  In this paper we construct a family of complex structures on a complex flag
manifold that converge to the real polarization coming from the Gelfand-Cetlin
integrable system, in the sense that holomorphic sections of a prequantum line
bundle converge to delta-function sections supported on the Bohr-Sommerfeld
fibers. Our construction is based on a toric degeneration of flag varieties and
a deformation of K\"ahler structure on toric varieties by symplectic
potentials.
