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

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

2007-09-19

arxiv.org/abs/0709.3115v2

Mathematics

Differential Geometry

31 Pages; typos corrected, including Equation 13 and Lemma 6.1; revised exposition; references added

Scientific paper

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.

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