Characterizes metrics on Lie groups, proving non-simultaneous existence of balanced and pluriclosed metrics.
problem Existence of balanced and pluriclosed metrics on real semisimple Lie groups.
method Characterization using Vogan diagrams and revisiting complex structure classification.
result Complex manifolds cannot simultaneously admit balanced and pluriclosed metrics.
The paper finds conditions for smooth curves of balanced metrics in Hermitian non-Kähler settings.
problem Balanced metrics under deformations of complex structures.
method Necessary conditions for smooth curves of balanced metrics.
result Existence of smooth curves of balanced metrics starting from a fixed balanced metric.
Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.
problem Classify balanced Hermitian structures on almost abelian Lie algebras.
method Classify six-dimensional almost abelian Lie algebras with balanced structures, investigate flow of balanced metrics and anomaly flow.
result Prove conjecture for compact almost abelian solvmanifolds with left-invariant complex structures.
No radial balanced metrics found on Kepler manifold unit ball with mild boundary conditions.
problem Finding radial balanced metrics on the unit ball of the Kepler manifold.
method Analyzing boundary behavior and weights of metrics.
result Explicit weights for radial metrics satisfying balanced condition identified.
Blowing up flat metrics yields balanced ones with constant curvature.
problem Constructing balanced metrics with constant curvature on orbifolds.
method Blowing up a compact orbifold with balanced Chern-Ricci flat metrics.
result Blown-up orbifolds admit balanced metrics with constant Chern scalar curvature.
Balanced metrics found on Lie groups and their quotients.
problem Existence of balanced metrics on Lie groups and quotients.
method Proved existence of invariant complex structures and Hermitian balanced metrics on Lie groups and quotients.
result Existence of balanced metrics on Lie groups and quotients, and no pluriclosed metrics.
Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
Study confirms asymptotic behavior of logarithmic balanced metric near infinity.
problem Asymptotic behavior of logarithmic balanced metric near infinity.
method Non-trivial refinement of tools from previous work.
result Partial confirmation of conjecture on asymptotic behavior.
Study shows convergence of anticanonically balanced metrics to Kähler-Einstein metrics on Fano manifolds.
problem Finding anticanonically balanced metrics on Fano manifolds.
method Simplification of Donaldson's proof using Berezin-Toeplitz quantization.
result Sequence of anticanonically balanced metrics converges to Kähler-Einstein metric.
Study non-Kähler metrics on complex nilmanifolds, proving torus structure under certain conditions.
problem Understanding special non-Kähler metrics on complex nilmanifolds.
method Analyzing locally conformally Kähler, k-Gauduchon, balanced, and locally conformally balanced metrics on compact complex manifolds. result Compact complex nilmanifolds with balanced or k-Gauduchon metrics are tori, extending previous results. Paper proves non-existence of certain balanced metrics on six-manifolds.
problem Existence of balanced metrics on six-manifolds with specific properties.
method Analysis of cohomogeneity one actions and properties of balanced structures.
result Non-existence of balanced non-Kähler SU(3)-structures on six-manifolds.
The paper constructs flat metrics on orbifolds and resolutions.
problem Finding flat metrics on orbifolds and their resolutions.
method Gluing construction and analysis of singularities.
result All crepant resolutions of non-Kähler Calabi-Yau orbifolds with Chern-Ricci flat balanced metrics admit such metrics.
This paper consists of two results dealing with balanced metrics (in S. Donaldson terminology) on nonconpact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan-Hartogs domain which admits a balanced metric is the complex hyperbolic spac…
Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle …
We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature Kähler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a coro…
Flow stabilizes on non-Kähler metrics near Calabi-Yau.
problem Stability of conformally balanced metrics flow near Calabi-Yau manifolds.
method Proving stability of the anomaly flow around Calabi-Yau metrics.
result The flow can converge on non-Kähler metrics near Calabi-Yau.
Paper surveys balanced metrics and proves a geodesic convexity result.
problem Understanding balanced metrics and stability in algebraic geometry.
method Survey and proof of geodesic convexity result.
result Geodesically convex function on a complete Riemannian manifold admits a critical point if and only if its asymptotic slope at infinity is positive.
On a complex manifold an Hermitian metric which is simultaneously SKT and balanced has to be necessarily Kähler. It has been conjectured that if a compact complex manifold (M,J) has an SKT metric and a balanced metric both compatible with J, then (M, J) is necessarily Kähler. We show that the conjecture is true for nil…
The paper broadens a mathematical correspondence to include more balanced metrics.
problem Extending a mathematical correspondence to a broader class of metrics.
method Using key observations and known theorems to apply results to a new class of metrics.
result The known results can be applied to a larger class of metrics, including those arising from multipolarizations.
Characterizes complex Finsler metrics and their properties.
problem Characterize complex Finsler metrics and their geometric properties.
method Defined the canonical connection and investigated holomorphic sectional curvature tensors and Ricci curvatures.
result Characterizes balanced complex Finsler metrics and provides sufficient and necessary conditions.
We determine the Christoffel's symbols for the Siegel-Jacobi ball endowed with the balanced metric. We study the equations of geodesics on the Siegel-Jacobi ball. We calculate the covariant derivative of one-forms in the variables in which is expressed the balanced metric on the Siegel-Jacobi ball.
New metrics solve complex equations on special 3D shapes.
problem Finding metrics on complex 3D shapes.
method Gluing construction to solve equations.
result Solves dilatino equation on small resolutions.
We construct balanced metrics on the family of non-Kähler Calabi-Yau threefolds that are obtained by smoothing after contracting (−1,−1)-rational curves on Kähler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to connected sum of k≥2 copies of $S^3\time…
Non-Archimedean balanced metrics approximate cscK metrics for totally degenerate abelian varieties
problem Non-Archimedean balanced metrics for polarized abelian varieties
method Non-Archimedean analogue of the cscK metric
result Uniform estimate for Calabi-Yau metrics on fibers
We consider a notion of balanced metrics for triples (X,L,E) which depend on a parameter α, where X is smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X. For generic choice of α, we prove that the limit of a convergent sequence of balanced metrics leads to a Hermitian-Einst…
We provide an algebraic framework for quantization of Hermitian metrics that are solutions of the Hitchin equation for Higgs bundles over a projective manifold. Using Geometric Invariant Theory, we introduce a notion of balanced metrics in this context. We show that balanced metrics converge at the quantum limit toward…
Proves conjecture about compatible SKT and balanced metrics on compact solvmanifolds.
problem Compact complex manifolds with both SKT and balanced metrics.
method Shear construction and classification of two-step solvable Lie algebras.
result Proves conjecture for compact two-step solvmanifolds with invariant complex structures.
We study the intrinsic geometrical structure of hypersurfaces in 6-manifolds carrying a balanced Hermitian SU(3)-structure, which we call {\em balanced} SU(2)-{\em structures}. We provide conditions which imply that such a 5-manifold can be isometrically embedded as a hypersurface in a manifold with a balanced SU(3)-st…
A manifold (M,I,J,K) is called hypercomplex if I,J,K are complex structures satisfying quaternionic relations. A quaternionic Hermitian metric is called HKT (hyperkaehler with torsion) if IdωI=JdωJ=KdωK, where ωI,ωJ,ωK are Hermitian forms associated with I, J, K. A Hermitian metric ω on a complex manifo…
Let E→M be a holomorphic vector bundle over a compact Kaehler manifold (M,ω). We prove that if E admits a ω-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L. Biliotti and A. Ghigi implies the existence and uniqueness of ω-balanced metrics of certain dir…
We study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, SKT and astheno-Kähler metrics. We prove that the twistor spaces of compact hyperkähler and negative quaternionic-Kähler manifolds do not admit astheno-Kähler metrics. Then w…
In this paper we study the set of balanced metrics (in Donaldson's terminology) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this set is finite when M admits…
The abstract conjectures and verifies a flow on balanced manifolds converging to Kähler metrics.
problem The convergence of pluriclosed flow on balanced manifolds with c1=0. method Analyzes specific cases of compact quotients of Lie groups, verifying the conjecture for invariant metrics.
result The pluriclosed flow on compact balanced manifolds with c1=0 converges to Kähler metrics. In \cite{D3}, Donaldson defines a dynamical system on the space of Fubini-Study metrics on a polarized compact Kähler manifold. Sano proved that if there exists a balanced metric for the polarization, then this dynamical system always converges to the balanced metric (\cite{S}). In \cite{DKLR}, Douglas, et. al., conjec…
Study on existence of balanced metrics on non-Kähler manifolds.
problem Existence of balanced metrics on non-Kähler complex manifolds.
method Analyzes obstructions and constructs examples, focusing on compact quotients of Lie groups.
result Proves non-existence on certain non-Kähler complex parallelizable manifolds and solvmanifolds.
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.
The paper analyzes systoles of complex projective spaces under various metrics.
problem Behavior of systoles in complex projective spaces for different metrics.
method Integral geometric techniques and careful analysis of systole functional.
result Balanced metrics locally minimize the systole on volume-normalized metrics.
Paper generalizes balanced metrics existence to singular cases using Quot-scheme limit.
problem Existence of balanced metrics and Gieseker stability of vector bundles.
method Quot-scheme limit of Fubini-Study metrics and Bergman 1-parameter subgroups.
result Existence of balanced metrics equivalent to Gieseker stability for singular cases.
Study on existence of p-Kähler structures on nilmanifolds with nilpotent complex structures.
problem Existence of p-Kähler structures on nilmanifolds with nilpotent complex structures. method Determine optimal p for existence of p-Kähler structures and analyze the relationship between balanced metrics and degeneracy steps of the Frölicher spectral sequence. result No p-Kähler structures exist for an optimal p on nilmanifolds with nilpotent complex structures. 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.
Quantizes symplectic fibrations to analyze vector bundles and metrics.
problem Quantizing higher rank vector bundles and understanding their metrics.
method Relates Berezin-Toeplitz quantization to hybrid systems and symplectic fibrations.
result Established refined estimates for computing balanced metrics on Kähler manifolds.
The abstract discusses conjectures about metrics on complex manifolds.
problem The abstract tackles the conjectures about metrics on complex manifolds, specifically balanced, SKT, and LCK.
method The abstract uses complex Hermitian manifolds, closed 1-forms, and conjectures to explore these metrics.
result The abstract verifies a conjecture about the Bott--Chern homology for all known classes of LCK manifolds.
We study the quantization of coupled Kähler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the ``balanced metrics''. We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern clas…
Introduces new Hermitian metrics linking to Gauduchon and balanced metrics.
problem Finding conditions for compact complex manifolds to be Kähler.
method Introducing pluriclosed star split metrics and studying their properties.
result Affirmative answer to Fino-Vezzoni conjecture under extra assumptions.
In this paper, by generalizing the concept of balanced metrics, we shall show that Donaldson's asymptotic approximation of balanced metrics for constant scalar curvature cases can be generalized to extremal Kaehler cases.
Proves openness of balanced HKT cone and studies hyperholomorphic vector fields.
problem Understanding balanced HKT structures on compact hypercomplex manifolds.
method Analyzes Lie algebra of hyperholomorphic vector fields and proves harmonicity properties.
result Proves openness of balanced HKT cone and non-existence of certain fields.
In this note we present, for every n≥4, a non-Kähler compact complex manifold X of complex dimension n admitting a balanced metric and an astheno-Kähler metric which is in addition k-th Gauduchon for any 1≤k≤n−1.
Investigates special metrics in hypercomplex geometry.
problem Characterizing and understanding special hyperhermitian metrics.
method Characterization of hypercomplex structures with Obata holonomy, investigation of quaternionic Gauduchon and balanced metrics, incompatibility results, and introduction of Einstein-type conditions.
result Joyce's manifolds always admit special metrics.