On the existence of balanced metrics on six-manifolds of cohomogeneity one

We consider balanced metrics on complex manifolds with holomorphically trivial canonical bundle, most commonly known as balanced SU(n)-structures. Such structures are of interest for both Hermitian geometry and string theory, since they provide the ideal setting for the Hull–Strominger system. In this paper, we provide a non-existence result for balanced non-Kähler $$\text {SU}(3)$$ SU ( 3 ) -structures which are invariant under a cohomogeneity one action on simply connected six-manifolds.

Basis divisors and balanced metrics

Using log canonical thresholds and basis divisors Fujita–Odaka introduced purely algebro-geometric invariants δ m {\delta_{m}} whose limit in m is now known to characterize uniform K-stability on a Fano variety. As shown by Blum–Jonsson this carries over to a general polarization, and together with work of Berman, Boucksom, and Jonsson, it is now known that the limit of these δ m {\delta_{m}} -invariants characterizes uniform Ding stability. A basic question since Fujita–Odaka’s work has been to find an analytic interpretation of these invariants. We show that each δ m {\delta_{m}} is the coercivity threshold of a quantized Ding functional on the mth Bergman space and thus characterizes the existence of balanced metrics. This approach has a number of applications. The most basic one is that it provides an alternative way to compute these invariants, which is new even for ℙ n {{\mathbb{P}}^{n}} . Second, it allows us to introduce algebraically defined invariants that characterize the existence of Kähler–Ricci solitons (and the more general g-solitons of Berman–Witt Nyström), as well as coupled versions thereof. Third, it leads to approximation results involving balanced metrics in the presence of automorphisms that extend some results of Donaldson.

Locally conformally balanced metrics on almost abelian Lie algebras

We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.