Abstract. An lc-trivial fibration is the data of a pair (X, B) and a fibration such that (K X + B)| F is torsion where F is the general fibre. For such a fibration we have an equality K X + B + 1/r(ϕ) = f * (K Z + B Z + M Z ) where B Z is the discriminant of f , M Z is the moduli part and ϕ is a rational function. It has been conjectured by Prokhorov and Shokurov that there exists an integer m = m(r, dim F ) such that mM Z ′ is base-point-free on some birational model Z ′ of Z. In this work we reduce this conjecture to the case where the base Z has dimension one. Moreover in the case where M Z ≡ 0 we prove the existence of an integer m, that depends only on the middle Betti number of a canonical covering of the fibre, such that mM Z ∼ 0.
IntroductionThe canonical bundle formula is a tool for studying the properties of the canonical bundle of a variety X, such that there exists a fibration f : X → Z, in terms of the properties of the canonical bundle of Z, of the singularities of the fibration and of the birational variation of the fibres of f .More precisely we consider a pair (X, B) such that there exists an lc-trivial fibration f : (X, B) → Z, that is a fibration such that (K X + B)| F is a torsion divisor, where F is a general fibre (see Section 2 for a complete definition). Then we can writewhere ϕ is a rational function and r is the minimum integer such that r(K X + B)| F ∼ 0. The divisor B Z is called the discriminant and it is defined in terms of some log-canonical thresholds with respect to the pair (X, B). Precisely we have B Z = (1 − γ p )p whereThe divisor M Z , called the moduli part, is a Q-Cartier divisor and it is nef on some birational modification of Z by [2, Theorem 0.2]. The moduli part should be related to the birational variation of the fibres of f . This is true for instance in the case of elliptic fibrations. Indeed, if f : X → Z is an elliptic fibration, we have Kodaira's canonical bundle formulaDate: July 30, 2017.where j : Z → P 1 = M 1 is the application induced by f to the moduli space of elliptic curves. In particular M Z is semiample.In [20] Prokhorov and Shokurov state the following conjecture (in our statement we specify the dimension of the base).EbS(k) 1.1 (Effective b-Semiampleness, Conjecture 7.13.3, [20]). There exists an integer number m = m(d, r) such that for any lc-trivial fibration f : (X, B) → Z with dimension of the generic fibre F equal to d, dimension of Z equal to k and Cartier index of (F, B| F ) equal to r there exists a birational morphism ν : Z ′ → Z such that mM Z ′ is base point free.The relevance of the above conjecture is well illustrated for instance by a result due to X. It is worth noticing that the proofs of semiampleness in these cases use the existence of a moduli space for the fibres.The main goal of this work is to develop an inductive approach to the conjecture EbS. Our first result is the following.
Theorem 1.2. EbS(1) implies EbS(k).An inductive approach on the dimension of the base as in Theorem 1.2 allows us to prove a result of effective semiam...
