Twistor Geometry of Null Foliations in Complex Euclidean Space

Arman Taghavi-Chabert

Symmetry, integrability and geometry: methods and applications, Tome 13 (2017)

Résumé

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.

Détail

Keywords: twistor geometry; complex variables; foliations; spinors.

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