Let $(M,J,g,ω)$ be a Kähler manifold. We prove a $W^{1,2}$ weak Bott-Chern decomposition and a $W^{1,2}$ weak Dolbeault decomposition, following the $L^2$ weak Kodaira decomposition on Riemannian manifolds. Moreover, if the Kähler metric is complete and the sectional curvature is bounded, the $W^{1,2}$ Bott-Chern decomposition is strictly related to the space of $W^{1,2}$ Bott-Chern harmonic forms, i.e., $W^{1,2}$ smooth differential forms which are in the kernel of an elliptic differential operator of order $4$, called Bott-Chern Laplacian. We also generalize to the non compact case the well known property that on compact Kähler manifolds the kernel of the Dolbeault Laplacian and the kernel of the Bott-Chern Laplacian coincide.
Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A crucial step in the proof is to show a vanishing theorem of Dolbeault-Morse-Novikov cohomology.
It is proved that the properties of being Dolbeault formal and geometrically-Bott-Chern-formal are not closed under holomorphic deformations of the complex structure. Further, we construct a compact complex manifold which satisfies the $\partial\overline{\partial}$-lemma but admits a non vanishing Aeppli-Bott-Chern-Massey product.
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 $$ K^{p,q}=\frac{ker( \partial ) \cap ker( {\bar{\partial}}) }{im( \partial )\cap ker( {\bar{\partial}} )+im( {\bar{\partial}})\cap ker( \partial )}$$ and ${\check{H}}^1({\mathcal{PH}})$. (Here $\mathcal{PH}$ is the sheaf of phuri-harmonic functions.) We then work out the complete Bott-Chern-Aeppli cohomology in some examples. We give the Bott-Chern-Aeppli cohomology for a hypothetical complex structure on $S^6$ in ter…
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.
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.…
We consider the primitive decomposition of $\bar \partial, \partial$, Bott-Chern and Aeppli-harmonic $(k,k)$-forms on compact almost Kähler manifolds $(M,J,ω)$. For any $D \in \{\bar\partial, \partial, BC, A\}$, we prove that the $L^k P^0$ component of $ψ\in \mathcal{H}_{D}^{k,k}$, is a constant multiple of $ω^k$. Focusing on dimension 8, we give a full description of the spaces $\mathcal{H}_{BC}^{2,2}$ and $\mathcal{H}_{A}^{2,2}$, from which follows $\mathcal{H}^{2,2}_{BC}\subseteq\mathcal{H}^{2,2}_{\partial}$ and $\mathcal{H}^{2,2}_{A}\subseteq\mathcal{H}^{2,2}_{\bar\partial}$. We also provi…
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 $Δ^k$ 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.
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.
Let $(M,J,g,ω)$ be a $2n$-dimensional almost Hermitian manifold. We extend the definition of the Bott-Chern Laplacian on $(M,J,g,ω)$, 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 $(M,J,g,ω)$ is a compact almost Kähler manifold, providing an explicit almost Kähler structure on the Kodaira-Thurston manifold. Furthermore, if $(M,J,g,ω)$ is a connected compact almost Hermitian $4$-manifold, denoting by $h^{1,1}_{BC}$ the dimension of the space of Bott…
$\del\delbar$-Lemma and Bott-Chern cohomology of twistor spaces
Dipartimento di Matematica “G. Peano”, Università degli studi di Torino（都灵大学）；Department of Mathematics and Statistics, Florida International University（佛罗里达国际大学）；Dipartimento di Scienze Matematiche, Fisiche e Informatiche Unità di Matematica e Informatica, Università degli Studi di Parma（帕尔马大学）
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 particular that either the $\partial\overline{\partial}$-Lemma or the property that the Hodge and Frölicher spectral sequence degenerates at the first level are not closed under deformations.
Let $(X,J,ω)$ be a compact $2n$-dimensional almost Kähler manifold. We prove primitive decompositions for Bott-Chern and Aeppli harmonic forms in special bidegrees and show that such bidegrees are optimal. We also show how the spaces of primitive Bott-Chern, Aeppli, Dolbeault and $\partial$-harmonic forms on $(X,J,ω)$ are related.
The goal of the memoir is to develop a new cohomology theory which encompasses De Rham and Dolbeault cohomology as well as Deligne Beilinson cohomology, in the context of general complex analytic manifolds. The special case of the Iwasawa manifold is investigated as a typical example of what occurs in the non Kähler case. Elementary applications to the Kodaira-Spencer deformation theory and to the calculation of Chern classes are given.
We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of harmonic spinors isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms between other types of harmonic spinors and Bott-Chern, Aeppli and twisted cohomology.
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 equality in such an inequality à la Frölicher characterizes the validity of the so-called cohomological property of satisfying the $\partial\overline{\partial}$-Lemma. As an application, we study cohomological properties of compact either complex, or symplectic, or, more in general, …
