Gauge theory for embedded surfaces, II

Gauge theory for embedded surfaces, II 47
Conjecture 8.1. There is a one-to-one correspondence between the irreducible
anti-self-dual connections in the moduli space M α k,l (X,Σ) (defined using the
Kähler metric) and the stable parabolic bundles (E, L,α)withc2(E)[X]=kand
c1(L)[Σ]=−l.
Some remarks are in order concerning these definitions and the conjecture.
First of all, the phrase ‘parabolic bundle’, though it now seems to be in common
usage, is less accurate than the original terminology [Se] which refers to
a parabolic structure along Σ. For bundles of arbitrary rank, such a parabolic
structure consists of a decreasing filtration of E|Σ together with an increasing
sequence of weights, usually constrained to lie in the interval (0, 1), though any
open interval of length 1 would serve as well. For the SL(2, C) case, we find it
tidier to use the interval (− 1
1
2 , 2
); the line-subbundle L provides the filtration
of E|Σ, and the weights are effectively −α and α. In the general case, a map
between parabolic bundles E and F is a holomorphic map j with the constraint
that, if j|Σ is non-zero on any term in the filtration of F, thenjmaps this term
into a term of the filtration of E whose weight is at least as large. (The condition
that the weights lie in an interval of length 1 could be dropped if we altered this
definition appropriately.)
In our SL(2, C) case, the weight α will usually be fixed in advance, so the
difference between quasi-parabolic and parabolic structures disappears. It is also
convenient then to note that, when testing the stability of (E, L), it is sufficient
to consider parabolic line bundle (F,β)withweightβ=±α(because the degree
of Σ is a positive quantity, so we should only consider the largest possible β for
each j). So we can reformulate the test for stability as follows. Given a (quasi-)
parabolic pair (E, L) andamapj:F→Efrom an (ordinary) holomorphic line
bundle F, define the α-degree of F, or more properly of the pair (F,j), to be
α-deg(F)=degF±α ω,[Σ] , (8.2)
where the sign is + if j|Σ maps F into L and − otherwise. Then we define (E, L)
to be α-stable if the α-degree of every (F,j) is negative.
The conjecture is stated rather loosely because, without a little more exposition,
we are not in a position to state how the correspondence is supposed to
go, even in one direction.
Although we shall not prove the conjecture, we shall prove a version sufficient
for our needs, and in section 9 we shall apply this to the ruled surface. The
conjecture above was formulated with a smooth metric in mind, but it can be
stated also for a cone-like Kähler metric. We shall prove the result in the special
case that the cone-angle is 2π/ν for some ν ∈ N and the weight α is a rational
number a/ν with denominator ν. Such a path was followed in [FuS] for the case
of complex dimension 1, and more recently for complex dimension 2 in [W] and
[SW]; our results here are phrased only for SU(2) connections and are therefore
a little less general.
The correspondence goes in two steps. First, for α = a/ν, the moduli space
M α is can be regarded as a moduli space of orbifold connections, so by applying