Symplectic geometry on moduli spaces of holomorphic bundles over complex
surfaces

Boris Khesin and Alexei Rosly

October 1998

We give a comparative description of the Poisson structures on the moduli
spaces of flat connections on {\it real} surfaces and holomorphic Poisson
structures on the moduli spaces of holomorphic bundles on {\it complex}
surfaces.  The symplectic leaves of the latter are classified by
restrictions of the bundles to certain divisors. This can be regarded as
fixing a ``complex analogue of the holonomy" of a connection along a
``complex analogue of the boundary" in analogy with the real case.