Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck

ISBN-13: 9783031272332 / Angielski / Twarda / 2023 / 183 str.

Jean-Michel Bismut; Shu Shen; Zhaoting Wei

Opis

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. 

Spis treści

Introduction.- Bott-Chern Cohomology and Characteristic Classes.- The Derived Category ${\mathrm{D^_{\mathrm}}}$.- 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 $\overline{\partial}^\partial^\omega^=0$..- The Hypoelliptic Superconnections.- The Hypoelliptic Superconnection Forms.-  The Hypoelliptic Superconnection Forms when $\overline{\partial}^\partial^\omega^=0$.-  Exotic Superconnections and Riemann-Roch-Grothendieck.- Subject Index.- Index of Notation.- Bibliography.