KSB moduli spaces parametrizes canonically polarized varieties endowed with a Kaehler-Einstein metric. The existence of such a metric is equivalent to various stability conditions, such as indeed KSB stability. Once the dimension and the volume of the parametrized varieties are fixed, the coarse KSB moduli spaces is known to be a projective.
I will give a quantitative description of a portion of the ample cone of KSB moduli spaces which depends only on the dimension of the parametrized varieties, not on the volume. Under the further assumption that at least one parametrized variety is klt, I will also prove a lower bound on the volume of these moduli spaces. A variant of some of these results also apply to moduli space of K-stable Fano varieties.
The proofs rely on a careful study of the Harder-Narasimhan filtration of vector bundles associated to one-parameter families of KSB-stable and K-stable varieties.
The talk is based on a joint work and a work in progress with L. Tasin and F. Viviani.