bott–Chern Chern character for coherent sheaves on compact complex manifolds with a finite group action, taking values in Bott–Chern cohomology on the fixed-point set and satisfying functorial, multiplicative, and RR–Grothendieck properties. It extends antiholomorphic superconnection techniques to the –equivariant setting, proves a category equivalence between equivariant antiholomorphic superconnections and equivariant derived category objects, and defines an equivariant Chern character that behaves well under pullbacks, tensor products, and cones. The main RR–Grothendieck theorems establish formulas for equivariant embeddings and projections, reducing to and generalizing classical RR results through deformation to the normal cone and Koszul resolutions, with a rigorous unicity theorem. The work also develops submersion, hypoelliptic, and exotic superconnection frameworks, showing convergence of hypoelliptic forms to elliptic counterparts and extending the RR–Grothendieck theory to more general geometric settings, including localization to fixed-point strata. Overall, this provides a robust, functorial, and unique equivariant RR machinery in Bott–Chern cohomology with potential applications to orbifolds and equivariant index theory in complex geometry.

Abstract

In this paper, we develope an equivariant theory of Chern characters for coherent sheaves on compact complex manifolds with finite group actions, taking values in Bott-Chern cohomology classes. Furthermore, we establish the corresponding Riemann-Roch-Grothendieck theorem in this context.

Paper Structure

This paper contains 82 sections, 57 theorems, 304 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The main results
  3. Earlier work
  4. The construction of equivariant Chern character
  5. The antiholomorphic superconnections
  6. The Chern character of coherent sheaves
  7. Our construction of the equivariant Chern character
  8. The Riemann-Roch-Grothendieck theorem
  9. The Riemann-Roch-Grothendieck for equivariant embeddings
  10. The Riemann-Roch-Grothendieck for equivariant projections
  11. The unicity theorem
  12. The organization of the article
  13. Preliminaries
  14. Group algebra
  15. Equivariant holomorphic vector bundles
  16. ...and 67 more sections

Key Result

Theorem 1.1

There is a map satisfies the following properties.

Figures (1)

  • Figure 1: Deformation to the normal cone

Theorems & Definitions (132)

  • Theorem 1.1: See Sections \ref{['EquiChernCharacter']}, \ref{['Section7-5']} and Theorem \ref{['MainTheorem']}
  • Theorem 1.2: See Theorem \ref{['Thm8-1']}
  • Definition 1.1
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Definition 1.2
  • Theorem 1.6
  • Proposition 1.7: See Proposition \ref{['LocalResolution']}
  • Proposition 1.8: See Proposition \ref{['Illusie']}
  • ...and 122 more