Twistor interpretation of slice regular functions

Abstract

Abstract. Given a slice regular function f : Ω ⊂ H → H, with Ω ∩ R = ∅, it is possible to lift it to a surface in the twistor space CP 3 of S 4 ≃ H ∪ {∞} (see [14]). In this paper we show that the same result is true if one removes the hypothesis Ω ∩ R = ∅ on the domain of the function f . Moreover we find that if a surface S ⊂ CP 3 contains the image of the twistor lift of a slice regular function, then S has to be ruled by lines. Starting from these results we find all the projective classes of algebraic surfaces up to degree 3 in CP 3 that contain the lift of a slice regular function. In addition we extend and further explore the so-called twistor transform, that is a curve in Gr 2 (C 4 ) which, given a slice regular function, returns the arrangement of lines whose lift carries on. With the explicit expression of the twistor lift and of the twistor transform of a slice regular function we exhibit the set of slice regular functions whose twistor transform describes a rational line inside Gr 2 (C 4 ), showing the role of slice regular functions not defined on R. At the end we study the twistor lift of a particular slice regular function not defined over the reals. This example shows the effectiveness of our approach and opens some questions.

Algebraic Surfaces With Infinitely Many Twistor Lines

“…As stated in the introduction, the theory of quaternionic slice regularity can be exploited in this case. In fact, as explained in [1,4,10], any slice regular function f : Ω → H where Ω ⊂ H can be lifted, with an explicit parametrisation, to a holomorphic curvef :…”

“…Take homogeneous coordinates t 1 , t 5 , t 4 , t 6 on F CP 3 . Note that E is the smooth quadric surface of F with t 1 t 6 − t 2 5 − t 2 4 = 0 as its equation and, over R, the quadric E has signature (1,3). So E has many real points, but it is not projectively isomorphic to RP 1 × RP 1 .…”

Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere

“…Let us denote by G(2, 4) the Grassmannian of lines in CP 3 and by σ : S 4 → G(2, 4) the C ∞ embedding of twistor lines, i.e. σ(q) := π −1 (q) (see [1,13] for an explicit definition of σ). We recall that dim(G(2, 4)) = 4.…”

Twistor lines on algebraic surfaces

“…We recall that the Grassmannian Gr(2, 4) can be identified with the Klein quadric K = {t 1 t 6 − t 2 t 3 + t 4 t 5 = 0} ⊂ CP 5 via Plücker embedding. The map j induces, then, a map on CP 5 (that will be also denoted by j) defined as j([t 1 : t 2 : t 3 : t 4 : t 5 : t 6 ]) = [t 1 : t 5 : −t 4 : −t 3 : t 2 : t 6 ], (see [1,12]). This new map j identifies twistor lines in Λ and it is an anti-holomorphic involution.…”