On the global generation of higher direct images of pluricanonical bundles

Abstract

Given a fibration f between two projective manifolds X and Y , we discuss the effective generation of the higher direct images R i f * (K m X ), where K m X is the m-th tensor power of the canonical bundle of X. In particular, we answer two questions posed by Popa-and therefore globally generated, for l m(n + 1).Popa and Schnell then posed a question whether the similar result holds for higher direct images. See Question in [PS14] after Corollary 2.10. In this paper, we give a positive answer to this question in some sense. Indeed, since by Proposition 2.1, it is reasonable to involve the asymptotic multiplier ideal when we consider higher direct images. So our main result is as follows, which implies Theorem 1.1 when i = 0.Theorem 1.2. Let f : X → Y be a fibration between two projective manifolds with dim Y = n, and A be an ample and globally generated line bundle on Y . If m 1 is an integer, then the sheaf