Hard Lefschetz actions in Riemannian geometry with special holonomy

TL;DR

This paper explores the geometric origins of the hard Lefschetz action in Riemannian geometry, generalizing it to manifolds with various holonomy groups and enlarging symmetries for special cases.

Contribution

It explains the geometric origin of Lefschetz actions and extends these symmetries to manifolds with different holonomy groups, including semi-flat Calabi-Yau and hyperk"ahler manifolds.

Findings

Identifies the Lie superalgebra actions on differential forms.

Generalizes Lefschetz actions to broader classes of manifolds.

Enlarges symmetries for semi-flat Calabi-Yau and hyperk"ahler manifolds.

Abstract

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.