The subadditivity of the Kodaira Dimension for Fibrations of Relative Dimension One in Positive Characteristics

Iitaka Conjecture for 3-Folds Over Finite Fields

“…The case C 3,2 follows from [10], so the main result here is C 3,1 . Our main tools are the log minimal model program for 3-folds developed recently by Hacon, Xu, and Birkar [12][3] [37], birational geometry of log surfaces over nonclosed fields (see below), and the semi-positivity results of Patakfalvi [28].…”

“…(of Theorem 1.2) We can assume κ(K Z ) ≥ 0 and κ(K F ) ≥ 0. As pointed out in the introduction C 3,2 follows from [10], so we will assume n = 3 and m = 1. Replacing X with a minimal model over Z, we can assume K X is nef/Z.…”

“…Over the complex numbers, the conjecture has been studied by Kawamata [18][15] [16], Kollár [22], Viehweg [34] [35] [35], Birkar [2], Chen and Hacon [9], Cao and Pǎun [6], etc. We refer to [10] for a collection of results over C. In positive characteristic, Chen and Zhang proved the conjecture for fibrations of relative dimension one [10], and Patakfalvi proved it when Z is of general type and the generic geometric fibre satisfies certain properties [27,Theorem 1.1] (see also [28,Corollary 4.6]).…”

Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension

“…Hence, our approach has a good chance to be portable to positive characteristic when the appropriate semi-positivity results (and other ingredients such as the mmp) become available in that setting. See [Pat12b] for the currently available semi-positivity results in positive characteristic, and [CZ13,Pat13] for results on subadditivty of Kodaira-dimension.…”

Birational characterization of Abelian varieties and ordinary Abelian varieties in characteristic p>0

“…If F is the generic fiber of q, then in (1.1) by a subadditivity of the Kodaira dimension type argument we show that F is a smooth genus 1 curve (c.f. [CZ13], although since Z is not smooth and K Z is not even Q-Cartier in general, we use an alternative self-contained argument). Then, let…”