On the global generation of direct images of pluri-adjoint line bundles

Abstract

We study the Fujita-type conjecture proposed by Popa and Schnell. We obtain an effective bound on the global generation of direct images of pluri-adjoint line bundles on the regular locus. We also obtain an effective bound on the generic global generation for a Kawamata log canonical Q-pair. We use analytic methods such as L 2 estimates, L 2 extensions and injective theorems of cohomology groups.

On the Effective Freeness of the Direct Images of Pluricanonical Bundles

“…In a joint work with Murayama [4], using the weak positivity of f * O Y (k(K X/Y + ∆)), the author proved effective global generation at general points with a bound of l k(n+1)+n 2 −n for log-canonical pairs. In the same paper and also in a work of Iwai [10], slightly better quadratic bound was shown for klt Q-pairs, improving the results of the current paper in high dimensions. In the situation of Theorem C, Iwai showed this generation at regular values without any assumptions on relative freeness of ω ⊗k X , improving a similar statement by Deng [3].…”

“…This suffices however in order to deduce the next Theorem, where assuming semiampleness of the canonical bundle along the smooth fibres, we prove that the global generation holds at the smooth (regular) values of f in Y . The relative semiampleness hypothesis was removed by Deng [3], later was improved by Iwai [10] when dim Y 5 (see 1.4 below).…”

Direct images of pluricanonical bundles and Frobenius stable canonical rings of fibers

“…If f is smooth over the complement of a normal crossing divisor on Y , then we can remove the above additional assumption by using a theorem of Kawamata [Kaw02, Theorem 1.7] when m = 1 and dim Y ⩽ 4. Deng [Den21], Dutta [Dut20], Dutta-Murayama [DM19] and Iwai [Iwa20] have studied Conjecture 1.1, and they have given sufficient conditions, in terms of lower bounds on l, for the sheaf f * ω m X ⊗ L l to be (generically) globally generated. Recently, Fujino [Fuj22] proposed a new generalization of Fujita's freeness conjecture, as follows.…”