Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck

Jean-Michel Bismut, Shu Shen, Zhaoting Wei

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English

Birkhauser Verlag AG
14 November 2023

Algebra; Differential calculus & equations; Differential & Riemannian geometry; Algebraic topology

Series: Progress in Mathematics

This monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group of coherent sheaves of a compact complex manifold with values in its Bott-Chern cohomology, and the proof of a corresponding Riemann-Roch-Grothendieck theorem. One main tool used is the equivalence of categories established by Block between the derived category of bounded complexes with coherent cohomology and the homotopy category of antiholomorphic superconnections. Chern-Weil theoretic techniques are then used to construct forms that represent the Chern character. The main theorem is then established using methods of analysis, by combining local index theory with the hypoelliptic Laplacian. Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck is an important contribution to both the geometric and analytic study of complex manifolds and, as such, it will be a valuable resource formany researchers in geometry, analysis, and mathematical physics.

By: Jean-Michel Bismut, Shu Shen, Zhaoting Wei
Imprint: Birkhauser Verlag AG
Country of Publication: Switzerland
Edition: 2023 ed.
Volume: 347
Dimensions: Height: 235mm, Width: 155mm,
Weight: 465g
ISBN: 9783031272332
ISBN 10: 3031272331
Series: Progress in Mathematics
Pages: 184
Publication Date: 14 November 2023
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Jean-Michel Bismut is a French mathematician who is a professor in the Mathematics Department in Orsay. He is known for his contributions to index theory, geometric analysis and probability theory. Together with Gilles Lebeau, he has developed the theory of the hypoelliptic Laplacian, to which he found applications in various fields of mathematics. He shared the Shaw Prize in Mathematical Sciences 2021 with Jeff Cheeger. Shu Shen is a maître de conférences at Sorbonne University in Paris. His research focuses on the fields of analysis, geometry, and representation theory. Zhaoting Wei is an assistant professor in mathematics at Texas A&M University-Commerce, USA. His research interests include noncommutative geometry and higher category theory.