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On the direct image of the adjoint big and nef line bundles

Yuta Watanabe, Yongpan Zou

Abstract

We investigate the positivity properties of the direct image $f_{\ast}(K_{X/Y} \otimes L)$ of the adjoint line bundle associated with a big and nef line bundle $L$, under a smooth fibration $f: X\to Y$ between projective varieties. We show that the vector bundle $f_{\ast}(K_{X/Y} \otimes L)$ is big.

On the direct image of the adjoint big and nef line bundles

Abstract

We investigate the positivity properties of the direct image of the adjoint line bundle associated with a big and nef line bundle , under a smooth fibration between projective varieties. We show that the vector bundle is big.
Paper Structure (7 sections, 20 theorems, 51 equations)

This paper contains 7 sections, 20 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Positivity in complex geometry
  3. Positivity of induced metrics on the tautological line bundle
  4. The bigness of direct image
  5. Set-up of the metric on $L$
  6. Nakano positivity via the optimal $L^2$-estimate
  7. Set-up of case $Z$

Key Result

Theorem 1.1

Mou97MT07Bo09BLNN23 Let $f: X\to Y$ be a smooth fibration of smooth projective varieties and denote the relative canonical line bundle by $K_{X/Y}$. For any ample or positive line bundle $L$ on $X$, the direct image $f_{\ast}(K_{X/Y} \otimes L)$ is either zero or an ample vector bundle.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: =Theorem \ref{['main']}
  • Corollary 1.4
  • Definition 2.1: Positive vector bundle
  • Definition 2.2: Singular metric and curvature current on line bundles
  • Definition 2.3: Multiplier ideal sheaves
  • Theorem 2.4
  • Lemma 2.5: Kodaira lemma
  • Definition 2.6
  • ...and 28 more