Stable Degenerations of log Fano Fibration Germs

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.

💡 Research Summary

The paper establishes the Stable Degeneration Conjecture for log Fano fibration germs, a conjecture formulated by Sun and Zhang. A log Fano fibration germ consists of a klt pair ((X,\Delta)) together with a projective morphism (f:X\to Z) such that (-\bigl(K_X+\Delta\bigr)) is (f)-ample, where (Z) is affine and a closed point (o\in Z) is fixed. The authors introduce a non‑Archimedean functional, the (\mathbf H)-invariant, defined on filtrations of the anti‑canonical section ring (R=\bigoplus_{m\ge0}H^0!\bigl(X,-m(K_X+\Delta)\bigr)). For a filtration (\mathcal F) they set
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