  We study the problem of classifying the holomorphic $(m,n)$-subharmonic
morphisms in complex space. This determines which holomorphic mappings
preserves $m$-subharmonicity in the sense that the composition of the
holomorphic mapping with a $m$-subharmonic functions is $n$-subharmonic. We
show that there are three different scenarios depending on the underlying
dimensions, and the model itself. Either the holomorphic mappings are just the
constant functions, or up to composition with a homotethetic map, canonical
orthogonal projections. Finally, there is a more intriguing case when
subharmonicity is gained in the sense of the Caffarelli-Nirenberg-Spruck
framework.
