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The Resource Hypoelliptic Laplacian and Bott–Chern Cohomology : A Theorem of Riemann–Roch–Grothendieck in Complex Geometry, by JeanMichel Bismut, (electronic resource)
Hypoelliptic Laplacian and Bott–Chern Cohomology : A Theorem of Riemann–Roch–Grothendieck in Complex Geometry, by JeanMichel Bismut, (electronic resource)
Resource Information
The item Hypoelliptic Laplacian and Bott–Chern Cohomology : A Theorem of Riemann–Roch–Grothendieck in Complex Geometry, by JeanMichel Bismut, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Oklahoma Libraries.This item is available to borrow from all library branches.
Resource Information
The item Hypoelliptic Laplacian and Bott–Chern Cohomology : A Theorem of Riemann–Roch–Grothendieck in Complex Geometry, by JeanMichel Bismut, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Oklahoma Libraries.
This item is available to borrow from all library branches.
 Summary
 The book provides the proof of a complex geometric version of a wellknown result in algebraic geometry: the theorem of Riemann–Roch–Grothendieck for proper submersions. It gives an equality of cohomology classes in Bott–Chern 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 McKean–Singer 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
 eng
 Edition
 1st ed. 2013.
 Extent
 1 online resource (210 p.)
 Note
 Description based upon print version of record
 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 Riemann–Roch–Grothendieck
 Bibliography
 Subject Index
 Index of Notation.
 Isbn
 9783319001289
 Label
 Hypoelliptic Laplacian and Bott–Chern Cohomology : A Theorem of Riemann–Roch–Grothendieck in Complex Geometry
 Title
 Hypoelliptic Laplacian and Bott–Chern Cohomology
 Title remainder
 A Theorem of Riemann–Roch–Grothendieck in Complex Geometry
 Statement of responsibility
 by JeanMichel Bismut
 Language

 eng
 eng
 Summary
 The book provides the proof of a complex geometric version of a wellknown result in algebraic geometry: the theorem of Riemann–Roch–Grothendieck for proper submersions. It gives an equality of cohomology classes in Bott–Chern 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 McKean–Singer 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
 http://library.link/vocab/creatorName
 Bismut, JeanMichel
 Dewey number
 514.23
 http://bibfra.me/vocab/relation/httpidlocgovvocabularyrelatorsaut
 ONqjlhwsMYA
 Language note
 English
 LC call number
 QA612.33
 Literary form
 non fiction
 Nature of contents
 dictionaries
 Series statement
 Progress in Mathematics,
 Series volume
 305
 http://library.link/vocab/subjectName

 Ktheory
 Differential equations, partial
 Global analysis
 KTheory
 Partial Differential Equations
 Global Analysis and Analysis on Manifolds
 Label
 Hypoelliptic Laplacian and Bott–Chern Cohomology : A Theorem of Riemann–Roch–Grothendieck in Complex Geometry, by JeanMichel Bismut, (electronic resource)
 Note
 Description based upon print version of record
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code
 cr
 Content category
 text
 Content type code
 txt
 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 Riemann–Roch–Grothendieck  Bibliography  Subject Index  Index of Notation.
 Dimensions
 unknown
 Edition
 1st ed. 2013.
 Extent
 1 online resource (210 p.)
 Form of item
 online
 Isbn
 9783319001289
 Media category
 computer
 Media type code
 c
 Other control number
 10.1007/9783319001289
 Specific material designation
 remote
 System control number

 (CKB)2670000000371274
 (EBL)1317088
 (SSID)ssj0000904256
 (PQKBManifestationID)11474233
 (PQKBTitleCode)TC0000904256
 (PQKBWorkID)10920116
 (PQKB)10914553
 (DEHe213)9783319001289
 (MiAaPQ)EBC1317088
 (EXLCZ)992670000000371274
 Label
 Hypoelliptic Laplacian and Bott–Chern Cohomology : A Theorem of Riemann–Roch–Grothendieck in Complex Geometry, by JeanMichel Bismut, (electronic resource)
 Note
 Description based upon print version of record
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code
 cr
 Content category
 text
 Content type code
 txt
 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 Riemann–Roch–Grothendieck  Bibliography  Subject Index  Index of Notation.
 Dimensions
 unknown
 Edition
 1st ed. 2013.
 Extent
 1 online resource (210 p.)
 Form of item
 online
 Isbn
 9783319001289
 Media category
 computer
 Media type code
 c
 Other control number
 10.1007/9783319001289
 Specific material designation
 remote
 System control number

 (CKB)2670000000371274
 (EBL)1317088
 (SSID)ssj0000904256
 (PQKBManifestationID)11474233
 (PQKBTitleCode)TC0000904256
 (PQKBWorkID)10920116
 (PQKB)10914553
 (DEHe213)9783319001289
 (MiAaPQ)EBC1317088
 (EXLCZ)992670000000371274
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