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On the superadditivity of anticanonical Iitaka dimension

Marta Benozzo, Iacopo Brivio, Chi-Kang Chang

TL;DR

This work proves a sharp Iitaka-type inequality for the anticanonical divisor in fibrations over fields of arbitrary characteristic: κ(X,-K_X) ≤ κ(X_y,-K_{X_y}) + κ(Y,-K_Y) under the hypothesis that the Q-linear series |-K_X|_{Q} has good singularities on the general fibre. The authors extend Chang’s result to positive characteristic by developing a robust Frobenius-based toolkit, including global F-splitting/regularity, trace maps of Frobenius, and a Calabi–Yau canonical bundle formula in settings where fibres are globally F-split. A key part of the strategy is to descend positivity of -K_X to the base and control the fibre via an injectivity theorem, flattened base changes, and an intertwined analysis of asymptotic multiplier/test ideals. The paper also provides a detailed comparison of characteristic zero and positive characteristic phenomena, along with explicit examples illustrating the applicability and limitations of the theory. The results deepen understanding of how the anticanonical Iitaka dimension behaves under fibrations, with potential implications for the broader study of positivity in algebraic geometry across characteristics.

Abstract

Given a fibration $f: X \to Y$ with normal general fibre $X_y$, over a field of any characteristic, we establish the Iitaka-type inequality $κ(X,-K_X) \leq κ(X_y,-K_{X_y})+κ(Y,-K_Y)$ whenever the $\mathbb{Q}$-linear series $|-K_X|_{\mathbb{Q}}$ has good singularities on $X_y$.

On the superadditivity of anticanonical Iitaka dimension

TL;DR

This work proves a sharp Iitaka-type inequality for the anticanonical divisor in fibrations over fields of arbitrary characteristic: κ(X,-K_X) ≤ κ(X_y,-K_{X_y}) + κ(Y,-K_Y) under the hypothesis that the Q-linear series |-K_X|_{Q} has good singularities on the general fibre. The authors extend Chang’s result to positive characteristic by developing a robust Frobenius-based toolkit, including global F-splitting/regularity, trace maps of Frobenius, and a Calabi–Yau canonical bundle formula in settings where fibres are globally F-split. A key part of the strategy is to descend positivity of -K_X to the base and control the fibre via an injectivity theorem, flattened base changes, and an intertwined analysis of asymptotic multiplier/test ideals. The paper also provides a detailed comparison of characteristic zero and positive characteristic phenomena, along with explicit examples illustrating the applicability and limitations of the theory. The results deepen understanding of how the anticanonical Iitaka dimension behaves under fibrations, with potential implications for the broader study of positivity in algebraic geometry across characteristics.

Abstract

Given a fibration with normal general fibre , over a field of any characteristic, we establish the Iitaka-type inequality whenever the -linear series has good singularities on .
Paper Structure (19 sections, 42 theorems, 83 equations)

This paper contains 19 sections, 42 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Proof idea
  3. Preliminaries
  4. Conventions and notation
  5. Weil divisors and divisorial sheaves
  6. Iitaka dimension and Iitaka fibration
  7. Characteristic
  8. Preliminaries
  9. Proof of
  10. -singularities
  11. Traces of Frobenius morphisms
  12. Globally -split and globally -regular varieties
  13. General fibres and -singularities of
  14. Globally -split varieties and canonical bundle formula
  15. Injectivity Theorem
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $f\colon X\to Y$ be a fibration of smooth projective complex varieties, of dimensions $n$ and $m$ respectively, let $y\in Y$ be a general point, and assume $\mathbf{B}(-K_X)$ does not dominate $Y$. Then

Theorems & Definitions (103)

  • Conjecture 1: Iitaka's Conjecture, $C_{n,m}$, Iitaka
  • Theorem 1.1: $C_{n,m}^-$, Chang
  • Theorem 1.2: $C_{n,m}^-$, $\textup{char}$ $p>0$, see \ref{['t-main']}
  • Theorem 1.3: $C_{n,m}^-$, $\textup{char}$ 0, see \ref{['t-main_char0']}
  • Conjecture 2: See \ref{['c-ourconj1']}
  • Example 1: Chang
  • Remark 1
  • Lemma 1: Flattening lemma, flattening
  • Remark 2
  • Definition 1
  • ...and 93 more