NONLINEAR SYMPLECTIC GRASSMANNIANS AND HAMILTONIAN ACTIONS IN PREQUANTUM LINE BUNDLES

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In this paper we extend to the Fréchet setting the following well-known fact about finite-dimensional symplectic geometry: if a Lie group G acts on a symplectic manifold in a Hamiltonian fashion with momentum map μ, given x ∈ M the isotropy group Gxacts linearly on the tangent space in a Hamiltonian fashion, with momentum map the Taylor expansion of μ up to degree 2. We use this result to give a conceptual explanation for a formula of Donaldson in [Scalar curvature and projective embeddings. I, J. Differential Geom.59(3) (2001) 479–522], which describes the momentum map of the Hamiltonian infinitesimal action of the Lie algebra of the group of Hamiltonian diffeomorphisms of a closed integral symplectic manifold, on sections of its prequantum line bundle.