Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Moishezon manifolds with no nef and big classes

Jia Jia, Sheng Meng

TL;DR

The paper studies nef and big $(1,1)$-classes on compact complex manifolds within Fujiki's class ${\\mathcal{C}}$. It proves that for a bimeromorphic map $f:X\\dashrightarrow Y$ that is isomorphic in codimension $1$, if $X$ is Kähler with $h^{1,1}(X,\\mathbb{R})=1$ and $f$ is not an isomorphism, then $Y$ carries no nontrivial nef $(1,1)$-classes. This is then used to construct infinite families of smooth Moishezon threefolds $Y_d$ (and related examples) with no nef and big $(1,1)$-classes, by explicit birational modifications starting from a quintic threefold (and from $\\ abla\\mathbb{P}^3$), producing Calabi–Yau Moisheyon threefolds in some cases. These results provide counterexamples to a claimed universal nef-positivity statement in Kim22 and illuminate the limitations of nef positivity under codimension-one birational transformations, with implications for automorphism group analyses and the geometry of non-Kähler Moishezon manifolds.

Abstract

We show that a compact complex manifold $X$ has no non-trivial nef $(1,1)$-classes if there is a non-isomorphic bimeromorphic map $f\colon X\dashrightarrow Y$ isomorphic in codimension $1$ to a compact Kähler manifold $Y$ with $h^{1,1}=1$. In particular, there exist infinitely many isomorphic classes of smooth compact Moishezon threefolds with no nef and big $(1,1)$-classes. This contradicts a recent paper (Strongly Jordan property and free actions of non-abelian free groups, Proc. Edinb. Math. Soc., (2022): 1--11).

Moishezon manifolds with no nef and big classes

TL;DR

The paper studies nef and big -classes on compact complex manifolds within Fujiki's class . It proves that for a bimeromorphic map that is isomorphic in codimension , if is Kähler with and is not an isomorphism, then carries no nontrivial nef -classes. This is then used to construct infinite families of smooth Moishezon threefolds (and related examples) with no nef and big -classes, by explicit birational modifications starting from a quintic threefold (and from ), producing Calabi–Yau Moisheyon threefolds in some cases. These results provide counterexamples to a claimed universal nef-positivity statement in Kim22 and illuminate the limitations of nef positivity under codimension-one birational transformations, with implications for automorphism group analyses and the geometry of non-Kähler Moishezon manifolds.

Abstract

We show that a compact complex manifold has no non-trivial nef -classes if there is a non-isomorphic bimeromorphic map isomorphic in codimension to a compact Kähler manifold with . In particular, there exist infinitely many isomorphic classes of smooth compact Moishezon threefolds with no nef and big -classes. This contradicts a recent paper (Strongly Jordan property and free actions of non-abelian free groups, Proc. Edinb. Math. Soc., (2022): 1--11).
Paper Structure (2 sections, 3 theorems, 5 equations)

This paper contains 2 sections, 3 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

Let $f\colon X\dashrightarrow Y$ be a bimeromorphic map of compact complex manifolds which is isomorphic in codimension $1$. Suppose $X$ is Kähler with $h^{1,1}(X,\mathbb{R})=1$ and $f$ is non-isomorphic. Then any nef $(1,1)$-class on $Y$ is trivial. In particular, $Y$ is a non-Kähler manifold in Fu

Theorems & Definitions (12)

  • Theorem 1.1
  • Example 1.2
  • Theorem 1.3
  • Example 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 2.1: cf. Gol21*Theorem 4.5 and \ref{['remark1']}
  • proof
  • Claim 2.2
  • proof
  • ...and 2 more