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Progress in Mathematics #305: Hypoelliptic Laplacian and Bott Chern Cohomology: A Theorem of Riemann Roch Grothendieck in Complex Geometryby Jeanmichel Bismut
Synopses & ReviewsPublisher Comments:The book provides the proof of a complex geometric version of a wellknown result in algebraic geometry: the theorem of RiemannRochGrothendieck for proper submersions. It gives an equality of cohomology classes in BottChern cohomology, which is a refinement for complex manifolds of de Rham cohomology. When the manifolds are Kähler, our main result is known. A proof can be given using the elliptic Hodge theory of the fibres, its deformation via Quillen's superconnections, and a version in families of the 'fantastic cancellations' of McKeanSinger in local index theory. In the general case, this approach breaks down, because the cancellations do not occur any more.
Synopsis:This book provides the proof of a complex geometric version of a wellknown result in algebraic geometry: the theorem of RiemannRochGrothendieck for proper submersions. It gives an equality of cohomology classes in BottChern cohomology.
About the AuthorJeanMichel Bismut is
Table of ContentsIntroduction. 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.
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