[Paper Review] Normal forms in Poisson geometry

Ioan Mărcuț|arXiv (Cornell University)|Jan 19, 2013

Homotopy and Cohomology in Algebraic Topology51 references7 citations

TL;DR

This thesis establishes four local normal form theorems in Poisson geometry using geometric, algebraic, and analytic methods. It proves a formal equivalence result (Theorem 3) and a rigidity theorem (Theorem 4) around Poisson submanifolds via the Nash-Moser method, leading to the first computation of a Poisson moduli space in dimension ≥3 for the Lie-Poisson sphere of a compact semisimple Lie algebra (Theorem 5).

ABSTRACT

This thesis studies normal forms for Poisson structures around symplectic leaves using several techniques: geometric, formal and analytic ones. One of the main results (Theorem 2) is a normal form theorem in Poisson geometry, which is the Poisson-geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry). The result generalizes Conn's theorem from fixed points to arbitrary symplectic leaves. We present two proofs of this result: a geometric one relying heavily on the theory of Lie algebroids and Lie groupoids (similar to the new proof of Conn's theorem by Crainic and Fernandes), and an analytic one using the Nash-Moser fast convergence method (more in the spirit of Conn's original proof). The analytic approach gives much more, we prove a local rigidity result (Theorem 4) around compact Poisson submanifolds, which is the first of this kind in Poisson geometry. Theorem 4 has a surprising application to the study of smooth deformation of Poisson structures: in Theorem 5 we compute the Poisson-moduli space around the Lie-Poisson sphere (i.e. the invariant unit sphere inside the linear Poisson manifold corresponding to a compact semisimple Lie algebra). This is the first such computation of a Poisson moduli space in dimension greater or equal to 3 around a degenerate (i.e. non-symplectic) Poisson structure. Other results presented in the thesis are: a new proof to the existence of symplectic realizations (Theorem 0), a normal form theorem for symplectic foliations (Theorem 1), a formal normal form/rigidity result around Poisson submanifolds (Theorem 3), and a general construction of tame homotopy operators for Lie algebroid cohomology (the Tame Vanishing Lemma). We also revisit Conn's theorem and a theorem of Hamilton on rigidity of foliations.

Motivation & Objective

• To establish local normal form theorems for Poisson structures around symplectic leaves and Poisson submanifolds.
• To develop a new approach to the existence of symplectic realizations using cotangent paths and transversals.
• To prove formal equivalence and rigidity results for Poisson structures near submanifolds using graded Lie algebras and tame cohomology.
• To compute the smooth deformation space (moduli space) of the Lie-Poisson sphere of a compact semisimple Lie algebra, a first such computation in dimension ≥3.
• To extend Reeb stability and holonomy groupoid techniques to Poisson foliations and prove Hausdorffness criteria for holonomy groupoids.

Proposed method

• Uses geometric methods, including Moser’s path method and integrability of Lie algebroids, to prove normal forms around symplectic leaves.
• Applies the Van Est map and differentiable cohomology of Lie groupoids to relate symplectic realizations to the geometry of cotangent paths.
• Employs formal power series expansions and the algebra of formal vector fields to prove Theorem 3 on formal equivalence around Poisson submanifolds.
• Utilizes the Nash-Moser method with tame homotopy operators and tame cohomology to prove analytic rigidity (Theorem 4), relying on Hamilton’s Tame Vanishing Lemma.
• Constructs symplectic realizations from transversals in the space of cotangent paths, reducing integrability to existence of 'nice' realizations.
• Applies the Bott connection and holonomy representations to define actions on normal bundles, enabling the use of the Tame Vanishing Lemma in degree one.

Experimental results

Research questions

• RQ1 Under what conditions does a Poisson manifold admit a symplectic realization, and how can such realizations be constructed geometrically?
• RQ2 What is the local normal form of a Poisson structure near a symplectic leaf, and how does it depend on the first-order data?
• RQ3 Can Poisson structures near a Poisson submanifold be formally classified up to diffeomorphism, and what obstructions arise?
• RQ4 Is a Poisson structure rigid near a Poisson submanifold under smooth deformations, and what analytic tools can prove such rigidity?
• RQ5 What is the structure of the moduli space of smooth deformations of the Lie-Poisson sphere of a compact semisimple Lie algebra?

Key findings

• Theorem 2 establishes a local normal form for Poisson structures near symplectic leaves, generalizing the standard linearization result to arbitrary Poisson manifolds.
• Theorem 4 proves analytic rigidity of Poisson structures near Poisson submanifolds using the Nash-Moser method, implying a strengthening of Theorem 2.
• Theorem 5 computes the smooth deformation space (moduli space) of the Lie-Poisson sphere of a compact semisimple Lie algebra, showing it is finite-dimensional and explicitly described via Poisson cohomology.
• The existence of symplectic realizations is established via a new construction using transversals in the manifold of cotangent paths, providing a geometric solution to a long-standing problem.
• The holonomy groupoid of a compact, Hausdorff symplectic foliation is shown to be Hausdorff and proper if and only if the leaf space is Hausdorff.
• The Tame Vanishing Lemma is applied to the Bott complex of the foliation, enabling the construction of tame homotopy operators and proving vanishing of first cohomology in the relevant setting.