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 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.
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.
The study classifies flat Gauduchon connections on compact Hermitian manifolds.
problem Understanding when compact Hermitian manifolds can have flat Gauduchon connections.
method Investigates the s-Gauduchon connections and their flatness conditions. result For certain values of s, the metric g is Kähler. 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.
Study on extremal metrics in conformal geometry.
problem Finding unique extremal metrics in conformal classes.
method Analysis of Euler-Lagrange equations for functionals involving the Lee form.
result Existence and uniqueness of extremal metrics on compact complex surfaces.
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.
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…
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.
We describe explicitly the moduli spaces Mgpst(S,E) of polystable holomorphic structures E with detE≅K on a rank 2 vector bundle E with c1(E)=c1(K) and c2(E)=0 for all minimal class VII surfaces S with b2(S)=1 and with respect to all possible Gauduchon metrics g. These surfaces S are …
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.
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 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.
This work shows all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces are known families.
problem Characterize all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces.
method Unified construction using axi-symmetric harmonic functions and methods from scalar-flat Kähler metrics.
result All such metrics are ALF and belong to known families.
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 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. 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.
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 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.
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.
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.
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.
Paper uses flow to prove theorem on Higgs bundles.
problem Proving generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles.
method Affine Hermitian-Yang-Mills flow
result Generalized Donaldson-Uhlenbeck-Yau theorem proved.
The paper explores applications of Gauduchon metrics in complex geometry.
problem Implications of Gauduchon metrics in complex geometry.
method Existence and properties of Gauduchon metrics.
result Non-existence of holomorphic sections and restrictions on ∂∂ˉ-closedness. Study Gauduchon connections on Kähler-like metrics in 6D solvmanifolds.
problem Investigate Hermitian metrics with Gauduchon connections having symmetries similar to Levi-Civita and Chern connections.
method Analyze 6-dimensional solvmanifolds with invariant complex structures and trivial canonical bundles. result Evidence supports two conjectures about Kähler-like conditions for different Gauduchon connections.
Researchers find a way to estimate potential functions for quaternionic metrics.
problem Existence of quaternionic Gauduchon metrics with prescribed volume form.
method Reframed as a fully nonlinear elliptic equation and established a uniform estimate.
result Uniform estimate for the potential function.
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.
Extends Gauduchon's result to higher dimensions, showing balanced metrics.
problem Understanding critical metrics in higher-dimensional Hermitian manifolds.
method Analyzes the functional of L2-norm of torsion 1-form and full Chern torsion. result Critical metrics are balanced in all dimensions.
The paper constructs metrics on Hirzebruch surfaces and ruled surfaces.
problem Existence of Hermitian metrics with constant Chern scalar curvature.
method Using Page--Bérard-Bergery's ansatz to construct metrics on Hirzebruch surfaces.
result Construction of Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces.
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.
We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics generalizing the Kahler-Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a K…
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.
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 …
We describe the positive cone and the pseudo-effective cone of a non-Kählerian surface. We use these results for two types of applications: - Describe the set σ(X) of possible total Ricci scalars associated with Gauduchon metrics of fixed volume 1 on a fixed non-Kähhlerian surface, and decide whether the assignment $…
Unified flow approach to curvature problem on specific manifolds.
problem Prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree.
method Unified flow approach with conditions on curvature function f. result Flow converges to a conformal Hermitian metric with specified curvature.
Existence of metrics on non-Kähler varieties, generalizing previous work.
problem Existence of metrics on non-Kähler varieties.
method Definition of slope stability and existence of singular Hermite-Einstein metrics.
result Existence and uniqueness of singular Hermite-Einstein metrics for slope-stable sheaves.
The study characterizes finite vector bundles on specific complex manifolds.
problem Characterizing finite vector bundles on complex manifolds.
method Proves a theorem for compact complex manifolds with Gauduchon astheno-Kahler metrics.
result Establishes a condition for finite vector bundles similar to Nori's theorem.
Investigates stability on toric surfaces with quadrilateral moment polytopes.
problem Analyzing stability and extremal Kähler metrics on toric surfaces.
method Introduces symplectic potentials and uses constructions from Apostolov-Calderbank-Gauduchon and Legendre.
result Provides a computable criterion for stability with 0 weights on edges of quadrilaterals, leading to log-stable regions.
Paper proves a theorem for Higgs bundles on certain non-compact manifolds.
problem Proving a generalized Donaldson-Uhlenbeck-Yau theorem for Higgs bundles.
method Generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles over non-compact Gauduchon manifolds.
result Proved a generalized Donaldson-Uhlenbeck-Yau theorem for Higgs bundles.
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} C manifolds. Our main idea is to exp…
We describe a family of locally conformal Kaehler metrics on class 1 Hopf surfaces H containing some recent metrics constructed by P. Gauduchon and L. ornea. We study some canonical foliations associated to these metrics, in particular a 2-dimensional foliation E that is shown to be independent of the metric. We elemen…
Conformal vector fields on LCP manifolds are orthogonal and Killing.
problem Understanding conformal vector fields on specific geometric manifolds.
method Analyzing properties of conformal vector fields on compact locally conformally product manifolds.
result Conformal vector fields are orthogonal to the flat distribution and Killing.
Study of Chern-Ricci flow on Hopf surfaces, showing finite-time volume collapse and uniform bounds.
problem Understanding the Chern-Ricci flow on Hopf surfaces, especially minimal non-Kähler ones.
method Construction of locally conformally Kähler metrics and analysis of Chern-Ricci flow.
result Finite-time volume collapse and uniform upper bounds on the metric tensor.