On the coordinate ring of a projective Poisson scheme

D. Kaledin

paper

On the coordinate ring of a projective Poisson scheme

arXiv:math/0312134

Abstract

The projective coordinate ring of a projective Poisson scheme $X$ does not usually admit a structure of a Poisson algebra. We show that when $H^{1} (X, O_{X}) = H^{2} (X, O_{X}) = 0$ , this can be corrected by embedding $X$ into a canonical one-parameter deformation. The scheme $X$ then becomes the Hamiltonian reduction of the spectrum of the deformed projective coordinate ring with respect to $G_{m}$ . The projection into the base of the deformation is the moment map.

Final version, slightly expanded; to appear in MRL