arXiv research

This paper discusses partial answers and a proof for conjectures about Gauduchon connections on Hermitian manifolds.

Complete classification of Hermitian manifolds with flat Gauduchon connections.

The paper explores properties of Gauduchon curvature in Hermitian manifolds.

Study of Hermitian and Gauduchon connections on Lie groups with almost Hermitian structures.

Given a Hermitian manifold $(M^n,g)$, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call $\nabla^s = (1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$ the $s$-Gauduchon connection of $M$, where $\nabla^c$ and $\nabla^b$ are r…

We study Hermitian metrics with a Gauduchon connection being "Kähler-like", namely, satisfying the same symmetries for curvature as the Levi-Civita and Chern connections. In particular, we investigate $6$-dimensional solvmanifolds with invariant complex structures with trivial canonical bundle and with invariant Hermit…

Study proves properties of compact Hermitian surfaces with specific curvature conditions.

Surveying locally homogeneous almost-Hermitian spaces with formulas for curvature.

This text is dedicated to the real Killing equation on 3-dimensional Weyl manifolds. Any manifold admitting a real Killing spinor of weight 0 satisfies the conditions of a Gauduchon-Tod geometry. Conversely, any simply connected Gauduchon-Tod geometry has a 2-dimensional space of solutions of the real Killing equation …

Enhanced Schwarz lemma for Hermitian manifolds with new curvature constraints.

Study extends continuity equation for Gauduchon metrics.

Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.

We show that a unipotent vector bundle on a non-Kaehler compact complex manifold does not admit a flat holomorphic connection in general. We also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.

We define strongly Gauduchon spaces and the class SG which are generalization of strongly Gauduchon manifolds in complex spaces. Comparing with the case of Kahlerian, the strongly Gauduchon space and the class SG are similar to the Kahler space and the Fujiki class C respectively. Some properties about these complex sp…

The Lee-Gauduchon cone is a convex cone of cohomology classes for complex manifolds.

On an almost Hermitian manifold, we have two Hermitian scalar curvatures with respect to any canonical Hermitian connection defined by P. Gauduchon. Explicit formulas of these two Hermitian scalar curvatures are obtained in terms of Riemannian scalar curvature, norms of decompositions of covariant derivative of the fun…

In this paper, we study strongly Gauduchon metrics on compact complex manifolds. We study the cohomology cones SG in the de Rham cohomology groups generated by all strongly Gauduchon metrics and its direct images under proper modifications. We also study the moduli of strongly Gauduchon manifolds. We prove an existence…

We pursuit the research line proposed in \cite{YZ-Gflat} about the classification of Hermitian manifolds whose $s$-Gauduchon connection $\nabla^s =(1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$ is flat, where $s \in \mathbb{R}$ and $\nabla^c$ and $\nabla^b$ are the Chern and the Bismut connections, respectively. We foc…

Paper solves Gauduchon scalar curvature problem on almost Hermitian manifolds.

Study on stable vector bundles over Gauduchon manifolds.

Gauduchon's theorem extended to singular spaces with smoothing.

The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.

We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.

Study Hull-Strominger system and Anomaly flow on specific solvmanifolds.

We show existence of unique smooth solutions to the Monge-Ampere equation for (n-1)-plurisubharmonic functions on Hermitian manifolds, generalizing previous work of the authors. As a consequence we obtain Calabi-Yau theorems for Gauduchon and strongly Gauduchon metrics on a class of non-Kahler manifolds: those satisfyi…

Paper solves a singular version of Gauduchon's conjecture.

Study on Gauduchon manifolds finds metrics for projectively flat bundles.

In this paper, we characterize Riemannian 4-manifold in terms of its almost Hermitian twistor spaces $(Z,g_t,\mathbb{J}_{\pm})$. Some special metric conditions (including Balanced metric condition, first Gauduchon metric condition) on $(Z,g_t,\mathbb{J}_{\pm})$ are studied. For the first Chern form of a natural unitary…

The study proves conditions for Hermitian metrics on compact almost complex manifolds.

We construct several natural connections and Dirac type operators on a general metric contact manifold which are more sensitive to the geometric background. In the special case of CR manifolds these connections are also compatible with the CR structure and include among them the Webster connection. We also describe sev…

The paper defines two types of hyperbolicity for complex manifolds and proves related results.

Study on prescribing curvature on specific manifolds with negative Gauduchon degree.

The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.

In this paper, we prove a generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles over a class of non-compact Gauduchon manifolds.

The paper explores applications of Gauduchon metrics in complex geometry.

In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the $γ_k$ functions on the space of its hermitian metrics.

Researchers find a way to estimate potential functions for quaternionic metrics.

Extends Gauduchon's result to higher dimensions, showing balanced metrics.

In this paper, we use the affine Hermitian-Yang-Mills flow to prove a generalized Donaldson-Uhlenbeck-Yau theorem on flat Higgs bundles over a class of non-compact affine Gauduchon manifolds.

In this paper, we consider the existence of approximate Hermitian-Einstein structure and the semi-stability on Higgs bundles over compact Gauduchon manifolds. By using the continuity method, we show that they are equivalent.

The paper studies LCAK metrics on complex manifolds and their properties.

The paper studies harmonic complex structures and special metrics on Sasakian manifolds.

Unified flow approach to curvature problem on specific manifolds.

Existence of metrics on non-Kähler varieties, generalizing previous work.

We propose the study of a Monge-Ampère-type equation in bidegree $(n-1,\,n-1)$ rather than $(1,\,1)$ on a compact complex manifold $X$ of dimension $n$ for which we prove uniqueness of the solution subject to positivity and normalisation restrictions. Existence will hopefully be dealt with in future work. The aim is to…

We investigate connections between the sGG property of compact complex manifolds, defined in earlier work by the second author and L. Ugarte by the requirement that every Gauduchon metric be strongly Gauduchon, and a possible degeneration of the Frölicher spectral sequence. In the first approach that we propose, we pro…

This paper is intended as the first step of a programme aiming to prove in the long run the long-conjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact Kähler manifolds, known as Fujiki {\it class} ${\cal C}$ manifolds. Our main idea is to exp…

Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat princip…