Study quaternionic Bott-Chern cohomology and HKT metrics existence.
problem Existence of HKT metrics on hypercomplex manifolds.
method Adapt results from complex geometry to quaternionic setting and prove a criterion.
result Criterion for existence of HKT metrics on compact hypercomplex manifolds of real dimension 8.
Proposes a new Hodge conjecture in Bott-Chern cohomology.
problem Hodge conjecture in Bott-Chern cohomology.
method Characterization of real holomorphic chains, atomic section theory, refined Bott-Chern classes.
result Proof of a new Hodge conjecture in Bott-Chern cohomology.
Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.
problem Characterize Bott-Chern cohomology of Vaisman manifolds.
method Explicit description via basic cohomology, infer relationships between cohomology groups, show invariants are unbounded, cohomological characterization of formality.
result Bott-Chern and Dolbeault numbers determine each other for Vaisman manifolds, and cohomological invariants are unbounded.
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.
problem Deformation theory and cohomology of complex manifolds.
method Introduce a double complex structure and study its Bott-Chern cohomology.
result Established a deformation theory for Bott-Chern cohomology and computed deformed cohomology for specific manifolds.
The paper calculates cohomology for complex 3-folds using new cohomology types.
problem Calculating cohomology for compact complex 3-folds.
method Defined new cohomology types Kp,q and used them to calculate cohomology. result Complete Bott-Chern-Aeppli cohomology for various complex 3-folds.
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 ∂∂-Lemma. This follows from a quantitativ…
This paper extends Bott-Chern cohomology to coherent sheaves using superconnections.
problem Extending Bott-Chern cohomology to coherent sheaves.
method Using superconnections and the theory of superconnections in the sense of Quillen.
result Characteristic classes are independent of Hermitian metrics and depend on the homotopy class of the coherent module.
The paper develops a theory of Bott-Chern cohomology and proves a residue theorem.
problem Localization of Bott-Chern classes and Hermitian residues.
method Developed a unified theory of Cech-Bott-Chern cohomology, defined cup product and integration, and introduced local and global duality homomorphisms.
result Proved a residue theorem for vector bundles with compatible Hermitian connections.
Introduces new cohomologies on complex manifolds, extending classical Bott-Chern and Aeppli.
problem Characterizing and studying cohomologies on complex manifolds.
method Introducing Er-Bott-Chern and Er-Aeppli cohomologies, extending classical cohomologies. result Provides analogues of Serre duality and characterizes page-(r−1)-∂∂ˉ-manifolds. 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…
Study new invariants in complex geometry using Bott-Chern hypercohomology.
problem Understanding geometry through Bott-Chern hypercohomology and bimeromorphic invariants.
method Construct new invariants involving sheaf cohomology, establish blow-up formula and canonical morphism.
result Compute invariants for specific complex threefolds like Iwasawa manifolds and quintic threefolds.
We study a geometric notion related to formality for Bott-Chern cohomology on complex manifolds.
The paper proves a blow-up formula for Bott-Chern cohomology and shows the bimeromorphic invariance of non-Kählerness degrees on threefolds.
problem The bimeromorphic invariance of the ∂∂ˉ-Lemma on threefolds. method Proves a blow-up formula for Bott-Chern cohomology and applies it to show bimeromorphic invariance of non-Kählerness degrees.
result The non-Kählerness degrees are bimeromorphic invariants on compact complex threefolds, implying the bimeromorphic invariance of the ∂∂ˉ-Lemma. New cohomological obstruction found for astheno-Kahler metrics.
problem Existence of astheno-Kahler metrics
method New cohomological obstruction
result Found a new cohomological obstruction
Vanishing result for cohomology leads to extension theorem for pluriharmonic functions.
problem Extension of pluriharmonic functions on complex manifolds.
method Vanishing result for Bott-Chern cohomology combined with Ehrenpreis technique.
result Hartogs extension theorem for pluriharmonic functions on cohomologically (n−1)-complete manifolds. Inequalities for symplectic cohomology groups are derived.
problem Symplectic cohomology inequalities
method Morse-type inequalities for symplectic Bott-Chern and Aeppli cohomology groups
result Derived inequalities for symplectic cohomology groups
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.
problem Understanding cohomologies and harmonic forms on almost complex manifolds.
method Introducing new cohomologies (Bott-Chern and Aeppli) and studying associated harmonic forms.
result Bott-Chern cohomology of 1-forms is finite-dimensional on compact manifolds and provides an invariant.
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 …
The paper characterizes when the ∂∂-lemma holds for twistor spaces.
problem Characterizing the ∂∂-lemma for twistor spaces. method Study Bott-Chern and Aeppli cohomologies of twistor spaces.
result Explicit computation of Dolbeault cohomology for flat torus twistor space.
We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class VII and for compact complex surfaces diffeomorphic to solvmanifolds.
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…
Left-invariant forms solve Dolbeault cohomology for complex nilmanifolds.
problem Computing Dolbeault cohomology for complex nilmanifolds.
method Inducing isomorphisms in cohomology using left-invariant forms.
result Invariant forms compute Dolbeault cohomology in every bidegree.
Study harmonic representatives and cohomology of Oeljeklaus-Toma manifolds.
problem Dolbeault and Bott-Chern cohomology of Oeljeklaus-Toma manifolds.
method Explicit harmonic representatives and geometric analysis.
result Showed geometric Dolbeault formality and studied Angella-Tomassini inequality.
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 ∂∂-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 shows Dolbeault cohomology is unchanged by complex Lie group actions.
problem Understanding how complex Lie group actions affect Dolbeault cohomology.
method Analyzes Dolbeault cohomology of compact complex manifolds with group actions.
result Induced action on Dolbeault cohomology is trivial.
The paper studies cohomologies of hypercomplex manifolds and their dimensions.
problem Understanding cohomologies and dimensions of invariant and anti-invariant subgroups.
method Proving a compact hypercomplex manifold is C∞-pure-and-full under certain conditions and studying dimensions of subgroups. result Characterization of hyperkähler with torsion metrics in terms of the dimension of the Jˉ-invariant subgroup. Study L2 Hilbert complexes on complex manifolds.
problem Analyse L2 Hilbert complexes on complex manifolds. method Define and study L2 Aeppli-Bott-Chern Hilbert complex; examine properties on various manifolds; use self-adjoint extensions of differential operators. result Kernels of operators on compact Hermitian manifolds are isomorphic to Aeppli or Bott-Chern cohomology.
Study cohomological properties of complex manifolds, extending a result to higher dimensions.
problem Upper bound for Bott-Chern cohomology in terms of Betti numbers for compact complex surfaces.
method Suitable metric conditions, extending a result by A. Teleman to higher dimensions.
result Upper bound for Bott-Chern cohomology in terms of Betti numbers for compact complex manifolds.
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.
problem Understanding canonical deformations and cohomology of complex manifolds.
method Analyzes canonical Aeppli deformations and their relations to deformed cohomology.
result Proves the jumping formula for deformed Aeppli cohomology and constant dimension conditions.
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…
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…
Introduces positivity for (1,1) classes in foliated manifolds.
problem Understanding positivity in foliated manifolds.
method Introduces positivity for a real basic (1,1) class in basic Bott-Chern cohomology group and studies its relationship with negativity of transverse holomorphic sectional curvature. result Establishes the relationship between positivity and negativity in foliated manifolds.
Paper constructs Chern character for coherent sheaves.
problem Chern character for coherent sheaves with values in Bott-Chern cohomology.
method Based on Block's fundamental construction, constructs Chern character.
result Proves Riemann-Roch-Grothendieck formula for coherent sheaves.
The study examines cohomological aspects on complex and symplectic manifolds.
problem Understanding qualitative properties through quantitative cohomological information.
method Analysis of Bott-Chern and Aeppli cohomology groups.
result Established comparisons among cohomology group dimensions and proved the ∂∂-lemma and Hard-Lefschetz condition. Develops equivariant Chern characters for coherent sheaves with group actions.
problem Computing Chern characters for coherent sheaves on manifolds with group actions.
method Introduces equivariant Chern characters and proves Riemann-Roch-Grothendieck theorem in Bott-Chern cohomology.
result Establishes a Riemann-Roch-Grothendieck theorem for coherent sheaves with finite group actions.
The paper extends Bott-Chern Laplacian definition and explores its properties on almost Hermitian manifolds.
problem Exploring the properties of Bott-Chern Laplacian on almost Hermitian manifolds.
method Extending the definition of Bott-Chern Laplacian, proving ellipticity, and analyzing kernels on different types of manifolds.
result The dimensions of Bott-Chern and Dolbeault harmonic forms differ on almost complex 4-manifolds with specific metrics.
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.
problem Exploring the moduli space of the heterotic system.
method Considering two dual deformation paths starting from a Kähler solution, one along Bott-Chern cohomology class and the other along Aeppli cohomology class. Using the implicit function theorem to prove local existence of heterotic solutions.
result Established an initial step to construct local moduli coordinates around a Kähler solution.
The study explores cohomological invariants and decomposes them into irreducible parts, focusing on zigzags.
problem Finding cohomological invariants and their decomposition into irreducible parts.
method Investigates various cohomological invariants on double complexes, focusing on the multiplicities of zigzags.
result The multiplicities of zigzags in double complexes are not sufficient to distinguish non-isomorphic double complexes.
Study cohomology of quaternionic foliations and orbifolds.
problem Understanding cohomology of quaternionic foliations and orbifolds.
method Definition and proof of foliated versions of classical results for quaternionic Kähler manifolds.
result Formulation and proof of foliated versions of classical results for quaternionic Kähler manifolds.
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.
problem Solving a nonlinear elliptic equation on compact Hermitian manifolds.
method Investigation of the deformed Hermitian-Yang-Mills equation.
result Analysis of eigenvalues and their relation to the equation.