Chern character for infinity vector bundles

Glass, Cheyne¹ ; Miller, Micah² ; Tradler, Thomas³ ; Zeinalian, Mahmoud⁴

¹ Department of Mathematics, State University of New York at New Paltz, New Paltz, NY, United States
² Department of Mathematics, Borough of Manhattan Community College, The City University of New York, New York, NY, United States
³ Department of Mathematics, College of Technology, City University of New York, Brooklyn, NY, United States
⁴ Department of Mathematics, Lehman College, The City University of New York, Bronx, NY, United States, Department of Mathematics, Graduate Center, The City University of New York, New York, NY, United States

Algebraic and Geometric Topology, Tome 24 (2024) no. 9, pp. 4939-4990

Résumé

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 modern language, such objects are obtained by the ∞–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 recover the Chern character of Toledo and Tong. As a consequence, our construction extends O’Brian, Toledo and 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.

Détail

DOI : 10.2140/agt.2024.24.4939

Keywords: coherent sheaves, simplicial sheaves, Chern–Simons invariant, Atiyah class