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On twisted Kawamata's semi-positivity and finite generation of generalized canonical rings

Yoshinori Gongyo, Shigeharu Takayama

TL;DR

The paper proves a twisted version of Kawamata's semi-positivity for fibrations, incorporating a nef ${\mathbb Q}$-line bundle with a semi-positive metric and vanishing Lelong numbers, and a ${\mathbb Q}$-divisor with carefully controlled coefficients. It defines and analyzes the discriminant and moduli parts on the base via a toroidal, well-prepared setup, showing the moduli part $M_Y$ is nef through a relative Bergman kernel construction. These analytic and algebro-geometric ingredients yield finite generation results for generalized canonical rings of Birkar's generalized pairs, using Iitaka fibrations and birational reductions together with BCHM-type arguments. The work also clarifies how the generalized setting differs from the classical one and provides concrete corollaries for anti-canonical rings and low-dimensional cases, highlighting the role of zero Lelong number metrics in establishing positivity and finite generation.

Abstract

We give a twisted version of the Kawamata semi-positivity theorem by the $\mathbb{Q}$-line bundle with a vanishing Lelong number at every point. Moreover, we apply the result to the finite generation problem for canonical rings of Birkar's generalized pairs.

On twisted Kawamata's semi-positivity and finite generation of generalized canonical rings

TL;DR

The paper proves a twisted version of Kawamata's semi-positivity for fibrations, incorporating a nef -line bundle with a semi-positive metric and vanishing Lelong numbers, and a -divisor with carefully controlled coefficients. It defines and analyzes the discriminant and moduli parts on the base via a toroidal, well-prepared setup, showing the moduli part is nef through a relative Bergman kernel construction. These analytic and algebro-geometric ingredients yield finite generation results for generalized canonical rings of Birkar's generalized pairs, using Iitaka fibrations and birational reductions together with BCHM-type arguments. The work also clarifies how the generalized setting differs from the classical one and provides concrete corollaries for anti-canonical rings and low-dimensional cases, highlighting the role of zero Lelong number metrics in establishing positivity and finite generation.

Abstract

We give a twisted version of the Kawamata semi-positivity theorem by the -line bundle with a vanishing Lelong number at every point. Moreover, we apply the result to the finite generation problem for canonical rings of Birkar's generalized pairs.
Paper Structure (10 sections, 18 theorems, 104 equations)

This paper contains 10 sections, 18 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Algebraic preliminary
  4. Analytic preliminary
  5. Fiber integral
  6. Basic estimate: Smooth total space
  7. Quasi-smooth total space
  8. Comparison of fiber integrals
  9. Semi-positivity of the moduli part
  10. On finite generation problem of log canonical rings for generalized pairs

Key Result

Theorem 1.1

Let $f: X \to Y$ be a surjective morphism of projective varieties with connected fibers. Suppose $X$ is quasi-smooth and $Y$ is smooth. Let $P=\sum_{j} P_j$ and $Q=\sum_{l} Q_l$ be reduced effective divisors on $X$ and $Y$, respectively, such that $f^{-1}(Q)\subseteq P$. Suppose that $f$ is a smooth Let (5) By construction, the pair $(Y,{\Delta}_Y)$ is sub-klt and $K_X+D+M\sim_{\mathbb{Q}} f^*(K_

Theorems & Definitions (38)

  • Theorem 1.1: Twisted Kawamata's semi-positivity theorem
  • Remark 1.2
  • Theorem 1.3: =Theorem \ref{['fg-canonical']}
  • Definition 2.1.2: Singularities of pairs
  • Theorem 2.1.3
  • proof : Proof of Theorem \ref{['wp']}
  • Lemma 2.1.4
  • proof
  • Definition 2.2.1
  • Proposition 2.2.2
  • ...and 28 more