Format:
Online-Ressource (XV, 203 p. 1 illus. in color, digital)
ISBN:
9783319001289
Series Statement:
Progress in Mathematics 305
Content:
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. .
Content:
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.
Note:
Includes bibliographical references and index
,
Preface; Contents; Chapter 0 Introduction; 0.1 The main result; 0.2 Background; 0.3 Local index theorem and Kähler fibrations; 0.4 The elliptic superconnections for arbitrary ωM; 0.5 The form 0: the case where ∂ -M ∂MωM=0; 0.6 The hypoelliptic superconnections; 0.7 The exotic superconnections; 0.8 The hidden role of functional integration; 0.9 Operators and characteristic forms; 0.10 The hypoelliptic Laplacian; 0.11 Applications; 0.12 The organization of the book; 0.13 Acknowledgments; Chapter 1 The Riemannian adiabatic limit; 1.1 A smooth submersion
,
1.2 The limit of the Levi-Civita connection as e→01.3 The trilinear form ρ0; Chapter 2 The holomorphic adiabatic limit; 2.1 A holomorphic fibration; 2.2 The limit as → 0 of the connection ∇TM; 2.3 The Riemannian and holomorphic adiabatic limits; 2.4 The case where ωM is closed; 2.5 The exotic connections on TRM; 2.6 The exotic connections on TRX; 2.7 The asymptotics of the exotic connections on TRM; Chapter 3 The elliptic superconnections; 3.1 The Clifford algebra; 3.2 The ∂ operator on M; 3.3 The antiholomorphic superconnections; 3.4 The variation of the volume form on X
,
3.5 The adjoint superconnections3.6 The elliptic superconnections A,B,C; 3.7 The Levi-Civita superconnection; 3.8 A formula relating B and ALC; 3.9 The curvature of the superconnection B; 3.10 A curvature identity when ∂M ∂M ωM=0; Chapter 4 The elliptic superconnection forms; 4.1 Bott-Chern cohomology and characteristic classes; 4.2 The scaling of the form ωM; 4.3 A compact Lie group action; 4.4 Supertraces; 4.5 The elliptic superconnection forms αg,t; 4.6 The expansion of the forms αg,t, γg,t at t→0; 4.7 The dependence of the forms αg,t on ωM, gF; 4.8 The direct image as a sheaf
,
4.9 The elliptic Quillen metric4.10 The case where R·p*F is locally free; 4.11 A non-explicit formula for chg,BC (R·p*F); Chapter 5 The elliptic superconnections forms when ∂M∂MωM=0; 5.1 An evaluation of αg,0 when -∂M∂MωM=0; 5.2 A proof of Theorem 0.1.1 when -∂M∂MωM=0; Chapter 6 The hypoelliptic superconnections; 6.1 The total space of TX and its superconnections; 6.2 A holomorphic section of π*TX; 6.3 The superconnections on Ω·(X,I); 6.4 A formula for AZ; 6.5 The Hermitian forms ,η; 6.6 Another expression for,; 6.7 The fibrewise connections on p*Λ·(T*CS) ⊗F
,
6.8 A formula for the curvature of AZ6.9 Hypoelliptic and elliptic superconnections; Chapter 7 The hypoelliptic superconnection forms; 7.1 The 2-parameter hypoelliptic superconnections; 7.2 Supertraces; 7.3 The hypoelliptic superconnection forms αg,b,t; 7.4 The expansion of the forms αg,b,t, γg,b,t as t→0; 7.5 The dependence of the forms αg,b,t on ωM,gTX,gF; 7.6 The dependence on b of the forms αg,b,t; 7.7 The hypoelliptic Quillen metrics; Chapter 8 The hypoelliptic superconnection forms of vector bundles; 8.1 Two holomorphic structures
,
8.2 A hypoelliptic superconnection over the total space of E
Additional Edition:
ISBN 9783319001272
Additional Edition:
Erscheint auch als Druck-Ausgabe Bismut, Jean-Michel, 1948 - Hypoelliptic laplacian and Bott-Chern cohomology Cham [u.a.] : Birkhäuser, 2013 ISBN 9783319001272
Language:
English
Subjects:
Mathematics
Keywords:
Elliptische Differentialgleichung
;
Mannigfaltigkeit
;
Elliptische Differentialgleichung
;
Mannigfaltigkeit
DOI:
10.1007/978-3-319-00128-9
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Author information:
Bismut, Jean-Michel 1948-
