Answer:
To construct an explicit biholomorphic map from the surface obtained by gluing two copies of the Riemann sphere, CP1, cut along a segment connecting points λ1, λ2 ∈ C, to CP1, we can use a fractional linear transformation.
First, let's denote the two copies of CP1 as U and V. On each copy, we can use the coordinates z and w, respectively.
For U, we have z ∈ CP1 with z ≠ λ1, and for V, we have w ∈ CP1 with w ≠ λ2.
Now, we can define the biholomorphic map f: X → CP1 as follows:
On U: f(z) = λ1 + (1/(z - λ1))
On V: f(w) = λ2 + (1/(w - λ2))
This map f takes points on U and V and maps them to points on CP1. The formulas provided above ensure that the resulting map is a bijection and preserves the holomorphic structure, making it a biholomorphic map.
It's important to note that the biholomorphic map f is not unique and depends on the specific choice of λ1 and λ2. The formulas above assume that the points λ1 and λ2 are distinct, as z ≠ λ1 and w ≠ λ2, ensuring the denominator is nonzero in each case.
Explanation: