arXiv research

In studying the Bott-Chern and Aeppli cohomologies for q-complete manifolds, we introduce the class of cohomologically Bott-Chern q-complete manifolds.

Haefliger cohomology characterizes taut foliated manifolds by Haefliger's theorem. We show that Haefliger cohomology characterizes strongly tense foliated manifolds, namely, foliated manifolds which admit a Riemannian metric such that the mean curvature form of the leaves is closed and basic. We show that Haefliger coh…

Vanishing result for cohomology leads to extension theorem for pluriharmonic functions.

In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic $(0,q)$-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we e…

We prove that, for some classes of complex nilmanifolds, the Bott-Chern cohomology is completely determined by the Lie algebra associated to the nilmanifold with the induced complex structure. We use these tools to compute the Bott-Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations, com…

In this article, we first consider the $L^{2}$ \textit{Morse-Novikov cohomology} on a complete Riemannian manifold $M$ equipped with a parallel $1$-form which includes Vaisman manifold. Based on a vanishing theorem of $L^{2}$ \textit{Morse-Novikov cohomology}, we prove that the $L^{2}$-harmonic forms on $M$ are identic…

We prove a non-vanishing result for the $L_{q,p}$-cohomology of complete simply-connected Riemannian manifolds with pinched negative curvature.

The study shows strong formality in certain complex manifolds.

In this paper we extend previous results concerning the behaviour of JSJ decompositions of closed 3-manifolds with respect to the profinite completion to the case of compact 3-manifolds with boundary. We also illustrate an alternative and perhaps more natural approach to part of the original theorem, using relative coh…

A vanishing theorem for a convex cocompact hyperbolic manifold is established, which relates the L2 cohomology to the Hausdorff dimension of the limit set. The borderline case is shown to characterize the manifold completely.

We show vanishing theorems of $L^2$-cohomology groups of Kodaira-Nakano type on complete Hessian manifolds. We obtain further vanishing theorems of $L^2$-cohomology groups $L^2H^{p,q}(Ω)$ on a regular convex cone $Ω$ with the Cheng-Yau metric for $p>q$.

We give a short proof of the duality theorem for the reduced $L_p$-cohomology of a complete oriented Riemannian manifold.

We prove a theorem of Leray-Hirsch type and give an explicit blow-up formula for Dolbeault cohomology on (\emph{not necessarily compact}) complex manifolds. We give applications to strongly $q$-complete manifolds and the $\partial\bar{\partial}$-lemma.

Let V be a holomorphic bundle over a complex manifold M, and s be a holomorphic section of V. We study different types of cohomology associated to the Koszul complex induced by s. When M is complete, these cohomologies are isomorphic to each other and have self duality.

Let $(X,J,ω,g)$ be a complete $n$-dimensional Kähler manifold. A Theorem by Gromov \cite{G} states that the if the Kähler form is $d$-bounded, then the space of harmonic $L_2$ forms of degree $k$ is trivial, unless $k=\frac{n}{2}$. Starting with a contact manifold $(M,α)$ we show that the same conclusion does not hold …

The primitive cohomology of Calabi-Yau intersections is described using a twisted de Rham complex.

Consider a complete orientable manifold with countably many components of bounded dimension. Suppose that its rational homology is infinitely generated in some degree. Then there is no choice of weight function for which the natural map from weighted L^2 cohomology to de Rham cohomology is surjective in that degree.

Let G --> G' be an embedding of semisimple complex Lie groups, let B and B' be a pair of nested Borel subgroups, and let f:G/B --> G'/B' be the associated equivariant embedding of flag manifolds. We study the pullbacks of cohomologies of invertible sheaves on G'/B' along the embedding f. Let O' be a G'-equivariant inve…

The main result of the paper is a Borel type description of the $Sp(1)^n$-equivariant cohomology ring of the manifold $Fl_n(\mathbb{H})$ of all complete flags in $\mathbb{H}^n$. To prove this, we obtain a Goresky-Kottwitz-MacPherson type description of that ring.

We give a counter example to a conjecture of E. Bueler stating the equality between the DeRham cohomology of complete Riemannian manifold and a weighted $L^2$ cohomology where the weight is the heat kernel.

Investigates Künneth formula for foliated de Rham cohomology, overcoming non-Hausdorff issues.

Explains Hodge theory and Kodaira embedding theorem for complex manifolds.

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in 1982. In order to study its main properties, Gromov himself initiated the dual theory of bounded cohomology, that developed into an active and independent research field. Gromov's theory of bounded cohomology was based on the use of mult…

For a complete hyperbolic three manifold M, we consider the representations of its fundamental group obtained by composing a lift of the holonomy with complex finite dimensional representations of SL(2,C). We prove a vanishing result for the cohomology of M with coefficients twisted by these representations, using tech…

Develops Lefschetz theory for noncompact manifolds.

We relate the L^2 cohomology of a complete hyperbolic manifold to the invariant currents on its limit set.

We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to…

Cylindrical contact homology linked to Ehrhart polynomials and Chen-Ruan cohomology.

We study certain foliated complex manifolds that behave similarly to complete nonsingular toric varieties. We classify them by combinatorial objects that we call marked fans. We describe the basic cohomology algebras of them in terms of corresponding marked fans. We also study the basic Dolbeault cohomology algebras of…

We define exotic twisted $S^1$-equivariant cohomology for the loop space $LZ$ of a smooth manifold $Z$ via the invariant differential forms on $LZ$ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Cher…

We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings with semisimple symplectic cohomologies. Manifolds with these properties can be constructed by generalizing the boundary connected sum operation to our setting, and are related to certain birational surgeries like blow-d…

We give the complete Bott-Chern-Aeppli cohomology for compact complex 3-folds in terms of Dolbeault, Frolicher, a bi-degree DeRham-like type of cohomology, $K^{p,q}$, defined as $$ K^{p,q}=\frac{ker( \partial ) \cap ker( {\bar{\partial}}) }{im( \partial )\cap ker( {\bar{\partial}} )+im( {\bar{\partial}})\cap ker( \part…

We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomology. We use the connection to generate the complete set of conservation laws (as elements of the characteristic cohomology) for the Tzitzei…

Let $M$ be a complete connected Riemannian manifold of finite volume. In this paper we present a new method of constructing classes in bounded cohomology of transformation groups such as $Homeo_0(M,μ)$, $Diff_0(M,vol)$ and $Symp_0(M,ω)$ (in case $M$ is symplectic). As an application we show that, under certain conditio…

Study cohomology of Bigolin complex on complex manifolds.

The paper explores holonomy, zeta functions, and cohomology in foliated manifolds with stratified boundaries.

We describe the Cartan and Weil models of twisted equivariant cohomology together with the Cartan homomorphism among the two, and we extend the Chern-Weil homomorphism to the twisted equivariant cohomology. We clarify that in order to have a cohomology theory, the coefficients of the twisted equivariant cohomology must…

The paper bounds Betti numbers of complex-hyperbolic manifolds.

Study on deformation theory of nearly G2 manifolds with obstructions.

Analytic torsion and Reidemeister torsion help show exponential growth of torsion in cohomology.

We study conditions under which sub-complexes of a double complex of vector spaces allow to compute the Bott-Chern cohomology. We are especially aimed at studying the Bott-Chern cohomology of special classes of solvmanifolds, namely, complex parallelizable solvmanifolds and solvmanifolds of splitting type. More precise…

We show that for closed orientable manifolds the $k$-dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree $k$ that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles…

Develops obstruction theory for a specific 4-manifold index.

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$, and $θ$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy space $\mathcal{E}^p(X,θ)$, making it a complete geodesic metric space.

The study limits the cohomological dimension of certain affine manifolds with partially hyperbolic holonomy groups.

We introduce a new class of zero-dimensional weighted complete intersections, by abstracting the essential features of rational cohomology algebras of equal rank homogeneous spaces of compact connected Lie groups. We prove that, on a 1-connected closed manifold M whose rational cohomology algebra belongs to this class,…

Ends and cohomology theory for noncompact spaces.

Study $L^2$-cohomology in unbounded geometry manifolds.