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Twistor spaces of hyperkähler manifolds with S1-actions

✍ Scribed by Birte Feix

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
142 KB
Volume
19
Category
Article
ISSN
0926-2245

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✦ Synopsis

We shall describe the twistor space of a hyperkähler 4n-manifold with an isometric S 1 -action which is holomorphic for one of the complex structures, scales the corresponding holomorphic symplectic form and whose fixed point set has complex dimension n.

We deduce that any hyperkähler metric on the cotangent bundle of a real-analytic Kähler manifold which is compatible with the canonical holomorphic symplectic structure, extends the given Kähler metric and for which the S 1 -action by scalar multiplication in the fibres is isometric is unique in a neighbourhood of the zero section. These metrics have been constructed independently by the author and Kaledin.

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## Abstract We display a family of compact complex 3‐manifolds which yields all twistor spaces containing a pencil of surfaces transversal to the twistor lines. This also yields explicit families of self‐dual metrics on connected sums of complex projective planes. This yields a completely alternati