Gabriel Frey | Searcharxiv

arXiv2026

A lower bound on the Calabi functional for a degeneration of polarized varieties

We prove a lower bound on the Calabi functional for degenerations of polarized varieties, involving the difference of CM degrees between generically isomorphic families. This may be viewed as a discretely valued version of Donaldson's lower bound for models, in the sense of non-Archimedean geometry. In particular, this generalizes a result of Donaldson, who considered a single polarized variety. As a main tool, we develop the theory of GIT height, introduced by Wang, and apply it to the family GIT problem of the Chow variety. Using the GIT height, we also give a numerical proof of separatedness of GIT quotients for general and special linear actions, strengthening prior work of Wang--Xu.

math.AG↗

arXiv2025

Equivariant CM minimization for extremal manifolds

We prove an equivariant version of the CM minimization conjecture for extremal K\"ahler manifolds. This involves proving that, given an equivariant punctured family of polarized varieties, a relative version of the CM degree is strictly minimized by an extremal filling. This generalizes a result by Hattori for cscK manifolds with discrete automorphism group by allowing automorphisms and extremal metrics. As a main tool, we extend results by Sz\'ekelyhidi on asymptotic filtration Chow stability of cscK manifolds with discrete automorphism group to the extremal setting.

math.AG↗