Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry
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The work Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry represents a distinct intellectual or artistic creation found in Bowdoin College Library. This resource is a combination of several types including: Work, Language Material, Books.
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Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry
Resource Information
The work Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry represents a distinct intellectual or artistic creation found in Bowdoin College Library. This resource is a combination of several types including: Work, Language Material, Books.
 Label
 Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry
 Title remainder
 a theorem of RiemannRochGrothendieck in complex geometry
 Statement of responsibility
 JeanMichel Bismut
 Language
 eng
 Summary
 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. One tool used in the book is a deformation of the Hodge theory of the fibres to a hypoelliptic Hodge theory, in such a way that the relevant cohomological information is preserved, and 'fantastic cancellations' do occur for the deformation. The deformed hypoelliptic Laplacian acts on the total space of the relative tangent bundle of the fibres. While the original hypoelliptic Laplacian discovered by the author can be described in terms of the harmonic oscillator along the tangent bundle and of the geodesic flow, here, the harmonic oscillator has to be replaced by a quartic oscillator. Another idea developed in the book is that while classical elliptic Hodge theory is based on the Hermitian product on forms, the hypoelliptic theory involves a Hermitian pairing which is a mild modification of intersection pairing. Probabilistic considerations play an important role, either as a motivation of some constructions, or in the proofs themselves. 
 Assigning source
 Source other than Library of Congress
 Cataloging source
 OHX
 Illustrations
 illustrations
 Index
 index present
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 Series statement
 Progress in mathematics
 Series volume
 v. 305
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