Stable Degenerations of log Fano Fibration Germs

TL;DR

This paper proves the stable degeneration conjecture for log Fano fibration germs by introducing an invariant that identifies a unique valuation leading to a special, K-semistable degeneration.

Contribution

It introduces the $ extbf{H}$-invariant for filtrations over log Fano fibration germs and establishes the existence and uniqueness of a valuation minimizing this invariant.

Findings

Existence of a unique quasi-monomial valuation minimizing the $ extbf{H}$-invariant.

Finitely generated associated graded ring of the valuation induces a special degeneration.

The degeneration leads to a K-semistable and K-polystable log Fano fibration germ.

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.