Lecture 5. Moduli Spaces of Vector Bundles over Riemann Surfaces
5.1. Prototype: The Jacobian
Let :E be a compact Riemann surface of genus g, and let 7r =< x 1 , ... , x 29 :
n;=l [X2j-li X21l = 1 > be its fundamental group. Let us first study a prototype
example: the space Jac(:E) of representations of 7r in U(l).
The space Jac(:E) has three descriptions:1. Jac(:E) = Hom (7r, U(l)) = H^1 (:E, U(l)) = U(l)^29
- Jac(:E) = Z^1 (:E)/Q(:E)
Here A(:E) = D^1 (:E) is the space of all connections on a line bundle over :E,
Z^1 (:E) ={A E A(:E) : dA = O} is the space offiat connections and the gauge
group Q(:E) = c=(:E, U(l)) acts on A(:E) = D^1 (:E) by u: Ar-+ A+ u-^1 du.
- Up to this point we have made no use of the complex structure on·:E, treating
:E simply as a two-manifold. Now we use the complex structure to prescribe
a holomorphic structure on a topologically trivial complex line bundle [, =
:E x C. Once we have specified the complex structure on :E, the connection
A decomposes asA= A(1,o) + A(o,1),
where A (l,O) is a (1, 0) form and A (O,l) a (0, 1) form. We define sectionss : :E ---+ C to be holomorphic if and only if 8As = 0 where [)A = [) + A^0 •^1
for A E A(:E) satisfying dA = 0. Then Jac(:E) parametrizes isomorphism
classes of holomorphic line bundles on :E.Let us choose a basis u 1 , ... , u 29 of H 1 (:E) that is symplectic (i.e. the intersection
form is 0'2j-l · u2 1 = 1 and u21-1 · u1 = O'k • u21 = 0 fork i=-2j -1 and l i=-2j). Then
choosing our identification of Jac(:E) = Hom (H1(:E), U(l)) with U(l)^29 using this
symplectic basis, we may define a symplectic structure on Jac(:E) by
(5.1)
g
w = L dB2j-l /\ dB21
j=lfor ( ei^1 h, ... , ei^62 u) giving coordinates on J ac(:E).
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