There is a connection between representations of the fundamental group
π_1(X) of a compact Riemann surface X, and holomorphic vector bundles on X. More speciﬁcally, for every representation ρ, there exists an associated holomorphic vector bundle V_ρ. The relation was explored in 1938 by André Weil and later M. S. Narasimhan and C. S. Seshadri in 1965. In particular, any vector bundle that comes from a representation has a holomorphic connection, and any bundle with such a connection comes from a representation.
An interesting problem is describing the interactions between this relation and deformation theory: How do the vector bundles associated to deformations of ρ relate to deformations of the associated bundle V_ρ? Providing a complete answer to this question is difficult, but we can give a complete description of the deformations of holomorphic vector bundles with holomorphic connections.
We first give an introduction to vector bundles in general, and then develop and introduce tools of deformation theory and homological algebra. The decription of deformation of vector bundles with holomorphic connections is given in terms of the total cohomology of a double complex, which we describe.