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Notes on rational chain connectedness

Osamu Fujino

Abstract

We extend Hacon--M\textsuperscript{c}Kernan's rational chain connectedness theorem to the complex analytic setting. As a consequence, we prove that the fibers of any resolution of singularities of complex analytic kawamata log terminal singularities are rationally chain connected. In contrast to the original approach, we avoid the use of extension theorems and instead rely on the minimal model program.

Notes on rational chain connectedness

Abstract

We extend Hacon--M\textsuperscript{c}Kernan's rational chain connectedness theorem to the complex analytic setting. As a consequence, we prove that the fibers of any resolution of singularities of complex analytic kawamata log terminal singularities are rationally chain connected. In contrast to the original approach, we avoid the use of extension theorems and instead rely on the minimal model program.
Paper Structure (6 sections, 22 theorems, 87 equations)

This paper contains 6 sections, 22 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inequality for the Kodaira Dimension
  4. A Criterion due to Hacon--McKernan
  5. The Hacon--McKernan Rational Chain Connectedness Theorem
  6. Proofs of the Results

Key Result

Theorem 1.1

Let $X$ be a normal complex variety and let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier. Let $f \colon X \to S$ be a projective morphism between complex analytic spaces. Assume that $-K_X$ is $f$-big and that $-(K_X+\Delta)$ is $f$-semiample. Let

Theorems & Definitions (51)

  • Theorem 1.1: hacon-mckernan
  • Corollary 1.2: hacon-mckernan
  • Corollary 1.3: hacon-mckernan
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6: Rational chain connectedness, see fujino-hyp and fujino-quasi-log
  • Corollary 1.7: hacon-mckernan
  • Definition 2.1: Boundary part
  • Definition 2.2: Non-klt locus
  • Definition 2.3
  • ...and 41 more