Cayley cones ruled by 2-planes: desingularization and implications of the twistor fibration

TL;DR

This paper explores Cayley cones in octonions ruled by 2-planes, linking them to pseudoholomorphic curves in Grassmannians and using twistor fibrations to demonstrate the existence of complex structures and their geometric implications.

Contribution

It establishes a correspondence between Cayley cones and pseudoholomorphic curves, utilizing twistor fibrations to prove existence results and analyze their geometric properties.

Findings

Existence of immersed higher-genus pseudoholomorphic curves in G(2,8).

Cayley cones with links as S^1-bundles over Riemann surfaces.

Asymptotic cones of non-conical Cayley 4-folds for large degree curves.

Abstract

Cayley cones in the octonions $\mathbb{O}$ that are ruled by oriented 2-planes are equivalent to pseudoholomorphic curves in the Grassmannian of oriented 2-planes G(2,8). The well known twistor fibration $G(2,8) -> S^6$ is used to prove the existence of immersed higher-genus pseudoholomorphic curves in $\gro$. Equivalently, this produces Cayley cones whose links are $S^1$-bundles over genus-$g$ Riemann surfaces. When the degree of an immersed pseudoholomorphic curve is large enough, the corresponding 2-ruled Cayley cone is the asymptotic cone of a non-conical 2-ruled Cayley 4-fold.