Varieties fibred over abelian varieties with fibres of log general type

Abstract

Let (X, B) be a complex projective klt pair, and let f : X → Z be a surjective morphism onto a normal projective variety with maximal albanese dimension such that K X + B is relatively big over Z. We show that such pairs have good log minimal models.

Log canonical pairs over varieties with maximal Albanese dimension

“…7 We point out that a crucial step in the argument in [5] relies on the extension theorem from [16] which was obtained via an analytic method. So, both Theorem 1.1 and [5] do not have a pure algebraic proof at this point. One wonders if these theorems can be argued in a parallel way as in [13].…”

Albanese map of special manifolds: A correction

“…While the proofs given there for the first two properties (which generalise earlier results by Y. Kawamata and Kawamata-Viehweg in the case when κ(X) = 0) are complete, the proof of the third property, even in the projective case, is not (as pointed out to me by K. Yamanoi and E. Rousseau 1 ). The aim of this note is to correct it in the projective case (but only partially in the compact Kähler case) by using the main result of [3], itself based on [1] (or, alternatively, [4]).…”

“…If X is special, then: Date: September 16, 2021. 1 The problem comes from the potential multiplicity-one exceptional divisors of a X which are no longer f -exceptional if g = q • f , where q : A → B is a non-trivial torus quotient sending D f to an ample divisor of B. We overcome this difficulty in the second step of the proof, by cutting the fibration by means of Poincaré reducibility.…”

“…1. Although the assumption that A is an Abelian variety in Claim 3 is certainly not necessary, our proof uses [1] (or alternatively [4]), presently known only for A projective. Even with a Kähler version of [1] or [4], we could not treat the Kähler case in general with the prsent arguments, because we use Poincaré reducibility.…”

“…The statement of[3] is only given for X projective, and f the Albanese map of X, but it is easy to check that the proof, derived from[1], or[4], through[3], 2.6, applies for X compact and any map to an Abelian variety.…”