Abstract
In this note, relying on a variant of Grauert–Remmert extension theorem for plurisubharmonic functions on locally irreducible complex spaces, we show that if [Formula: see text] is a holomorphic surjection from an irreducible complex space [Formula: see text] to a locally irreducible complex space [Formula: see text] and [Formula: see text] is an upper semicontinuous function on [Formula: see text], then [Formula: see text] if and only if [Formula: see text]. Moreover, by concrete examples, we also present that both of the irreducibility assumptions on [Formula: see text] and [Formula: see text] are sharp.
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