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Albanese map for Kähler manifolds with nef anticanonical bundle

Philipp Naumann, Xiaojun Wu

Abstract

We study the structure of the Albanese map for Kähler manifolds with nef anticanonical bundle. First, we give a result for fourfolds whose Albanse torus is an elliptic curve. In the general case of any dimension, we look at two cases: The general fiber of the Albanese map is a Calabi-Yau manifold or a projective space. In the first case, we show that the manifold itself must be Calabi-Yau. In the second case, we give a more topological proof of a result by Cao and Höring which says that the manifold must be a projectivization of a numerically flat vector bundle.

Albanese map for Kähler manifolds with nef anticanonical bundle

Abstract

We study the structure of the Albanese map for Kähler manifolds with nef anticanonical bundle. First, we give a result for fourfolds whose Albanse torus is an elliptic curve. In the general case of any dimension, we look at two cases: The general fiber of the Albanese map is a Calabi-Yau manifold or a projective space. In the first case, we show that the manifold itself must be Calabi-Yau. In the second case, we give a more topological proof of a result by Cao and Höring which says that the manifold must be a projectivization of a numerically flat vector bundle.
Paper Structure (6 sections, 19 theorems, 50 equations)

This paper contains 6 sections, 19 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Structure of the Albanese map for 4-folds when the base is a curve
  3. First Case: Fibers are Calabi-Yau
  4. Second case: Fibers are projective spaces
  5. The general uniruled case when fibers are surfaces
  6. Algebraic approximation

Key Result

Theorem A

Let $X$ be a non-projective compact Kähler fourfold with $-K_X$ nef such that $\tilde{q}(X)=q(X)=1$. Then, up to a finite étale cover, $X$ is either the product of a K3 surface and a projectivization of a rank two numerically flat vector bundle over an elliptic curve or the product of a simply conne

Theorems & Definitions (45)

  • Conjecture 1
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Conjecture 2
  • Theorem E
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • ...and 35 more