Webs and the Pleba\'nski equation

TL;DR

This paper generalizes the Plebański equation to higher dimensions within hyper-para-complex structures and characterizes compatible Segre structures through differential equations, advancing the understanding of complex geometric structures.

Contribution

It introduces a higher-dimensional generalization of the Plebański equation and characterizes Segre structures compatible with hyper-para-complex structures.

Findings

Generalization of the Plebański equation to higher dimensions

Characterization of Segre structures via differential equations

Insights into hyper-para-complex structures and integrable systems

Abstract

We consider 3-webs, hyper-para-complex structures and integrable Segre structures on manifolds of even dimension and generalise the second heavenly Pleba\'nski equation in the context of higher-dimensional hyper-para-complex structures. We also characterise the Segre structures admitting a compatible hyper-para-complex structure in terms of systems of ordinary differential equations.