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On quasi-Albanese maps

Osamu Fujino

Abstract

We discuss Iitaka's theory of quasi-Albanese maps in details. We also give a detailed proof of Kawamata's theorem on the quasi-Albanese maps for varieties of the logarithmic Kodaira dimension zero. Note that Iitaka's theory is an application of Deligne's mixed Hodge theory for smooth algebraic varieties.

On quasi-Albanese maps

Abstract

We discuss Iitaka's theory of quasi-Albanese maps in details. We also give a detailed proof of Kawamata's theorem on the quasi-Albanese maps for varieties of the logarithmic Kodaira dimension zero. Note that Iitaka's theory is an application of Deligne's mixed Hodge theory for smooth algebraic varieties.
Paper Structure (13 sections, 50 theorems, 193 equations)

This paper contains 13 sections, 50 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Logarithmic Kodaira dimensions and irregularities
  4. Quasi-abelian varieties in the sense of Iitaka
  5. Quasi-Albanese maps due to Iitaka
  6. Basic properties of quasi-abelian varieties
  7. Characterizations of abelian varieties and complex tori
  8. On subadditivity of the logarithmic Kodaira dimensions
  9. Remarks on semipositivity theorems
  10. Weak positivity theorems revisited
  11. Finite covers of quasi-abelian varieties
  12. Quasi-Albanese maps for varieties with $\overline \kappa =0$
  13. Proof of Theorem \ref{['p-thm1.3']} and Corollaries \ref{['p-cor1.4']} and \ref{['p-cor1.5']}

Key Result

Theorem 1.1

Let $X$ be a smooth algebraic variety defined over $\mathbb C$. Then there exists a morphism $\alpha\colon X\to A$ to a quasi-abelian variety $A$ such that

Theorems & Definitions (116)

  • Theorem 1.1: see iitaka1 and Theorem \ref{['p-thm3.16']}
  • Theorem 1.2: see kawamata-abelian and Theorem \ref{['p-thm10.1']}
  • Theorem 1.3: see mendes, fmpt, and cdy
  • Corollary 1.4: see mendes
  • Corollary 1.5
  • Definition 2.1: Logarithmic Kodaira dimension
  • Definition 2.2: Logarithmic irregularity
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 106 more