Minimal models of compact symplectic semitoric manifolds

TL;DR

This paper introduces the helix, a new symplectic invariant for semitoric manifolds, enabling classification of minimal models and extending existing results on focus-focus singularities.

Contribution

It constructs the helix invariant for symplectic semitoric manifolds and applies it to classify minimal models and analyze focus-focus singularities.

Findings

Helix is a symplectic analogue of the toric fan.

Classification of minimal models without blowdowns.

Extension of constraints on focus-focus singularities.

Abstract

A symplectic semitoric manifold is a symplectic $4$-manifold endowed with a Hamiltonian $(S^1 \times \mathbb{R})$-action satisfying certain conditions. The goal of this paper is to construct a new symplectic invariant of symplectic semitoric manifolds, the helix, and give applications. The helix is a symplectic analogue of the fan of a nonsingular complete toric variety in algebraic geometry, that takes into account the effects of the monodromy near focus-focus singularities. We give two applications of the helix: first, we use it to give a classification of the minimal models of symplectic semitoric manifolds, where "minimal" is in the sense of not admitting any blowdowns. The second application is an extension to the compact case of a well known result of V\~{u} Ngoc about the constraints posed on a symplectic semitoric manifold by the existence of focus-focus singularities. The helix permits to translate a symplectic geometric problem into an algebraic problem, and the paper describes a method to solve this type of algebraic problem.