Twistor Theory Lectures: solutions to selected exercises

U+ = {[λ1̇ : λ2̇]|λ1̇ 6= 0}, φ+ : [λ1̇ : λ2̇] 7→ λ2̇ λ1̇ = λ+, U− = {[λ1̇ : λ2̇]|λ2̇ 6= 0}, φ− : [λ1̇ : λ2̇] 7→ λ1̇ λ2̇ = λ−. We can immediately see that on the overlap U+ ∩U− we have that λ+ = λ−1 − . We can immediately compute that the map ψ+− = φ+ ◦ φ−1 − : φ−(U+ ∩ U−)→ φ+(U+ ∩ U−) satisfies ψ+−(λ−) = φ+ ◦ φ−1 − ◦ φ−([λ1̇ : λ2̇]) = φ0([λ1̇ : λ2̇]) = λ+. (1.2) To understand the bundle O(−1) we consider the two trivialisations t+ : O(−1)|U+ → U+ × C, t− : O(−1)|U− → U− × C. More explicitly the maps act as follows t+ : ([1 : λ+], μ(λ1̇, λ2̇)) 7→ (λ+, μλ1̇), t− : ([λ− : 1], μ(λ1̇, λ2̇)) 7→ (λ−, μλ2̇) on the overlap O(−1)|U+∩U− we compute the transition function f+− using Ψ+− = t+ ◦ t− with Ψ+−(λ−, μλ2̇) = (ψ+−(λ−), f+−(λ−)μλ2̇) = (λ+, f+−(λ−)μλ2̇). (1.3)