Moishezon manifolds with no nef and big classes

Abstract We show that a compact complex manifold X has no non-trivial nef $(1,1)$ -classes if there is a non-biholomorphic bimeromorphic map $f\colon X\dashrightarrow Y$ , which is an isomorphism 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., 65 (3) (2022), 736–746).