Bott-Chern Cohomology Research Articles

Bott–Chern cohomology of solvmanifolds

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 precisely, we can construct explicit finite-dimensional double complexes that allow to compute the Bott–Chern cohomology of compact quotients of complex Lie groups, respectively, of some Lie groups of the type $$\mathbb {C}^n\ltimes _\varphi N$$ where N is nilpotent. As an application, we compute the Bott–Chern cohomology of the complex parallelizable Nakamura manifold and of the completely solvable Nakamura manifold. In particular, the latter shows that the property of satisfying the $$\partial \overline{\partial }$$ -Lemma is not strongly closed under deformations of the complex structure.

QUATERNIONIC BOTT–CHERN COHOMOLOGY AND EXISTENCE OF HKT METRICS

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 Angella-Dloussky-Tomassini [3] for compact complex surfaces.

Quantitative and qualitative cohomological properties for non-Kähler manifolds

We introduce a "qualitative property" for Bott-Chern cohomology of complex non-K\"ahler 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 quantitative study of Bott-Chern cohomology. In this context, we also prove a new bound on the dimension of the Bott-Chern cohomology in terms of the Hodge numbers. We also give a generalization of this upper bound, with applications to symplectic cohomologies.

Aeppli–Bott-Chern cohomology and Deligne cohomology from a viewpoint of Harvey–Lawson’s spark complex

By comparing Deligne complex and Aeppli–Bott-Chern complex, we construct a differential cohomology \(\widehat{H}^*(X, *, *)\) that plays the role of Harvey–Lawson spark group \(\widehat{H}^*(X, *)\), and a cohomology \(H^*_{\mathrm{ABC}}(X; \mathbb Z(*, *))\) that plays the role of Deligne cohomology \(H^*_{\mathcal {D}}(X; \mathbb Z(*))\) for every complex manifold X. They fit in the short exact sequence $$\begin{aligned} 0\rightarrow H^{k+1}_{\mathrm{ABC}}(X; \mathbb Z(p, q)) \rightarrow \widehat{H}^k(X, p, q) \overset{\delta _1}{\rightarrow } Z^{k+1}_I(X, p, q) \rightarrow 0 \end{aligned}$$ and \(\widehat{H}^{\bullet }(X, \bullet , \bullet )\) possess ring structure and refined Chern classes, acted by the complex conjugation, and if some primitive cohomology groups of X vanish, there is a Lefschetz isomorphism. Furthermore, the ring structure of \(H^{\bullet }_{\mathrm{ABC}}(X; \mathbb Z(\bullet , \bullet ))\) inherited from \(\widehat{H}^{\bullet }(X, \bullet , \bullet )\) is compatible with the one of the analytic Deligne cohomology \(H^{\bullet }(X; \mathbb Z(\bullet ))\). We compute \(\widehat{H}^*(X, *, *)\) for X the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.

Coeffective basic cohomologies of K-contact and Sasakian 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 Betti numbers of $M$ are obtained. In the case of Sasakian manifolds, we define and study coeffective Dolbeault and Bott-Chern cohomologies for basic forms. Also, in this case, we prove some Hodge decomposition theorems for coeffective basic de Rham cohomology, relating this cohomology with coeffective basic Dolbeault or Bott-Chern cohomology. The notions are introduced in a similar manner with the case of symplectic and K\"ahler manifolds.

On Bott–Chern cohomology and formality

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

An analytic proof for the formula of the first order obstruction making the dimensions of Bott-Chern cohomology groups and Aeppli cohomology groups jumping

An analytic proof for the formula of the first order obstruction making the dimensions of Bott-Chern cohomology groups and Aeppli cohomology groups jumping

Aeppli and Bott–Chern cohomology for bi-generalized Hermitian manifolds and d′d″-lemma

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 cohomology groups. Some calculations of Bott–Chern cohomology groups of some bi-generalized Hermitian manifolds are given.

Degree of non-Kählerianity for 6-dimensional nilmanifolds

We use Bott-Chern cohomology to measure the non-K\"ahlerianity of 6-dimensional nilmanifolds endowed with the invariant complex structures in M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa's classification, [Invariant Complex Structures on 6-Nilmanifolds: Classification, Fr\"olicher Spectral Sequence and Special Hermitian Metrics, J. Geom. Anal. (2014)]. We investigate the existence of pluriclosed metric in connection with such a classification.

On Bott-Chern cohomology of compact complex surfaces

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

A parabolic flow of balanced metrics

Abstract We prove a general criterion to establish existence and uniqueness of a short-time solution to an evolution equation involving “closed” sections of a vector bundle, generalizing a method used by Bryant and Xu [8] for studying the Laplacian flow in G 2 $G_{2}$ -geometry. We apply this theorem in balanced geometry introducing a natural extension of the Calabi flow to the balanced case. We show that this flow has always a unique short-time solution belonging to the same Bott–Chern cohomology class of the initial balanced structure and that it preserves the Kähler condition. Finally, we study explicitly the flow on the Iwasawa manifold.

On the Bott–Chern cohomology and balanced Hermitian nilmanifolds

The Bott–Chern cohomology of six-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 [Formula: see text]-lemma condition and defined in terms of the Bott–Chern cohomology, and show that the vanishing of some of these invariants is not a closed property under holomorphic deformations. In the balanced case, we determine the spaces that parametrize deformations in type IIB supergravity described by Tseng and Yau in terms of the Bott–Chern cohomology group of bidegree (2, 2).

Cohomologies of certain orbifolds

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

Bott–Chern cohomology and q-complete domains

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

On the $\partial\overline{\partial}$ -Lemma and Bott-Chern cohomology

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.

The Cohomologies of the Iwasawa Manifold and of Its Small Deformations

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, completing the computations in arXiv:0709.3528v1 [math.AG] by M. Schweitzer.

Laplacien hypoelliptique et cohomologie de Bott–Chern

Let p : M → S be a proper submersion of complex manifolds, and let F be a holomorphic vector bundle on M. When R ⋅ p ⁎ F is locally free, we establish a Riemann–Roch–Grothendieck theorem in Bott–Chern cohomology. When M is equipped with a ∂ ¯ ∂ -closed ( 1 , 1 ) form inducing a Hermitian metric along the fibres, the proof is obtained by using elliptic superconnections. In the general case, we construct an exotic version of the hypoelliptic superconnections which we introduced in previous work.