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Progress in Mathematics #305: Hypoelliptic Laplacian and Bott Chern Cohomology: A Theorem of Riemann Roch Grothendieck in Complex Geometry

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Progress in Mathematics #305: Hypoelliptic Laplacian and Bott Chern Cohomology: A Theorem of Riemann Roch Grothendieck in Complex Geometry Cover

 

Synopses & Reviews

Publisher Comments:

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.

Synopsis:

This 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.

About the Author

Jean-Michel Bismut is

Table of 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.​

Product Details

ISBN:
9783319001272
Author:
Bismut, Jean-michel
Publisher:
Birkhauser
Author:
Bismut, Jean-Michel
Location:
Cham
Subject:
Algebra - Abstract
Subject:
Riemann-Roch theorems and Chern characters
Subject:
analytic torsion
Subject:
determinants and determinant bundles
Subject:
heat and other parabolic equation methods
Subject:
hypoelliptic equations
Subject:
index theory and related fixed point theorems
Subject:
K-theory
Subject:
PARTIAL DIFFERENTIAL EQUATIONS
Subject:
Global Analysis and Analysis on Manifolds
Subject:
Mathematics-Abstract Algebra
Subject:
Global A
Subject:
nalysis and Analysis on Manifolds
Subject:
Mathematics
Subject:
B
Subject:
mathematics and statistics
Subject:
Differential equations, partial
Subject:
Global analysis.
Copyright:
Edition Description:
2013
Series:
Progress in Mathematics
Series Volume:
305
Publication Date:
20130531
Binding:
HARDCOVER
Language:
English
Pages:
220
Dimensions:
235 x 155 mm

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Related Subjects


Science and Mathematics » Mathematics » Algebra » Abstract Algebra
Science and Mathematics » Mathematics » Analysis General
Science and Mathematics » Mathematics » Differential Equations
Science and Mathematics » Mathematics » Geometry » Algebraic Geometry
Science and Mathematics » Mathematics » Topology

Progress in Mathematics #305: Hypoelliptic Laplacian and Bott Chern Cohomology: A Theorem of Riemann Roch Grothendieck in Complex Geometry New Hardcover
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Product details 220 pages Birkhauser - English 9783319001272 Reviews:
"Synopsis" by , This 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.
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