Hodge theory is a powerful tool that connects topological and holomorphic/algebraic information about complex manifolds. In particular, there is a close connection between singular and coherent cohomology of complex manifolds. The natural embedding of constant functions in the structure sheaf induces a natural morphism from singular cohomology of a complex manifold with complex coefficients to the coherent cohomology of its structure sheaf. Hodge theory tells us, among many other things, that this natural map is surjective. This may be interpreted as saying that the coherent cohomology of the structure sheaf has topological origin. Kollár's principle states that when that happens, it leads to vanishing theorems for certain (coherent) cohomology groups. I will explain how this principle leads to an elegant proof (due to Kollár) of the Kodaira vanishing theorem. Time permitting, I will mention how the same principle can be used to prove more general vanishing theorems.