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On the Kodaira dimension of some algebraic fiber spaces

Yongpan Zou

TL;DR

The paper develops an analytic framework to study the descent of positivity for the canonical bundle along algebraic fiber spaces, centering on the canonical bundle formula and the moduli part M_Y. By constructing semipositive singular Hermitian metrics and carefully analyzing fiber integrals, it proves a key descent theorem: if K_X is pseudoeffective, then K_Y+M_Y+B_D+C_D is pseudoeffective, with C_D supported on the base divisor. This structural result enables the proof of Schnell's conjecture by relating Campana–Peternell's generalized non-vanishing to the positivity of the descended canonical data, and yields new direct proofs of classical Iitaka-type results. Overall, the work clarifies the role of the moduli divisor in positivity descent and provides analytic tools for broader applications in the geometry of fiber spaces.

Abstract

In this paper, we study the descent of positivity of the canonical bundle along fiber spaces. As a consequence, we prove a conjecture of Schnell, establishing the equivalence between the Non-vanishing Conjecture and its generalized version proposed by Campana and Peternell.

On the Kodaira dimension of some algebraic fiber spaces

TL;DR

The paper develops an analytic framework to study the descent of positivity for the canonical bundle along algebraic fiber spaces, centering on the canonical bundle formula and the moduli part M_Y. By constructing semipositive singular Hermitian metrics and carefully analyzing fiber integrals, it proves a key descent theorem: if K_X is pseudoeffective, then K_Y+M_Y+B_D+C_D is pseudoeffective, with C_D supported on the base divisor. This structural result enables the proof of Schnell's conjecture by relating Campana–Peternell's generalized non-vanishing to the positivity of the descended canonical data, and yields new direct proofs of classical Iitaka-type results. Overall, the work clarifies the role of the moduli divisor in positivity descent and provides analytic tools for broader applications in the geometry of fiber spaces.

Abstract

In this paper, we study the descent of positivity of the canonical bundle along fiber spaces. As a consequence, we prove a conjecture of Schnell, establishing the equivalence between the Non-vanishing Conjecture and its generalized version proposed by Campana and Peternell.
Paper Structure (13 sections, 17 theorems, 150 equations)

This paper contains 13 sections, 17 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Positive metrics
  4. Extend sigular metric
  5. Main theorem
  6. Setup II
  7. Complex structure
  8. Moduli divisor in canonical bundle formula
  9. The proof of the main theorem
  10. Part $1$
  11. Part $2$
  12. The proof of the Schnell conjecture
  13. Miscellaneous results

Key Result

Theorem 1.1

In addition to Setup I, let $D = \sum d_i P_i$ be a $\mathbb{Q}$-divisor on $X$ satisfying the following conditions: Let By construction, the pair $(Y, B_D)$ is sub-klt. Let $M_Y = L - B_D$, we have Then the $\mathbb{Q}$-divisor $M_Y$ is nef.

Theorems & Definitions (30)

  • Theorem 1.1: Kaw98, Theorem 2
  • Theorem 1.2: = Theorem \ref{['claim']}
  • Theorem 1.3: =Theorem \ref{['schnell-2']}
  • Corollary 1.4: =Theorem \ref{['G']}
  • Corollary 1.5: = Theorem \ref{['A']}
  • Corollary 1.6: =Theorem \ref{['VZ']}
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • ...and 20 more