arXiv research

New findings on complex manifold properties under deformations.

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 bigraded formality and Aeppli-Bott-Chern-Massey products on complex manifolds.

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

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

Study geometric formal metrics and Massey products on Kähler manifolds with torsion.

Study of deformed Bott-Chern cohomology on complex manifolds.

Study on properties of special Kähler metrics and their interplay.

Study harmonic representatives and cohomology of Oeljeklaus-Toma manifolds.

Compute Dolbeault and Bott-Chern cohomologies of complex solvmanifolds.

Study on geometrically formal metrics on complex manifolds.

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…

Formula derived for Bott-Chern classes in complex blow-ups.

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

Study on Kähler manifolds proves weak decompositions and relates harmonic forms.

The paper proves Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.

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

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

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

Study primitive decompositions for harmonic forms on almost Kähler manifolds.

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 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…

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 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…

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

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 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 …

Study shows gap between de Rham and symplectic-Bott-Chern harmonic forms for specific almost-Kähler 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 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…

The paper introduces new cohomologies and studies harmonic forms on almost 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…

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

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.

Study volumes of Bott-Chern classes on complex manifolds.

New cohomological obstruction found for astheno-Kahler metrics.

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.

Inequalities for symplectic cohomology groups are derived.

In this paper, we introduce six axioms for relative Bott-Chern secondary characteristic classes and prove the uniqueness and existence theorem for them. Such a work provides us a natural way to understand and hence to prove the arithmetic Grothendieck-Riemann-Roch theorem.

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

Generalizes double transgression formulas on complex manifolds.

Study categorizes Vaisman manifolds with vanishing first Chern class and finds canonical metrics.

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…

Develops equivariant Chern characters for coherent sheaves with group actions.

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…