Positive-characteristic boundary of Iitaka's subadditivity conjecture C_{n,m}

Iitaka's conjecture $C_{n,m}$ predicts that a fibration $f:X\to Y$ of smooth projective varieties satisfies $\kappa(X)\ge\kappa(Y)+\kappa(F)$, where $F$ is the general fiber and $\kappa$ is the Kodaira dimension. Cascini–Ejiri–Kollár–Zhang proved that over any algebraically closed field of positive characteristic subadditivity fails in its strong form (formulated with the generic fiber), through fibrations with uniruled bases and singular geometric generic fibers built from Tango–Raynaud curves. The open problem is to determine exactly where the boundary lies: the weak form - replacing the fiber by a smooth birational model of the geometric generic fiber - is expected to hold in positive characteristic and is open in general, as is the strong form over non-uniruled bases. Known positive regimes include relative dimension one (Chen–Zhang), threefolds in characteristic $p>5$ with smooth geometric generic fiber (Ejiri–Zhang), all fibrations of threefolds over $\bar{\mathbb F}_p$ with $p>5$ (Birkar–Chen–Zhang), and general-type bases under Frobenius-stability hypotheses (Patakfalvi, Ejiri, L. Zhang).

Background

Let $k$ be a field. A fibration is a surjective projective morphism $f:X\to Y$ between normal projective varieties with $f_*\mathcal O_X=\mathcal O_Y$, so the fibers form a family of varieties parametrized by the base $Y$. The Kodaira dimension $\kappa(X)\in\{-\infty,0,\dots,\dim X\}$ measures the growth rate of the spaces of pluricanonical forms $H^0(X,mK_X)$ and is the fundamental birational invariant by which projective varieties are classified. Writing $\bar\eta$ for a geometric generic point of $Y$ and $X_{\bar\eta}$ for the geometric generic fiber, Iitaka conjectured in the early 1970s that for a fibration of smooth projective varieties over $\mathbb C$ with $\dim X=n$ and $\dim Y=m$ the Kodaira dimensions are subadditive: $\kappa(X)\ \ge\ \kappa(Y)+\kappa(X_{\bar\eta}).$ Over $\mathbb C$ this is known in major cases - bases of dimension at most one (Kawamata) and two (Cao), fibers of general type (Kollár), bases of general type (Viehweg), and bases of maximal Albanese dimension, meaning that the Albanese map (the universal morphism to an abelian variety) is generically finite on the base (Cao–Păun) - but it remains open in full generality, where it would follow from the existence of good minimal models, that is, minimal models whose canonical divisor is semi-ample. In positive characteristic the situation differs in two structural ways. First, Frobenius (the morphism raising functions to the $p$-th power, $p=\mathrm{char}\,k$) allows inseparable extensions of function fields, so the geometric generic fiber $X_{\bar\eta}$ can be singular or even non-reduced even when $X$ and $Y$ are smooth; one therefore distinguishes a strong form of subadditivity, using the generic fiber $X_\eta$, from a weak form, using a smooth birational model of $X_{\bar\eta}$. Second, the positivity tools behind the complex proofs - Kodaira vanishing and weak positivity of the direct images $f_*\omega_{X/Y}^l$ - fail in positive characteristic (Raynaud's counterexamples). Cascini–Ejiri–Kollár–Zhang showed the failure is intrinsic: over every algebraically closed field of characteristic $p>0$ they constructed fibrations violating the strong form, built from Tango–Raynaud curves (smooth projective curves $C$ carrying a line bundle $L$ with $L^p\simeq\omega_C$ for which the Frobenius-induced map $H^1(C,L^{-1})\to H^1(C,L^{-p})$ is not injective) and Raynaud surfaces (surfaces constructed from such curves whose direct-image sheaves of pluricanonical forms are anti-ample), taking fiber products over the base curve. The examples have uniruled bases (bases covered by rational curves), singular geometric generic fibers, and total-space dimension at most $2p-1$ for $p\ge3$ and four for $p=2$; a companion example of Tanaka shows a logarithmic version (the analogue for pairs, where a boundary divisor is retained as part of the data) fails already for surfaces over non-algebraically-closed fields. The same authors state that it is still reasonable to expect subadditivity to hold either when the base is not uniruled, or for the weak form. What is salvaged is confined to restricted regimes. For separable fibrations of relative dimension one between smooth projective varieties the weak form holds in every characteristic, with $\kappa$ of the normalization of the geometric generic fiber (Chen–Zhang). For threefolds in characteristic $p>5$, $C_{3,1}$ holds when the geometric generic fiber is smooth (Ejiri–Zhang), and over the algebraic closure $\bar{\mathbb F}_p$ of a finite field with $p>5$ the full conjecture $C_{n,m}$ holds for threefolds (Birkar–Chen–Zhang), the proof running through the minimal model program (the MMP, which contracts extremal rays until the canonical class becomes nef) and relative good minimal models, which are available over $\bar{\mathbb F}_p$. With singular geometric generic fibers only conditional results are known, under Frobenius-stability hypotheses: a base of general type together with a non-nilpotent Hasse–Witt matrix of the geometric generic fiber (the matrix of Frobenius acting on $H^1(F,\mathcal O_F)$; Patakfalvi), sufficiently large Frobenius stable canonical rings of the fibers, meaning the part of the pluricanonical ring of the geometric generic fiber consisting of sections invariant under pullback by powers of Frobenius (Ejiri), semi-ample canonical divisor of the total space over a base of general type (Ejiri), or relative bigness of $K_{X/Y}$ over a non-uniruled base (L. Zhang).

Problem Statement

Over an algebraically closed field $k$ of characteristic $p>0$, determine for which fibrations $f:X\to Y$ between smooth (or, where resolution is unavailable, normal) projective varieties the subadditivity of Kodaira dimension survives - that is, locate the exact boundary between the known counterexamples and the known positive regimes. Two precise surviving targets, both posed in the counterexample paper as expected to hold: (1) the weak form: does $\kappa(X)\ge\kappa(Y)+\kappa(Z)$ hold for every fibration, where $Z$ is a smooth projective birational model of the geometric generic fiber $X_{\bar\eta}$ - or, where a smooth model is not available, where $\kappa(Z)$ is understood resolution-independently, following Definition 1.3 of [1], as the minimum of $\kappa(Z',K_{Z'})$ over projective normal birational models $Z'$ of the reduction of $X_{\bar\eta}$? (2) the strong form over a non-uniruled base: does $\kappa(X)\ge\kappa(Y)+\kappa(X_\eta)$ hold, where $X_\eta$ is the fiber over the generic point of $Y$, whenever the base $Y$ is not covered by rational curves? Concrete frontier instances of these targets, none of which is settled: $C_{3,1}$ and $C_{3,2}$ for fibrations of smooth projective threefolds over an arbitrary algebraically closed field of characteristic $p\ge5$ when the geometric generic fiber is singular; subadditivity for threefolds in characteristic $5$ (the minimal model program is available there, but no Iitaka result is known) and in characteristics $2$ and $3$; and the removal or extension of the Frobenius-stability hypotheses under which general-type bases are currently handled. The two targets above are posed as separate questions.

References

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