In 1977, Gauduchon proved that on every compact hermitian manifold (X, omega) there exists a conformally equivalent hermitian metric omega(G) which satisfies dd(c)omega(n-1)(G) = 0. In this note, we extend this result to irreducible compact singular hermitian varieties which admit a smoothing.
The author is grateful to his thesis advisors Vincent Guedj and Henri Guenancia for their support, suggestions and encouragement. The author thanks Tat Dat To for pointing out the reference [ACS17] and Tsung-Ju Lee for helpful comments. The author would like to thank the anonymous referees for useful comments and suggestions. This work has benefited from state aid managed by the ANR under the 'PIA' program bearing the reference ANR-11-LABX-0040 (research project HERMETIC). The author is also partially supported by the ANR project PARAPLUI and the EUR MINT project ANR-18-EURE-0023.