# Rank three representations of Painleve systems: III. Dolbeault structure, spectral correspondence

## TL;DR

This paper establishes a holomorphic symplectic isomorphism between rank 2 and 3 Painleve system representations in the Dolbeault structure, revealing their moduli spaces are hyperKähler isometric, with explicit elliptic fibration descriptions.

## Contribution

It provides the first explicit Dolbeault isomorphism between different rank Painleve representations and links their moduli spaces through hyperKähler geometry.

## Key findings

- Existence of a holomorphic symplectic isomorphism between rank 2 and 3 representations.
- Explicit descriptions of elliptic fibrations associated with these representations.
- Moduli spaces are hyperKähler isometric in the Dolbeault structure.

## Abstract

We prove that there exists a holomorphic symplectic isomorphism between the rank 2 and 3 representations of the Painleve systems in the Dolbeault complex structure, and give explicit descriptions of the corresponding elliptic fibrations. This, combined with the de Rham description given in part II, implies that the corresponding moduli spaces are hyperKaehler isometric to each other.