Journal of Differential Geometry

Compact Self-Dual Manifolds with Torus Actions

Akira Fujiki

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We show that a compact self-dual four-manifold with a smooth action of a two-torus and with non-zero Euler characterestic is necessarily diffeomorphic to a connected sum of copies of complex projective planes, and furthermore the self-dual structure is isomorphic to one of those constructed by Joyce in [11]. This settles a conjecture of Joyce [11] affirmatively. Our method of proof is to show, by complex geometric techniques, that the associated twistor space, which is a compact complex threefold with the induced holomorphic action of algebraic two-torus, has a very special structure and is indeed determined by a certain invariant which is eventually identified with the invariant associated with the Joyce's construction of his self-dual manifolds.

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J. Differential Geom., Volume 55, Number 2 (2000), 229-324.

First available in Project Euclid: 20 July 2004

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Fujiki, Akira. Compact Self-Dual Manifolds with Torus Actions. J. Differential Geom. 55 (2000), no. 2, 229--324. doi:10.4310/jdg/1090340879.

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