The Resource Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut, (electronic resource)
Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut, (electronic resource)
Resource Information
The item Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in Bowdoin College Library.This item is available to borrow from 1 library branch.
Resource Information
The item Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in Bowdoin College Library.
This item is available to borrow from 1 library branch.
 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. 
 Language
 eng
 Extent
 1 online resource (xv, 203 pages)
 Contents

 Introduction
 The Riemannian adiabatic limit
 The holomorphic adiabatic limit
 The elliptic superconnections
 The elliptic superconnection forms
 The elliptic superconnections forms
 The hypoelliptic superconnections
 he hypoelliptic superconnection forms
 The hypoelliptic superconnection forms of vector bundles
 The hypoelliptic superconnection forms
 The exotic superconnection forms of a vector bundle
 Exotic superconnections and RiemannRochGrothendieck
 Isbn
 9783319001272
 Label
 Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry
 Title
 Hypoelliptic Laplacian and BottChern cohomology
 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
 http://library.link/vocab/creatorName
 Bismut, JeanMichel
 Illustrations
 illustrations
 Index
 index present
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 Series statement
 Progress in mathematics
 Series volume
 v. 305
 http://library.link/vocab/subjectName

 Cohomology operations
 Geometry, Algebraic
 Label
 Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut, (electronic resource)
 Bibliography note
 Includes bibliographical references (pages 191195) and indexes
 Contents
 Introduction  The Riemannian adiabatic limit  The holomorphic adiabatic limit  The elliptic superconnections  The elliptic superconnection forms  The elliptic superconnections forms  The hypoelliptic superconnections  he hypoelliptic superconnection forms  The hypoelliptic superconnection forms of vector bundles  The hypoelliptic superconnection forms  The exotic superconnection forms of a vector bundle  Exotic superconnections and RiemannRochGrothendieck
 Control code
 ssj0000904256
 Dimensions
 unknown
 Extent
 1 online resource (xv, 203 pages)
 Form of item
 online
 Governing access note
 Access restricted to subscribing institutions
 Isbn
 9783319001272
 Isbn Type
 (alk. paper)
 Lccn
 2013939622
 Other control number
 10.1007/9783319001289
 Specific material designation
 remote
 System control number
 (WaSeSS)ssj0000904256
 Label
 Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut, (electronic resource)
 Bibliography note
 Includes bibliographical references (pages 191195) and indexes
 Contents
 Introduction  The Riemannian adiabatic limit  The holomorphic adiabatic limit  The elliptic superconnections  The elliptic superconnection forms  The elliptic superconnections forms  The hypoelliptic superconnections  he hypoelliptic superconnection forms  The hypoelliptic superconnection forms of vector bundles  The hypoelliptic superconnection forms  The exotic superconnection forms of a vector bundle  Exotic superconnections and RiemannRochGrothendieck
 Control code
 ssj0000904256
 Dimensions
 unknown
 Extent
 1 online resource (xv, 203 pages)
 Form of item
 online
 Governing access note
 Access restricted to subscribing institutions
 Isbn
 9783319001272
 Isbn Type
 (alk. paper)
 Lccn
 2013939622
 Other control number
 10.1007/9783319001289
 Specific material designation
 remote
 System control number
 (WaSeSS)ssj0000904256
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<div class="citation" vocab="http://schema.org/"><i class="fa faexternallinksquare fafw"></i> Data from <span resource="http://link.bowdoin.edu/portal/HypoellipticLaplacianandBottCherncohomology/ewRZpbXmyBs/" typeof="Book http://bibfra.me/vocab/lite/Item"><span property="name http://bibfra.me/vocab/lite/label"><a href="http://link.bowdoin.edu/portal/HypoellipticLaplacianandBottCherncohomology/ewRZpbXmyBs/">Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut, (electronic resource)</a></span>  <span property="potentialAction" typeOf="OrganizeAction"><span property="agent" typeof="LibrarySystem http://library.link/vocab/LibrarySystem" resource="http://link.bowdoin.edu/"><span property="name http://bibfra.me/vocab/lite/label"><a property="url" href="https://link.bowdoin.edu/">Bowdoin College Library</a></span></span></span></span></div>