Twistor geometry of the Flag manifold

Abstract

A study is made of algebraic curves and surfaces in the flag manifold F = SU (3)/T 2 , and their configuration relative to the twistor projection π from F to the complex projective plane 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 P 3 of S 4 . Deformations of twistor fibres project to real surfaces in P 2 , whose metric geometry is investigated. Attention is then focussed on toric Del Pezzo surfaces that are the simplest type of surfaces in F of bidegree (1, 1). These surfaces define orthogonal complex structures on specified dense open subsets of 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 fibres that are contained in more general algebraic surfaces in F.

Surfaces in the flag threefold containing smooth conics and twistor fibers

“…In the whole paper we will take advantage of notations and basic facts already introduced in [3]. Nevertheless, we recall here briefly what is needed in order to be as more self-contained as possible.…”

“…Remark 2.4. Any curve C of bidegree (0, 1) is obtained as the intersection of two surfaces of bidegree (1, 0) (see [3,Section 3…”

“…In this paper we investigate integral (i.e. irreducible and reduced) smooth curves of bidegree (1,1), called smooth conics, contained in a surface S ⊂ F. The interest in this subject emerges from the fact that this family of curves contains the set of fibers of the standard twistor projection π : F → P 2 [4,9,3]. In fact, from twistor theory, the fibers of π can be characterized to be the j-invariant rational smooth curves with normal bundle O(1) ⊕ O (1), where j : F → F is an antiholomorphic involution involved in the twistor construction (see Formula (2)).…”

“…Continuing from [3] and inspired from [1,2], we aim to obtain results on the maximum number of smooth conics that can be contained in a smooth surface, on the geometry of the generic surface containing a certain amount of smooth conics and a description of the surfaces containing infinitely many twistor fibers.…”

“…In particular, we define the bidegree of a curve and of a surface in F, we define smooth conics and twistor fibers and we present a detailed study of the linear sections, the (1, 0) and (0, 1) surfaces, which are Hirzebruch surfaces of type 1. For other relevant geometry of the flag we refer to [3].…”