We use Chern-Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every del-dellbar exact real form of the type (k,k) on an n-dimensional complex manifold X arises as a difference of the Chern character forms of trivial Hermitian vector bundles with canonical connections, and that (modulo the image of del and delbar) every real form of type (k,k), k<n, arises as a Bott-Chern form for two Hermitian metrics on some trivial vector bundle over X. The latter result is a complex manifold analogue of Proposition 2.6 in the paper arXiv: 0810.4935 by J. Simons and D. Sullivan. As an application, we obtain an explicit formula for the Bott-Chern form of a short exact sequence of holomorphic vector bundles, considered by Bott and Chern in classic 1965 paper, for the case when the first term is a line bundle. We also present a very simple explicit formula for the total Chern form of a hypersurface in the complex projective space.
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 Bott-Chern classes for holomorphic vector bundles in this differential cohomology. These refined Bott-Chern classes transform naturally to standard Chern classes, Bott-Chern classes and Cheeger-Simons' refined Chern classes.
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.
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 …
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…
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.
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.
Given a compact complex manifold $X$ and a integrable Beltrami differential $ϕ\in A^{0,1}(X, T_{X}^{1,0})$, we introduce a double complex structure on $A^{\bullet,\bullet}(X)$ 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 $\partial\bar{\partial}_ϕ$-lemma is studied and we show a compact complex manifold satisfying $\partial\bar{\partial}_ϕ$-lemma is formal.
本调查总结了2015年7月在比勒费尔德大学举行的“比勒费尔德几何与拓扑日”会议上讨论的结果。我们感兴趣的是Bott-Chern上同调的定量和定性性质。我们公布了[D. Angella，N. Tardini，Quantitative and Qualitative cohomological properties for non-Kähler manifold，arXiv：1507.07108] jointly with Nicoletta Tardini.特别地，我们证明了Bott-Chern上同调维数的一个上界。我们还介绍了一个概念，称为Schweitzer定性性质，它编码的存在一个非退化配对的Bott-Chern上同调，如庞加莱对偶的德-拉姆上同调。我们证明了这个性质刻画了$\partial\overline\partial$ -引理。
The aim of this article is to study the geometry of Bott-Chern hypercohomology from the bimeromorphic point of view. We construct some new bimeromorphic invariants involving the cohomology for the sheaf of germs of pluriharmonic functions, the truncated holomorphic de Rham cohomology, and the de Rham cohomology. To define these invariants, using a sheaf-theoretic approach, we establish a blow-up formula together with a canonical morphism for the Bott-Chern hypercohomology. In particular, we compute the invariants of some compact complex threefolds, such as Iwasawa manifolds and quintic threefo…
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…
Let $X$ be a cohomologically $(n-1)$-complete complex manifold of dimension $n\geq 2$. We prove a vanishing result for the Bott-Chern cohomology group of type $(1, 1)$ with compact support in $X$, which combined with the well-known technique of Ehrenpreis implies a Hartogs type extension theorem for pluriharmonic functions on $X$.
$\del\delbar$-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）
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. Math. 192, (2013), 71-81], are bimeromorphic invariants. Consequently, the $\partial\bar{\partial}$-Lemma on threefolds admits the bimeromorphic invariance.
We present a general framework for obtaining currential double transgression formulas on complex manifolds which can be seen as manifestations of Bott-Chern Duality. These results complement on one hand the simple transgression formulas obtained by Harvey -Lawson and on the other hand the double transgression formulas of Bismut-Gillet-Soulé. Among the applications we mention a Gysin isomorphism for Bott-Chern cohomology, an abstract Poincaré-Lelong formula for sections of holomorphic and Hermitian vector bundles implying Andersson's generalization of the standard Poincaré-Lelong, a Bott-Chern …
It is shown that the singular set for the Yang-Mills flow on unstable holomorphic vector bundles over compact Kaehler manifolds is completely determined by the Harder-Narasimhan-Seshadri filtration of the initial holomorphic bundle. We assign a multiplicity to irreducible top dimensional components of the singular set of a holomorphic bundle with a filtration by saturated subsheaves. We derive a singular Bott-Chern formula relating the second Chern form of a smooth metric on the bundle to the Chern current of an admissible metric on the associated graded sheaf. This is used to show that the mu…
For every positive integer $r$, we introduce two new cohomologies, that we call $E_r$-Bott-Chern and $E_r$-Aeppli, on compact complex manifolds. When $r=1$, they coincide with the usual Bott-Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when $r\geq 2$. They provide analogues in the Bott-Chern-Aeppli context of the $E_r$-cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page-$(r-1)$-$\partial\bar\partial$-manifolds that we introduced very recently. We also…
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 connect…
We prove that on a compact almost Hermitian 4-manifold the space of $\bar\partial$-harmonic $(1,1)$-forms always has dimension $h_{\bar\partial}^{1,1} = b_- +1$ or $b_-$, whilst the space of Bott-Chern harmonic $(1,1)$-forms always has dimension $h_{BC}^{1,1} = b_- +1$. We also perform calculations of $h^{2,1}_{BC}$ and $h^{1,2}_{BC}$ on the Kodaira-Thurston manifold, thereby providing a full account of when $h^{p,q}_{BC}$ is or is not invariant of the choice of almost Hermitian metric. Finally, we introduce a decomposition of the space of $L^2$ functions on all torus bundles over $S^1$, which…
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 study an analogue of the Calabi flow in the non-Kähler setting for compact Hermitian manifolds with vanishing first Bott-Chern class. We prove a priori estimates for the evolving metric along the flow given a uniform bound on the Chern scalar curvature. If the Chern scalar curvature remains uniformly bounded for all time, we show that the flow converges smoothly to the unique Chern-Ricci-flat metric in the $\partial\bar{\partial}$-class of the initial metric.
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 show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern numbers of such a holomorphic vector bundle are nonnegative and can be bounded below and above respectively by two special Chern numbers. As applications, we obtain a family of new results on compact connected complex manifolds which are homogeneous or can be holomorphically imm…
