Mathematical Research Letters

Volume 15 (2008)

Number 4

Hard Lefschetz actions in Riemannian geometry with special holonomy

Pages: 683 – 698

DOI: http://dx.doi.org/10.4310/MRL.2008.v15.n4.a7

Authors

Naichung Conan Leung (The Chinese University of Hong Kong)

Changzheng Li (The Chinese University of Hong Kong)

Abstract

It is known that the hard Lefschetz action, together with Kähler identities for Kähler (resp. hyperkähler) 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. hyperkähler) manifolds, these symmetries can be enlarged to a $\mathfrak{so}(2,2)_{sup}$ (resp. $\mathfrak{su}(2,2)_{sup}$) action.

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