Moishezon manifolds with no nef and big classes

TL;DR

The paper demonstrates that certain Moishezon manifolds lack non-trivial nef classes, especially when related to specific bimeromorphic maps to Kähler manifolds with minimal Hodge number, challenging previous assumptions.

Contribution

It establishes the existence of infinitely many Moishezon threefolds with no nef and big classes, providing new examples that counter recent conjectures.

Findings

Existence of Moishezon threefolds with no nef and big classes.

Construction of manifolds with specific bimeromorphic properties.

Contradiction of previous assumptions in complex geometry.

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 K\"ahler 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).