Bott-Chern Cohomology Research Articles

The L2 Aeppli-Bott-Chern Hilbert complex

We analyse the L2 Hilbert complexes naturally associated to a non-compact complex manifold, namely the ones which originate from the Dolbeault and the Aeppli-Bott-Chern complexes. In particular we define the L2 Aeppli-Bott-Chern Hilbert complex and examine its main properties on general Hermitian manifolds, on complete Kähler manifolds and on Galois coverings of compact complex manifolds. The main results are achieved through the study of self-adjoint extensions of various differential operators whose kernels, on compact Hermitian manifolds, are isomorphic to either Aeppli or Bott-Chern cohomology.

Balanced and Aeppli parameters for the heterotic moduli

In this paper, we fix the complex structure and explore the moduli space of the heterotic system by considering two different yet “dual” deformation paths starting from a Kähler solution. They correspond to deformation along the Bott–Chern cohomology class and the Aeppli cohomology class, respectively. Using the implicit function theorem, we prove the stability of the existence of heterotic solutions under these two deformations and hence establish an initial step in constructing local moduli coordinates around a Kähler solution.

Bott-Chern cohomology and the Hartogs extension theorem for pluriharmonic functions

Bott-Chern cohomology and the Hartogs extension theorem for pluriharmonic functions

On Bott–Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms

In this paper we introduce several new cohomologies of almost complex manifolds, among which stands a generalization of Bott–Chern and Aeppli cohomologies defined using the operators d, dc. We explain how they are connected to already existing cohomologies of almost complex manifolds and we study the spaces of harmonic forms associated to d, dc, showing their relation with Bott–Chern and Aeppli cohomologies and to other well-studied spaces of harmonic forms. Notably, Bott–Chern cohomology of 1-forms is finite-dimensional on compact manifolds and provides an almost complex invariant hd+dc1 that distinguishes between almost complex structures. On almost Kähler 4-manifolds, the spaces of harmonic forms we consider are particularly well-behaved and are linked to harmonic forms considered by Tseng and Yau in the study of symplectic cohomology.

Bott-Chern cohomology of compact Vaisman manifolds

We give an explicit description of the Bott-Chern cohomology groups of a compact Vaisman manifold in terms of the basic cohomology. We infer that the Bott-Chern numbers and the Dolbeault numbers of a Vaisman manifold determine each other. On the other hand, we show that the cohomological invariants Δ k \Delta ^k introduced by Angella-Tomassini are unbounded for Vaisman manifolds. Finally, we give a cohomological characterization of the Dolbeault and Bott-Chern formality for Vaisman metrics.

Intersection theory and Chern classes in Bott–Chern cohomology

In this article, we investigate an axiomatic approach introduced by Grivaux for the study of rational Bott-Chern cohomology, and use it in that context to define Chern classes of coherent sheaves. This method also allows us to derive a Riemann-Roch-Grothendieck formula for a projective morphism between smooth complex compact manifolds. In the general case of complex spaces, the Poincar\'e and Dolbeault-Grothendieck lemmas are not always valid. For this reason, and to simplify the exposition, we only consider non singular complex spaces. The appendix presents a calculation of integral Bott-Chern cohomology in top degree for a connected compact manifold.

On the $$\partial {{\bar{\partial }}}$$-Lemma and Bott-Chern cohomology with local coefficients

On the $$\partial {{\bar{\partial }}}$$-Lemma and Bott-Chern cohomology with local coefficients

Bott–Chern Laplacian on almost Hermitian manifolds

Let (M,J,g,omega ) be a 2n-dimensional almost Hermitian manifold. We extend the definition of the Bott–Chern Laplacian on (M,J,g,omega ), proving that it is still elliptic. On a compact Kähler manifold, the kernels of the Dolbeault Laplacian and of the Bott–Chern Laplacian coincide. We show that such a property does not hold when (M,J,g,omega ) is a compact almost Kähler manifold, providing an explicit almost Kähler structure on the Kodaira–Thurston manifold. Furthermore, if (M,J,g,omega ) is a connected compact almost Hermitian 4-manifold, denoting by h^{1,1}_{BC} the dimension of the space of Bott–Chern harmonic (1, 1)-forms, we prove that either h^{1,1}_{BC}=b^- or h^{1,1}_{BC}=b^-+1. In particular, if g is almost Kähler, then h^{1,1}_{BC}=b^-+1, extending the result by Holt and Zhang (Harmonic forms on the Kodaira–Thurston manifold. arXiv:2001.10962, 2020) for the kernel of Dolbeault Laplacian. We also show that the dimensions of the spaces of Bott–Chern and Dolbeault harmonic (1, 1)-forms behave differently on almost complex 4-manifolds endowed with strictly locally conformally almost Kähler metrics. Finally, we relate some spaces of Bott-Chern harmonic forms to the Bott–Chern cohomology groups for almost complex manifolds, recently introduced in Coelho et al. (Maximally non-integrable almost complex structures: an h-principle and cohomological properties, arXiv:2105.12113, 2021).

On blow-up formula of integral Bott–Chern cohomology

On blow-up formula of integral Bott–Chern cohomology

Remarks on Dolbeault cohomology of Oeljeklaus-Toma manifolds and Hodge theory

We give explicit harmonic representatives of Dolbeault cohomology of Oeljeklaus-Toma manifolds and show that they are geometrically Dolbeault formal. We also give explicit harmonic representatives of Bott-Chern cohomology of Oeljeklaus-Toma manifolds of type $(s,1)$ and study the Angella-Tomassini inequality.

On the deformed Bott-Chern cohomology

Given a compact complex manifold X and a integrable Beltrami differential ϕ∈A0,1(X,TX1,0), we introduce a double complex structure on A•,•(X) naturally determined by ϕ and study its Bott-Chern cohomology. In particular, we establish a deformation theory for Bott-Chern cohomology and use it to compute the deformed Bott-Chern cohomology for the Iwasawa manifold and the holomorphically parallelizable Nakamura manifold. The ∂∂¯ϕ-lemma is studied and we show a compact complex manifold satisfying ∂∂¯ϕ-lemma is formal.

Left invariant generalized complex and Kähler structures on simply connected four dimensional Lie groups: Classification and invariant cohomologies

We give a complete classification of left invariant generalized complex structures of type 1 on four dimensional simply connected Lie groups and we compute for each class its invariant generalized Dolbeault cohomology, its invariant generalized Bott-Chern cohomology and its invariant generalized Aeppli cohomology. We classify also left invariant generalized Kähler structures on four dimensional simply connected Lie groups.

The Bott-Chern and Aeppli Cohomologies of Complex Structures on S3 × S3

We calculate the dimensions of Bott-Chern and Aeppli cohomologies associated to a complex structure on S3 × S3. We express them in terms mainly of Hogde numbers.

Bott–Chern blow-up formulae and the bimeromorphic invariance of the $\partial \partial $-Lemma for threefolds

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-K\"{a}hlerness degrees, introduced by Angella-Tomassini [Invent. Math. 192, (2013), 71-81], are bimeromorphic invariants. Consequently, the $\partial\bar{\partial}$-Lemma on threefolds admits the bimeromorphic invariance.

Dolbeault cohomology of compact complex manifolds with an action of a complex Lie group

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.

Localization of Bott‐Chern classes and Hermitian residues

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 the three. We then study the localization problem of characteristic classes in the relative Bott-Chern cohomology. For this we define the cup product and integration in our framework and we discuss local and global duality homomorphisms. After reviewing some materials on connections, we give a vanishing theorem relevant to our localization. With these, we prove a residue theorem for a vector bundle admitting a Hermitian connection compatible with an action of the non-singular part of a singular distribution. As a typical case, we discuss the action of a distribution on the normal bundle of an invariant submanifold (so-called the Camacho-Sad action) and give a specific example.

Symplectic Bott–Chern cohomology of solvmanifolds

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 remarks on symplectic Hodge theory.

Weighted Bott–Chern and Dolbeault cohomology for LCK-manifolds with potential

A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler form taking values in a local system $L$, called the conformal weight bundle. The $L$-valued cohomology of $M$ is called Morse-Novikov cohomology. It was conjectured that (just as it happens for Kahler manifolds) the Morse-Novikov complex satisfies the $dd^c$-lemma. If true, it would have far-reaching consequences for the geometry of LCK manifolds. Counterexamples to the Morse-Novikov $dd^c$-lemma on Vaisman manifolds were found by R. Goto. We prove that $dd^c$-lemma is true with coefficients in a sufficiently general power $L^a$ of $L$ on any LCK manifold with potential (this includes Vaisman manifolds). We also prove vanishing of Dolbeault and Bott-Chern cohomology with coefficients in $L^a$. The same arguments are used to prove degeneration of the Dolbeault-Frohlicher spectral sequence with coefficients in any power of $L$.

Hodge numbers of a hypothetical complex structure on S6

These are the notes for the talk “Hodge numbers of a hypothetical complex structure on S6” given by the author at the MAM1 “(Non)-existence of complex structures on S6” held in Marburg in March 2017. They are based on [12] and [27], where Hodge numbers and the dimensions of the successive pages of the Frölicher spectral sequence for S6 endowed with a hypothetical complex structure are investigated. We also add results from [20], where the Bott–Chern cohomology of hypothetical complex structures on S6 is studied. The material is not intended to be original.

On small deformations of balanced manifolds

We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the ∂∂‾-Lemma, is characterized in terms of the strongly Gauduchon cone and of the first ∂∂‾-degree measuring the difference of Aeppli and Bott–Chern cohomologies with respect to the Betti number b1.