  Holomorphic gauge fields in N=1 supersymmetric heterotic compactifications
can constrain the complex structure moduli of a Calabi-Yau manifold. In this
paper, the tools necessary to use holomorphic bundles as a mechanism for moduli
stabilization are systematically developed. We review the requisite deformation
theory -- including the Atiyah class, which determines the deformations of the
complex structure for which the gauge bundle becomes non-holomorphic and,
hence, non-supersymmetric. In addition, two equivalent approaches to this
mechanism of moduli stabilization are presented. The first is an efficient
computational algorithm for determining the supersymmetric moduli space, while
the second is an F-term potential in the four-dimensional theory associated
with vector bundle holomorphy. These three methods are proven to be rigorously
equivalent. We present explicit examples in which large numbers of complex
structure moduli are stabilized. Finally, higher-order corrections to the
moduli space are discussed.
