1,721,382 research outputs found
Sobolev Spaces and Bochner Laplacian on Complex Projective Varieties and Stratified Pseudomanifolds
Full text linkLet V⊂ CPn be an irreducible complex projective variety of complex dimension v and let g be the Kähler metric on reg (V) , the regular part of V, induced by the Fubini–Study metric of CPn. In (J Am Math Soc 8:857–877, 1995) Li and Tian proved that W01,2(reg(V),g)=W1,2(reg(V),g), that the natural inclusion W1 , 2(reg (V) , g) ↪ L2(reg (V) , g) is a compact operator and that the heat operator associated with the Friedrich extension of the scalar Laplacian Δ0:Cc∞(reg(V))→Cc∞(reg(V)), that is, e-tΔ0F:L2(reg(V),g)→L2(reg(V),g), is a trace class operator. The goal of this paper is to provide an extension of the above result to the case of Sobolev spaces of sections and symmetric Schrödinger type operators with potential bounded from below where the underlying Riemannian manifold is the regular part of a complex projective variety endowed with the Fubini–Study metric or the regular part of a stratified pseudomanifold endowed with an iterated edge metric
On the Laplace-Beltrami operator on compact complex spaces
Full text linkLet (X, h) be a compact and irreducible Hermitian complex space of complex dimension v > 1. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover, we provide some estimates for the growth of the corresponding eigenvalues, and we use these estimates to deduce that the associated heat operators are trace class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension 2
Poincaré duality, Hilbert complexes and geometric applications
Full text linkLet (M, g) be an open, oriented and incomplete riemannian manifold. The aim of this paper is to study the following two sequences of L2-cohomology groups:1.H2,m→Mi(M,g) defined as the image (H2,mini(M,g)→H2,maxi(M,g))2.H-2,m→Mi(M,g) defined as the image (H-2,mini(M,g)→H-2,maxi(M,g)). We show, under suitable hypothesis, that the first sequence is the cohomology of a Hilbert complex which contains the minimal one and is contained in the maximal one. In particular this leads us to prove a Hodge theorem for these groups. We also show that when the second sequence is finite dimensional then Poincaré duality holds and that, with the same assumptions, when dim(M)=4n then we can employ H-2,m→M2n(M,g) in order to define an L2-signature on M. We prove several applications to the intersection cohomology of compact smoothly stratified pseudomanifolds and we get some results about the Friedrichs extension δiF of δi
On the L 2-Poincaré duality for incomplete Riemannian manifolds: A general construction with applications
Full text linkLet (M,g) be an open, oriented and incomplete Riemannian manifold of dimension m. Under some general conditions we show the existence of a Hilbert complex (L2ωi(M,g),d,i) such that its cohomology groups, labeled with H2,i(M,g), satisfy the following properties: H2,i(M,g) =ker(dmax,i)/im(dmin,i) H2,i(M,g)=H 2,m-i(M,g) (Poincaré duality holds) There exists a well-defined and nondegenerate pairing: H2,i(M,g) × H 2,m-i(M,g), M If (L2ωi(M,g),d,i) is a Fredholm complex, then every closed extension of the de Rham complex (ωci(M),d i) is a Fredholm complex and, for each i = 0,..,m, the quotient (dmax,i)/(dmin,i) is a finite dimensional vector space
Degenerating Hermitian metrics, canonical bundle and spectral convergence
Full text linkLet (M, J) be a compact complex manifold of complex dimension m and let gs be a one-parameter family of Hermitian forms on M that are smooth and positive definite for each fixed s ∈ (0, 1] and that somehow degenerates to a Hermitian pseudometric h for s tending to 0. In this paper under rather general assumptions on gs we prove various spectral convergence type theorems for the family of Hodge-Kodaira Laplacians ∆∂,m,0,s associated to gs and acting on the canonical bundle of M. In particular we show that, as s tends to zero, the eigenvalues, the heat operators and the heat kernels corresponding to the family ∆∂,m,0,s converge to the eigenvalues, the heat operator and the heat kernel of ∆∂,m,0,abs, a suitable self-adjoint operator with entirely discrete spectrum defined on the limit space (A, h|A)
General perversities and L2 de Rham and Hodge theorems for stratified pseudomanifolds
Full text linkGiven a compact stratified pseudomanifold X with a Thom-Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the L2 de Rham and Hodge cohomology of the regular part of X and the intersection cohomology of X associated to some perversities. More precisely, to a kind of metric which we call quasi edge with weights, we associate two general perversities in the sense of G. Friedman, pg and its dual qg. We then show that:1.The absolute L2 Hodge cohomology is isomorphic to the maximal L2 de Rham cohomology and this is in turn isomorphic to the intersection cohomology associated to the perversity qg.2.The relative L2 Hodge cohomology is isomorphic to the minimal L2 de Rham cohomology and this is in turn isomorphic to the intersection cohomology associated to the perversity pg. Moreover we give a partial answer to the inverse question: given p, a general perversity in the sense of Friedman on X, is there a riemannian metric g on reg(X) such that an L2 de Rham and Hodge theorem holds for g and pα We then show that the answer is positive in the following two cases: if p is greater or equal to the upper middle perversity or if it is smaller or equal to the lower middle one. Finally we conclude giving several corollaries about the properties of these L2 Hodge and de Rham cohomology groups. © 2013 Elsevier Masson SAS
-cohomology, heat semigroup and stratified spaces
Full text linkLet be an incomplete Riemannian manifold of finite volume and let
. In the first part of this paper we prove that under certain
assumptions the inclusion of the space of -differential forms into that of
-differential forms gives rise to an injective/surjective map between the
corresponding and cohomology groups. Then in the second part we
provide various applications of these results to the curvature and the
intersection cohomology of compact Thom-Mather stratified pseudomanifolds and
complex projective varieties with only isolated singularities.Comment: Final version. To appear on The Journal of Geometric Analysi
Degenerating Hermitian metrics and spectral geometry of the canonical bundle
Full text linkLet (X,h) be a compact and irreducible Hermitian complex space of complex dimension m. In this paper we are interested in the Dolbeault operator acting on the space of L2 sections of the canonical bundle of reg(X), the regular part of X. More precisely let d ̅m,0:L2Ωm,0(reg(X),h)→L2Ωm,1(reg(X),h) be an arbitrarily fixed closed extension of ∂ ̅m,0:L2Ωm,0(reg(X),h)→L2Ωm,1(reg(X),h) where the domain of the latter operator is Ωcm,0(reg(X)). We establish various properties such as closed range of d ̅m,0, compactness of the inclusion D(d ̅m,0)↪L2Ωm,0(reg(X),h) where D(d ̅m,0), the domain of d ̅m,0, is endowed with the corresponding graph norm, and discreteness of the spectrum of the associated Hodge–Kodaira Laplacian d ̅m,0⁎∘d ̅m,0 with an estimate for the growth of its eigenvalues. Several corollaries such as trace class property for the heat operator associated to d ̅m,0⁎∘d ̅m,0, with an estimate for its trace, are derived. Finally in the last part we provide several applications to the Hodge–Kodaira Laplacian in the setting of both compact irreducible Hermitian complex spaces with isolated singularities and complex projective surfaces
Symplectic manifolds, L p -cohomology and q-parabolicity
Full text linkLet (M,ω,J,g) be a non-compact almost Kähler manifold. In this paper we provide various criteria that assure that ω k induces a non trivial class in the reduced L p maximal/minimal cohomology of (M,g). Furthermore in the last part we explore some topological applications of our results
q-parabolicity of stratified pseudomanifolds and other singular spaces
Full text linkThe main result of this paper is a sufficient condition to have a compact Thom–Mather stratified pseudomanifold endowed with a c^ -iterated edge metric on its regular part q-parabolic. Moreover, besides stratified pseudomanifolds, the q-parabolicity of other classes of singular spaces, such as compact complex Hermitian spaces, is investigated
- …
