Hodge structures of the surface of planes in a cubic 5-fold

TL;DR

This paper investigates the geometry and Hodge structures of surfaces of planes in a general cubic 5-fold, revealing their moduli space as a smooth projective surface with ample canonical bundle and maximal variation of Hodge structures.

Contribution

It provides a detailed analysis of the moduli space of planes in cubic 5-folds, demonstrating its smoothness, projectiveness, and the maximal variation of Hodge structures using hyper-Kahler geometry.

Findings

Moduli space is a smooth projective surface

Canonical bundle of the moduli space is ample

Variation of Hodge structures is maximal

Abstract

We study the geometry of the moduli space of planes in a general cubic 5-fold and its deformation. We show that this moduli space is a smooth projective surface whose canonical bundle is ample. We also show that the variation of degree 1 Hodge structures of a particular family of such surfaces is maximal. The main technical input is the hyper-Kahler geometry and an elaborated calculation of the Hodge numbers of such surfaces Keywords: Cubic hypersurfaces, Hyper-Kahler manifolds, Schubert calculus, Variation of Hodge structures