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Instantons on $${\mathbb{C}}{\mathbb{P}}_ 2$$. (English) Zbl 0586.32034
For the groups $$G=SU(n)$$, Sp(n), SO(n) and U(n) all G-instantons on $${\mathbb{C}}{\mathbb{P}}_ 2$$ are described and classified. The description is an analogue of the Atiyah-Drinfeld-Hitchin-Manin construction for instantons on $$S^ 4$$, utilizing the one-to-one correspondence between instantons and holomorphic vector bundles on the associated twistor space together with techniques from the classification theory of stable bundles on projective spaces. From this description the various moduli spaces of (topologically distinct) instantons are constructed, and precise conditions are given under which, and only under which, irreducible G- instantons of specified topological type exist on $${\mathbb{C}}{\mathbb{P}}_ 2$$. The moduli space of SU(2)-instantons of second Chern class -1 is constructed explicitly as an example.

##### MSC:
 32L05 Holomorphic bundles and generalizations 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results 32L25 Twistor theory, double fibrations (complex-analytic aspects) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 32G13 Complex-analytic moduli problems
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