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Chern character for infinity vector bundles

Cheyne Glass, Micah Miller, Thomas Tradler, Mahmoud Zeinalian

TL;DR

The paper constructs infinity-vector-bundle objects as simplicial presheaves ${\bf{IVB}}$ and a holomorphic-form target ${\bf \Omega}$, and defines a Chern character ${\bf Ch}:{\bf{IVB}}\to {\bf \Omega}$ whose MC-data and trace yield higher Chern characters. Through Čech sheafification, it elevates these to higher (hyper)descent objects, identifying 0-simplices with Toledo–Tong twisting cochains and recovering OTT’s Chern character on coherent sheaves via $\pi_0$. The framework extends the classical Chern character to stacks and the equivariant setting, and provides higher Chern–Simons-type invariants from the induced maps on higher homotopy groups. By working within the local projective model structure and height-bounded truncations, the authors obtain well-behaved sheaf/hyper-sheaf objects that faithfully encode coherent-sheaf data and their morphisms up to higher homotopies. This establishes a robust, metric-free, homotopy-theoretic approach to characteristic classes in complex-analytic geometry with potential for further equivariant and stack-theoretic extensions.

Abstract

Coherent sheaves on general complex manifolds do not necessarily have resolutions by finite complexes of vector bundles. However D. Toledo and Y.L.L. Tong showed that one can resolve coherent sheaves by objects analogous to chain complexes of holomorphic vector bundles, whose cocycle relations are governed by a coherent infinite system of homotopies. In the modern language such objects are obtained by the infinity-sheafification of the simplicial presheaf of chain complexes of holomorphic vector bundles. We define a Chern character as a map of simplicial presheaves, whereby the connected components of its sheafification recovers the Chern character of Toledo and Tong. As a consequence our construction extends Toledo Tong and O'Brian Toledo Tong's definition of the Chern character to the settings of stacks and in particular the equivariant setting. Even in the classical setting of complex manifolds, the induced maps on higher homotopy groups provide new Chern-Simons, and higher Chern-Simons, invariants for coherent sheaves.

Chern character for infinity vector bundles

TL;DR

The paper constructs infinity-vector-bundle objects as simplicial presheaves and a holomorphic-form target , and defines a Chern character whose MC-data and trace yield higher Chern characters. Through Čech sheafification, it elevates these to higher (hyper)descent objects, identifying 0-simplices with Toledo–Tong twisting cochains and recovering OTT’s Chern character on coherent sheaves via . The framework extends the classical Chern character to stacks and the equivariant setting, and provides higher Chern–Simons-type invariants from the induced maps on higher homotopy groups. By working within the local projective model structure and height-bounded truncations, the authors obtain well-behaved sheaf/hyper-sheaf objects that faithfully encode coherent-sheaf data and their morphisms up to higher homotopies. This establishes a robust, metric-free, homotopy-theoretic approach to characteristic classes in complex-analytic geometry with potential for further equivariant and stack-theoretic extensions.

Abstract

Coherent sheaves on general complex manifolds do not necessarily have resolutions by finite complexes of vector bundles. However D. Toledo and Y.L.L. Tong showed that one can resolve coherent sheaves by objects analogous to chain complexes of holomorphic vector bundles, whose cocycle relations are governed by a coherent infinite system of homotopies. In the modern language such objects are obtained by the infinity-sheafification of the simplicial presheaf of chain complexes of holomorphic vector bundles. We define a Chern character as a map of simplicial presheaves, whereby the connected components of its sheafification recovers the Chern character of Toledo and Tong. As a consequence our construction extends Toledo Tong and O'Brian Toledo Tong's definition of the Chern character to the settings of stacks and in particular the equivariant setting. Even in the classical setting of complex manifolds, the induced maps on higher homotopy groups provide new Chern-Simons, and higher Chern-Simons, invariants for coherent sheaves.
Paper Structure (14 sections, 38 theorems, 83 equations)

This paper contains 14 sections, 38 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. The Simplicial Presheaves ${\bf{IVB}}$ and ${\bf \Omega}$
  3. Chern Character ${\bf{Ch}}:{\bf{IVB}}\to {\bf \Omega}$
  4. A Higher Chern Character for Coherent Sheaves
  5. Čech Sheafification of ${\bf{IVB}}$ as Twisting Cochains
  6. Čech sheafification of the Chern map ${\bf{Ch}}$
  7. The Induced Map on Classifying Stacks
  8. Sheaves in the local projective model structure
  9. A Weak Equivalence $s\mathcal{S}et({\widehat{\Delta}}^\bullet,K)\to s\mathcal{S}et(\Delta^\bullet,K)$
  10. Explicit Description of Totalization
  11. Totalization
  12. Simplicies of $\Delta^k\times \Delta^\ell$
  13. Totalization for the case $K=s\mathcal{S}et({\widehat{\Delta}},\tilde{K})$
  14. Totalization and Fibrant Objects

Key Result

Theorem \ref{THM:CH-is-map-of-simplicial-presheaves}

The Chern character ${\bf{Ch}}:{\bf{IVB}}\to {\bf \Omega}$ defined above is a map of simplicial presheaves.

Theorems & Definitions (98)

  • Theorem \ref{THM:CH-is-map-of-simplicial-presheaves}
  • Theorem \ref{THM:CH(IVB)=CH(TwCoch)}
  • Theorem \ref{THM:IVBStack}
  • Definition 2.1
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Example 2.6
  • Lemma 2.7
  • proof
  • ...and 88 more