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On generalized canonical bundle formula and boundedness of complements in complex analytic setting

Kenta Hashizume

Abstract

We establish the generalized canonical bundle formula for generalized lc-trivial fibrations with irrational coefficients over non-compact bases in the complex analytic setting, and we show that the discriminant b-divisor and moduli b-divisor are compatible with restriction to arbitrary open subsets. We also discuss the boundedness of complements in this setting.

On generalized canonical bundle formula and boundedness of complements in complex analytic setting

Abstract

We establish the generalized canonical bundle formula for generalized lc-trivial fibrations with irrational coefficients over non-compact bases in the complex analytic setting, and we show that the discriminant b-divisor and moduli b-divisor are compatible with restriction to arbitrary open subsets. We also discuss the boundedness of complements in this setting.
Paper Structure (9 sections, 27 theorems, 81 equations)

This paper contains 9 sections, 27 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Divisors and morphisms
  4. Generalized pairs
  5. Generalized canonical bundle formula
  6. Boundedness of complements
  7. Statements
  8. Extension of complements
  9. Proofs of statements

Key Result

Theorem 1.1

Let $\pi \colon (X,B+M) \to S$ be a generalized sub-pair with the nef part $\boldsymbol{\rm M}$, and let $f\colon (X,B+M) \to Z$ be a generalized lc-trivial fibration over $S$. Let $W \subset S$ be a compact subset such that $\pi \colon X \to S$ and $W$ satisfy (P). Suppose that Then, after replacing $S$ with a suitable open neighborhood of $W$, we have the following properties. In particular, t

Theorems & Definitions (68)

  • Theorem 1.1: = Theorem \ref{['thm--gen-can-bundle-formula-lc-main']}
  • Theorem 1.2: = Theorem \ref{['thm--generalized-compl-main-1']}
  • Theorem 1.3: = Theorem \ref{['thm--generalized-compl-main-2']}
  • Corollary 1.4: cf. birkar-compl
  • Corollary 1.5
  • Definition 2.1: Property (P), see fujino-analytic-bchm
  • Definition 2.2
  • Definition 2.3: chhx-compl
  • Definition 2.4
  • Definition 2.5
  • ...and 58 more