Twistor geometry of the Flag manifold

TL;DR

This paper explores the geometry of algebraic curves and surfaces in the flag manifold related to twistor theory, focusing on their projections, deformations, and special toric Del Pezzo surfaces, revealing their complex structures and discriminant loci.

Contribution

It provides a detailed analysis of algebraic surfaces in the flag manifold, especially toric Del Pezzo surfaces, and their relation to twistor fibers and complex structures on projective planes.

Findings

Discriminant loci of bidegree (1,1) surfaces are characterized.

Bounds are established on the number of twistor fibers in algebraic surfaces.

Deformations of twistor fibers relate to real surfaces in CP^2.

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 $\mathbb{F}$.