How the existence of holomorphic sections depends on the choice of complex structure

Question

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?