38 research outputs found
Hodge theory on Cheeger spaces
We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary operators and the notion of mezzoperversity, which intermediates between the standard lower and upper middle perversities in intersection theory, as interpreted in this de Rham setting, and show that the de Rham operator with these boundary conditions is Fredholm and has compact resolvent. We also prove an isomorphism between the resulting Hodge and L2 de Rham cohomology groups, and that these are independent of the choice of iterated edge metric. On spaces which admit ideal boundary conditions of this type which are also self-dual, which we call ‘Cheeger spaces’, we show that these Hodge/de Rham cohomology groups satisfy Poincare' Duality
Infinitesimal deformations of a formal symplectic groupoid
Given a formal symplectic groupoid over a Poisson manifold ,
we define a new object, an infinitesimal deformation of , which can be
thought of as a formal symplectic groupoid over the manifold equipped with
an infinitesimal deformation of the Poisson bivector
field . The source and target mappings of a deformation of are
deformations of the source and target mappings of . To any pair of natural
star products having the same formal symplectic groupoid
we relate an infinitesimal deformation of . We call it the deformation
groupoid of the pair . We give explicit formulas for the
source and target mappings of the deformation groupoid of a pair of star
products with separation of variables on a Kaehler- Poisson manifold. Finally,
we give an algorithm for calculating the principal symbols of the components of
the logarithm of a formal Berezin transform of a star product with separation
of variables. This algorithm is based upon some deformation groupoid.Comment: 22 pages, the paper is reworked, new proofs are adde
A local families index formula for d-bar operators on punctured Riemann surfaces
Using heat kernel methods developed by Vaillant, a local index formula is
obtained for families of d-bar operators on the Teichmuller universal curve of
Riemann surfaces of genus g with n punctures. The formula also holds on the
moduli space M{g,n} in the sense of orbifolds where it can be written in terms
of Mumford-Morita-Miller classes. The degree two part of the formula gives the
curvature of the corresponding determinant line bundle equipped with the
Quillen connection, a result originally obtained by Takhtajan and Zograf.Comment: 47 page
L^2 rho form for normal coverings of fibre bundles
We define the secondary invariants L^2- eta and -rho forms for families of
generalized Dirac operators on normal coverings of fibre bundles. On the
covering family we assume transversally smooth spectral projections, and
Novikov--Shubin invariants bigger than 3(dim B+1) to treat the large time
asymptotic for general operators. In the particular case of a bundle of spin
manifolds, we study the L^2- rho class in relation to the space of positive
scalar curvature vertical metrics.Comment: 21 pages, revised versio
Pointwise Bounds for Steklov Eigenfunctions
Let (Ω,g) be a compact, real-analytic Riemannian manifold with real-analytic boundary ∂Ω. The harmonic extensions of the boundary Dirichlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfunctions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp h-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosphere bundle S∗∂Ω. These estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations Pu=0 near the characteristic set {σ(P)=0}
Oscilatory modules
Developing the ideas of Bressler and Soibelman and of Karabegov, we introduce
a notion of an oscillatory module on a symplectic manifold which is a sheaf of
modules over the sheaf of deformation quantization algebras with an additional
structure. We compare the category of oscillatory modules on a torus to the
Fukaya category as computed by Polishchuk and Zaslow.Comment: To appear in the proceedings of Moshe Flato Memorial Conference,
November, 2008, Ben Gurion Universit
Groupoids etale, eta invariants and index theory
Let G be a discrete finitely generated group. We consider a G- equivariant fibration, with fibers di¤eomorphic to a fixed even dimensional manifold with boundary Z and with base B. We assume that M is a Galois covering of a compact manifold with boundary. We consider a G-equivariant family of Dirac-type operators. Undery the assumption that the boundary family is L2 -invertible, we define an index class in the K-theory of the algebra obtained by taking the cross-product of C(B) and of G. If, in addition, G is of polynomial growth, we define higher indices by pairing the index class with suitable cyclic cocycles. Our main result is then a formula for these higher indices: the structure of the formula is as in the seminal work of Atiyah, Patodi and Singer, with an interior geometric contribution and a boundary contribution in the form of a higher eta invariant associated to the boundary family. Under similar assump- tions we extend our theorem to any G-proper manifold, with G an etale groupoid. We employ this generalization in order to establish a higher Atiyah-Patodi-Singer index formula on certain foliations with boundary. Fundamental to our work is a suitable generalization of Melrose b-pseudodi¤erential calculus as well as the superconnection proof of the index theorem on G-proper manifolds given by Gorokhovsky and Lott