Communications in Analysis and Geometry

Volume 16 (2008)

Number 5

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

Pages: 937 – 968

DOI: http://dx.doi.org/10.4310/CAG.2008.v16.n5.a2

Author

Daniel Fox (Mathematical Institute, St Giles’, Oxford)

Abstract

Cayley cones in the octonions $\Oc$ that are ruled byoriented 2-planes are equivalent to pseudoholomorphiccurves in the Grassmannian of oriented $2$-planes $\gro$.The well known twistor fibration $\gro \to \s{6}$ is usedto prove the existence of immersed higher-genuspseudoholomorphic curves in $\gro$. Equivalently, thisproduces Cayley cones whose links are $\s{1}$-bundles overgenus-$g$ Riemann surfaces. When the degree of an immersedpseudoholomorphic curve is large enough, the corresponding$2$-ruled Cayley cone is the asymptotic cone of anon-conical $2$-ruled Cayley $4$-fold.

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