Real Forms of Holomorphic Hamiltonian Systems

The paper proves that, for an R-compatible holomorphic integrable system μ on a holomorphic symplectic manifold (M, Ω), the fixed-point set MR of a real-symplectic structure R carries a real integrable system under the action of the real form Gρ, with momentum map μ̂: MR → (gρ)* (Theorem 4.5). Its argument relies on: (i) MR is a real-symplectic form (hence ωI|MR = 0 and ω̂R := ωR|MR is symplectic), (ii) R-compatibility of the G-action (Definition 2.2) so Gρ preserves MR, and (iii) independence of the differentials on an open dense subset (Lemma 4.1), yielding integrability on MR . The candidate solution reproduces these points with more explicit details: it verifies MR is symplectic for ωR, shows fundamental fields are tangent to MR using the infinitesimal equivariance R∗(Xξ) = Xρ∗ξ (4.2), constructs μ̂ via ⟨μ̂, ξ⟩ = Re⟨μ, ξ⟩, checks Poisson-commutativity, and establishes independence of differentials on a dense open set—precisely the content of the paper’s sketch, expanded step-by-step .