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 $$\mathbb {P}^{2}$$ , 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 $$\mathbb {P}^3$$ of the 4-dimensional sphere $$S^4$$ . Deformations of twistor fibers project to real surfaces in $$\mathbb {P}^{2}$$ , 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 $$\mathbb {P}^{2}$$ 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 fibers that are contained in more general algebraic surfaces in $$\mathbb {F}$$ .