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The Resource Hypoelliptic Laplacian and Bott-Chern cohomology : a theorem of Riemann-Roch-Grothendieck in complex geometry, Jean-Michel Bismut, (electronic resource)

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

Creator
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. --
Language
eng
Work
Publication
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 Riemann-Roch-Grothendieck
Isbn
9783319001272
Instance
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
Assigning source
Source other than Library of Congress
Cataloging source
OHX
http://library.link/vocab/creatorName
Bismut, Jean-Michel
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 Bott-Chern cohomology : a theorem of Riemann-Roch-Grothendieck in complex geometry, Jean-Michel Bismut, (electronic resource)
Instantiates
Publication
Bibliography note
Includes bibliographical references (pages 191-195) 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 Riemann-Roch-Grothendieck
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
10.1007/978-3-319-00128-9
Specific material designation
remote
System control number
(WaSeSS)ssj0000904256
Label
Hypoelliptic Laplacian and Bott-Chern cohomology : a theorem of Riemann-Roch-Grothendieck in complex geometry, Jean-Michel Bismut, (electronic resource)
Publication
Bibliography note
Includes bibliographical references (pages 191-195) 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 Riemann-Roch-Grothendieck
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
10.1007/978-3-319-00128-9
Specific material designation
remote
System control number
(WaSeSS)ssj0000904256

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