Null K\"ahler geometry and isomonodromic deformations

TL;DR

This paper explores null-Kähler metrics, their normal forms, and their connection to integrable systems and Painlevé equations, revealing new geometric structures and their relation to moduli spaces and twistor theory.

Contribution

It constructs normal forms of null-Kähler metrics and links cohomogeneity-one anti-self-dual cases to Painlevé equations using twistor methods.

Findings

Null-Kähler metrics admit a compatible parallel nilpotent endomorphism.

Cohomogeneity-one anti-self-dual null-Kähler metrics are characterized by Painlevé I or II solutions.

The work connects null-Kähler geometry with dispersionless integrability and moduli space stability conditions.

Abstract

We construct the normal forms of null-K\"ahler metrics: pseudo-Riemannian metrics admitting a compatible parallel nilpotent endomorphism of the tangent bundle. Such metrics are examples of non-Riemannian holonomy reduction, and (in the complexified setting) appear in the Bridgeland stability conditions of the moduli spaces of Calabi-Yau three-folds. Using twistor methods we show that, in dimension four - where there is a connection with dispersionless integrability - the cohomogeneity-one anti-self-dual null-K\"ahler metrics are generically characterised by solutions to Painlev\'e I or Painlev\'e II ODEs.