Twistor Geometry of Null Foliations in Complex Euclidean Space

TL;DR

This paper explores the geometric relationship between complex projective quadrics and their twistor spaces, demonstrating how null foliations in complex Euclidean space relate to submanifolds in twistor space, with applications to conformal Killing structures.

Contribution

It provides a detailed geometric correspondence between quadrics and twistor spaces, and constructs null foliations in complex Euclidean space via complex submanifolds of twistor space, especially in odd dimensions.

Findings

Null foliations correspond to complex submanifolds of twistor space.

Explicit examples involve conformal Killing spinors and Yano 2-forms.

Focus on odd-dimensional complex Euclidean spaces.

Abstract

We give a detailed account of the geometric correspondence between a smooth complex projective quadric hypersurface $\mathcal{Q}^n$ of dimension $n \geq 3$, and its twistor space $\mathbb{PT}$, defined to be the space of all linear subspaces of maximal dimension of $\mathcal{Q}^n$. Viewing complex Euclidean space $\mathbb{CE}^n$ as a dense open subset of $\mathcal{Q}^n$, we show how local foliations tangent to certain integrable holomorphic totally null distributions of maximal rank on $\mathbb{CE}^n$ can be constructed in terms of complex submanifolds of $\mathbb{PT}$. The construction is illustrated by means of two examples, one involving conformal Killing spinors, the other, conformal Killing-Yano $2$-forms. We focus on the odd-dimensional case, and we treat the even-dimensional case only tangentially for comparison.