This book presents the analytic foundations to the theory of the hypoelliptic Laplacian. The hypoelliptic Laplacian, a second-order operator acting on the cotangent bundle of a compact manifold, is supposed to interpolate between the classical Laplacian and the geodesic flow. Jean-Michel Bismut and Gilles Lebeau establish the basic functional analytic properties of this operator, which is also studied from the perspective of local index theory and analytic torsion.
The book shows that the hypoelliptic Laplacian provides a geometric version of the Fokker-Planck equations. The...
This book presents the analytic foundations to the theory of the hypoelliptic Laplacian. The hypoelliptic Laplacian, a second-order operator acting...
This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation,...
This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. ...
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 Kahler, 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,...
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 ...
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 Kahler, 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,...
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 ...
This monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group of coherent sheaves of a compact complex manifold with values in its Bott-Chern cohomology, and the proof of a corresponding Riemann-Roch-Grothendieck theorem. One main tool used is the equivalence of categories established by Block between the derived category of bounded complexes with coherent cohomology and the homotopy category of antiholomorphic superconnections. Chern-Weil theoretic techniques are then used to construct forms that...
This monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group of coh...
