Notice bibliographique
- Notice
000 02690n0 m 22000271 45as
001 FRBNF446742490000003
008 170801s 2013 deeng b 001
009 a
009 sx 1 x
017 .. $o OCoLC $a 892786440 $k ZWZ $l fre $n ncafnor
020 .. $a 9783319001289
051 .. $a txt $b c
245 1. $a Hypoelliptic Laplacian and Bott-Chern Cohomology $d Texte électronique $e A Theorem of Riemann-Roch-Grothendieck in Complex Geometry $f by Jean-Michel Bismut
260 .. $a Heidelberg $c Springer International Publishing $c Springer e-books $c Imprint: Birkhäuser $d 2013
280 .. $a 1 online resource
295 1. $a Progress in Mathematics $v 305
330 .. $a 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
