By E. Ramirez De Arellano
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Extra resources for Algebraic Geometry and Complex Analysis
In particular the invariant polynomial of the trivial structure on an afline Poisson variety of dimension s is S'. 51 For the Poisson structures on C2, which axe defined by a single polynomial jx, yj, with W:A 0 we have p RS2 + kS, where k is the number of components of W(x, y) the plane curve defined by W(x, y) 0. Its invariant matrix is thus given by = = = ( It follows in 0 k 0 0 0). 1 particular that the polynomial invariant is not a complete invariant: all nonpolynomials W(x, y) lead to a Poisson structure on C2 with invariant constant irreducible p = RS2 + S.
For simplicity let us take the case n 4 with a constant Poisson structure of rank 4. g. a =A 0 and look for a Jp1 P21 linear function G alql + b'q2 + 41 + dIP2 which is in involution with F. Replacing G by = = I I i = = = 7 = G - Fa'/a if necessary we may assume G = b1q2 that a' + (db' == - 0 and bd)pl we + find dP2 general solution (up to adding multiples of F). Here Y, d' EE C are arbitrary, so essentially a one-paxameter family of possibilities for G (paxametrized by d1b'), The Poisson bracket of two of these all leading to an integrable subalgebra A of O(C4) for is by G, given possibilities as the most that we have .
8) integrable Hamiltonian system and the projection maps 7ri are morphisms. 21. involutivity, firi Ai (2) 7r2* A2 7r,*1 Al , (9 7r;2 A2 I mi . m, 58 -":::: 7ri* 1 J& A111 + 1r*2JA2, A212 2 0- As for 3. r2*A2 is complete and involutive with respect to the product bracket, this computation shows that 7r,*Ai (8),7r2*A2 is integrable. Since for earch of the projection maps iri The fibers of the one has -7ri*Ai C 7r1*A1 0 7rM2, these projection maps are morphisms. momentum map are given by the fibers of M, x M2 -+ Spec(7r,*Al 0 lr2*A2), that is, of the product map M, x M2 -+ Spec A, x Spec A2 hence all fibers are products of level sets of A, and A2.