Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck

Beschreibung

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 manyresearchers in geometry, analysis, and mathematical physics. 
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. 

Presents a solution to a long-standing problem in complex algebraic geometry Proves an RRG theorem for coherent sheaves of a compact complex manifold Offers a valuable resource for many researchers in geometry, analysis, and mathematical physics

Autor*in

Jean-Michel Bismut

Themen in »Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck«

Derived Categories Riemann-Roch Theorems Chern characters Hypoelliptic equations Coherent sheaves

Stimmen zu »Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck«

“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)

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