Resolutions of compact balanced spaces

Abstract

Let $X$ be a compact reduced balanced complex space of pure dimension $n\geq2$. We prove that every resolution of singularities of $X$ is a balanced manifold; in particular, $X$ belongs to the class $\mathscr{B}$. This removes the additional cohomological assumptions appearing in earlier results of Fu-Meng-Xia. The main new ingredient is the linear independence of the Aeppli classes of the exceptional prime divisors of a projective bimeromorphic morphism onto a normal complex space. This follows from the vanishing of curve degrees for Aeppli-trivial divisors and the relative negativity lemma. For non-normal spaces, strict positivity is used to eliminate divisors mapping onto codimension-one subsets before passing to the normalization. We also recall a classical surface example of Grauert showing that the inclusion of compact balanced spaces in $\mathscr{B}$ is strict.