# Transverse Hilbert Schemes and Completely Integrable Systems

## TL;DR

This paper constructs a class of holomorphic completely integrable systems from surfaces with symplectic forms and projections, characterizing all such systems arising from the transverse Hilbert scheme approach.

## Contribution

It introduces a novel method linking transverse Hilbert schemes to integrable systems and fully characterizes the systems generated by this construction.

## Key findings

- Holomorphic integrable systems are associated with surfaces via transverse Hilbert schemes.
- A complete classification of these integrable systems is provided.
- The construction generalizes known examples and offers new insights into their structure.

## Abstract

Via the transverse Hilbert scheme construction, we associate a holomorphic completely integrable system to a surface $S$ endowed with a holomorphic symplectic form $\omega$ and a projection onto $\mathbb{C}$. We provide a full characterization of the completely integrable systems that arise in this way.