This paper constructs a Hodge theory of noncompact topologically tame manifolds $M$. The main result is an isomorphism between the de Rham cohomology with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut Laplacian $\lap_μ$ associated to a measure $dμ$ which has sufficiently rapid growth at infinity on $M$. This follows from the construction of a space of forms associated to $\lap_μ$ which satisfy an ``extension by zero'' property. The ``extension by zero'' property is proved for manifolds with cylindrical ends possessing gaussian growth measures.
We give explicit harmonic representatives of Dolbeault cohomology of Oeljeklaus-Toma manifolds and show that they are geometrically Dolbeault formal. We also give explicit harmonic representatives of Bott-Chern cohomology of Oeljeklaus-Toma manifolds of type $(s,1)$ and study the Angella-Tomassini inequality.
A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bi-complex where one of the two operators is the classical $\overline\partial$-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex m…
We prove the Hodge symmetry type result on the Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in the direct sum of holomorphic line bundles. Consequently, we show the vanishing and non-vanishing of Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in holomorphic line bundles.
This article constructs Von Neumann invariants for constructible complexes and coherent D-modules on compact complex manifolds, generalizing the work of the author on coherent L 2-cohomology. We formulate a conjectural generalization of Dingoyan's L 2-Mixed Hodge structures in terms of Saito's Mixed Hodge Modules and give partial results in this direction. 2020 AMS Classification: 32J27.
On a smoothly stratified space, we identify intersection cohomology of any given perversity with an associated weighted $L^2$ cohomology for iterated fibred cusp metrics on the smooth stratum. In particular given a Witt space, we identify the $L^2$ cohomology of iterated fibred cusp metrics with the middle perversity intersection cohomology of the corresponding stratified space.
We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown-Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson's unstable splitting. As a consequence, we obtain an analog of Quillen's theorem for Hodge filtered Brown-Peterson cohomology for complex manifolds.
A systematic study of the contributions at infinity for the cohomology of variations of polarized Hodge structures over quasicompact Kähler manifolds. Several isomorphisms between different cohomologies given.
In this paper we show that every rational cohomology class of type $(p,p)$ on a compact Kähler manifold can be representated as a differential $(p,p)$-form given by an explicit formula involving a Čech cocycle. First we represent Chern characters of smooth vector bundles by Čech cocycles with values in the sheaf of differential forms. We then consider the behavior of these cocycles with respect to the Hodge structure on cohomology when the base manifold is compact Kähler.
We introduce a Hopf algebroid associated to a proper Lie group action on a smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is equal to the de Rham cohomology of invariant differential forms. When the action is cocompact, we develop a generalized Hodge theory for the de Rham cohomology of invariant differential forms. We prove that every cyclic cohomology class of the Hopf algebroid is represented by a generalized harmonic form. This implies that the space of cyclic cohomology of the Hopf algebroid is finite dimensional. As an application of the techniques developed …
A measured solenoid is a compact laminated space endowed with a transversal measure. The De Rham $L^2$-cohomology of the solenoid is defined by using differential forms which are smooth in the leafwise directions and $L^2$ in the transversal direction. We develop the theory of harmonic forms for Riemannian measured solenoids, and prove that this computes the De Rham $L^2$-cohomology of the solenoid. This implies in particular a Poincare duality result.
Given a compact Riemannian manifold $(M^n,g)$ and a fixed cohomology class, $[α^*] \in H^k(M)$, we consider the existence of a minimizer $α\in [α^*]$ of the generalized minimal surface energy $\int_M \sqrt{1+|α|^2} dV_g$. When $k = 1$, we prove the existence of unique minimizers for every cohomology class $[α^*]$. Next, when $k > 1$, we construct examples of singular solutions for finite cohomology class $[α^*] \in H^k(S^k \times S^k,g)$, where $g$ is conformal to the standard metric on $S^k \times S^k$. Additionally, we show that when $k=2$, these singular solutions are also solutions to the …
The purpose of this work is to propose a mixed Hodge structure over a CR manifold. As you know, for a CR manifold, Kohn-Rossi cohomology is naturally introduced. However, the relation between Kohn-Rossi cohomology and De Rham cohomology is not so well understood, even in Tanaka's work. We discuss this point.
We present a method to develop a Hodge theory for tangential cohomology of foliations by mimicing Witten's approach to ordinary Morse theory by perturbations of the Laplacian
We propose a version of the Hodge conjecture in Bott-Chern cohomology and using results from characterizing real holomorphic chains by real rectifiable currents to provide a proof for this question. We define a Bott-Chern differential cohomology and use atomic section theory of Harvey and Lawson to construct refined Bott-Chern classes for holomorphic vector bundles in this differential cohomology. These refined Bott-Chern classes transform naturally to standard Chern classes, Bott-Chern classes and Cheeger-Simons' refined Chern classes.
We compute the Dolbeault cohomology of geodesically convex domains contained in Cousin groups which satisfy a strong dispersiveness condition. As a consequence we obtain a description of the Dolbeault cohomology of Oeljeklaus-Toma manifolds and in particular the fact that the Hodge decomposition holds for their cohomology.
Let $V$ be a complex projective variety with isolated singularities. Let the smooth part be given the metric induced by a projective imbedding. Then we develop the $L_2$ harmonic theory and construct a pure Hodge structure on the $L_2$-cohomology of $V$. If the dimension of $V$ is two, we put a cohomological Hodge structure on its complex of sheaves of $L_2$ forms.
We introduce new finite-dimensional cohomologies on symplectic manifolds. Each exhibits Lefschetz decomposition and contains a unique harmonic representative within each class. Associated with each cohomology is a primitive cohomology defined purely on the space of primitive forms. We identify the dual currents of lagrangians and more generally coisotropic submanifolds with elements of a primitive cohomology, which dualizes to a homology on coisotropic chains.
Given a compact stratified pseudomanifold with a Thom-Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the $L^{2}$ de Rham and Hodge cohomology and the intersection cohomology of $X$ associated to some perversities. More precisely, to a kind of metric which we call \emph{quasi edge with weights}, we associate two general perversities in the sense of G. Friedman, $p_{g}$ and its dual $q_{g}$. We then show that the absolute $L^{2}$ Hodge cohomology is isomorphic to the maximal $L^{2}$ de Rham cohomology and this is in turn isomorph…
In this article we use the combinatorial and geometric structure of manifolds with embedded cylinders in order to develop an adiabatic decomposition of the Hodge cohomology of these manifolds. We will on the one hand describe the adiabatic behaviour of spaces of harmonic forms by means of a certain Čech-de Rham complex and on the other hand generalise the Cappell-Lee-Miller splicing map to the case of a finite number of edges, thus combining the topological and the analytic viewpoint. In parts, this work is a generalisation of works of Cappell, Lee and Miller in which a single-edged graph is c…
We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and $\A^1$-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms.
In this expository article, we outline the theory of harmonic differential forms and its consequences. We provide self-contained proofs of the following important results in differential geometry: (1) Hodge theorem, which states that for a compact complex manifold, the de Rham cohomology group is isomorphic to the group of harmonic forms, (2) Hodge decomposition theorem, which states that for a Kähler manifold, the de Rham cohomology group decomposes into the Dolbeault cohomology groups, and (3) The Kodaira Embedding theorem, which gives a criterion of when a compact complex manifold is in fac…
We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS `span' the space of all infinitesimal VHS; and (3) show that the cohomology classes dual the Schubert VHS form a basis of the invariant characteristic cohomology associated to the infinitesimal period relation (a.k.a. Griffiths transversality).
The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential $d_ω=d+ω\wedge$, where $ω$ is a closed $1$-form. We study Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy invariance and then extend it to more general type of forms on a Riemannian foliation. We study the Laplacian and Hodge decompositions for the corresponding differential operators on reduced leafwise Morse-Novikov complexes. In the…
We construct a simply-connected compact complex non-Kähler manifold satisfying the $\partial\bar\partial$-Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the $\partial\bar\partial$-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in \cite{rao-yang-yang, yang-yang, stelzig-blowup, stelzig-doublecomplex} with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology…
This is a sequel to our work in tropical Hodge theory. Our aim here is to prove a tropical analogue of the Clemens-Schmid exact sequence in asymptotic Hodge theory. As an application of this result, we prove the tropical Hodge conjecture for smooth projective tropical varieties which are rationally triangulable. This provides a partial answer to a question of Kontsevich who suggested the validity of the tropical Hodge conjecture could be used as a test for the validity of the Hodge conjecture.
The basic Dolbeault cohomology groups of a Sasakian manifold M are invariants of its characteristic foliation F (the orbit foliation of the Reeb flow). We show some fundamental properties of this cohomology, which are useful for its computation. In the first part of the article, we show that the basic Hodge numbers, the dimensions of the basic Dolbeault cohomology groups, only depend on the isomorphism class of the underlying CR structure. Equivalently, we show that the basic Hodge numbers are invariant under deformations of type I. This result reduces their computation to the quasi-regular ca…
We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms and therefore lead directly to the construction of primitive cohomologies on symplectic manifolds. Using these operators, we introduce new primitive cohomologies that are analogous to the Dolbeault cohomology in the complex theory. Interestingly, the finiteness of these primitive cohomologies follows directly from an elliptic complex. We calcul…
We construct real polarizable Hodge structures on the reduced leafwise cohomology of Kähler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's Kählerian analogue of the Weil conjectures carries over as well. Generalizing a construction of Looijenga and Lunts one obtains possibly infinite dimensional Lie algebras attached to Kähler-Riemann foliations. Finally using $(\mathfrak{g},K)$-cohomology we discuss a class of examples obtained by dividing a product of symmetric spaces by a cocompact lattice and considering t…
For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of $L^2$ harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the present paper, we obtain a generalization of this result to situations where, rather than a fibration at infinity, there is a Riemannian foliation with compact leaves admitting a resolution by a fibration. If the associated stratified space (obtained now by collapsing the leaves of…
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered cohomologies give a two-sided resolution of Lefschetz maps, and thereby, they are directly related to the kernels and cokernels of the Lefschetz maps. We also introduce a novel, non-associative product operation on differential forms for symplectic manifolds. This product generates…
We consider a compact, oriented, smooth Riemannian manifold $M$ (with or without boundary) and we suppose $G$ is a torus acting by isometries on $M$. Given $X$ in the Lie algebra and corresponding vector field $X_M$ on $M$, one defines Witten's inhomogeneous coboundary operator $d_{X_M} = d+ι_{X_M}: Ω_G^\pm \toΩ_G^\mp$ (even/odd invariant forms on $M$) and its adjoint $δ_{X_M}$. In the 1980s Witten showed that the resulting cohomology classes have $X_M$-harmonic representatives (forms in the null space of $Δ_{X_M} = (d_{X_M}+δ_{X_M})^2$), and the cohomology groups are isomorphic to the ordinar…
In this paper we prove that the cohomology of smooth projective tropical varieties verify the tropical analogs of three fundamental theorems which govern the cohomology of complex projective varieties: Hard Lefschetz theorem, Hodge-Riemann relations and monodromy-weight conjecture. On the way to establish these results, we introduce and prove other results of independent interest. This includes a generalization of the results of Adiprasito-Huh-Katz, Hodge theory for combinatorial geometries, to any unimodular quasi-projective fan having the same support as the Bergman fan of a matroid, a tropi…
We study the space of L^2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including asymptotically locally Euclidean manifolds, the (known types of) gravitational instantons, and also Poincaré metrics on Q-rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in Kähler manifolds. The answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold. The L^2 signatur…
