Hypoelliptic Laplacian and BottChern Cohomology: A Theorem of RiemannRochGrothendieck in Complex Geometry free ebook download.

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Hypoelliptic Laplacian and BottChern Cohomology: A Theorem of RiemannRochGrothendieck in Complex Geometry free ebook download

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Author(s):	poelliptic Laplacian and Bott–Chern Cohomology: A Theorem of Riemann–Roch–Grothendieck in Complex Ge

Date:	2017	Format:	PDF	Language:	English	ISBN/ASIN:	3319001272
Pages:	1	OCR:		Quality:		ISBN13:
Uploader:	alex21s	Upload Date:	2/7/2019 5:14:07 AM

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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 RiemannRochGrothendieck for proper submersions. It gives an equality of cohomology classes in BottChern cohomology, which is a refinement for complex manifolds of de Rham cohomology. When the manifolds are Khler, 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 McKeanSinger 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.

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2408281	Book	http://k2s.cc/file/1503d84c0ac52/3319001...		3.00	alex21s

Tags: Cohomology Complex Geometry Hypoelliptic Laplacian Theorem

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