Defines strongly Gauduchon spaces and their properties.
problem Generalization of strongly Gauduchon manifolds in complex spaces.
method Definition and properties of strongly Gauduchon spaces and class SG.
result Similarities and properties of strongly Gauduchon spaces to Kahlerian spaces.
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…
The study proves conditions for Hermitian metrics on compact almost complex manifolds.
problem Conditions for Hermitian metrics on compact almost complex manifolds.
method Analyzes compact almost complex manifolds with Hermitian metrics and integral conditions involving ∂-harmonic (0,1)-forms. result The integral condition is automatically satisfied for strongly Gauduchon metrics, and equivalent to being strongly Gauduchon for integrable almost complex structures.
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…
The study proves estimates for transverse nonlinear equations on Sasakian manifolds with applications in geometry.
problem Estimating transverse fully nonlinear equations on Sasakian manifolds.
method Proving a priori estimates for transverse fully nonlinear equations.
result The study proves estimates for transverse fully nonlinear equations on Sasakian manifolds and gives geometric applications.
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…
The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
problem Proving the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
method Using the ∂∂-class, the study proves the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces. result Uniform convergence of the normalized Chern-Ricci flow starting at any Gauduchon metric on all Inoue-Bombieri surfaces, with smooth convergence and bounded curvature for initial metrics in the ∂∂-class of the Tricerri/Vaisman metric. Study on special metrics on complex nilmanifolds, proving existence and properties.
problem Existence and properties of special metrics on complex nilmanifolds.
method Analysis of astheno-Kähler, strongly Gauduchon, and balanced metrics.
result Existence of astheno-Kähler metrics implies specific properties of nilmanifolds.
We review the relations between compact complex manifolds carrying various types of Hermitian metrics (Kähler, balanced or {\it strongly Gauduchon}) and those satisfying the ∂∂ˉ-lemma or the degeneration at E1 of the Frölicher spectral sequence, as well as the behaviour of these properties under h…
Inspired by a construction due to Hitchin, we produce strongly bihermitian metrics on certain Hopf complex surfaces, which integrate the locally conformally Kaehler metrics found by Gauduchon and Ornea. We also show that the Inoue complex surfaces with zero second Betti number do not admit bihermitian metrics. This com…
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 shows compact Vaisman manifolds cannot have certain special Hermitian metrics.
problem Compact Vaisman manifolds and their compatibility with special Hermitian structures.
method Proof of non-existence of specific Hermitian metrics on compact Vaisman manifolds.
result Compact Vaisman manifolds cannot admit special Hermitian metrics like special k-Gauduchon metrics or pluriclosed metrics. We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied.…
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} C manifolds. Our main idea is to exp…
Study on complex manifolds' Gauduchon metrics and spectral sequences under deformations.
problem Understanding Gauduchon metrics and spectral sequences in complex manifold deformations.
method Two approaches: partial degeneration of Frölicher spectral sequence and h-∂∂ˉ-property. result Introduction of a positivity cone and its lower semicontinuity under deformations.
Gauduchon's theorem extended to singular spaces with smoothing.
problem Extending Gauduchon's theorem to singular spaces.
method Using smoothing techniques for singular spaces.
result Existence of conformally equivalent metrics on singular spaces.
Adiabatic limit proves deformation of complex structures.
problem Proving the deformation of complex structures using adiabatic limit.
method Adapted adiabatic limit to complex structures, introduced new vector bundles and Gauduchon metrics.
result Deformation limiting manifold of holomorphic families of Moishezon manifolds is Moishezon.
Study of complex structures on specific solvmanifolds, proving existence and non-existence results.
problem Classification and properties of complex structures on six-dimensional solvmanifolds.
method Classification of Lie algebras, analysis of complex structures, study of Hermitian metrics.
result Determination of new balanced solvmanifolds and confirmation of conjectures.
We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the ∂∂-Lemma, is characterized in terms of the strongly Gauduchon cone and of the first $\partial\overl…
The paper defines two types of hyperbolicity for complex manifolds and proves related results.
problem Defining and studying hyperbolicity for a broader class of complex manifolds.
method Introducing SKT hyperbolicity and Gauduchon hyperbolicity, proving results using SKT and Gauduchon metrics.
result Every SKT hyperbolic manifold is also Kobayashi/Brody hyperbolic and every Gauduchon hyperbolic manifold is divisorially hyperbolic.
In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the γk functions on the space of its hermitian metrics.
New hyperbolicity concepts expand manifold study.
problem Studying hyperbolicity on complex manifolds.
method Introducing sG-hyperbolicity, weakly p-Kähler hyperbolic structures, and pluriclosed star split hyperbolic metrics.
result Expands the class of divisorially hyperbolic manifolds.
The Bott-Chern cohomology of 6-dimensional nilmanifolds endowed with invariant complex structure is studied with special attention to the cases when balanced or strongly Gauduchon Hermitian metrics exist. We consider complex invariants introduced by Angella and Tomassini and by Schweitzer, which are related to the $\pa…
The paper explores spectral sequences of complex manifolds with special metrics.
problem Understanding spectral sequences of compact complex manifolds with special metrics.
method Investigation of Frölicher spectral sequences and special metrics (balanced, SKT, Gauduchon) on manifolds.
result Found compact manifolds where spectral sequences do not degenerate at the second page, providing counterexamples and new families.
Surveying locally homogeneous almost-Hermitian spaces with formulas for curvature.
problem Understanding the geometry of locally homogeneous almost-Hermitian spaces.
method Using the framework of varying Lie brackets to compute curvature of Gauduchon connections.
result Explicit formulas and examples for curvature of Gauduchon connections on locally homogeneous almost-Hermitian spaces.
Study extends continuity equation for Gauduchon metrics.
problem Continuity equation for Gauduchon metrics.
method Solution to Gauduchon conjecture by Székelyhidi, Tosatti, and Weinkove.
result Extended interval of maximal existence for continuity equation.
Study of Hermitian and Gauduchon connections on Lie groups with almost Hermitian structures.
problem Characterizing connections on Lie groups with almost Hermitian structures.
method Analyzing left-invariant Hermitian and Gauduchon connections on Lie groups equipped with almost Hermitian structures.
result Explicit formulas for torsion components and curvature of Gauduchon connections on Lie groups.
The Lee-Gauduchon cone is a convex cone of cohomology classes for complex manifolds.
problem Understanding the Lee-Gauduchon cone for complex manifolds.
method Analyzing the Lee-Gauduchon cone as a convex cone of cohomology classes.
result The Lee-Gauduchon cone is a bimeromorphic invariant.
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 …
Given a (smooth) complex analytic family of compact complex manifolds, we prove that the central fibre must be Moishezon if the other fibres are Moishezon. Using a "strongly Gauduchon metric" on the central fibre whose existence was proved in our previous work on limits of projective manifolds, we show that the irreduc…
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…
The paper proves a continuity theorem for sheaves on complex surfaces.
problem Describing Donaldson-Uhlenbeck compactifications on arbitrary Gauduchon surfaces.
method Proving a continuous map on the semi-stable locus to the compactification of instanton moduli space.
result Provides an efficient tool for the explicit description of compactifications.
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 …
The paper explores properties of Gauduchon curvature in Hermitian manifolds.
problem Investigating properties of Gauduchon curvature in Hermitian manifolds.
method Analyzing the Ricci curvature of Gauduchon connections and proving existence of metrics.
result Monotonicity theorem for Gauduchon holomorphic sectional curvature.
Paper solves Gauduchon scalar curvature problem on almost Hermitian manifolds.
problem Prescribed Gauduchon scalar curvature problem on almost Hermitian manifolds.
method Reduced to solving a semi-linear partial differential equation with exponential nonlinearity using super and sub-solution method.
result Existence of solution depends on the sign of a constant associated to Gauduchon degree.
Study on stable vector bundles over Gauduchon manifolds.
problem Existence and stability of vector bundles over Gauduchon manifolds.
method Uhlenbeck--Yau's continuity method for approximate Hermitian--Einstein structures.
result Equivalence of semi-stability and existence of Hermitian--Einstein structures.
The study discovers a family of complex nilmanifolds with infinitely many real homotopy types.
problem Finding infinitely many real homotopy types of complex nilmanifolds.
method Investigates a one-parameter family of non-isomorphic nilpotent Lie algebras of dimension eight with complex structures, restricting parameters to rational values.
result Existence of infinitely many real homotopy types of 8-dimensional nilmanifolds with complex structures.
The Fubini-Study metric minimizes a volume-normalized holomorphic systole in CPn.
problem Finding metrics with minimal holomorphic systoles in complex projective spaces.
method Introduced holomorphic k-systole and used Gauduchon metrics to establish minimization. result The Fubini-Study metric locally minimizes the volume-normalized holomorphic (n−1)-systole. The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.
problem Deformations of Calabi-Yau manifolds under co-polarised conditions.
method Analyzes local deformations of Calabi-Yau ∂∂ˉ-manifolds using Gauduchon metrics and constructs a new hp-HS form. result Proves the p-SKT h-∂∂ˉ-property is deformation open. This paper discusses partial answers and a proof for conjectures about Gauduchon connections on Hermitian manifolds.
problem Conjectures about Gauduchon connections and Hermitian metrics on compact manifolds.
method Analyzes partial answers to conjectures and provides a proof for a related conjecture, discovering a duality phenomenon.
result Proof of the second conjecture about two Kähler-like Gauduchon connections implying a Kähler metric.
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.
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…
Study proves properties of compact Hermitian surfaces with specific curvature conditions.
problem Characterizing compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature.
method Analyzes surfaces with Gauduchon connections and Lichnerowicz holomorphic sectional curvature.
result Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature are either Kähler or isosceles Hopf surfaces.
Characterizes almost abelian Lie algebras with integrable complex structure
problem Classifying almost abelian Lie algebras
method Using presentations consisting of a real number, an element in a vector space, and an endomorphism
result Classifies p-Kähler, p-pluriclosed, Kähler, balanced, pluriclosed, and Gauduchon metrics Paper solves a singular version of Gauduchon's conjecture.
problem Finding Gauduchon metrics with prescribed Ricci curvature on compact complex manifolds.
method Study of the Monge-Ampère equation for (n−1)-plurisubharmonic functions with a gradient term, adapted to singular settings. result Obtained a C0-estimate for the singular problem, proving smoothness of solutions on holomorphic Kähler families. Study on Gauduchon manifolds finds metrics for projectively flat bundles.
problem Existence of Hermitian-Poisson metrics on projectively flat bundles.
method Heat flow techniques and continuity methods.
result Established a correspondence between Hermitian-Poisson metrics and semi-simplicity.
Study on prescribing curvature on specific manifolds with negative Gauduchon degree.
problem Prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree.
method Analysis of geometric flow convergence to obtain existence results.
result Existence results for curvature functions that are nonzero and nonpositive, and sign-changing cases.
Complete classification of Hermitian manifolds with flat Gauduchon connections.
problem Classifying compact Hermitian manifolds with flat Gauduchon connections.
method Analyzing properties of Hermitian manifolds and using Gauduchon connections.
result Established a conjecture about Kähler-like conditions and flatness.