The weak form of Iitaka's $C_{3,m}$ conjecture for threefold fibrations over algebraically closed fields of positive characteristic

Iitaka's $C_{n,m}$ conjecture predicts that for a fibration $f:X\to Z$ between smooth projective varieties of dimensions $n$ and $m$, the Kodaira dimensions satisfy $\kappa(X)\ge\kappa(Z)+\kappa(F)$, where $F$ is the general fibre. In positive characteristic the geometric generic fibre can be singular or non-reduced even when $X$ and $Z$ are smooth, so the conjecture survives in a weak form, with $\kappa(F)$ computed on a smooth model of the geometric generic fibre. Birkar, Chen and Zhang proved this weak form for threefolds over $\overline{\mathbb{F}}_p$ with $p>5$; Ejiri and Zhang later proved it over arbitrary algebraically closed fields of characteristic $p>5$ when the geometric generic fibre is smooth, and Cascini, Ejiri, Kollár and Zhang refuted the alternative strong form over any algebraically closed field of positive characteristic. The problem is the surviving core: prove or refute the weak form of $C_{3,m}$ for fibrations from a smooth projective threefold to a smooth projective variety of dimension at most two over every algebraically closed field of positive characteristic — in particular over non-finite fields of characteristic $p>5$ when the geometric generic fibre is singular, which is the case where the original proof depends on finite-field semi-ampleness, and in characteristics $p\le 5$, where the threefold minimal model program is only partially available.

Background

Let $X$ be a smooth projective variety over an algebraically closed field and $K_X$ its canonical divisor. The Kodaira dimension $\kappa(X)\in\{-\infty,0,1,\dots,\dim X\}$ is the maximal dimension of the image of the rational maps defined by the pluricanonical linear systems $|mK_X|$ for $m\ge 1$ (with $\kappa(X)=-\infty$ when all these systems are empty); it is the fundamental birational invariant in the classification of algebraic varieties. A fibration (algebraic fiber space) is a surjective morphism $f:X\to Z$ of projective varieties with connected fibres, normalized by $f_*\mathcal{O}_X=\mathcal{O}_Z$. Iitaka's $C_{n,m}$ conjecture, formulated in characteristic zero, predicts for such an $f$ between smooth projective varieties with $\dim X=n$, $\dim Z=m$ and general fibre $F$ the subadditivity

$$\kappa(X)\ \ge\ \kappa(Z)+\kappa(F).$$

In characteristic zero this is known when the base has dimension at most two (Kawamata for curve bases, Junyan Cao for surface bases) and in a number of other cases; the general conjecture remains open.

In positive characteristic the statement must be reformulated with care. Even when $X$ and $Z$ are smooth, the geometric generic fibre — the fibre of $f$ over the generic point of $Z$, base changed to an algebraic closure of the function field $k(Z)$ — can be singular or non-reduced (that is, contain nilpotent elements in its structure sheaf), because the field extension $k(Z)\subset k(X)$ need not be separable; the fibration is called separable when this extension is. One therefore distinguishes the strong form $\kappa(X)\ge\kappa(Z)+\kappa(X_\eta)$, where the Kodaira dimension of the generic fibre is computed on normal birational models, from the weak form $\kappa(X)\ge\kappa(Z)+\kappa(\tilde F)$, where $\tilde F$ is a smooth projective model of the geometric generic fibre — a smooth projective variety birational to it, which exists by resolution of singularities in the dimensions occurring here; the two coincide when that fibre is smooth. The weak form is the faithful positive-characteristic analogue of Iitaka's question and is the one expected to hold.

Progress in positive characteristic runs through the minimal model program (MMP) and abundance: for a pair, consisting of a variety together with a boundary divisor $B$ (an effective $\mathbb Q$-divisor with coefficients in $[0,1]$), the MMP produces birational models on which the log canonical divisor $K_X+B$ is nef (numerically effective: of non-negative degree on every curve), and abundance asserts that such a nef divisor with non-negative Kodaira dimension is semi-ample, meaning that some positive multiple is basepoint free and defines a morphism. For threefolds the log MMP is available in characteristic $p>5$ and, more recently, in characteristic $5$, but not in characteristics $2$ and $3$. Before the source paper, subadditivity was known for separable fibrations of relative dimension one — that is, whose general fibre is a curve — over any algebraically closed field of positive characteristic (Chen–Zhang), and for bases of general type whose geometric generic fibre has non-nilpotent Hasse–Witt matrix, a condition on the Frobenius action on cohomology (Patakfalvi).

The source paper (Birkar–Chen–Zhang) proved the weak form of $C_{n,m}$ for threefolds over the algebraic closure $\overline{\mathbb{F}}_p$ of a finite field with $p>5$: for a contraction $f:X\to Z$ from a smooth projective threefold to a smooth projective curve over $\overline{\mathbb{F}}_p$ with $p>5$, and $\tilde F$ a smooth model of the geometric generic fibre, $\kappa(K_X)\ge\kappa(K_Z)+\kappa(K_{\tilde F})$; together with the relative-dimension-one theorem this settles $C_{3,m}$ over $\overline{\mathbb{F}}_p$ for $p>5$. The proof runs the threefold log MMP relatively, develops the birational geometry of log surfaces over the nonclosed (imperfect) function field of the base, and uses semi-positivity results of Patakfalvi. The restriction $k=\overline{\mathbb{F}}_p$ is essential at exactly one point: when the restriction of the canonical divisor to the generic fibre has Kodaira dimension one, the argument needs that a nef and big divisor on a surface over $k$ — nef, and big in the sense that its spaces of sections grow as fast as those of an ample divisor — is semi-ample. This is true over $\overline{\mathbb{F}}_p$ by a theorem of Keel, essentially because numerically trivial divisors there are torsion; over other algebraically closed fields this semi-ampleness is not known in the needed generality, since Keel's positive-characteristic criterion reduces it to a torsion condition on the curves the divisor annihilates, a condition automatic over $\overline{\mathbb{F}}_p$ but not over general fields, and the analogous statement is false in characteristic zero (Keel's example). The authors state that the corresponding conclusions over other fields would perhaps require substantial effort; the restriction $p>5$ reflects the availability of the threefold MMP at the time.

Problem Statement

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $f:X\to Z$ be a fibration (a surjective morphism with $f_*\mathcal{O}_X=\mathcal{O}_Z$) between smooth projective varieties over $k$ with $\dim X=3$ and $\dim Z\le 2$. Let $\tilde F$ be a smooth projective model of the geometric generic fibre of $f$ — that is, a smooth projective variety birational to the geometric generic fibre, the fibre of $f$ over the generic point of $Z$ base changed to an algebraic closure of the function field $k(Z)$. Determine whether

$$\kappa(X)\ \ge\ \kappa(Z)+\kappa(\tilde F)$$

always holds; that is, either prove this inequality for every such fibration over every algebraically closed field of positive characteristic, or construct an explicit fibration satisfying these hypotheses for which the inequality fails.

The decisive open cases include: (i) $\dim Z=1$ over a field of characteristic $p>5$ that is not the algebraic closure of a finite field, with singular (for example geometrically non-reduced) geometric generic fibre — the case in which the source proof's finite-field semi-ampleness input has no known substitute; (ii) characteristics $p\le 5$ (over any algebraically closed field), where the threefold MMP and abundance inputs are available only in part; and (iii) inseparable fibrations of relative dimension one (necessarily with $\dim Z=2$ and non-reduced geometric generic fibre), which the settled relative-dimension-one theorem does not cover because it assumes separability. The cases of separable fibrations of relative dimension one (any characteristic) and of fibrations with smooth geometric generic fibre in characteristic $p>5$ are already settled in the published literature, and an answer may rely on those theorems; the acceptance boundary is nevertheless the full statement above, not a subfamily of fibrations.

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