Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it?
I guess maybe there are two competing settings a priori: a complex symplectic manifold $(M, \omega)$ and an action of a Lie group $G$ preserving $\omega$, where either:
- $G$ is a complex Lie group and the action $G \times M \to M$ is holomorphic
- $G$ is merely a real Lie group acting on $M$ by biholomorphisms (actually this is automatic when the action preserves $\omega$)
I'm thinking that the two settings are more or less equivalent, but I'm not sure it's worth expanding my thoughts.
In any case, I think that one can define a complex (holomorphic) moment map and a complex symplectic quotient $M//G$ naturally equipped with a reduced complex symplectic structure (provided of course the appropriate conditions allowing symplectic reduction are satisfied, the action is Hamiltonian, etc). Right?