Progress in Mathematics

Hypoelliptic Laplacian and Bott–Chern Cohomology

A Theorem of Riemann–Roch–Grothendieck in Complex Geometry

Authors: Bismut, Jean-Michel

  • Gives an important application of the theory of the hypoelliptic Laplacian in complex algebraic geometry  
  • Provides an introduction to applications of Quillen's superconnections in complex geometry with hypoelliptic operators  
  • Presents several techniques partly inspired from physics, which concur to the proof of a result in complex algebraic geometry  
  • The method of hypoelliptic deformation of the classical Laplacian was developed by the author during the last ten years  
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About this book

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.

About the authors

Jean-Michel Bismut is Professor of Mathematics at Université Paris-Sud (Orsay) and a member of the Académie des Sciences. Starting with a background in probability, he has worked extensively on index theory. With Gillet, Soulé, and Lebeau, he contributed to the proof of a theorem of Riemann–Roch–Grothendieck in Arakelov geometry. More recently, he has developed a theory of the hypoelliptic Laplacian, a family of operators that deforms the classical Laplacian, and provides a link between spectral theory and dynamical systems.

Table of contents (12 chapters)

Buy this book

eBook $79.99
price for USA (gross)
  • ISBN 978-3-319-00128-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $109.00
price for USA
  • ISBN 978-3-319-00127-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $109.00
price for USA
  • ISBN 978-3-319-03389-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Hypoelliptic Laplacian and Bott–Chern Cohomology
Book Subtitle
A Theorem of Riemann–Roch–Grothendieck in Complex Geometry
Authors
Series Title
Progress in Mathematics
Series Volume
305
Copyright
2013
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel
eBook ISBN
978-3-319-00128-9
DOI
10.1007/978-3-319-00128-9
Hardcover ISBN
978-3-319-00127-2
Softcover ISBN
978-3-319-03389-1
Series ISSN
0743-1643
Edition Number
1
Number of Pages
XV, 203
Topics