Twistor geometry of the Flag manifold

Abstract

A study is made of algebraic curves and surfaces in the flag manifold $\mathbb{F}=SU(3)/T^2$, and their configuration relative to the twistor projection $\pi$ from $\mathbb{F}$ to the complex projective plane ${\mathbb{CP}}^2$, defined with the help of an anti-holomorphic involution $j$. This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space ${\mathbb{CP}}^3$ of $S^4$. Deformations of twistor fibres project to real surfaces in ${\mathbb{CP}}^2$, whose metric geometry is investigated. Attention is then focussed on toric Del Pezzo surfaces that are the simplest type of surfaces in $\mathbb{F}$ of bidegree $(1,1)$. These surfaces define orthogonal complex structures on specified dense open subsets of ${\mathbb{CP}}^2$ relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree $(1,1)$ are determined, and bounds given on the number of twistor fibres that are contained in more general algebraic surfaces in $\mathbb{F}$.

34 pages; 7 figures. Comments about v2: we added a couple of remarks, clarified some proof, removed several typos and updated the reference list. To appear on Mathematische Zeitschrift

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• Surfaces in the flag threefold containing smooth conics and twistor fibers