Hard Lefschetz actions in Riemannian geometry with special holonomy

Details Hard Lefschetz actions in Riemannian geometry with special holonomy Hard Lefschetz actions in Riemannian geometry with special holonomy

2008-08-04

arxiv.org/abs/0808.0393v1

Math. Res. Lett. 15 (2008), no.4, 683-698

Mathematics

Differential Geometry

16 pages

Scientific paper

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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