Stable Degenerations of log Fano Fibration Germs

Abstract

We prove the stable degeneration conjecture of log Fano fibration germs formulated by Sun-Zhang. Precisely, we introduce the $\mathbf{H}$-invariant for filtrations over a log Fano fibration germ, and show that there exists a unique quasi-monomial valuation $v_0$ minimizing the $\mathbf{H}$-invariant. Moreover, we prove that the associated graded ring of $v_0$ is finitely generated and induces a special degeneration to a K-semistable polarized log Fano fibration germ, which further admits a unique K-polystable special degeneration.

65 pages, comments are welcome! v2: Minor changes. Added Theorem 4.30 on the largest weighted volume