Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck

Download or read book Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck written by Jean-Michel Bismut and published by Springer Nature. This book was released on 2023-12-15 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.