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Noether inequality for irregular threefolds of general type

Yong Hu, Tong Zhang

Abstract

Let $X$ be a smooth irregular $3$-fold of general type over $\mathbb{C}$. We prove that the optimal Noether inequality $$ \mathrm{vol}(X) \ge \frac{4}{3}p_g(X) $$ holds if $p_g(X) \ge 16$ or if $X$ has a Gorenstein minimal model. Moreover, when $X$ attains the equality and $p_g(X) \ge 16$, its canonical model can be explicitly described.

Noether inequality for irregular threefolds of general type

Abstract

Let be a smooth irregular -fold of general type over . We prove that the optimal Noether inequality holds if or if has a Gorenstein minimal model. Moreover, when attains the equality and , its canonical model can be explicitly described.
Paper Structure (33 sections, 48 theorems, 191 equations)

This paper contains 33 sections, 48 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Sketch of the proof
  5. Notation and conventions
  6. Preliminaries
  7. Linear systems on $3$-folds
  8. Positivity of divisors on fibred varieties
  9. Vector bundles on elliptic curves
  10. Irregular $3$-folds of general type with $q \ge 2$
  11. The case when ${\rm alb} \dim X = 1$
  12. The case when ${\rm alb} \dim X \ge 2$
  13. $\dim \Sigma = 3$
  14. $\dim \Sigma = 2$
  15. $\dim \Sigma = 1$
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $X$ be a smooth irregular $3$-fold of general type. Then we have the following optimal Noether inequality: provided one of the following conditions holds: Moreover, if $X$ satisfies the equality in eq: Irregular Noether dim 3 as well as any of the above three conditions (1)--(3), then $1 \le q(X) \le 2$.

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 75 more