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Let $X$ be a symplectic (affine) variety over $\mathbb{C}$, that is, a normal variety with a non-degenerate closed (algebraic) 2-form on the smooth locus. How can we deduce that the Poisson center is 1-dimensional?

If $\mathrm{dim}_\mathbb{C}X = n$, I guess Hamiltonian flows by suitably chosen $n$ functions will suffice to force a central element to be constant, just like usual $C^\infty$ manifolds, but since we don't have Darboux coordinates in algebraic setting (see this MO post), I couldn't go further.

Algebraic geometry is not my major, and I'd like to see any suggestions as well as (hopefully introductory) references regarding symplectic/Poisson varieties. Thank you.

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There are lecture notes by Roubtsov-Suchánek on Poisson algebras, where the subject is rendered very algebraically. I believe section 4 has an answer to your question, especially Proposition 4.1., almost by definition.

Suppose that our Poisson structure comes from a non-degenerate closed skew-form $\omega$. The map $\varphi :H\mapsto -X_H$ is an homomorphism of Lie algebras and the center is its kernel (this is where the space of functions gets its structure of a Lie algebra). But this map is nothing more than plugging the differential $dH$ of a function to the isomorphism $T^*X\to TX$ induced by $\omega$, hence the differential of an element of the kernel should be identically zero.

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  • $\begingroup$ Section 5.5 there was helpful. I think I had misunderstood the definition of non-degeneracy; since it is defined so that 1-forms and derivations are 1-to-1, central elements are indeed locally constant. $\endgroup$
    – Qwert Otto
    Commented Jul 30 at 4:49

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