Descript 
xv, 203 p. ; 25 cm 
Series 
Progress in Mathematics ; 305 

Progress in Mathematics ; 305

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 RiemannRochGrothendieck  Bibliography  Subject Index  Index of Notation. 

The book provides the proof of a complex geometric version of a wellknown 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 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 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 
Subject 
Mathematics


Ktheory


Global analysis


Differential equations, Partial


KTheory


Partial Differential Equations


Global Analysis and Analysis on Manifolds

