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On the equivariant formality of Kähler manifolds with finite group action. (English) Zbl 0805.55009

For a given finite group \(G\) denote by \({\mathcal O}_ G\) the category of canonical orbits, having the orbits \(G/K\) as objects and the \(G\)- equivariant maps between them as morphisms. As is well-known the functors from \({\mathcal O}_ G\) to abelian groups play a crucial role in equivariant homotopy theory. The second-named author has succeeded in [Trans. Am. Math. Soc. 274, 509-532 (1982; Zbl 0516.55010)] to provide an algebraic description of rational \(G\)-homotopy types by weak equivalence classes of injective \(G\)-systems of dga’s. Here a \(G\)-system of dga’s means a (covariant) functor form \({\mathcal O}_ G\) into the category of graded- commutative differential \(\mathbb{Q}\)-algebras and the weak equivalence is the relation generated by the existence of a natural transformation inducing a cohomology isomorphism at each subgroup \(K\) of \(G\). For technical reasons it was necessary to work only with systems of dga’s which are injective objects when considered by neglect of structure as objects in the abelian category of functors from \({\mathcal O}_ G\) to \(\mathbb{Q}\)-vector spaces. The main result of the paper under review removes this inconvenience, by establishing that any \(G\)-system of dga’s is weakly equivalent by means of a functorial embedding to an injective one. For example, this enables the authors to generalize in a straightforward manner the result by P. Deligne, P. Griffiths, J. Morgan and D. Sullivan on the formality of 1-connected compact Kähler manifolds [Invent. Math. 29, 245-274 (1975; Zbl 0312.55011)] to the equivariant case.

MSC:

55P62 Rational homotopy theory
57S17 Finite transformation groups
55P91 Equivariant homotopy theory in algebraic topology
32Q15 Kähler manifolds
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