Construction of Hyperkähler Metrics for Complex Adjoint Orbits

3.3 Regular Nahm Solutions and Line bundles on S 34
ai,l(ζ) = ci,l exp( k−1
j=1 (λi) jq +
j
(ζ)) exp(λis),
bi,l(ζ −1 ) = ci,l exp(− k−1
j=1 (λi) j q −
j (ζ−1 )).
Comparing with (3.13), we have
fi = exp( k−1
j=1 (λi) j q +
j (ζ)) exp(λis) ( k−1
l=0 ci,lζ l ), (3.15)
˜fi = exp(− k−1
j=1 (λi) j q −
j (˜ ζ)) ( k−1
l=0 ci,l ˜ ζ k−1−l ). (3.16)
Now we shall see that a holomorphic section of ML s (k − 1) is in fact
uniquely determined by its restriction to one of the components of the spec-
tral curve, C1 say. Consider the exact sequence
ML s (k − 3) η−λ1ζ
−→ ML s (k − 1) −→ ML s (k − 1) |C1 . (3.17)
Taking the exact sequence in cohomology, we have in particular
H 0 (ML s (k − 3)) −→ H 0 (ML s (k − 1))
γ
−→ H 0 (ML s (k − 1) |C1) (3.18)
Now by hypothesis H 0 (ML s (k−2)) = 0, from which it follows that H 0 (ML s (k−
3)) = 0. Thus the map γ in (3.18) is an injection. However since on
C1 ∼ = P 1 we have ML s (k − 1) ∼ = O(k − 1), the space H 0 (ML(k − 1) |C1)
is k-dimensional; since H 0 (ML s (k − 1)) is also k-dimensional, it follows that
the map γ is a bijection, thus proving our claim.
We shall now give the conditions the f ′ is must satisfy for σ 0 to be a
holomorphic function on a neighbourhood of the singularity (η, ζ) = (0, 0) in
S. We shall see that these conditions depend only on the (k − 2)−jets of the
f ′ is. This was already observed in [Hit83] in the analysis of axially-symmetric
monopoles (whose spectral curves also split into linear factors); however we
want to write these conditions more explicitly here. First observe that since
the spectral curve is defined by a monic polynomial in η of degree k, it follows
that if σ 0 is holomorphic it may be uniquely written in the form
σ 0 = g0(ζ) + g1(ζ)η + ... + gk−1(ζ)η k−1 , (3.19)