Hypoelliptic Laplacian and Bott-Chern Cohomology

This book provides the proof of a complex geometric version of a well-known result in algebraic geometry: the theorem of Riemann-Roch-Grothendieck for proper submersions. It gives an equality of cohomology classes in Bott-Chern cohomology.

Gives an important application of the theory of the hypoelliptic Laplacian in complex algebraic geometry Provides an introduction to applications of Quillen's superconnections in complex geometry with hypoelliptic operators Presents several techniques partly inspired from physics, which concur to the proof of a result in complex algebraic geometry The method of hypoelliptic deformation of the classical Laplacian was developed by the author during the last ten years Includes supplementary material: sn.pub/extras

Autorentext
Jean-Michel Bismut is Professor of Mathematics at Université Paris-Sud (Orsay) and a member of the Académie des Sciences. Starting with a background in probability, he has worked extensively on index theory. With Gillet, Soulé, and Lebeau, he contributed to the proof of a theorem of RiemannRochGrothendieck in Arakelov geometry. More recently, he has developed a theory of the hypoelliptic Laplacian, a family of operators that deforms the classical Laplacian, and provides a link between spectral theory and dynamical systems.

Inhalt
Introduction.- 1 The Riemannian adiabatic limit.- 2 The holomorphic adiabatic limit.- 3 The elliptic superconnections.- 4 The elliptic superconnection forms.- 5 The elliptic superconnections forms.- 6 The hypoelliptic superconnections.- 7 The hypoelliptic superconnection forms.- 8 The hypoelliptic superconnection forms of vector bundles.- 9 The hypoelliptic superconnection forms.- 10 The exotic superconnection forms of a vector bundle.- 11 Exotic superconnections and RiemannRochGrothendieck.- Bibliography.- Subject Index.- Index of Notation.