Pushforwards of pluricanonical bundles under morphisms to abelian varieties

Kodaira dimension of fibrations over abelian varieties

“…Recall that an (algebraic) fiber space is a surjective projective morphism with connected fibers. We remark, as could have been done already in [LPS20], that Theorem 2.1 implies a strengthening of the subadditivity of the Kodaira dimension for algebraic fiber spaces over abelian varieties proved in [CP17] and later also in [HPS18].…”

“…On the other hand, Iitaka's C n,m conjecture on the subadditivity of the Kodaira dimension is known for algebraic fiber spaces over abelian varieties by [CP17] (see also [HPS18]), giving κ(X) ≥ κ(F ) for any such. Thus the key point is the last statement in (2), for which we employ techniques from [LPS20] regarding the Chen-Jiang decomposition, as well as a hyperbolicity-type result from [PS17]. We prove in fact a stronger statement: Theorem B.…”

“…In a different but related direction, we remark that using the klt version provided in [Men21] (see also [Jia21] for related results) of the global generation result for direct images in [LPS20] used for the results above, one can go beyond the results of [CP17] and [HPS18]. Concretely, we show the validity of the log Iitaka conjecture over an abelian variety A, in the case of divisors that do not dominate A (though the statement can be phrased slightly more generally).…”

“…The most general case of the log Iitaka conjecture allows for divisors E that dominate A, in other words replacing κ(F ) by κ(F, K F +E| F ). It would be solved with similar methods if extensions of the results in [LPS20] were available in the setting of log canonical pairs; this is however beyond our reach at the moment. Note also that if E = (f * D) red , it is predicted in [Pop21, Conjecture 4.1] that the inequality in Theorem E becomes an equality if f is smooth away from D, which would extend Theorem A(2).…”

“…Background. We will make use of the following two results from [LPS20], which are shown in loc. cit.…”