Related papers: Deformations of balanced metrics
Existence of strong K\"ahler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth…
The property of admitting an astheno-K\"ahler metric is not stable under the action of small deformations of the complex structure of a compact complex manifold. In this paper, we prove necessary cohomological conditions for the existence…
On a complex manifold an Hermitian metric which is simultaneously SKT and balanced has to be necessarily K\"ahler. It has been conjectured that if a compact complex manifold (M,J) has an SKT metric and a balanced metric both compatible with…
We consider balanced metrics on complex manifolds with holomorphically trivial canonical bundle, most commonly known as balanced $\rm{SU}(n)$-structures. Such structures are of interest for both Hermitian geometry and string theory, since…
We study the stability at blow-up and deformations of a class of Hermitian metrics whose fundamental two-form $\omega$ satisfies the condition $\partial \bar \partial \omega^k=0$, for any $k$ between 1 and $n-1$ (where $n$ is the complex…
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 $\partial\overline\partial$-Lemma, is…
We investigate compact complex manifolds endowed with SKT or balanced metrics. In each case we define a new functional whose critical points are proved to be precisely the K\"ahler metrics, if any, on the manifold. As general manifolds of…
We show that Hermitian metrics with vanishing holomorphic curvature on compact complex manifolds with pseudoeffective canonical bundle are conformally balanced. Pluriclosed metrics with vanishing holomorphic curvature on compact K\"ahler…
On a 4-dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-K\"ahler metric with zero or negative Hermitian scalar…
In this paper we discuss the smoothness conditions for metrics on a cohomogeneity one manifold, i.e. metrics invariant under a Lie group whose generic orbits are hypersurfaces. Along these hypersurfaces one describes the metrics in terms of…
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…
A Hermitian metric on a complex manifold of complex dimension $n$ is called {\em astheno-K\"ahler} if its fundamental $2$-form $F$ satisfies the condition $\partial \overline \partial F^{n - 2} =0$. If $n =3$, then the metric is {\em strong…
In the paper we study the existence of balanced metrics of Hodge-Riemann type on non-K\"ahler complex manifolds. We first find some general obstructions, for instance that a Hodge-Riemann balanced manifold of complex dimension $n$ has to be…
Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ which agrees with $g$ outside a neighborhood of $\partial M$ and…
We consider a notion of balanced metrics for triples (X,L,E) which depend on a parameter \alpha, 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 \alpha, we…
The definition of balanced metrics was originally given by Donaldson in the case of a compact polarized K\"{a}hler manifold in 2001, who also established the existence of such metrics on any compact projective K\"{a}hler manifold with…
We give an algebraic criterion for the existence of projectively Hermitian-Yang-Mills metrics on a holomorphic vector bundle $E$ over some complete non-compact K\"ahler manifolds $(X,\omega)$, where $X$ is the complement of a divisor in a…
We continue our study on Hermitian manifolds that are {\em Bismut torsion parallel,} or {\em BTP} for brevity, which means that the Bismut connection has parallel torsion tensor. For $n\geq 3$, BTP metrics can be balanced (and…
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…
Given a compact complex $n$-fold $X$ satisfying the $\partial\bar\partial$-lemma and supposed to have a trivial canonical bundle $K_X$ and to admit a balanced (=semi-K\"ahler) Hermitian metric $\omega$, we introduce the concept of…