arXiv research

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…

Gauduchon's theorem extended to singular spaces with smoothing.

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

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.

The paper explores properties of Gauduchon curvature in Hermitian manifolds.

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

We prove a priori estimates for a class of transverse fully nonlinear equations on Sasakian manifolds and give some geometric applications such as the transversion Calabi-Yau theorem for transverse balanced and (strongly) Gauduchon metrics. We also explain that similar results hold on compact oriented, taut, transverse…

A vector bundle E on a projective variety X is called finite if it satisfies a nontrivial polynomial equation with integral coefficients. A theorem of Nori implies that E is finite if and only if the pullback of E to some finite etale Galois covering of X is trivial. We prove the same statement when X is a compact comp…

In this paper, we prove a Liouville theorem for holomorphic functions on a class of complete Gauduchon manifolds. This generalizes a result of Yau for complete Kähler manifolds to the complete non-Kähler case.

We introduce a natural map from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the infinitesimal deformations of this complex manifold. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the first author. As direct corollar…

A C^2 function on C^n is called (n-1)-plurisubharmonic in the sense of Harvey-Lawson if the sum of any n-1 eigenvalues of its complex Hessian is nonnegative. We show that the associated Monge-Ampere equation can be solved on any compact Kahler manifold. As a consequence we prove the existence of solutions to an equatio…

Study extends continuity equation for Gauduchon metrics.

Enhanced Schwarz lemma for Hermitian manifolds with new curvature constraints.

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.

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…

Paper solves Gauduchon scalar curvature problem on almost Hermitian manifolds.

Study on stable vector bundles over Gauduchon manifolds.

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

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

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.

Paper solves a singular version of Gauduchon's conjecture.

Study on Gauduchon manifolds finds metrics for projectively flat bundles.

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

We prove that any flat family $(\mathcal{ F}_u)_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$ with values in the Donaldson-Uhlenbeck compactification of the corresponding in…

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

Complete classification of Hermitian manifolds with flat Gauduchon connections.

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

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.

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.

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

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…

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…

Conformal vector fields on LCP manifolds are orthogonal and Killing.

The Fubini-Study metric minimizes a volume-normalized holomorphic systole in $\mathbb{C}P^n$.

The equality between the balanced and the Gauduchon cones is discussed in several situations. In particular, it is shown that equality does not hold on many twistor spaces, and it holds on Moishezon manifolds. Moreover, it is proved that a SKT manifold of dimension three on which the balanced cone equals the Gauduchon …

Study of $t$-Gauduchon Ricci-flat condition under Chern-Ricci flow on non-Kähler manifolds.

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 …

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

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

We classify invariant complex structures on 6-dimensional nilmanifolds up to equivalence. As an application, the behaviour of the associated Frölicher sequence is studied as well as its relation to the existence of strongly Gauduchon metrics. We also show that the strongly Gauduchon property and the balanced property a…

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…

Conformal vector fields on lcK manifolds are shown to be Killing or holomorphic.

Let $X$ be a compact Hermitian surface, and $g$ be any fixed Gauduchon metric on $X$. Let $E$ be an Hermitian holomorphic vector bundle over $X$. On the bundle $E$, Donaldson's heat flow is gauge equivalent to a flow of holomorphic structures. We prove that this flow converges, in the sense of Uhlenbeck, to the double …