000			03883nam a22004815i 4500
001			978-3-319-00128-9
003			DE-He213
005			20140220082837.0
007			cr nn 008mamaa
008			130524s2013 gw | s |||| 0|eng d
020			_a9783319001289 _9978-3-319-00128-9
024	7		_a10.1007/978-3-319-00128-9 _2doi
050		4	_aQA612.33
072		7	_aPBPD _2bicssc
072		7	_aMAT002010 _2bisacsh
082	0	4	_a512.66 _223
100	1		_aBismut, Jean-Michel. _eauthor.
245	1	0	_aHypoelliptic Laplacian and Bott–Chern Cohomology _h[electronic resource] : _bA Theorem of Riemann–Roch–Grothendieck in Complex Geometry / _cby Jean-Michel Bismut.
264		1	_aHeidelberg : _bSpringer International Publishing : _bImprint: Birkhäuser, _c2013.
300			_aXV, 203 p. 1 illus. in color. _bonline resource.
336			_atext _btxt _2rdacontent
337			_acomputer _bc _2rdamedia
338			_aonline resource _bcr _2rdacarrier
347			_atext file _bPDF _2rda
490	1		_aProgress in Mathematics ; _v305
505	0		_aIntroduction -- 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.  .
520			_aThe 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.
650		0	_aMathematics.
650		0	_aK-theory.
650		0	_aGlobal analysis.
650		0	_aDifferential equations, partial.
650	1	4	_aMathematics.
650	2	4	_aK-Theory.
650	2	4	_aPartial Differential Equations.
650	2	4	_aGlobal Analysis and Analysis on Manifolds.
710	2		_aSpringerLink (Online service)
773	0		_tSpringer eBooks
776	0	8	_iPrinted edition: _z9783319001272
830		0	_aProgress in Mathematics ; _v305
856	4	0	_uhttp://dx.doi.org/10.1007/978-3-319-00128-9
912			_aZDB-2-SMA
999			_c96360 _d96360