Cubic Fourfolds with an Involution

Abstract: Cubic fourfolds are one of the most intensely studied objects in Algebraic Geometry, especially due to their role in the construction of hyperkähler manifolds. A big open problem surrounding hyperkähler manifolds is the construction of new examples: currently there are only 4 known deformation types. One approach is to consider finite symplectic group actions of a known hyperkähler manifold, and study the symplectic resolution (if it exists) of the quotient. In this talk, we will relate automorphisms of hyperkähler manifolds of OG10 type to automorphisms of cubic fourfolds, and use the geometry of the cubic in order to complete the classification of involutions Hodge theoretically. More specifically, we compute the algebraic sublattice of the middle cohomology of a cubic fourfold with a certain involution explicitly. If time permits, we discuss several consequences; namely a cubic fourfold with a specific involution is rational.

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