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Deformations of hyperk\"ahler cones
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We use twistor methods to promote Namikawa's universal Poisson deformations of conic affine symplectic singularities to families of hyperk\"ahler structures deforming hyperk\"ahler cone metrics. The metrics we produce are generally incomplete, but for specific classes of hyperk\"ahler cones, even these incomplete metrics have some interest and applications: we study in detail the case of the nilpotent cone of a simple complex Lie algebra, with applications to hyperk\"ahler metrics with symmetries and hyperk\"ahler quotients, and the case of Kleinian singularities, with applications to codimension-4 singularities of G2-holonomy metrics and their dual description in theoretical physics in terms of 3-dimensional gauge theory.
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A Concrete Variant of the Twistor Theorem
Given a family of holomorphic symplectic forms of a specific shape on a real manifold, a pseudo-hyper-Kähler structure exists directly on that manifold, and it agrees with the classical twistor construction when both apply.
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