The Resource Hypoelliptic Laplacian and Bott–Chern Cohomology : A Theorem of Riemann–Roch–Grothendieck in Complex Geometry, by Jean-Michel Bismut, (electronic resource)

Hypoelliptic Laplacian and Bott–Chern Cohomology : A Theorem of Riemann–Roch–Grothendieck in Complex Geometry, by Jean-Michel Bismut, (electronic resource)

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 Jean-Michel Bismut
Creator
Author
Author
Subject
Language
  • eng
  • eng
Summary
The book provides the proof of a complex geometric version of a well-known 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
Member of
http://library.link/vocab/creatorName
Bismut, Jean-Michel
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
  • K-theory
  • Differential equations, partial
  • Global analysis
  • K-Theory
  • 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 Jean-Michel Bismut, (electronic resource)
Instantiates
Publication
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
10.1007/978-3-319-00128-9
Specific material designation
remote
System control number
  • (CKB)2670000000371274
  • (EBL)1317088
  • (SSID)ssj0000904256
  • (PQKBManifestationID)11474233
  • (PQKBTitleCode)TC0000904256
  • (PQKBWorkID)10920116
  • (PQKB)10914553
  • (DE-He213)978-3-319-00128-9
  • (MiAaPQ)EBC1317088
  • (EXLCZ)992670000000371274
Label
Hypoelliptic Laplacian and Bott–Chern Cohomology : A Theorem of Riemann–Roch–Grothendieck in Complex Geometry, by Jean-Michel Bismut, (electronic resource)
Publication
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
10.1007/978-3-319-00128-9
Specific material designation
remote
System control number
  • (CKB)2670000000371274
  • (EBL)1317088
  • (SSID)ssj0000904256
  • (PQKBManifestationID)11474233
  • (PQKBTitleCode)TC0000904256
  • (PQKBWorkID)10920116
  • (PQKB)10914553
  • (DE-He213)978-3-319-00128-9
  • (MiAaPQ)EBC1317088
  • (EXLCZ)992670000000371274

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