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 for many researchers in geometry, analysis, and mathematical physics.
Arvustused
Throughout the book, the authors demonstrate a mastery of advanced techniques in differential geometry, algebra, and analysis. The proofs are rigorous and often involve intricate calculations and estimates. ... Researchers and advanced graduate students in complex geometry, algebraic geometry, and related areas will find this work to be a valuable resource, albeit one that requires careful study and a strong background in the subject. (Byungdo Park, Mathematical Reviews, May, 2025)
Introduction.- Bott-Chern Cohomology and Characteristic Classes.- The
Derived Category ${\mathrm{D^{b}_{\mathrm{coh}}}}$.- Preliminaries on Linear
Algebra and Differential Geometry.- The Antiholomorphic Superconnections of
Block.- An Equivalence of Categories.- Antiholomorphic Superconnections and
Generalized Metrics.- Generalized Metrics and Chern Character Forms.- The
Case of Embeddings.- Submersions and Elliptic Superconnections.- Elliptic
Superconnection Forms and Direct Images.- A Proof of Theorem 10-1 when
$\overline{\partial}^{X}\partial^{X}\omega^{X}=0$..- The Hypoelliptic
Superconnections.- The Hypoelliptic Superconnection Forms.- The Hypoelliptic
Superconnection Forms when
$\overline{\partial}^{X}\partial^{X}\omega^{X}=0$.- Exotic Superconnections
and Riemann-Roch-Grothendieck.- Subject Index.- Index of Notation.-
Bibliography.
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.
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