On the pluricanonical map and the canonical volume of projective 4-folds of general type

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For nonsingular projective 4-folds V of general type with plurigenus P m 0 ( V ) ≥ 2 for some positive integer m 0, we show that φ m is birational onto its image for all integers m ≥ 76 m 0 + 77 and the canonical volume Vol ( V ) has the lower bound 1 1680 m 0 ( m 0 + 1 ) 3 . This improves earlier results of Meng Chen.