Construction of Hyperkähler Metrics for Complex Adjoint Orbits

2.2 Nahm’s Equations and Spectral Curves 12
One can show that H 0 (S, O(2)) is spanned by η d/dζ, d/dζ, ζd/dζ, ζ 2 d/dζ,
which we abbreviate to η, 1, ζ, ζ 2 . We shall write Ks = ker(ms), Vs =
H 0 (S, ML s (k − 1)). We may write an element of Ks uniquely as η ⊗ σ0 +
1 ⊗ σ1 + ζ ⊗ σ2 + ζ 2 ⊗ σ3 where σi ∈ Vs satisfy ησ0 + σ1 + ζσ2 + ζ 2 σ3 = 0.
One then shows as in [Hit83] (Proposition 4.8) that the map h : Ks −→ Vs
defined by
h(η ⊗ σ0 + 1 ⊗ σ1 + ζ ⊗ σ2 + ζ 2 ⊗ σ3) = σ0
is an isomorphism for each s ∈ (−∞, 0]. Letting for i = 1, 2, 3, hi : Ks → Vs
be defined by
hi(η ⊗ σ0 + 1 ⊗ σ1 + ζ ⊗ σ2 + ζ 2 ⊗ σ3) = σi,
we consider the endomorphisms Ã0(s), Ã1(s), Ã2(s) of Vs defined by Ãi−1(s) =
hi ◦ h −1 . We clearly have
for all σ ∈ Vs.
(η + Ã0(s) + ζ Ã1(s) + ζ 2 Ã2(s))σ = 0
Next we consider the line bundle N over C×S whose fibre at (z, w) ∈ C×S
is ML z (k − 1)w. Denoting by p the projection C × S → C, the direct
image sheaf p∗N is locally free and thus defines a vector bundle V over C
of rank k, whose fibre at s ∈ (−∞, 0] is Vs = H 0 (S, ML s (k − 1)). If σ is a
holomorphic section of V , we can represent σ(z) by holomorphic functions
f0 : U0 → C k , f1 : U∞ → C k , such that f0 = e zη/ζ e c ζ k−1 f1 on U0 ∩ U∞, where
U0 = Ũ0 ∩ S, U∞ = Ũ∞ ∩ S.
Following [Hit83], we define a covariant derivative ∇ on V over (−∞, 0]
as follows. ∇σ is the section of V defined by
g0 = ∂f0/∂s + Ã+f0 over U0,
g1 = ∂f1/∂s + Ã−f1 over U∞.