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A footnote to a theorem of Kawamata

Margarida Mendes Lopes, Rita Pardini, Sofia Tirabassi

Abstract

Kawamata has shown that the quasi-Albanese map of a quasi-projective variety with log-irregularity equal to the dimension and log-Kodaira dimension 0 is birational. In this note we show that under these hypotheses the quasi-Albanese map is proper in codimension 1 as conjectured by Iitaka.

A footnote to a theorem of Kawamata

Abstract

Kawamata has shown that the quasi-Albanese map of a quasi-projective variety with log-irregularity equal to the dimension and log-Kodaira dimension 0 is birational. In this note we show that under these hypotheses the quasi-Albanese map is proper in codimension 1 as conjectured by Iitaka.
Paper Structure (7 sections, 7 theorems, 12 equations)

This paper contains 7 sections, 7 theorems, 12 equations.

Table of Contents

  1. Preliminaries
  2. Logarithmic ramification formula
  3. Quasi-abelian varieties and their compactifications
  4. The quasi-Albanese map
  5. Proof of theorem \ref{['thm:main']}
  6. Special case: all the $E_i$'s are components of $\overline{R}_f$
  7. Conclusion of the proof

Key Result

Theorem 1

Let $V$ be a smooth complex algebraic variety of dimension $n$. If $\overline{\kappa}(V)=0$ and $\overline{q}(V)=n$ then the quasi-Albanese morphism $a_V\colon V\rightarrow A(V)$ is birational.

Theorems & Definitions (10)

  • Theorem : Kawamata, Theorem 28 of Ka81
  • Theorem 1
  • Corollary 2
  • Lemma 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • proof