Equivariant CM minimization for extremal manifolds

We prove an equivariant version of the CM minimization conjecture for extremal Kähler 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 Székelyhidi on asymptotic filtration Chow stability of cscK manifolds with discrete automorphism group to the extremal setting.

💡 Research Summary

The paper establishes an equivariant version of the CM minimization conjecture for extremal Kähler manifolds, extending earlier work of Hattori that dealt with constant scalar curvature Kähler (cscK) manifolds under the assumption of a discrete automorphism group. The author allows non‑trivial automorphism groups and replaces the cscK condition by the more general extremal metric condition (∂grad^{1,0} scal(ω)=0).

The setting is a discrete valuation ring (DVR) R (for example the formal power series ring C