[Paper Review] Polysymplectic spaces, s-Kahler manifolds and lagrangian fibrations

Michele Grassi|ArXiv.org|Jun 21, 2000

Geometry and complex manifolds3 references3 citations

TL;DR

This paper introduces polysymplectic manifolds and s-Kähler manifolds as generalizations of symplectic and Kähler geometry, tailored for the geometric study of partial differential equations (PDEs) with multiple independent variables. It establishes a natural generalization of the Legendre transformation and Hamiltonian formalism for PDEs, and shows that s-Kähler manifolds—especially via semi-flat special Lagrangian fibrations of Calabi-Yau manifolds—provide rich geometric models with generalized Hodge theory and Lefschetz operators.

ABSTRACT

In the first part of this paper we begin the study of polysymplectic manifolds, and of their relationship with PDE's. This notion provides a generalization of symplectic manifolds which is very well suited for the geometric study of PDE's with values in a smooth manifold. Some of the standard tools of analytical mechanics, such as the Legendre transformation and Hamilton's equations, are shown to generalize to this new setting. There is a strong link with lagrangian fibrations, which can be used to build polysymplectic manifolds. We then provide the definition and some basic properties of s-Kahler and almost s-Kahler manifolds. These are a generalization of the usual notion of Kahler and almost Kahler manifold, and they reduce to them for s=1. The basic properties of Kahler manifolds, and their Hodge theory, can be generalized to s-Kahler manifolds, with some modifications. The most interesting examples come from semi-flat special lagrangian fibrations of Calabi-Yau manifolds.

Motivation & Objective

• To develop a geometric framework for PDEs with multiple independent variables using polysymplectic structures.
• To generalize classical analytical mechanics tools—like the Legendre transformation and Hamilton’s equations—to the polysymplectic setting.
• To define and study s-Kähler and almost s-Kähler manifolds as higher-rank generalizations of Kähler geometry.
• To explore the role of semi-flat special Lagrangian fibrations in constructing s-Kähler manifolds and their cohomological properties.
• To extend classical Hodge-theoretic and Lefschetz-type theorems to the s-Kähler setting, including the action of $\mathbf{sl}(s+1,\mathbf{R})$ on cohomology.

Proposed method

• Define polysymplectic manifolds via $s$ closed 2-forms $\omega_1, \dots, \omega_s$ locally modeled as $\sum_i dx_i \wedge dy^j_i$.
• Construct a canonical polysymplectic structure on the $s$-fold fiber product $^sT^*(M)$, generalizing the cotangent bundle in classical mechanics.
• Introduce an $s$-Poisson structure on polysymplectic manifolds, generalizing the Poisson bracket to $s$ compatible brackets.
• Define s-Kähler manifolds as manifolds with $s$ 2-forms $\omega_j$ and a Riemannian metric $\mathbf{g}$ such that $\omega_j = \sum_i dx_i \wedge dy^j_i$ and $\mathbf{g}_{\alpha\beta} = \delta_{\alpha\beta} + \mathcal{O}(2)$ in local coordinates.
• Establish the existence of almost s-Kähler structures on any polysymplectic manifold via compatible metrics.
• Use the $\mathbf{sl}(s+1,\mathbf{R})$-action on cohomology to define primitive forms and prove s-Lefschetz theorems for s-Kähler manifolds.

Experimental results

Research questions

• RQ1 How can the geometric machinery of symplectic and Hamiltonian mechanics be generalized to PDEs with multiple independent variables?
• RQ2 What is the role of the Legendre transformation in the polysymplectic setting, and how does it relate to the canonical $s$-Poisson structure?
• RQ3 To what extent do classical Kähler invariants—such as Hodge theory and Lefschetz operators—generalize to s-Kähler manifolds?
• RQ4 Can semi-flat special Lagrangian fibrations of Calabi-Yau manifolds be used to construct non-trivial examples of s-Kähler manifolds?
• RQ5 What are the cohomological consequences of the $\mathbf{sl}(s+1,\mathbf{R})$-action on s-Kähler manifolds, and how do they relate to the Lefschetz theorems?

Key findings

• Polysymplectic manifolds generalize symplectic manifolds for PDEs with $s$ independent variables, with a local model $\omega_j = \sum_i dx_i \wedge dy^j_i$.
• The Legendre transformation generalizes to a local diffeomorphism between $^sT(M)$ and $^sT^*(M)$ when the Lagrangian is non-degenerate, mapping the induced polysymplectic structure to the canonical one on $^sT^*(M)$.
• A canonical $s$-Poisson structure exists on any polysymplectic manifold, allowing a generalization of Hamilton’s equations to $s$ evolution directions.
• s-Kähler manifolds generalize Kähler manifolds, reducing to them when $s=1$, and admit a rich Hodge-theoretic structure with generalized Lefschetz theorems.
• The $\mathbf{sl}(s+1,\mathbf{R})$-action on cohomology preserves the primitive decomposition, and the s-Lefschetz theorems hold for s-Kähler manifolds.
• Counterexamples show that $\mathbf{Prim}^q(M)$ need not vanish for $q > n$ in compact $s$-Kähler manifolds, e.g., the 3-torus $\mathbf{T}^3$ has $\dim_{\mathbf{R}}(\mathbf{Prim}^2(\mathbf{T}^3)) = 1$.