Hypoelliptic Laplacian and Bott-Chern cohomology a theorem of Riemann-Roch-Grothendieck in complex geometry Jean-Michel Bismut

Format:

Book

Author/Creator:

Bismut, Jean-Michel

Series:

Progress in mathematics (Boston, Mass.) v. 305

Progress in mathematics v. 305

Language:

English

Subjects (All):

Hypoelliptic operators.

Cohomology operations.

Geometry, Algebraic.

Mathematics.

K-Theory.

Partial Differential Equations.

Global Analysis and Analysis on Manifolds.

Local Subjects:

Mathematics.

K-Theory.

Partial Differential Equations.

Global Analysis and Analysis on Manifolds.

Physical Description:

1 online resource

Place of Publication:

Cham New York Birkhäuser ©2013

Summary:

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

Contents:

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 Riemann-Roch-Grothendieck

Bibliography

Subject Index

Index of Notation

Introduction The Riemannian adiabatic limit The holomorphic adiabatic limit The elliptic superconnections The elliptic superconnection forms The elliptic superconnections forms when ̄MMωM=0 The hypoelliptic superconnections The hypoelliptic superconnection forms The hypoelliptic superconnection forms of vector bundles The hypoelliptic superconnection forms when ̄MMωM=0 The exotic superconnection forms of a vector bundle Exotic superconnections and Riemann-Roch-Grothendieck

Notes:

Includes bibliographical references and indexes

Other Format:

Printed edition:

ISBN:

9783319001289

3319001280

OCLC:

846845296

Access Restriction:

Restricted for use by site license