Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Strictly nef divisors on K-trivial fourfolds

Haidong Liu, Shin-ichi Matsumura

TL;DR

This work proves the ampleness conjecture for strictly nef divisors on K-trivial fourfolds by combining non-vanishing and abundance arguments tailored to the $K_X\sim0$, $h^1(O_X)=0$ setting. It develops a robust framework, including almost strictly nef divisors, a specialized Hirzebruch–Riemann–Roch formula, and the study of prime Calabi–Yau divisors, to push through a dimension-four inductive strategy. The key results show that strictly nef $\mathbb{Q}$-Cartier divisors with nonnegative Iitaka dimension are actually ample, with a detailed treatment of the case where the numerical dimension is $3$ and a careful analytic approach for lower numerical dimensions. The findings advance understanding of Serrano-type conjectures in the K-trivial context and connect to log versions and abundance phenomena, offering structural constraints on the pseudoeffective and movable cones in these geometries.

Abstract

In this paper, we prove the ampleness conjecture and Serrano's conjecture for strictly nef divisors on K-trivial fourfolds. Specifically, we show that any strictly nef divisors on projective fourfolds with trivial canonical bundle and vanishing irregularity are ample.

Strictly nef divisors on K-trivial fourfolds

TL;DR

This work proves the ampleness conjecture for strictly nef divisors on K-trivial fourfolds by combining non-vanishing and abundance arguments tailored to the , setting. It develops a robust framework, including almost strictly nef divisors, a specialized Hirzebruch–Riemann–Roch formula, and the study of prime Calabi–Yau divisors, to push through a dimension-four inductive strategy. The key results show that strictly nef -Cartier divisors with nonnegative Iitaka dimension are actually ample, with a detailed treatment of the case where the numerical dimension is and a careful analytic approach for lower numerical dimensions. The findings advance understanding of Serrano-type conjectures in the K-trivial context and connect to log versions and abundance phenomena, offering structural constraints on the pseudoeffective and movable cones in these geometries.

Abstract

In this paper, we prove the ampleness conjecture and Serrano's conjecture for strictly nef divisors on K-trivial fourfolds. Specifically, we show that any strictly nef divisors on projective fourfolds with trivial canonical bundle and vanishing irregularity are ample.

Paper Structure

This paper contains 10 sections, 23 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Almost strictly nef divisors
  4. Hirzebruch--Riemann--Roch formula
  5. Relative terminal and canonical models
  6. Prime Calabi--Yau divisors
  7. Abundance for strictly nef divisors
  8. Non-vanishing for strictly nef divisors
  9. The case of $\nu(L)=3$
  10. The case of $\nu(L)\leq 2$

Key Result

Theorem 1.2

Any strictly nef $\mathbb Q$-Cartier divisor $L$ on a K-trivial fourfold $X$ is ample.

Theorems & Definitions (50)

  • Conjecture 1.1: Ampleness conjecture
  • Theorem 1.2
  • Theorem 1.3: Theorems \ref{['thm.num3']} and \ref{['thm-num=2']}
  • Theorem 1.4: Theorem \ref{['thm.abundance']}
  • Conjecture 1.5: Log version of Serrano's conjecture
  • Theorem 1.6: Theorems \ref{['thm.abundance.eff']}, \ref{['thm.ab.dim3']} and Corollary \ref{['cor.abundance.eff']}
  • Definition 2.1: ccp*Definition 1.1
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 40 more