Modified Shiffman–Zelditch Conjecture (Part a): Finiteness of Calabi and Mabuchi Volumes under Compact Automorphism Group

Determine whether, for a polarized manifold (M, L) with compact automorphism group Aut(M, L), there exists an integer k0 such that for all k ≥ k0 the Calabi volume μ_Ca(B_{M,L}^k) and the Mabuchi volume μ_Ma(B_{M,L}^k) are finite.

Background

Motivated by counterexamples demonstrating that μCa(B{M,L}^k) and μMa(B{M,L}^k) can be infinite when Aut(M,L) is non-compact, the authors propose a modified problem restricting to polarized manifolds with compact automorphism groups.

This reformulation asks whether finiteness of the Calabi and Mabuchi volumes might hold uniformly for sufficiently large k in the compact-automorphism setting, thereby salvaging a useful measure-theoretic framework on Bergman spaces.