# Complex symplectic reduction

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:

1. $G$ is a complex Lie group and the action $G \times M \to M$ is holomorphic

or

1. $G$ 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?

Any thoughts?

• Shouldn't this be similar to hyper-Kahler reduction? – Paul Reynolds Mar 12 '15 at 21:20
• Sort of, but I would rather say that hyperkähler reduction is a subreduction of complex symplectic reduction (or something like that). It's implied in the original paper of Hitchin , Karlhede , Lindström , Roček defining hyperkähler reduction. Anyway, I see no need to talk about hyperkähler structures here. – seub Mar 12 '15 at 21:28
• Are you looking for something like Chapter 1 of Chriss-Ginzburg (Representation Theory and complex geometry)? Or, say, places dealing with Kostant-Kirillov-Souriau. – Peter Dalakov Mar 12 '15 at 22:02
• Thanks for the reference. After a brief look, I think it does talk about the moment map in a possibly holomorphic setting, however reduction is not addressed. – seub Mar 12 '15 at 23:37
• Have you tried instead using the search term "holomorphic symplectic reduction"? – Qiaochu Yuan Mar 13 '15 at 8:00