The Poisson Embedding Problem

Transcription

1 The Poisson Embedding Problem Symposium in honor of Alan Weinstein Aaron McMillan Fraenkel Boston College July 2013

2 Notation / Terminology By a Poisson structure on a space X, we mean: A geometric space X with an algebra of functions A over a field F = R or C, and a poisson bracket {, } X on the algebra of functions A. Remark (One advantage to the Poisson point-of-view) One can consider Poisson structures on different categories of geometric spaces: Smooth manifold M, with a bracket on C (M). Complex analytic space X, with a sheaf of poisson algebras (O X, {, } X ) As well as algebraic varieties, schemes, formal schemes... all of which can be singular.

3 The Poisson Embedding Problem Question Given a singular Poisson space (X, {, } X ), when can one realize it as an embedded Poisson subspace of some nonsingular Poisson space (P, {, } P )? We call such a realization a Poisson embedding of (X, {, } X ) into (P, {, } P ). Singular refers to the differentiable structure. Definition (Poisson embedding dimension) Given a Poisson structure (X, {, } X ), the Poisson embedding dimension of X is the number: d = min k N {j : X Pk is a Poisson embedding and dim(p k ) = k} Question For a given X, is the Poisson embedding dimension finite?

4 History of the problem (R. Cushman, R. Sjamaar, E. Lerman 1991) Understanding singular symplectic reduction. Symplectic stratified spaces. Nice examples + a conjecture (A. Weinstein) Poisson embedding may give a nonsingular approximation to a singular symplectic space. (A. Egilsson ) Answered conjecture with counterexample(s) in dissertation. Related the obstruction to embedding to semi-group structures (toric varieties). (B. Davis 2001) Used formal techniques to prove smooth results. Improved Egilsson s result: increased lower bound on Poisson embedding dimension. (AF 2011 ) Related Poisson embedding to lie algebra cohomology. Example of a nontrivial, finite Poisson embedding dimension. Many low dimensional Poisson varieties have Poisson embeddings.

5 Poisson structures for a singular symplectic problem Quotients of Poisson structures are well-behaved: Observation If M is a Poisson manifold and G a compact group action such that φ g {f 1, f 2 } = {φ g f 1, φ g f 2 } for f i C (M) and φ g Diff(M) then M/G has a natural Poisson structure. Remarks: The Poisson bracket {, } restricts to the subalgebra C (M) G C (M). The quotient space doesn t need to be smooth as long as we can make sense of C (M/G).

6 Motivating Problem (Singular Reduction) The following was considered by Cushman and Sjamaar (1990): Let G be a compact group acting on a symplectic manifold M, with equivariant moment map µ : M g. Suppose 0 is a singular value of µ, so that X := µ 1 (0)/G is a singular quotient (and therefore not a symplectic manifold). (Lerman-Sjamaar, 1991) What we do know: The singular quotient X is a symplectic stratified space (and a Poisson space). The large singular quotient M/G is a Poisson space and the strata of X are symplectic leaves of M/G. When is there a Poisson embedding of M/G into R n for some Poisson structure on R n?

7 Motivating Problem An answer to this question may: Give a better understanding of how symplectic leaves fit together near a singular point (or a more complicated singular locus). Relate group-theoretic descriptions of the symplectic strata (e.g. via orbit type) to invariants of the Poisson structure on R n. Enable one to approximate the singular quotient X with a nearby nonsingular symplectic leaf in R n.

8 Focus on Singular Symplectic Spaces Finitely many symplectic leaves. Interesting local structure at finitely many singular points. Simple, but nontrivial! Setup: V a symplectic vector space (over F = R or C) G Sp(V ) a finite group with 0 as an isolated fixed point. The singular point 0 is a 0-dimensional symplectic leaf. V /G is a (semi)-algebraic variety allows one to use algebraic techniques. (G. Schwarz, 75) If G is a compact group acting linearly on V = F n, then there exist polynomials σ 1,..., σ k that generate C (V ) G. That is, every F C (V ) G may be written as F = F (σ 1,..., σ k ) for some F C (F k ).

9 An Example: R 2 /Z 2 Example Let V = R 2 and let Z 2 = {1, 1} act via multiplication. The Poisson algebra is C (R 2 ) with {x, y} = 1. The algebra of invariants C (R 2 ) G is generated by {σ 1, σ 2, σ 3 } = { 1 2 x 2, xy, 1 2 y 2 )}. The generators are subject to the relation σ 1 σ 3 4σ 2 2 = 0. The Poisson brackets on these generators are: {σ 2, σ 3 } = σ 3, {σ 2, σ 1 } = σ 1, {σ 1, σ 3 } = σ 2, Thus F : R 2 /Z 2 R 3 = sl 2 (R) defined by F (x, y) = (σ 1 (x, y), σ 2 (x, y), σ 3 (x, y)) is a realization of the quotient as a Poisson subvariety of the linear Poisson algebra sl 2 (R)

10 The quotient is the union of the half-cone and the origin. Graphic by B. Davis

11 Why did this example work? Reason # 1 C (R 2 ) Z2 is generated by quadratic invariants and the degree of {, } is 2. Example If G Sp(V ) acts on V and the invariants C (V ) G are generated by quadratic elements {σ 1,..., σ l }, then {, } has degree 2 {σi, σ j } is again a quadratic G-invariant function. {σi, σ j } = P k ck ij σ k where cij k F V /G Poisson embeds in g as a cone, where g is the Lie algebra with structure constants cij k.

12 Why did this example work? Reason #2 (R 2 /Z 2 is low dimensional.) 3 variables Jacobi identity is only one equation. Hypersurface X = V (φ). Example (2-dimensional real symplectic orbifolds) Each quotient R 2 /Z n is realizable as a Poisson subspace of R 3. The quotient is (a subset of) the vanishing locus of xy n 2 z n in R 3. The Poisson structure on R 3 is not linear for n > 2: {z, y} = y, {z, x} = x, {x, y} = z n 1, Example (2-dim. holomorphic symplectic orbifolds) The local models for the 2-dimensional holomorphic symplectic orbifolds are C 2 /G, where G = Z n, BD 4n, BT 24, BO 48, BI 120. Each embeds into C 3 as a Poisson subspace.

13 Nonembedding Results The case of quadratic invariants led to the following: Conjecture (Cushman, Sjamaar, 1991) If R 2n /G embeds in R N, then there is a Poisson structure on R N extending the Poisson structure on the quotient. A. Egilsson found a counter-example: Theorem (A. Egilsson,1995) T R 3 /S 1 S 1 acts on T R 3 = C 3 with weights (1, 1, ±2). T R 3 /S 1 algebraically embeds in R 11. T R 3 /S 1 does not Poisson embed in R 11 Extended result to the singular symplectic space µ 1 (0)/S 1 R 10. Partially extended the results to other weights (2000).

14 B. Davis strengthened this counter-example: Theorem (Davis, 2001) While T R 3 /S 1 embeds into R N for N 11, there is no Poisson structure on R N extending the Poisson structure on T R 3 /S 1 for N = 11, 12. There is a dimensional obstruction to extending the Poisson bracket that is purely Poisson theoretic. It is not known if the Poisson embedding dimension of T R 3 /S 1 is finite!

15 Recent Results 4-dimensional symplectic quotients The previous counter-example is typical: Theorem (AF, 2011) The 4-dimensional symplectic quotient R 4 /Z n is realizable as a subvariety of R 2(n+1)+4. However there is no Poisson bracket on R 2(n+1)+4 extending the Poisson bracket on the quotient. In particular, for the n = 3 case: Theorem (AF, 2011) The 4-dimensional symplectic quotient R 4 /Z 3 embeds as an algebraic variety in R 12. There is no Poisson structure on R 12 extending the Poisson bracket on R 4 /Z 3. There is a Poisson structure on R 78 extending the Poisson bracket on R 4 /Z 3. The Poisson embedding dimension is 13 d 78.

16 Question Can this obstruction (and its disappearance) be seen in any known invariants of the quotient X? The proofs of these theorems are technical and unenlightening. Macaulay2 is often used to show properties of the specific examples (Davis, AF). AF uses theory of normal forms (e.g. Levi decomposition) to refine what a potential extension must look like and show such a refined bracket is incompatible with the ideal defining X. For example, the bivector extending the Poisson structure on C 2 /Z 3 to R 78 is given by:

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20 Question Is there a more tractable, lower-dimensional example of this same phenomenon? For example, does there exist an embedded singular surface X F 3 and a Poisson bracket {, } X that does not extend to F 3? Theorem (AF 2013) Suppose X = V (φ) F 3 is a 2-dimensional affine Poisson variety with only isolated singularities. There exists a Poisson bracket on F 3 extending the Poisson bracket on X. Moreover, this bracket has the form: {, } π = f φ x y z + f φ y z x + f φ z x y Remark A similar statement for X = V (φ) F 4 is true, with the additional assumption that φ is a weight-homogeneous polynomial.

21 Extending an embedded Poisson hypersurface Suppose (X, {, } X ) is a Poisson variety, where X = V (φ) F n is a hypersurface with isolated singularities. K (φ) : 0 Ω 0 d φ d... φ Ω n 2 d φ Ω n 1 d φ Ω n 0 C (φ) : 0 X n d φ... d φ X 2 d φ X 1 d φ X 0 0 The differentials are given by d φ (α) = α dφ, where α Ω k, dφ (A) = ι dφ A = [A, φ], where A X k. The chain complexes are isomorphic via the standard pairing between Ω and X on F n. K is isomorphic to the Koszul complex of the Jacobian ideal of V (φ) H k (K ) 0 when k n.

22 Extending an embedded Poisson hypersurface If B X 2 extends {, } X, then d φ B = [B, φ] = φa 1, for some vector field A 1 X 1. [B, B] = φa 3, for some trivector field A 3 X 3. Therefore 0 = dφb 2 = d φ (φa 1 ) = φd φ (A 1 ), Since d φ (A 1 ) = 0, exactness A 2 X 2 s.t. A 1 = d φ A 2. d φ B = d φ (φa 2 ) d φ (B φa 2 ) = 0 (B φa 2 ) = d φ A 3, for some A 3 X 3. So any such extension is given by B = d φ A 3 + φa 2. In particular, B = d φ A 3 extends the Poisson structure on X, and in dimension 3, it is a Poisson structure.