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The Hodge Chern character of holomorphic connections as a map of simplicial presheaves

Cheyne Glass, Micah Miller, Thomas Tradler and Mahmoud Zeinalian

Algebraic & Geometric Topology 22 (2022) 1057–1112
Abstract

We define a map of simplicial presheaves, the Chern character, that assigns to every sequence of composable non-connection- preserving isomorphisms of vector bundles with holomorphic connections an appropriate sequence of holomorphic forms. We apply this Chern character map to the Čech nerve of a good cover of a complex manifold and assemble the data by passing to the totalization to obtain a map of simplicial sets. In simplicial degree 0, this map gives a formula for the Chern character of a bundle in terms of the clutching functions. In simplicial degree 1, this map gives a formula for the Chern character of bundle maps. In each simplicial degree beyond 1, these invariants, defined in terms of the transition functions, govern the compatibilities between the invariants assigned in previous simplicial degrees. In addition to this, we apply this Chern character to complex Lie groupoids to obtain invariants of bundles on them in terms of the simplicial data. For group actions, these invariants land in suitable complexes calculating various Hodge equivariant cohomologies. In contrast, the de Rham Chern character formula involves additional terms and will appear in a sequel paper. In a sense, these constructions build on a point of view of “characteristic classes in terms of transition functions” advocated by Raoul Bott, which has been addressed over the years in various forms and degrees, concerning the existence of formulas for the Hodge and de Rham characteristic classes of bundles solely in terms of their clutching functions.

Keywords
simplicial presheaf, Chern character, Chern–Simons, cosimplicial simplicial set, totalization, Čech complex
Mathematical Subject Classification 2010
Primary: 18G30, 19L10, 58J28
References
Publication
Received: 17 November 2019
Revised: 22 February 2021
Accepted: 5 April 2021
Published: 25 August 2022
Authors
Cheyne Glass
Department of Mathematics
St Joseph’s College Long Island
Patchogue, NY
United States
Micah Miller
Department of Mathematics
Borough of Manhattan Community College
The City University of New York
New York, NY
United States
Thomas Tradler
Department of Mathematics
New York City College of Technology
The City University of New York
Brooklyn, NY
United States
Mahmoud Zeinalian
Department of Mathematics
Lehman College
The City University of New York
Bronx, NY
United States