Null Kähler geometry and isomonodromic deformations

Abstract

We construct the normal forms of null-Kähler 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-Kähler metrics are generically characterised by solutions to Painlevé I or Painlevé II ODEs.

arXiv commentDedicated to Jenya Ferapontov on the occasion of his 60th birthday. Final version, to appear in Communications In Math Physics

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