Tha's all right. It is still a cute observation. I completely forgot that curvature reduces for submanifolds.

Diverietti, I am looking at the curvature operator (i.e. I want the analogue of Nakano-semi negativity).

I see. Indeed...Thanks! I suppose the action can be chosen so that the resulting beast is not the product of two surfaces eh ? A follow up question - do you think such a beast will have non-trivial deformations of complex structure ?

Obviously one is referring to the Chern character classes (which are non trivial) in the line bundles case !