  In this paper, we establish a deformation theory for Dolbeault cohomology
classes valued in holomorphic tensor bundles. We prove the extension equation
which will play the role of Maurer-Cartan equation. Following the classical
theory of Kodaira-Spencer-Kuranishi, we construct a canonical complete family
of deformations by using the power series method. We also prove a simple
relation between the existence of deformations and the varying of the
dimensions of Dolbeault cohomology. The deformations of $(p,q)$-forms is shown
to be unobstructed under some mild conditions. By analyzing Nakamura's example
of complex parallelizable manifolds, we will see that the deformation theory
developed in this work provides precise explanations to the jumping phenomenon
of Dolbeault cohomology.
