# Convexity of singular affine structures and toric-focus integrable Hamiltonian systems

## TL;DR

This paper investigates the convexity properties of singular affine structures in integrable Hamiltonian systems with focus-focus singularities, revealing conditions for local and global convexity and introducing the concept of an 'integral affine black hole' as a counterexample.

## Contribution

It provides a systematic analysis of convexity in singular affine structures, including new results on local and global convexity and the construction of a novel example with non-trivial monodromy.

## Key findings

- Base spaces are locally convex near focus singularities.
- Global convexity holds under certain conditions.
- Existence of an 'integral affine black hole' example.

## Abstract

This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), similarly to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently big then the local-global convexity principle breaks down, and the base spaces can be globally non-convex even for compact manifolds. As one of surprising examples, we construct a 2-dimensional "integral affine black hole", which is locally convex but for which a straight ray from the center can never escape.