On the Kodaira dimension of base spaces of families of manifolds

Abstract

We prove that the variation in a smooth projective family of varieties admitting a good minimal model forms a lower bound for the Kodaira dimension of the base, if the dimension of the base is at most five and its Kodaira dimension is nonnegative. This gives an affirmative answer to the conjecture of Kebekus and Kovács for base spaces of dimension at most five.Theorem 1.2. Conjecture 1.1 holds when dim(V ) ≤ 5.

Birational geometry of smooth families of varieties admitting good minimal models

“…We resolve this problem by showing that the construction of such invertible sheaves are in a sense functorial. More precisely we show that the Hodge theoretic constructions in [Taj18], from which these line bundle arise, verify various functorial properties that are sufficiently robust for the construction of the line bundle L in Theorem 1.3, using the one constructed at the level of moduli stacks. This forms the main content of Section 2.…”

“…Clearly ⋆⋆ is equivalent to ⋆ when variation is maximal, in which case the result is due to [PS17], and [VZ01] when the base is of dimension one. But as it is shown in [Taj18] and [PS17] the discrepancy between the two statements poses a major obstacle in proving Kebekus-Kovács Conjecture in its full generality. In this paper we close this gap and prove the following result.…”

“…In [Taj18,Sect. 5] it was shown that, thanks to Campana's results on the orbifold C n.m conjecture, once Theorem 1.1 is established the following conjecture of Kebekus-Kovács [KK08, Conj.…”

“…Establishing ⋆ in the more general context of projective manifolds with good minimal models has been an important goal in this topic. In its absence, a weaker result was established in [Taj18] where it was shown that for projective families with good minimal models we have: ⋆⋆ There are k ∈ N, a pseudo-effective line bundle B and a line bundle L on B, with (L ⊗ B) ⊆ Ω 1 B (log D) ⊗k , such that κ(L ) ≥ Var(f U ).…”

Hyperbolicity of bases of log Calabi-Yau families

“…ι : Ω Y (log B) ֒→ Ω Y log(B + T ) denotes the natural inclusion. Let us mention that the similar construction as p+q=ℓ F p,q , p+q=ℓ ϕ p,q is made by Taji in his work [Taj18] on a conjecture of Kebekus-Kovács.…”