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Algebraic Hamiltonian actions

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Abstract

In this paper we deal with a Hamiltonian action of a reductive algebraic group G on an irreducible normal affine Poisson variety X. We study the quotient morphism \({\mu_{G,X}//G : X//G \rightarrow \mathfrak{g} //G}\) of the moment map \({\mu_{G,X} : X\rightarrow \mathfrak{g}}\) . We prove that for a wide class of Hamiltonian actions (including, for example, actions on generically symplectic varieties) all fibers of the morphism μ G,X //G have the same dimension. We also study the “Stein factorization” of μ G,X //G. Namely, let C G,X denote the spectrum of the integral closure of \({\mu_{G,X}^{*}(\mathbb{K}[\mathfrak{g}]^G)}\) in \({\mathbb{K}(X)^G}\) . We investigate the structure of the \({\mathfrak{g} //G}\) -scheme C G,X . Our results partially generalize those obtained by F. Knop for the actions on cotangent bundles and symplectic vector spaces.

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  1. Department of Mathematics, Massachusetts Institute of Technology, 2-101 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Ivan V. Losev

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  1. Ivan V. Losev
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Correspondence to Ivan V. Losev.

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Supported by RFBR grant 05-01-00988.

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Losev, I.V. Algebraic Hamiltonian actions. Math. Z. 263, 685–723 (2009). https://doi.org/10.1007/s00209-009-0587-7

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