arXiv research

Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.

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 Bo…

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

Study of deformed Bott-Chern cohomology on complex manifolds.

We introduce a "qualitative property" for Bott-Chern cohomology of complex non-Kähler manifolds, which is motivated in view of the study of the algebraic structure of Bott-Chern cohomology. We prove that such a property characterizes the validity of the $\partial\overline\partial$-Lemma. This follows from a quantitativ…

We study the symplectic Bott-Chern cohomology by L.-S. Tseng and S.-T. Yau for solvmanifolds endowed with left-invariant symplectic structures. Our results are applicable to cohomology with values in local systems. Studying symplectic Bott-Chern cohomology of solvmanifolds with values in local systems, we give some rem…

Introduces new cohomologies on complex manifolds, extending classical Bott-Chern and Aeppli.

Study new invariants in complex geometry using Bott-Chern hypercohomology.

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 study a geometric notion related to formality for Bott-Chern cohomology on complex manifolds.

New cohomological obstruction found for astheno-Kahler metrics.

Inequalities for symplectic cohomology groups are derived.

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

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…

The paper introduces new cohomologies and studies harmonic forms on almost complex manifolds.

This survey summarizes the results discussed in a talk at "Bielefeld Geometry & Topology Days" held at Bielefeld University in July 2015. We are interested in quantitative and qualitative properties of Bott-Chern cohomology. We announce new results obtained in [D. Angella, N. Tardini, Quantitative and qualitative cohom…

We define Aeppli and Bott-Chern cohomology for bi-generalized complex manifolds and show that they are finite dimensional for compact bi-generalized Hermitian manifolds. For totally bounded double complexes $(A, d', d'')$, we show that the validity of $d'd''$-lemma is equivalent to having the same dimension of several …

We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class $\text{VII}$ and for compact complex surfaces diffeomorphic to solvmanifolds.

The paper characterizes when the $\partial \overline{\partial}$-lemma holds for twistor spaces.

The Bott-Chern cohomology of 6-dimensional nilmanifolds endowed with invariant complex structure is studied with special attention to the cases when balanced or strongly Gauduchon Hermitian metrics exist. We consider complex invariants introduced by Angella and Tomassini and by Schweitzer, which are related to the $\pa…

We develop a theory of Cech-Bott-Chern cohomology and in this context we naturally come up with the relative Bott-Chern cohomology. In fact Bott-Chern cohomology has two relatives and they all arise from a single complex. Thus we study these three cohomologies in a unified way and obtain a long exact sequence involving…

Using the concept of a cohesive module defined by Block, we use the theory of superconnections in the sense of Quillen to construct natural superconnections on Hermitian cohesive modules. By the Chern-Weil construction, we obtain characteristic classes with values in Bott-Chern cohomology which refines the usual deRham…

Study harmonic representatives and cohomology of Oeljeklaus-Toma manifolds.

On a compact complex manifold $X$, we prove a Frölicher-type inequality for Bott-Chern cohomology and we show that the equality holds if and only if $X$ satisfies the $\partial\overline{\partial}$-Lemma.

A notion of geometric formality in the context of Bott-Chern and Aeppli cohomologies on a complex manifold is discussed. In particular, by using Aeppli-Bott-Chern-Massey triple products, it is proved that geometric Aeppli-Bott-Chern formality is not stable under small deformations of the complex structure.

Study $L^2$ Hilbert complexes on complex manifolds.

The purpose of this paper is to study the bimeromorphic invariants of compact complex manifolds in terms of Bott-Chern cohomology. We prove a blow-up formula for Bott-Chern cohomology. As an application, we show that for compact complex threefolds the non-Kählerness degrees, introduced by Angella-Tomassini [Invent. Mat…

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…

Study canonical deformations of complex forms and their cohomology properties.

We study Bott-Chern and Aeppli cohomologies of a vector space endowed with two anti-commuting endomorphisms whose square is zero. In particular, we prove an inequality à la Frölicher relating the dimensions of the Bott-Chern and Aeppli cohomologies to the dimensions of the Dolbeault cohomologies. We prove that the equa…

Introduces positivity for $(1,1)$ classes in foliated manifolds.

Paper constructs Chern character for coherent sheaves.

We study quaternionic Bott-Chern cohomology on compact hypercomplex manifolds and adapt some results from complex geometry to the quaternionic setting. For instance, we prove a criterion for the existence of HKT metrics on compact hypercomplex manifolds of real dimension 8 analogous to the one given by Teleman [35] and…

We provide further techniques to study the Dolbeault and Bott-Chern cohomologies of deformations of solvmanifolds by means of finite-dimensional complexes. By these techniques, we can compute the Dolbeault and Bott-Chern cohomologies of some complex solvmanifolds, and we also get explicit examples, showing in particula…

Develops equivariant Chern characters for coherent sheaves with group actions.

The paper extends Bott-Chern Laplacian definition and explores its properties on almost Hermitian manifolds.

In this paper we define coeffective de Rham cohomology for basic forms on a $K$--contact or Sasakian manifold $M$ and we discuss its relation with usually basic cohomology of $M$. When $M$ is of finite type (for instance it is compact) several inequalities relating some basic coeffective numbers to classical basic Bett…

We study the Bott-Chern cohomology of complex orbifolds obtained as quotient of a compact complex manifold by a finite group of biholomorphisms.

The paper explores moduli space of heterotic system using two deformation paths.

The study explores cohomological invariants and decomposes them into irreducible parts, focusing on zigzags.

In this paper we generalize the basic Lichnerowicz cohomology on transversally locally conformally Kählerian foliations and we study its relation with basic Bott-Chern cohomology and $0$--th basic Dolbeault cohomology with values in the associated foliated weight bundle.

Study on a nonlinear elliptic equation on compact Hermitian manifolds.

Compute Dolbeault and Bott-Chern cohomologies of complex solvmanifolds.

Let $G$ be a complex Lie group acting on a compact complex Hermitian manifold $M$ by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott-Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Che…

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same …

The study proves stability of a flow on specific Lie groups.

The paper generalizes current constructions to cohesive modules and characteristic forms.