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We show existence of unique smooth solutions to the Monge–Ampère equation for(   n  -  1   )(n-1)   -plurisubharmonic functions on Hermitian manifolds, generalizing previous work of the authors. As a consequence we obtain Calabi–Yau theorems for Gauduchon and strongly Gauduchon metrics on a class of non-Kähler manifolds: those satisfying the Jost–Yau condition known as Astheno–Kähler. Gauduchon conjectured in 1984 that a Calabi–Yau theorem for Gauduchon metrics holds on  all  compact complex manifolds. We discuss another Monge–Ampère equation, recently introduced by Popovici, and show that the full Gauduchon conjecture can be reduced to a second-order estimate of Hou–Ma–Wu type.

Hermitian metrics, ( n -1, n -1) forms and Monge–Ampère equations