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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 |
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Algebraic & Geometric Topology 22 (2022) 1057–1112
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Abstract |
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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 , this map gives a formula for the Chern character of a bundle in terms of the clutching functions. In simplicial degree , this map gives a formula for the Chern character of bundle maps. In each simplicial degree beyond , 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. |
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Keywords
simplicial presheaf, Chern character, Chern–Simons,
cosimplicial simplicial set, totalization, Čech complex
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Mathematical Subject Classification 2010
Primary: 18G30, 19L10, 58J28
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References |
Publication
Received: 17 November 2019
Revised: 22 February 2021
Accepted: 5 April 2021
Published: 25 August 2022
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Authors |
| Cheyne Glass | |
| Department of Mathematics St Joseph’s College Long Island Patchogue, NY United States |
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| Micah Miller | |
| Department of Mathematics Borough of Manhattan Community College The City University of New York New York, NY United States |
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| Thomas Tradler | |
| Department of Mathematics New York City College of Technology The City University of New York Brooklyn, NY United States |
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| Mahmoud Zeinalian | |
| Department of Mathematics Lehman College The City University of New York Bronx, NY United States |
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