Construction of Hyperkähler Metrics for Complex Adjoint Orbits

3.3 Regular Nahm Solutions and Line bundles on S 40
For j = 0, 1, 2, 3, we denote the restriction of σj to Ci ∩ U0 by fji. As in
(3.15) and (3.16), we may write
fji = exp( k−1
j=1 (λi) j q +
j (ζ)) exp(λis) ( k−1
l=0 cj
i,l ζl ), (3.31)
˜fji = exp(− k−1
j=1 (λi) j q −
j (˜ ζ)) ( k−1
l=0 cj
i,l ˜ ζ k−1−l ). (3.32)
The relation (3.30) on Ci ∩ U0 may be written as
λiζf01 + f1i + ζf2i + ζ 2 f3i = 0, (3.33)
for i = 1, . . . , k; we have similar relations for the restrictions to Ci ∩ U∞. In
order to obtain the endomorphism Ãj−1(s), for j = 1, 2, 3, which is defined
by the equation Ãj−1(s)σ0 = σj, we have to solve for the c j
i,l ′ s in terms of the
c 0 i,l ′ s. It is clear from the discussion in the previous paragraph and from the
relations (3.33) that we may choose a basis of H 0 (S; ML s (k−1)) with respect
to which the matrix [ Ãj−1(s)] has coefficients which are rational functions of
e λ1s , . . . , e λks . We shall come back to this is the discussion of the Kähler
potential later on.
3.3.4 The Theta Divisor Condition
We now consider the condition
H 0 (S, ML s (k − 2)) = 0 for all s ∈ (−∞, 0], (3.34)
for a line bundle M of degree 0 on the spectral curve S. In general, this is
a difficult condition to verify, as observed in [Hit83]. On the other hand, in
our situation the simple expression of the spectral curve allows us to obtain
an explicit description of the theta divisor, which in turn shows that the
condition (3.34) is not very restrictive: if a line bundle M does not satisfy
this condition, one may translate it by a line bundle of the form L a in order
for (3.34) to hold.