Let $X$ be an irreducible reduced compact complex space of dimension $n$, and let $ω$ be a Hermitian metric on $X$. For each $1\le k\le n-1$, we prove that there exist a constant $c_k$ and a smooth positive function $ρ$ on $X_{\mathrm{reg}}$ such that $\ii\partial\bar\partial(ρω^k)\wedgeω^{n-k-1}=c_kρω^n$. The function $ρ$ is bounded above and bounded away from zero. The constant $c_k$ is unique, and $ρ$ is unique up to a positive multiple among bounded smooth positive solutions. We also prove that the full sign set $\{-1,0,1\}$ of the generalized Gauduchon constants $c_k$ on a compact complex manifold is preserved by finite sequences of blow-ups with smooth centers.
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…
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.
Compact lcK manifolds with Gauduchon parallel vector fields
Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay（巴黎萨克雷大学、法国国家科学研究中心、奥赛数学实验室）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西米恩·斯托伊洛瓦数学研究所）；Institut for Matematik, Aarhus University（奥胡斯大学数学系）
We study compact almost complex manifolds admitting a Hermitian metric satisfying an integral condition involving $\overline \partial$-harmonic $(0,1)$-forms. We prove that this integral condition is automatically satisfied, if the Hermitian metric on the compact almost complex manifold is strongly Gauduchon. Under the further assumption that the almost complex structure is integrable, we show that the integral condition for a Gauduchon metric is equivalent to be strongly Gauduchon. In particular, a compact complex surface with a Gauduchon metric satisfying the integral condition is automatica…
We complete the classification of compact Hermitian manifolds admitting a flat Gauduchon connection. In particular, we establish a conjecture of Yang and Zheng, showing that apart from the cases of a flat Chern or Bismut connection, such manifolds are Kähler. More generally, we prove the same result holds when the flatness assumption is replaced by the so-called Kähler-like condition, proving a conjecture of Angella, Otal, Ugarte and Villacampa. We also treat the non-compact case.
In a paper by Angella, Otal, Ugarte, and Villacampa, the authors conjectured that on a compact Hermitian manifold, if a Gauduchon connection other than Chern or Strominger is Kähler-like, then the Hermitian metric must be Kähler. They also conjectured that if two Gauduchon connections are both Kähler-like, then the metric must be Kähler. In this paper, we discuss some partial answers to the first conjecture, and give a proof to the second conjecture. In the process, we discovered an interesting `duality' phenomenon amongst Gauduchon connections, which seems to be intimately tied to the questio…
In 1984, Gauduchon considered the functional of $L^2$-norm of his torsion $1$-form on a compact Hermitian manifold. He obtained the Euler-Lagrange equation for this functional, and showed that in dimension $2$ the critical metrics must be balanced (namely with vanishing torsion $1$-form). In this note we extend his result to higher dimensions, and show that critical metrics are balanced in all dimensions. We also consider the $L^2$-norm of the full Chern torsion, and show by examples that there are critical points of this functional that are not Kähler.
We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds: it locally conformally Kähler, $k$-Gauduchon, balanced and locally conformally balanced and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a $k$-Gauduchon metric is necessarily a torus. Combined with a result of Fino and Vezzoni from 2016, this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics: balanced, pluriclosed and locally conformally Kähler is a torus. Moreover, w…
A well known conjecture in complex geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler if the constant is non-zero and must be Chern flat if the constant is zero. The conjecture is confirmed 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 the class of complex nilmanifolds and confirm the conjecture in …
