Invariance of Plurigenera and Torsion-Freeness of Direct Image Sheaves of Pluricanonial Bundles

Abstract

The 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.