User:Bencwbrown/sandbox

Moduli space of Vector Bundles

Let $X$ be a connected Riemann surface. The Picard group $H^{1}(X,{\mathcal {O}})$ parametrising the isomorphism classes of holomorphic vector bundles on $X$, where the group structure is given by the tensor product of line bundles. If we fix the first Chern class -- which we will call degree from now on -- to be d then $Pic^{d}(X)$ is the set of equivalence classes of degree d.