Invariant hypercomplex structures and algebraic curves

TL;DR

This paper establishes a correspondence between $U(k)$-invariant hypercomplex structures on certain orbits in complex Lie algebras and algebraic curves of genus $(k-1)^2$ with specific geometric properties, linking differential geometry and algebraic geometry.

Contribution

It introduces a novel classification of hypercomplex structures via algebraic curves with flat projections and involutions, bridging geometric structures and algebraic curves.

Findings

Hypercomplex structures correspond to algebraic curves with genus $(k-1)^2$.

These curves have a flat projection of degree $k$ to ${ m P}^1$.

An antiholomorphic involution covers the antipodal map on ${ m P}^1$.

Abstract

We show that $U(k)$-invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in $\mathfrak{gl}(k,{\mathbb C})$ correspond to algebraic curves $C$ of genus $(k-1)^2$, equipped with a flat projection $\pi:C\to{\mathbb P}^1$ of degree $k$, and an antiholomorphic involution $\sigma:C\to C$ covering the antipodal map on ${\mathbb P}^1$.