The Resource Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut
Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut
Resource Information
The item Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Missouri Libraries.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 represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Missouri Libraries.
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.
 Contents

 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
 Isbn
 9783319001289
 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
 Subject

 Cohomology operations
 Cohomology operations
 Differential equations, partial.
 Geometry, Algebraic
 Geometry, Algebraic
 Geometry, Algebraic
 Global Analysis and Analysis on Manifolds
 Global Analysis and Analysis on Manifolds.
 Global analysis.
 Hypoelliptic operators
 Hypoelliptic operators
 Hypoelliptic operators
 KTheory
 Ktheory.
 MATHEMATICS  Functional Analysis
 Mathematics
 Mathematics.
 Partial Differential Equations
 Partial Differential Equations.
 Cohomology operations
 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
 Cataloging source
 GW5XE
 http://library.link/vocab/creatorName
 Bismut, JeanMichel
 Dewey number
 515/.7242
 Index
 index present
 LC call number
 QA329.42
 LC item number
 .B57 2013
 Literary form
 non fiction
 Nature of contents

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

 Hypoelliptic operators
 Cohomology operations
 Geometry, Algebraic
 Mathematics
 KTheory
 Partial Differential Equations
 Global Analysis and Analysis on Manifolds
 MATHEMATICS
 Cohomology operations
 Geometry, Algebraic
 Hypoelliptic operators
 Label
 Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references and indexes
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents
 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
 Control code
 846845296
 Dimensions
 unknown
 Extent
 1 online resource.
 File format
 unknown
 Form of item
 online
 Isbn
 9783319001289
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other control number
 10.1007/9783319001289
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number
 (OCoLC)846845296
 Label
 Hypoelliptic Laplacian and BottChern cohomology : a theorem of RiemannRochGrothendieck in complex geometry, JeanMichel Bismut
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references and indexes
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents
 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
 Control code
 846845296
 Dimensions
 unknown
 Extent
 1 online resource.
 File format
 unknown
 Form of item
 online
 Isbn
 9783319001289
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other control number
 10.1007/9783319001289
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number
 (OCoLC)846845296
Subject
 Cohomology operations
 Cohomology operations
 Differential equations, partial.
 Geometry, Algebraic
 Geometry, Algebraic
 Geometry, Algebraic
 Global Analysis and Analysis on Manifolds
 Global Analysis and Analysis on Manifolds.
 Global analysis.
 Hypoelliptic operators
 Hypoelliptic operators
 Hypoelliptic operators
 KTheory
 Ktheory.
 MATHEMATICS  Functional Analysis
 Mathematics
 Mathematics.
 Partial Differential Equations
 Partial Differential Equations.
 Cohomology operations
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