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Rosa Gini, Liviu Ornea, Maurizio Parton

Locally conformal Kähler reduction

We define reduction of locally conformal Kähler manifolds, considered as conformal Hermitian manifolds, and we show its equivalence with an unpublished construction given by Biquard and Gauduchon. We give two independent, equivalent definitions, the first via local charts, the second via lifting to Kähler reduction of the universal covering. By a recent result of Kamishima and the second author, in the Vaisman case (that is, when a metric in the conformal class has parallel Lee form) if the manifold is compact its universal covering comes equipped with the structure of Kähler cone over a Sasaki compact manifold. We show the compatibility between our reduction and Sasaki reduction, hence describing a subgroup of automorphisms whose action causes the quotient to bear a Vaisman structure. Then we apply this theory to construct a wide class of Vaisman manifolds.

Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter

Print ISSN: 0075-4102
Volume: 2005, 04/2005
Pages: 1 - 21

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