Hard Lefschetz actions in Riemannian geometry with special holonomy

• Autores: Naichung Conan Leung, Changzheng Li
• Localización: Mathematical research letters, ISSN 1073-2780, Vol. 15, Nº 4, 2008, págs. 683-698
• Idioma: inglés
• DOI: 10.4310/mrl.2008.v15.n4.a7
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• Resumen

  • 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.