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Compact self-dual manifolds with positive scalar curvature. (English) Zbl 0583.53054
Penrose’s twistor program translates the conformal geometry on a self dual manifold to the holomorphic geometry on a complex three-dimensional manifold, the associated twistor space. Using the machinery of algebraic geometry and exploiting Hitchin’s technique on studying Kählerian twistor spaces, we find that the intersection of two four-dimensional quadrics is a singular model of a compact twistor space. This twistor space is associated to a self-dual conformal class on the connected sum of two copies of the complex projective plane with canonical orientation. The uniqueness of this conformal class is discussed. So is the uniqueness of other known self-dual conformal classes on a manifold with prescribed topology. Our main interests are in self-dual metrics with positive scalar curvature. Also, we use the Ward correspondence to construct a moduli space of self-dual SO(3)-connections on the connected sum of two projective planes.

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
32Q99 Complex manifolds
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