### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 005, 42 pages      arXiv:1505.06938      https://doi.org/10.3842/SIGMA.2017.005

### Twistor Geometry of Null Foliations in Complex Euclidean Space

Arman Taghavi-Chabert
Università di Torino, Dipartimento di Matematica ''G. Peano'', Via Carlo Alberto, 10 - 10123, Torino, Italy

Received April 01, 2016, in final form January 14, 2017; Published online January 23, 2017

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.

Key words: twistor geometry; complex variables; foliations; spinors.

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