Firstly, we recall Siu and Demailly's proof of Grauert-Riemenschneider conjecture, and explain Berndtsson's refined estimates on the cohomology of semipositive line bundles on compact complex manifolds. Secondly, we discuss two questions on nef and pseudoeffective line bundles given by Demailly and Matsumura analogue to Berndtsson's estimates. In this direction, we introduce a pointview of Matsumura to prove vanishing theorems via asymptotic vanishing theorems and injectivity theorems. In the end, we will talk our recent progress on the growth of dimension of cohomology of semipositive line bundles on non-compact complex manifolds.
