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Zbl 0763.53034
Gauduchon, Paul
Structures de Weyl et théorèmes d'annulation sur une variété conforme autoduale. (Weyl structures and vanishing theorems on self-du- dual conformal manifolds). (French)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 18, No.4, 563-629 (1991). ISSN 0391-173X

The expression ``gauge theory'' is common enough nowadays in mathematics, but it is ironic that the mathematics where the words originated has been largely forgotten. It was Weyl's attempt to incorporate the electromagnetic field into Einstein's theory of gravity which led to the geometrical structure known as a Weyl structure. This consists of a manifold equipped with a conformal structure and a torsion-free connection which preserves it. The induced connection on the real line bundle of volume forms has a curvature form which is a closed 2-form, and which Weyl took to represent the electromagnetic field.\par The basis of the paper under review is a careful and lengthy account of Weyl structures. It is motivated, however, by an assortment of arguments relating to twistor theory and the geometry of self-dual 4-manifolds, and out of this general context, a number of interesting results arise. One of these concerns the algebraic dimension of twistor spaces --- if a self-dual compact 4-manifold admits a metric of negative total scalar curvature, then its twistor space contains no non-trivial divisor. Another is that a self-dual Weyl structure induces a metric on the twistor space which is standard in the author's terminology (it is what other authors would call a Gauduchon metric). This generalization of the Kähler condition has a role to play in the stability theory of holomorphic vector bundles, a fact which is brought out in a proof of one of the theorems in this paper.\par The article itself uses Weyl structures to give alternate, and sometimes more general, proofs of theorems in the literature on self-dual and twistor geometry, and focusses in particular on vanishing theorems of Weitzenböck type. These are traditionally stated for a Riemannian metric within a conformal class, but here it is the curvature of the Weyl structure which appears in the formula. A key component is the correspondence between Weyl structures in a conformal class and holomorphic structures on a certain line bundle over the twistor space.
[N.J.Hitchin (Coventry)]

MSC 2000:: *53C07 Special connections and metrics on vector bundles
53-02 Research monographs (differential geometry)
32L20 Vanishing theorems (analytic spaces)
32L25 Twistor theory, etc. (analytic spaces)

Keywords: conformal structure; twistor space; Gauduchon metric; Weitzenböck type; conformal class

Cited in: Zbl 0995.53036 Zbl 0923.53026 Zbl 0858.32026 Zbl 0799.53049

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