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Field name	Details
Dewey Class	512.66
Title	Hypoelliptic Laplacian and Bott-Chern Cohomology ([Ebook]) : A Theorem of Riemann-Roch-Grothendieck in Complex Geometry / by Jean-Michel Bismut.
Author	Bismut, Jean-Michel
Other name(s)	SpringerLink (Online service)
Publication	Heidelberg : Springer International Publishing
Publication	, 2013.
Physical Details	XV, 203 pages : 1 illus. in color. : online resource.
Series	Progress in mathematics ; 305
ISBN	9783319001289
Summary Note	The book provides the proof of a complex geometric version of a well-known result in algebraic geometry: the theorem of RiemannâRochâGrothendieck for proper submersions. It gives an equality of cohomology classes in BottâChern cohomology, which is a refinement for complex manifolds of de Rham cohomology. When the manifolds are KÃ¤hler, 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 McKeanâSinger in local index theory. In the general case, this approach breaks down because the cancellations do not occur any more.Â 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 Â 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.Â 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.:
Contents note	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 RiemannâRochâGrothendieck -- Bibliography -- Subject Index -- Index of Notation. Â .
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users).
Internet Site	http://dx.doi.org/10.1007/978-3-319-00128-9
Links to Related Works	Subject References: Differential equations, Partial . Global analysis . Global Analysis and Analysis on Manifolds . K-Theory . Mathematics . Partial differential equations . Authors: Bismut, Jean-Michel . Corporate Authors: SpringerLink (Online service) . Series: Progress in mathematics . Classification: 512.66 . 512.66 (DDC 22) .

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Hypoelliptic Laplacian and Bott-Chern Cohomology: A Theorem of Riemann-Roch-Grothendieck in Complex Geometry

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