Mathematical Research Letters

Volume 15, Issue 4, July 2008 pp. 683-698.

Hard Lefschetz actions in Riemannian geometry with special holonomy

Authors: Naichung Conan Leung (1) and Changzheng Li (2)
Author institution: The Chinese University of Hong Kong (1) and The Chinese University of Hong Kong (2)

Summary: It is known that the hard Lefschetz action, together with K\"ahler identities for K\"ahler (resp. hyperk\"ahler) manifolds, determines a $\mathfrak{su}(1,1)_{sup}$ (resp. $\mathfrak{sp}(1,1)_{sup}$) Lie superalgebra action on differential forms. In this paper, we explain the geometric origin of this action, and we also generalize it to manifolds with other holonomy groups. For semi-flat Calabi-Yau (resp. hyperk\"ahler) manifolds, these symmetries can be enlarged to a $\mathfrak{so}(2,2)_{sup}$ (resp. $\mathfrak{su}(2,2)_{sup}$) action.

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