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Positivity of vector bundles and Dominance

Laytimi Fatima, Werner Nahm

Abstract

Let $E$ be a vector bundle and $S_a$, $S_b$ the Schur functors associated to partitions $a$ and $b$. Previously we have shown that ampleness of $S_aE$ implies ampleness of $S_bE$ when $a$ is greater than $b$ in the dominance partial order. Here we prove that this result generalizes to $k$-ample, semiample and nef vector bundles. Our proof uses the common algebraic nature of these three properties and an investigation of the Littlewood-Richardson rules.

Positivity of vector bundles and Dominance

Abstract

Let be a vector bundle and , the Schur functors associated to partitions and . Previously we have shown that ampleness of implies ampleness of when is greater than in the dominance partial order. Here we prove that this result generalizes to -ample, semiample and nef vector bundles. Our proof uses the common algebraic nature of these three properties and an investigation of the Littlewood-Richardson rules.
Paper Structure (2 sections, 12 theorems, 31 equations)

This paper contains 2 sections, 12 theorems, 31 equations.

Table of Contents

  1. Algebraic Positivities
  2. Partitions and their algebra

Key Result

Theorem 1.2

Semiampleness, $k$-ampleness and nefness are algebraic properties.

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2
  • proof
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.7
  • ...and 17 more