[Paper Review] The Local Product Theorem for bihamiltonian structures

Francisco-Javier Turiel|arXiv (Cornell University)|Jul 12, 2011

Homotopy and Cohomology in Algebraic Topology20 references3 citations

TL;DR

This paper establishes a local product decomposition theorem for real analytic or holomorphic bihamiltonian structures, proving they split into a Kronecker and a symplectic component around regular points when a condition on the characteristic polynomial of the symplectic factor holds. The result fails in the smooth ($C^\infty$) category, as shown by a counterexample, resolving a long-standing problem in the geometric theory of bihamiltonian structures.

ABSTRACT

In this work one proves that, around each point of a dense open set (regular points), a real analytic or holomorphic bihamiltonian structure decomposes into a product of a Kronecker bihamiltonian structure and a symplectic one if a necessary condition on the characteristic polynomial of the symplectic factor holds. Moreover we give an example of bihamiltonian structure for showing that this result does not extend to the $C^\infty$-category. Thus a classical problem on the geometric theory of bihamiltonian structures is solved at almost every point.

Motivation & Objective

• To resolve the classical problem of whether bihamiltonian structures locally decompose into Kronecker and symplectic components.
• To identify necessary and sufficient conditions for such a decomposition in the real analytic and holomorphic categories.
• To demonstrate the failure of the decomposition in the $C^\infty$ category, highlighting the necessity of analyticity.
• To generalize the notion of Veronese webs to Veronese flags as a key tool for analyzing bihamiltonian structures.

Proposed method

• Introduces the concept of a Veronese flag as a quotient structure derived from a bihamiltonian system.
• Uses the Nijenhuis torsion and invariance properties of distributions to define weak Veronese flags.
• Applies complexification techniques to reduce real analytic problems to holomorphic ones.
• Employs differential systems and prolongations of tensor fields to analyze the structure of bihamiltonian operators.
• Constructs a decomposition of the tangent bundle into invariant subbundles via spectral projections of the $(1,1)$-tensor field.
• Uses symplectic and closed 2-forms to verify independence of components on complementary coordinates, ensuring product structure.

Experimental results

Research questions

• RQ1 Under what conditions does a bihamiltonian structure locally decompose into a Kronecker and a symplectic component?
• RQ2 Does the local product decomposition hold in the $C^\infty$ category, or are analytic/holomorphic assumptions necessary?
• RQ3 Can the theory of Veronese webs be generalized to handle higher-codimension foliations in bihamiltonian geometry?
• RQ4 How do the characteristic polynomials of the symplectic factor constrain the local structure of bihamiltonian systems?
• RQ5 What role do differential systems and prolongations of tensor fields play in proving the existence of local decompositions?

Key findings

• A real analytic or holomorphic bihamiltonian structure locally decomposes into a product of a Kronecker and a symplectic bihamiltonian structure around each regular point, provided a condition on the characteristic polynomial of the symplectic factor is satisfied.
• The decomposition is established via the local structure of Veronese flags, which are shown to split into a Veronese web and a pair of compatible symplectic forms.
• In the $C^\infty$ category, the result fails: a counterexample is constructed where the decomposition does not hold, demonstrating the necessity of analyticity.
• The proof relies on complexification for real analytic cases, transforming the problem into a holomorphic one where solutions to differential systems are guaranteed.
• The existence of solutions to certain systems of partial differential equations—potentially including Lewy-type systems—is shown indirectly, relying on the analytic category.
• The decomposition of the tangent bundle into invariant subbundles via spectral projections of the $(1,1)$-tensor field ensures the product structure of the bihamiltonian system.