Some applications of Grothendieck Duality Theorem

In this paper, we systematically apply Grothendieck duality theorem to simplify the proofs of several theorems in different papers: Including a vanishing theorem in KMM, a theorem of Koll'{a}r’s paper, a vanishing theorem due to Kov'{a}cs and a theorem of Fujino. We remark that all of the above are achieved by the same trick.