balanced metrics on complex manifolds
Работы про комплексные многообразия с эрмитовыми
метриками, где условие кэлеровости ослаблено до
"балансированности", замкнутости \omega^{n-1}
(n - размерность многообразия, \omega - эрмитова
(1,1)-форма).

Балансированных некэлеровых многобразий довольно
много, например, все пространства твисторов,
а также все многообразия из класса C Фуджики.
Для размерности 2, балансированность это кэлеровость,
из чего ясно, что небалансированных многообразий тоже
довольно много. Также не допускают балансированных метрик
многообразия Хопфа и Калаби-Экманна, в любой размерности
(и это весьма просто видеть).

Лекция Яу о балансированных метриках:
http://www.math.ucla.edu/~greene/YauTwister(8-9).pdf

Библиография.

Источник термина:

MR0688351 (84i:53063)
Michelsohn, M. L.
On the existence of special metrics in complex geometry.
Acta Math. 149 (1982), no. 3-4, 261--295.

In this article, the class of compact Hermitian manifolds
$X$ that one might term "co-KДhler", namely those for
which the KДhler form $\omega$ satisfies $d^*\omega=0$, is
studied. That name would be somewhat misleading, for $d^*$
also depends on the metric; one can rewrite the equation
as $d(\omega^{n-1})=0$, where $n=\dim_{C}X$. The author
uses instead the term "balanced" for such manifolds, as
the condition can also be expressed as the vanishing of
the trace of the Hermitian torsion tensor. There is, in
addition, a reformulation in terms of the Dirac
operator. It is immediate that KДhler manifolds are
balanced, and that the two notions coincide for
$n=2$. There exist balanced manifolds that admit no KДhler
metric for all $n>2$, as is shown by example in the last
section of the paper. Examples come from twistor spaces,
complex solvmanifolds, and one-dimensional families of
KДhler manifolds.

The main results of the paper are: A. (Characterization
theorem: 4.7) A compact complex manifold $X$ admits a
balanced metric ("is balanced") if and only if every
closed $(2n-2)$-dimensional current, whose
$(n-1,n-1)$-component is positive (and nonzero),
represents a nontrivial $C$-homology class. B. ("Inductive
criterion": 5.5) A compact connected complex manifold $X$
is balanced if it admits a mapping onto a complex curve
such that: (1) The smooth (i.e., general) fibers are
balanced. (2) Each fiber, as a codimension two current on
$X$, is not the $(n-1,n-1)$-component of a boundary (in
the sense of currents). It is observed that a converse
assertion is false: There exist balanced manifolds fibered
over a curve with nonbalanced fibers.

* * *

Вот тут доказывается, что многообразие,
бимероморфно эквивалентное балансированному, тоже
балансированное; в частности, класс C Фуджики.

MR1198601
Alessandrini, Lucia; Bassanelli, Giovanni
Metric properties of manifolds bimeromorphic to compact KДhler spaces.
J. Differential Geom. 37 (1993), no. 1, 95--121.

MR1122151
Alessandrini, Lucia; Bassanelli, Giovanni
Smooth proper modifications of compact KДhler manifolds.
Complex analysis (Wuppertal, 1991), 1--7,
Aspects Math., E17, Vieweg, Braunschweig, 1991.

Вот тут конструкцию Микельсона разнообразно обобщают

Egidi, Nadaniela
Special metrics on compact complex manifolds.
Differential Geom. Appl. 14 (2001), no. 3, 217--234.

In this paper, the author defines different special
metrics on compact complex manifolds. Let $\omega_h$ be
the KДhler form of a Hermitian metric on a compact complex
manifold of dimension $n$. $h$ is said to be balanced if
$d\omega_h^{n-1}=0$; pluriclosed if
$\partial\overline\partial\omega_h=0$; plurinegative if
$\partial\overline\partial\omega_h\leq 0$; and
1-symplectic if $\partial\omega_h=\overline\partial\phi$
and $\partial\phi=0$ for $\phi\in E^{2,0}(M)$. Of course,
KДhler manifolds are balanced, pluriclosed, plurinegative
and 1-symplectic. In Theorem 3.3, these metrics are
characterized using conditions on the space of positive
currents. The second part of this paper gives a quite
complete study of holomorphic submersion and
modifications. Some classical examples are given in the
last part of this paper.

В теории экстремальных метрик (Калаби, Мабучи, Дональдсон)
слово "balanced metric" тоже используется, но совершенно
в другом значении