Siu, Yum-Tong2022-12-14Siu YT. 2004. Invariance of Plurigenera and Torsion-Freeness of Direct Image Sheaves of Pluricanonial Bundles. In Finite or Infinite Dimensional Complex Analysis and Applications. Advances in Complex Analysis and Its Applications, vol 2, eds. Le H.S., Tutschke W., Yang C.C: 45-83. Boston, MA: Springer.978-1-4613-0221-6978-1-4020-7658-9978-1-4613-7948-5https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37373833The deformational invariance of plurigenera proved recently by the author is a special case of the torsion-free property of the first direct image sheaf of the pluricanonical line bundle when the target space of the proper surjective holomorphic map is the open unit 1-disk with nonsingular fibers. This article discusses the adaptation of the techniques used in the proof of the deformational invariance of plurigenera to the general problem of proving the torsion-free property of the first direct image sheaf of the pluricanonical line bundle. The discussion covers also the more general case of the torsion-free property of the first direct image of the pluricanonical line bundle after twisting by a Hermitian holomorphic line bundle with semipositive curvature current and by its multiplier ideal sheaf. A number of results are obtained for the torsion-free problems by the adaptation of the techniques of the proof of the deformational invariance of plurigenera.en-USMonograph or Book2017-03-2620032022-12-1410.1007/978-1-4613-0221-6_4