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Example text

Let d be the degree of u. The fibre bundle F = P(C⊕u∗ H 2 ⊗H −1 ) is isomorphic to F1 for d = 0 and to F2d−1 for d > 0. 1. 4) that deg(s∗ T E v ) > 0. If d = 0, s1 (z) = (c, z) for some c ∈ CP 1 . The same argument as before shows that deg(s∗ T E v ) > 0 unless s2 is the unique section of F such that deg(s∗2 T F v ) = −1. It is easy to write down this section explicitly. We conclude that any holomorphic section s of π with deg(s∗ T E v ) ≤ 0 belongs to the family S = {sc }c∈CP 1 , 1092 P. 5) sc (z) = [1 : 0] ∈ P(C ⊕ Hc2 ⊗ Hz−1 ) ⊂ Ez .

4) V 0 −→ s∗2 (T F v ) −→ s∗ T E v −→ s∗1 (ker(D pr2 )) −→ 0 Dπ of holomorphic vector bundles over CP 1 . Since s1 is a section of pr2 , it is given by s1 (z) = (u(z), z) for some u : CP 1 → CP 1 . Clearly, s∗1 (ker(D pr2 )) = u∗ T CP 1 . Let d be the degree of u. The fibre bundle F = P(C⊕u∗ H 2 ⊗H −1 ) is isomorphic to F1 for d = 0 and to F2d−1 for d > 0. 1. 4) that deg(s∗ T E v ) > 0. If d = 0, s1 (z) = (c, z) for some c ∈ CP 1 . The same argument as before shows that deg(s∗ T E v ) > 0 unless s2 is the unique section of F such that deg(s∗2 T F v ) = −1.

For γ ∈ Γ, let a1 ∗γ a2 ∈ H∗ (M ; Z/2) be the sum of a1 ∗A a2 over all classes A which can be represented by a smooth map w : S 2 → M with [w] = γ (only finitely many terms of this sum are nonzero). The quantum intersection product ∗ on QH∗ (M, ω) is defined by the formula a1 ⊗ γ1 ∗ a2 ⊗ γ2 = (a1 ∗γ a2 ) ⊗ γ1 + γ2 + γ , γ∈Γ extended to infinite linear combinations in the obvious way. ∗ is a bilinear Λ-module map with the following properties: (i) (ii) (iii) (iv) if a1 ∈ QHi (M, ω) and a2 ∈ QHj (M, ω), a1 ∗ a2 ∈ QHi+j−2n (M, ω).