Uniqueness of optimal symplectic connections

Consider a holomorphic submersion between compact Kähler manifolds, such that each fibre admits a constantscalar curvature Kähler metric. When the fibres admit continuous automorphisms, a choice of fibrewise constant scalarcurvature Kähler metric is not unique. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric satisfying a geometric partial differential equation. The condition generalises the Hermite-Einstein condition for a holomorphic vector bundle through the induced fibrewise Fubini-Study metric on the associated projectivisation. We prove various foundational analytic results concerning optimal symplectic connections. Our main result proves that optimal symplectic connections are unique, up to the action of the automorphism group of the submersion, when they exist. Thus optimal symplectic connections are canonical relatively Kähler metrics when they exist. In addition, we show that the existence of an optimal symplectic connection forces the automorphism group of the submersion to be reductive and that an optimal symplectic connection is automatically invariant under a maximal compact subgroup of this automorphism group. We also prove that when a submersion admits an optimal symplectic connection, it achieves the absolute minimum of a natural log norm functional, which we define.

Local Models of Isolated Singularities for Affine Special Kähler Structures in Dimension Two

We construct local models of isolated singularities for special Kähler structures in real dimension two assuming that the associated holomorphic cubic form does not have essential singularities. As an application we compute the holonomy of the flat symplectic connection, which is a part of the special Kähler structure.

Symplectic fibrations

This chapter begins with a general discussion of symplectic fibrations and symplectic forms on the total space. The next section describes in detail symplectic 2-sphere bundles over Riemann surfaces. Subsequent sections develop the notions of symplectic connection and holonomy, explain the Sternberg–Weinstein universal construction for fibre bundles, discuss Seidel’s construction of generalized Dehn twists, and introduce the Guillemin–Lerman–Sternberg coupling form. The final section studies Hamiltonian fibrations.

WIGNER FUNCTIONS, FRESNEL OPTICS, AND SYMPLECTIC CONNECTIONS ON PHASE SPACE

We prove that Wigner functions contain a symplectic connection. The latter covariantizes the symplectic exterior derivative on phase space. We analyze the role played by this connection and introduce the notion of local symplectic covariance of quantum-mechanical states. This latter symmetry is at work in the Schrödinger equation on phase space.

SYMPLECTIC CONNECTIONS WITH A PARALLEL RICCI CURVATURE

A symplectic connection on a symplectic manifold, unlike the Levi-Civita connection on a Riemannian manifold, is not unique. However, some spaces admit a canonical connection (symmetric symplectic spaces, Kähler manifolds, etc.); besides, some ‘preferred’ symplectic connections can be defined in some situations. These facts motivate a study of the symplectic connections, inducing a parallel Ricci tensor. This paper gives the possible forms of the Ricci curvature on such manifolds and gives a decomposition theorem (linked with the holonomy decomposition) for them.AMS 2000 Mathematics subject classification: Primary 53B05; 53B30; 53B35; 53C25; 53C55

ON THE GEOMETRY OF THE CHARACTERISTIC CLASS OF A STAR PRODUCT ON A SYMPLECTIC MANIFOLD

The characteristic class of a star product on a symplectic manifold appears as the class of a deformation of a given symplectic connection, as described by Fedosov. In contrast, one usually thinks of the characteristic class of a star product as the class of a deformation of the Poisson structure (as in Kontsevich's work). In this paper, we present, in the symplectic framework, a natural procedure for constructing a star product by directly quantizing a deformation of the symplectic structure. Basically, in Fedosov's recursive formula for the star product with zero characteristic class, we replace the symplectic structure by one of its formal deformations in the parameter ℏ. We then show that every equivalence class of star products contains such an element. Moreover, within a given class, equivalences between such star products are realized by formal one-parameter families of diffeomorphisms, as produced by Moser's argument.