arXiv research

The paper finds conditions for smooth curves of balanced metrics in Hermitian non-Kähler settings.

Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.

Introduces new Hermitian metrics linking to Gauduchon and balanced metrics.

Study shows compact Vaisman manifolds cannot have certain special Hermitian metrics.

Study on balanced Hermitian threefolds with parallel Bismut torsion.

The paper broadens a mathematical correspondence to include more balanced metrics.

Balanced metrics found on Lie groups and their quotients.

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 $Idω_I = Jd ω_J=Kdω_K$, where $ω_I,ω_J, ω_K$ are Hermitian forms associated with I, J, K. A Hermitian metric $ω$ on a complex manifo…

The abstract discusses conjectures about metrics on complex manifolds.

We first study the degeneration of a sequence of Hermitian-Yang-Mills metrics with respect to a sequence of balanced metrics on a Calabi-Yau threefold $\hat{X}$ that degenerates to the balanced metric constructed by Fu, Li, and Yau on the complement of finitely many (-1,-1)-curves in $\hat{X}$. Then under some assumpti…

In this article, we examine the behavior of the Riemannian and Hermitian curvature tensors of a Hermitian metric, when one of the curvature tensors obeys all the symmetry conditions of the curvature tensor of a Kähler metric. We will call such metrics G-Kähler-like or Kähler-like, for lack of better terminologies. Such…

Study of Hermitian structures on toric suspensions of balanced manifolds.

New metrics found on non-Kähler Calabi-Yau manifolds.

Verify conjecture for special Hermitian manifolds.

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…

A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, th…

Solutions to Strominger system found for square of Kähler class.

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…

Paper generalizes balanced metrics existence to singular cases using Quot-scheme limit.

The paper explores spectral sequences of complex manifolds with special metrics.

The study explores metrics with constant curvature on compact manifolds.

Unified flow approach to curvature problem on specific manifolds.

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

Study locally conformally balanced metrics on specific Lie algebras.

We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\infty}$ a priori estimates for a solution of the complex Monge-Ampere equation when the background metric is Hermitian (in complex dimension two) or balanced…

Study of complex structures on specific solvmanifolds, proving existence and non-existence results.

The paper solves a problem related to curvature in complex geometry.

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…

Paper proves non-existence of certain balanced metrics on six-manifolds.

The study classifies metrics with vanishing curvature on complex manifolds.

Study on balanced Hermitian structures on Lie algebras twisted by representations.

Study on special Hermitian metrics on cohomogeneity one manifolds.

We introduce a new geometric flow of Hermitian metrics which evolves an initial metric along the second derivative of the Chern scalar curvature. The flow depends on the choice of a background metric, it always reduces to a scalar equation and preserves some special classes of Hermitian structures, as balanced and Gaud…

The Streets-Tian conjecture is confirmed for specific types of Hermitian 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…

Compactifies moduli spaces of Hermitian-Yang-Mills connections on balanced manifolds.

Motivated from mathematical aspects of the superstring theory, we introduce a new equation on a balanced, hermitian manifold, with zero first Chern class. Solving the equation, one will obtain, in each Bott--Chern cohomology class, a balanced metric which is hermitian Ricci--flat. This can be viewed as a differential f…

Let (J,g) be a Hermitian structure on a compact nilmanifold M with invariant complex structure J and compatible metric g, which is not required to be invariant. We give classifications of 6-dimensional nilmanifolds M admitting strong Kähler with torsion, balanced or locally conformal Kähler structures (J,g).

Study non-Kähler metrics on complex nilmanifolds, proving torus structure under certain conditions.

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…

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…

Proves regularity of geodesic equation on Hermitian manifolds.

The paper analyzes systoles of complex projective spaces under various metrics.

We review some constructions and properties of complex manifolds admitting pluriclosed and balanced metrics. We prove that for a 6-dimensional solvmanifold endowed with an invariant complex structure J having holomorphically trivial canonical bundle the pluriclosed flow has a long time solution for every invariant init…

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…

The invariant balanced Hermitian geometry of nilmanifolds of dimension 6 is described. We prove that the holonomy group of the associated Bismut connection reduces to a proper subgroup of SU(3) if and only if the complex structure is abelian. As an application we show that if J is abelian then any invariant balanced J-…

The paper explores Hermitian structures on tangent bundles of affine manifolds with Riemannian metrics.