We know that the existence of positive line bundle is a topological property(more explicitly, it's mertic free).
When the line bundle is induced by a smooth hypersurface $V$ of the projective manifold $M$ with complex dimension $n$, the proof of Lefschetz hypersurface theorem tell us that the positivity can be viewed as existence of a Morse function with index $\geq n$.
But when the line bundle is induced by any divisor（even any line bundle）, can we find an analogue topological explaination？