In this paper, we study the semi-stable twisted holomorphic vector bundles over compact Gauduchon manifolds. By using Uhlenbeck--Yau's continuity method, we show that the existence of approximate Hermitian--Einstein structure and the semi-stability of twisted holomorphic vector bundles are equivalent over compact Gauduchon manifolds. As its application, we show that the Bogomolov type inequality is also valid for a semi-stable twisted holomorphic vector bundle.
Motivated by a recent work of Chen-Zheng [8] on Strominger space forms, we prove that a compact Hermitian surface with pointwise constant holomorphic sectional curvature with respect to a Gauduchon connection $\nabla^t $ is either Kähler, or an isosceles Hopf surface with an admissible metric and $t=-1$ or $t=3$. In particular, a compact Hermitian surface with pointwise constant Lichnerowicz holomorphic sectional curvature is Kähler. We further generalize the result to the case for the two-parameter canonical connections introduced by Zhao-Zheng [30], which extends a previous result by Apostol…
We apply the existence and special properties of Gauduchon metrics to give several applications. The first one is concerned with the implications of algebro-geometric nature under the existence of a Hermitian metric with nonnegative holomorphic sectional curvature. The second one is to show the non-existence of holomorphic sections on Hermitian vector bundles under certain conditions. The third one is to give a restriction on the $\partial\bar{\partial}$-closedness of some real $(n-1,n-1)$-forms on compact complex manifolds.
In this paper, we study numerically flat holomorphic vector bundles over a compact non-Kähler manifold $(X, ω)$ with the Hermitian metric $ω$ satisfying the Gauduchon and Astheno-Kähler conditions. We prove that numerically flatness is equivalent to numerically effectiveness with vanishing first Chern number, semistablity with vanishing first and second Chern numbers, approximate Hermitian flatness and the existence of a filtration whose quotients are Hermitian flat. This gives an affirmative answer to the question proposed by Demailly, Peternell and Schneider.
A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler when the constant is non-zero and must be Chern flat when the constant is zero. The conjecture is known in complex dimension $2$ by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to pluriclosed manifolds, and confirm the conjecture for the specia…
We prove the Kobayashi-Hitchin correspondence and the approximate Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if $X$ is a compact manifold and $g$ is a Gauduchon metric on $X$, a twisted holomorphic vector bundle on $X$ is $g-$polystable if and only if it is $g-$Hermite-Einstein, and if $X$ is a compact Kähler manifold and $g$ is a Kähler metric on $X$, then a twisted holomorphic vector bundle on $X$ is $g-$semistable if and only if it is approximate $g-$Hermite-Einstein.
In this paper, we establish the Hitchin--Kobayashi correspondence for the $I_\pm$-holomorphic quiver bundle $\mathcal{E}=(E,ϕ)$ over a compact generalized Kähler manifold $(X, I_+,I_-,g, b)$ such that $g$ is Gauduchon with respect to both $I_+$ and $I_-$, namely $\mathcal{E}$ is $(α,σ,τ)$-polystable if and only if $\mathcal{E}$ admits an $(α,σ,τ)$-Hermitian--Einstein metric.
We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following Bérard-Bergery which includes, among the others, the holomorphic line bundles on $\mathbb C\mathbb P^{m-1}$, the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, Kähler-like, pluriclosed, locally conformally Kähler, Vaisman, Gauduchon. Furthermore, we construct new examples …
Balanced Bismut torsion-parallel fourfolds with constant holomorphic sectional curvature
Halıcıoğlu Data Science Institute, University of California San Diego（加州大学圣地亚哥分校哈利奥格鲁数据科学研究所）；School of Mathematical Sciences, Chongqing Normal University（重庆师范大学数学科学学院）
Stable Higgs bundles and Hermitian-Einstein metrics on non-Kähler manifolds
设$X$是紧Gauduchon流形，$E$和$V_0$是$X$上的全纯向量丛.假设$E$是稳定的，当考虑由Higgs场$θ\in H^0($ End $(E)\otimes V_0)$保持的所有子层时。然后，修改版本的唐纳森热流收敛沿着一个子序列的时间的广义厄米-爱因斯坦方程的解决方案，由$iΛF+[θ,θ^\dagger]=λI$。
In this paper we establish partial structure results on the geometry of compact Hermitian manifolds of semipositive Griffiths curvature. We show that after appropriate arbitrary small deformation of the initial metric, the null spaces of the Chern-Ricci two-form generate a holomorphic, integrable distribution. This distribution induces an isometric, holomorphic, almost free action of a complex Lie group on the universal cover of the manifold. Our proof combines the strong maximum principle for the Hermitian Curvature Flow (HCF), new results on the interplay of the HCF and the torsion-twisted c…
We review a method to construct $\rm{G}_2$--instantons over compact $\rm{G}_2$--manifolds arising as the twisted connected sum of a matching pair of Calabi-Yau $3$-folds with cylindrical end, based on the series of articles [SE15, SEW15, JMPSE17, MNSE17] by the author and others. The construction is based on gluing $\rm{G}_2$--instantons obtained from holomorphic bundles over such building blocks, subject to natural compatibility and transversality conditions. Explicit examples are obtained from matching pairs of semi-Fano $3$-folds by an algorithmic procedure based on the Hartshorne-Serre cor…
In his Annals of Mathematics paper (2009), Berndtsson proves an important result on the Nakano positivity of holomorphic infinite-rank vector bundles whose fibers are Hilbert spaces consisting of holomorphic $L^2$-functions with respect to a family of weight functions $\left\{e^{-φ(\cdot,t)}\right\}_{t \in U}$, varying in $t \in U \in \mathbb{C}^m$, over a pseudoconvex domain. Using a variant of Hörmander's theorem due to Donnelly and Fefferman, we show that Berndtsson's Nakano positivity result holds under different (in fact, more general) curvature assumptions. This is of particular interest…
Let $π: \mathcal{X}\rightarrow Δ$ be a holomorphic family of compact complex manifolds over an open disk in $\mathbb{C}$. If the fiber $π^{-1}(t)$ for each nonzero $t$ in an uncountable subset $B$ of $Δ$ is Moishezon and the reference fiber $X_0$ satisfies the local deformation invariance for Hodge number of type $(0,1)$ or admits a strongly Gauduchon metric introduced by D. Popovici, then $X_0$ is still Moishezon. We also obtain a bimeromorphic embedding $\mathcal{X}\dashrightarrow\mathbb{P}^N\timesΔ$. Our proof can be regarded as a new, algebraic proof of several results in this direction pr…
Kähler流形$X$上的扭曲Higgs丛是由全纯向量丛$E$和某个全纯向量丛$M$的全纯丛态射$ϕ\colon M\otimes E \to E$组成的对$(E,ϕ)$。当$X$是一条曲线，$M$是$X$的切丛时，Hitchin首先考虑了这样的对象，Simpson也考虑了高维基。希钦-小林对应表明$(E,ϕ)$是多稳定的当且仅当$E$允许一个厄密度量解希钦方程。这种对应关系是判断方程是否存在解的有力工具，但它提供的关于实际解的信息很少。在本文中，我们研究了这个问题的量化表示在有限维数据和平衡度量，使近似解希钦方程。本研究的动机来自Donagi-Wijnholt在arXiv：1104.2610上关于Vafa-Witten方程的平衡度量的工作。
We propose a method to construct G_2-instantons over a compact twisted connected sum G_2-manifold, applying a gluing result of Sá Earp and Walpuski to instantons over a pair of 7-manifolds with a tubular end (see arXiv:1310.7933). In our example, the moduli spaces of the ingredient instantons are non-trivial, and their images in the moduli space over the asymptotic cross-section K3 surface intersect transversely. Such a pair of asymptotically stable holomorphic bundles is obtained using a twisted version of the Hartshorne-Serre construction, which can be adapted to produce other examples. More…
We introduce two notions of hyperbolicity for not necessarily Kähler $n$-dimensional compact complex manifolds $X$. The first, called {\it balanced hyperbolicity}, generalises Gromov's Kähler hyperbolicity by means of Gauduchon's balanced metrics. The second, called {\it divisorial hyperbolicity}, generalises the Brody hyperbolicity by ruling out the existence of non-degenerate holomorphic maps from $\C^{n-1}$ to $X$ that have what we term a subexponential growth. Our main result in the first part of the paper asserts that every balanced hyperbolic $X$ is also divisorially hyperbolic. We provi…
