On singular fibers of parabolic fibrations

TL;DR

The paper studies singular fibers of a parabolic fibration $f:X\to Y$ under the hypothesis that the moduli divisor $M_Y$ is numerically trivial and the discriminant divisor $B_Y$ vanishes. Using the Fujino–Mori canonical bundle formula $bK_X=f^*b(K_Y+M_Y+B_Y)+B_X$ and a relative MMP, it shows that if a good minimal model $X'$ over $Y$ exists, the induced fibration $f':X'\to Y$ is locally trivial; without this assumption, the vertical part $B_X$ is effective and fibers satisfy that $f^*P\setminus B_X$ is reduced and irreducible for prime divisors $P\subset Y$. It also derives, via a birational comparison and very exceptional divisors, that all $f$-exceptional divisors lie in $B_X$, and that $f^*P\setminus B_X$ remains irreducible and reduced on suitable models. Finally, through limit mixed Hodge structures and the Clemens–Schmid sequence, it proves the degeneration identity $\sum_i \dim H^0(K_{X_i})=\dim H^0(K_{X_t})$, clarifying how the components of the central fiber contribute to the global sections of the canonical bundle in semistable degenerations. Overall, the work advances understanding of singular fibers in higher-dimensional parabolic fibrations and clarifies when isotriviality and fiber-structure results hold under numerical constraints on the moduli part.

Abstract

We describe the singular fibers of a parabolic fibration $f:X\to Y$ whose moduli divisor $M_Y$ is numerically trivial and discriminant divisor $B_Y$ is zero.

Theorems & Definitions (21)

• Theorem 1.1: Fujino-Mori FujinoMori2000
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof