Coverart for item
The Resource Hypoelliptic Laplacian and Bott-Chern cohomology : a theorem of Riemann-Roch-Grothendieck in complex geometry, Jean-Michel Bismut

Hypoelliptic Laplacian and Bott-Chern cohomology : a theorem of Riemann-Roch-Grothendieck in complex geometry, Jean-Michel Bismut

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
Jean-Michel Bismut
Creator
Subject
Language
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
Cataloging source
GW5XE
http://library.link/vocab/creatorName
Bismut, Jean-Michel
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
  • K-Theory
  • Partial Differential Equations
  • Global Analysis and Analysis on Manifolds
  • MATHEMATICS
  • Cohomology operations
  • Geometry, Algebraic
  • Hypoelliptic operators
Label
Hypoelliptic Laplacian and Bott-Chern cohomology : a theorem of Riemann-Roch-Grothendieck in complex geometry, Jean-Michel Bismut
Instantiates
Publication
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 Riemann-Roch-Grothendieck -- 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
10.1007/978-3-319-00128-9
Quality assurance targets
not applicable
Reformatting quality
unknown
Sound
unknown sound
Specific material designation
remote
System control number
(OCoLC)846845296
Label
Hypoelliptic Laplacian and Bott-Chern cohomology : a theorem of Riemann-Roch-Grothendieck in complex geometry, Jean-Michel Bismut
Publication
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 Riemann-Roch-Grothendieck -- 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
10.1007/978-3-319-00128-9
Quality assurance targets
not applicable
Reformatting quality
unknown
Sound
unknown sound
Specific material designation
remote
System control number
(OCoLC)846845296

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