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Superconnection and Orbifold Chern character

Qiaochu Ma, Xiang Tang, Hsian-Hua Tseng, Zhaoting Wei

TL;DR

This work develops a comprehensive framework for orbifold Chern characters via antiholomorphic flat superconnections on complex orbifolds. It proves a fundamental equivalence $D^b_{ ext{coh}}(X)\simeq \underline{B}(X)$ between coherent sheaves and antiholomorphic superconnections, enabling a Bott-Chern orbifold Chern character $\mathrm{ch}_{\text{BC}}: K(X)\to H^{(=)}_{\text{BC}}(IX,\mathbb{C})$ that is functorial and Morita-invariant. A central achievement is the Riemann-Roch-Grothendieck theorem for embeddings of orbifolds, established by decomposing embeddings into iso-spatial and stabilizer-preserving types and deriving explicit pushforward formulas. The results yield a canonical and robust interpretation of the orbifold Chern character, compatible with operations such as pullbacks, tensor products, and mapping cones, and provide tools for computations in Bott-Chern cohomology on inertia orbifolds. Overall, the paper advances index theory and K-theory on complex orbifolds, with potential applications to moduli spaces and equivariant geometry.

Abstract

We use flat antiholomorphic superconnections to study orbifold Chern character following the method introduced by Bismut, Shen, and Wei. We show the uniqueness of orbifold Chern character by proving a Riemann-Roch-Grothendieck theorem for orbifold embeddings.

Superconnection and Orbifold Chern character

TL;DR

This work develops a comprehensive framework for orbifold Chern characters via antiholomorphic flat superconnections on complex orbifolds. It proves a fundamental equivalence between coherent sheaves and antiholomorphic superconnections, enabling a Bott-Chern orbifold Chern character that is functorial and Morita-invariant. A central achievement is the Riemann-Roch-Grothendieck theorem for embeddings of orbifolds, established by decomposing embeddings into iso-spatial and stabilizer-preserving types and deriving explicit pushforward formulas. The results yield a canonical and robust interpretation of the orbifold Chern character, compatible with operations such as pullbacks, tensor products, and mapping cones, and provide tools for computations in Bott-Chern cohomology on inertia orbifolds. Overall, the paper advances index theory and K-theory on complex orbifolds, with potential applications to moduli spaces and equivariant geometry.

Abstract

We use flat antiholomorphic superconnections to study orbifold Chern character following the method introduced by Bismut, Shen, and Wei. We show the uniqueness of orbifold Chern character by proving a Riemann-Roch-Grothendieck theorem for orbifold embeddings.

Paper Structure

This paper contains 44 sections, 63 theorems, 329 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Orbifolds
  4. Coherent sheaves
  5. Chern character
  6. Outlook
  7. Outline
  8. Acknowledgment
  9. A review of groupoids and orbifolds
  10. Lie groupoids
  11. Generalized morphisms
  12. Proper generalized morphisms
  13. Complex groupoids
  14. Orbifold embeddings
  15. Graph generalized morphisms
  16. ...and 29 more sections

Key Result

Theorem 1.1

Let $i_{X,Y}\colon X\hookrightarrow Y$ be an embedding of a compact complex orbifold groupoid. Let $\mathcal{F}\in D^b_{\text{coh}}(X)$ and $i_{X,Y,*}\mathcal{F}\in D^b_{\text{coh}}(Y)$ be its direct image. We have where $Ii_{X,Y}$ is the induced morphism between inertia groupoids.

Figures (1)

  • Figure 1: The total space $W$

Theorems & Definitions (195)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 185 more