
TL;DR
This paper introduces the Plebański complex, an elliptic differential complex derived from linearising equations for hyper-Kähler and Einstein 4-manifolds, revealing new geometric and analytical structures.
Contribution
It defines the Plebański complex, studies its properties, and shows its relation to known complexes, including its connection to Dirac operators and elliptic theory in 4-manifold geometry.
Findings
The Plebański complex is elliptic and encodes hyper-Kähler structures.
It extends to Einstein equations, fitting into the elliptic complex framework.
Operators in the complex assemble into Dirac operators, linking to spectral theory.
Abstract
As is very well-known, linearisation of the instanton equations on a 4-manifold gives rise to an elliptic complex of differential operators, the truncated (twisted) Hodge complex . Moreover, the linearisation of the full YM equations also fits into this framework, as it is given by the second map followed by its adjoint. We define and study properties of what we call the Pleba\'nski complex. This is a differential complex that arises by linearisation of the equations implying that a Riemannian 4-manifold is hyper-K\"ahler. We recall that these are most naturally stated as the condition that there exists a perfect triple of 2-forms that are closed . The Riemannian metric is encoded by the 2-forms . We show that what results…
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Algebraic and Geometric Analysis
