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Hypoelliptic Laplacian and Bott-Chern Cohomology - A Theorem of Riemann-Roch-Grothendieck in Complex Geometry (Paperback, 2013 ed.)
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Hypoelliptic Laplacian and Bott-Chern Cohomology - A Theorem of Riemann-Roch-Grothendieck in Complex Geometry (Paperback, 2013 ed.)
Series: Progress in Mathematics, 305
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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 Kahler, 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.
General
Imprint: |
Birkhauser Verlag AG
|
Country of origin: |
Switzerland |
Series: |
Progress in Mathematics, 305 |
Release date: |
June 2015 |
First published: |
2013 |
Authors: |
Jean-Michel Bismut
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Dimensions: |
235 x 155 x 12mm (L x W x T) |
Format: |
Paperback
|
Pages: |
203 |
Edition: |
2013 ed. |
ISBN-13: |
978-3-319-03389-1 |
Categories: |
Books >
Science & Mathematics >
Mathematics >
Topology >
Algebraic topology
|
LSN: |
3-319-03389-1 |
Barcode: |
9783319033891 |
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