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)
Resource Information
The item Hypoelliptic Laplacian and Bott-Chern cohomology : a theorem of Riemann-Roch-Grothendieck in complex geometry, Jean-Michel Bismut, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in Bowdoin College Library.This item is available to borrow from 1 library branch.
Resource Information
The item Hypoelliptic Laplacian and Bott-Chern cohomology : a theorem of Riemann-Roch-Grothendieck in complex geometry, Jean-Michel Bismut, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in Bowdoin College Library.
This item is available to borrow from 1 library branch.
- 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
- 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
- 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
- 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. --
- 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)
- 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)
- 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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<div class="citation" vocab="http://schema.org/"><i class="fa fa-external-link-square fa-fw"></i> Data from <span resource="http://link.bowdoin.edu/portal/Hypoelliptic-Laplacian-and-Bott-Chern-cohomology/ewRZpbXmyBs/" typeof="Book http://bibfra.me/vocab/lite/Item"><span property="name http://bibfra.me/vocab/lite/label"><a href="http://link.bowdoin.edu/portal/Hypoelliptic-Laplacian-and-Bott-Chern-cohomology/ewRZpbXmyBs/">Hypoelliptic Laplacian and Bott-Chern cohomology : a theorem of Riemann-Roch-Grothendieck in complex geometry, Jean-Michel Bismut, (electronic resource)</a></span> - <span property="potentialAction" typeOf="OrganizeAction"><span property="agent" typeof="LibrarySystem http://library.link/vocab/LibrarySystem" resource="http://link.bowdoin.edu/"><span property="name http://bibfra.me/vocab/lite/label"><a property="url" href="https://link.bowdoin.edu/">Bowdoin College Library</a></span></span></span></span></div>