In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has holomorphic sections. This caused me to think about how the existence of holomorphic sections depends on the choice of complex structure. I reason that it cannot be independent since, for complex projective space, a simple switching to the opposite complex structure will switch those line bundles with sections from positive to negative. Is these a neat answer here in general?