Shiffman–Zelditch Conjecture: Finiteness of the Calabi Volume of Bergman Spaces

Prove that for every polarized manifold (M, L) and every positive integer k, the Calabi volume μ_Ca(B_{M,L}^k) of the Bergman space B_{M,L}^k, equipped with the restriction of the Calabi metric, is finite.

Background

The paper studies the Calabi and Mabuchi metrics on the space of Bergman metrics B_{M,L}^k associated to a polarized manifold (M, L). The corresponding measures μ_Ca and μ_Ma define volume forms on these finite-dimensional spaces.

Shiffman and Zelditch conjectured that the Calabi volume of B_{M,L}^k is finite for each k. The present paper shows that in general, when Aut(M,L) is non-compact (e.g., for products with projective space), both the Calabi and Mabuchi volumes can be infinite, indicating that the original conjecture does not hold universally.