COMPLEX GEOMETRY Course notes

Similarly for a holomorphic vector bundle E −→ X with a Hermit metric h, one has the line bundle
i
L π −1 ˜π
(E) E
P(E)
π
X
Π
where P(E) = (E\{zero sections})/C ∗ and L is denoted by L = O P(E) (−1). The composition π ◦ i is called
the blow up of E at its zero section. Here π −1 (E) is the fibre product or pullback of π and Π. Let
F = (PE) x∈X be the fibre of π at x and f : F ↩→ PE the inclusion. Then f ∗ c 1 (| | 2 h
) is a positive (1, 1)-form,
where c 1 (| | 2 h ) = i
2π ∂∂log| |2 h . Hence c 1(| | 2 h
) is a (1, 1)-form on P(E) that is positive in the vertical direction
of π of X, where X is Kähler with Kähler form ω X . Hence P(E) is also Kähler.
Definition 3.5.3. O Eϕ (h) := L −k where L −k := (L ∨ ) ⊗k .
Note that given a vector bundle E −→ ϕ
X, then 1 ϕ = ϕ Eϕ (1) = L ∨ = L −1 is a holomorphic line bundle over
P(E).
Consider the compactification E = P(E ⊕ O) ⊇ E, E −→ X and E ⊕ O −→ ϕ
X are vector bundles over X,
and E is open in P(E ⊕ O). We see that the blow up of E (or E) along its zero section lies in P(ϕ −1 (E)) and
hence it is Kähler.
ϕ
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