gpt 6 astra pro attempt on Math-147

Iitaka Subadditivity Conjecture for Kodaira Dimension

Definitions and mathematical statement

For a smooth projective variety $Z$, its Kodaira dimension $\kappa(Z)$ is the growth exponent of the spaces $H^0(Z,mK_Z)$, with value $-\infty$ if all positive pluricanonical spaces vanish. For the surjective morphism and general fiber in the record, the target is \[ \kappa(X)\ge\kappa(Y)+\kappa(F). \] The usual fibration convention uses connected fibers and the characteristic-zero setting of the cited source; the catalogue's abbreviated sentence does not separately spell out those conventions. When one term on the right is $-\infty$, the inequality is interpreted in the extended ordered sense.

Formulation and concept references: [S1].

Short English statement

Is the Kodaira dimension of a projective fibration at least the sum of those of its base and general fiber?

Research attempt

Current frontier and source audit.

Birkar's introduction explicitly works over an algebraically closed field of characteristic zero and defines an algebraic fibre space to have connected fibres. Those are the conventions incorporated by the present entry. Theorem 1.3 proves the conjecture for $\dim X\leq6$; the introduction also records the good-minimal-model fibre case. Neither unrestricted existence of good minimal models nor abundance is assumed here. [S1]

Established methods include positivity of pluricanonical direct images; Cao--Păun prove the abelian-base case, with a logarithmic extension. A December 2025 manuscript by Ammar states the case $\dim\operatorname{alb}_Y(Y)\geq\dim Y-2$. That recent result is reported as a manuscript theorem, not used as an independently audited ingredient. The complete arbitrary-base, arbitrary-fibre claim is not supplied by these hypotheses. [E1][E2]

Attack route.

A basis of pluricanonical forms on the generic fibre extends meromorphically over the base. The central question is the cost of removing its poles. The route below turns this into a precise section-counting lemma, then asks whether positivity can bound that cost in directions supported by the base's canonical system. A second route tests descent through a finite étale product cover. A third, numerical-positivity-only route is rejected by a degree-zero line-bundle example.

Attempt: a pole-cost criterion.

Write $K_{X/Y}=K_X-f^*K_Y$ and \[ E_m=f_*\mathcal O_X(mK_{X/Y}),\qquad r_m=\operatorname{rank}E_m=h^0(F,mK_F). \] The last equality is generic base change; the general fibre is understood in that sense for each $m$. The sheaf is torsion-free, and all generic-rank arguments below take place on a dense open subset before being interpreted as morphisms of coherent sheaves on $Y$. Projection formula gives \begin{equation}\label{wsref-0-65-71-3a-33-30-30-30-35-31-36-38-3a-70-72-6f-6a-65-63-74-69-6f-6e} h^0(X,mK_X)=h^0(Y,E_m\otimes\mathcal O_Y(mK_Y)). \end{equation} If either Kodaira dimension on the right of the target is $-\infty$, the inequality is automatic. Assume from now on that both are nonnegative.

Lemma 1 (canonical-direction pole cost). Choose an integer $q>0$ and an effective divisor $D\sim qK_Y$. Suppose along an unbounded sequence of multiples $m$ of $q$ there are integers $b_m\geq0$ with $b_m=o(m)$ and morphisms \begin{equation}\label{wsref-0-65-71-3a-33-30-30-30-35-31-36-38-3a-69-6e-6a-65-63-74-69-6f-6e} \mathcal O_Y(-b_mD)^{\oplus r_m}\hookrightarrow E_m \end{equation} that have rank $r_m$ at the generic point. Choose $q$ sufficiently divisible that the standard lower plurigenus-growth bounds for $F$ and $Y$ hold on its multiples. Then $\kappa(X)\geq\kappa(F)+\kappa(Y)$.

Proof. A generically injective map from the displayed locally free sheaf is injective everywhere as a sheaf map: its kernel is torsion-free and zero at the generic point. Tensoring by $\mathcal O_Y(mK_Y)$ and taking sections gives \[ h^0(X,mK_X)\geq r_m\, h^0\left(Y,(m-qb_m)K_Y\right). \] Eventually $m-qb_m\geq m/2$ and it is still a multiple of $q$. Thus the right side is at least $c m^{\kappa(F)+\kappa(Y)}$ along the sequence. This yields the asserted Kodaira dimension. The lower growth estimates need no finite generation theorem: choose pluricanonical sections whose ratios realize the Iitaka dimension in a fixed degree; independent monomials in those ratios give the required polynomial lower bound in multiples of that degree. $\square$

The hypothesis is concretely testable: $r_m$ global sections of $E_m(b_mD)$ forming a generic basis yield (eq:30005168:injection). It is stronger than what the conjecture requests and has not been shown to hold in general. In particular, when $K_Y$ is torsion one may have $D=0$, leaving no ample twisting room at all.

Attempt: what sublinear ample twisting actually proves.

A more plausible output of generic generation is an injection \[ \mathcal O_Y(-b_mH)^{\oplus r_m}\hookrightarrow E_m, \qquad b_m=o(m), \] for a fixed ample effective divisor $H$. This suffices when $K_Y$ is big. Indeed, choose $q$ and $G\geq0$ with $qK_Y\sim H+G$. For $m=q\ell$, \[ mK_Y-b_mH\sim(\ell-b_m)H+\ell G. \] For sufficiently large $m$, $\ell-b_m\geq\ell/2$, so multiplication by the section of $\ell G$ and ample section growth give \[ h^0(Y,mK_Y-b_mH)\geq c m^{\dim Y}. \] The previous proof then gives $\kappa(X)\geq\kappa(F)+\dim Y$. This isolates, rather than re-proves from scratch, the geometric room available in the established general-type-base case. [S1]

That reasoning fails sharply for a non-big base. On an elliptic curve, $K_Y\sim0$ and an effective divisor $H$ of positive degree satisfies \[ h^0(Y,mK_Y-H)=h^0(Y,-H)=0 \] for every $m$. Even constant pole cost $b_m=1=o(m)$ cannot be absorbed. Thus "the poles grow sublinearly" is insufficient unless their direction is also controlled. Lemma 1 is one possible replacement, but its canonical-direction hypothesis is exactly an unresolved part of this route.

Attempt: finite étale descent without a false invariants count.

The following restricted case can be handled completely. Suppose a finite étale cover $Y'\to Y$ trivializes the fibration: \[ X'=X\times_Y Y'\simeq F\times Y'. \] Then $\kappa(X)=\kappa(F)+\kappa(Y)$.

Here are the descent details. For a finite Galois étale cover $\pi:Z\to W$ with group $G$, étaleness identifies $K_Z=\pi^*K_W$ and \[ R(W,K_W)=R(Z,K_Z)^G. \] For any element $s$ of the latter graded canonical ring, the monic polynomial $\prod_{g\in G}(T-gs)$ has invariant coefficients. Hence $R(Z,K_Z)$ is integral over its invariant subring. Their fraction fields have equal transcendence degree, so their Kodaira dimensions agree. If there are no positive-degree sections downstairs, the norm of a nonzero homogeneous section upstairs would produce one, so this also handles $\kappa=-\infty$. Passing through a Galois closure handles a non-Galois finite étale cover. None of this asserts that invariant sections occupy a fixed fraction of every individual graded piece.

On a product, the complete pluricanonical section space is the tensor product of the two factor spaces. Its rational map is the Segre product of the two pluricanonical maps wherever defined. Choosing a common sufficiently divisible degree gives $\kappa(F\times Y')=\kappa(F)+\kappa(Y')$, with the usual $-\infty$ convention. Applying étale invariance to $X'\to X$ and $Y'\to Y$ proves the restricted statement. This is a self-contained recovery of a familiar isotrivial situation, not a proof for arbitrary fibrations.

Stress test: positivity is not a supply of untwisted sections.

Let $L$ be a non-torsion degree-zero line bundle on an elliptic curve. It is nef and numerically trivial, yet $H^0(Y,L^m)=0$ for every $m>0$. A nonzero section would have an effective zero divisor of degree zero, hence no zeros, and would trivialize $L^m$, a contradiction. Consequently one cannot replace the generic-generation hypothesis in Lemma 1 by numerical semipositivity, even for line bundles. This example is not asserted to be a pluricanonical direct image of a counterexample fibration: the geometric origin of $E_m$ is exactly the additional structure a successful proof must exploit. The direct-image methods in Cao--Păun use much more than arbitrary numerical positivity. [E1]

Likewise, a generically chosen fibre basis can always be cleared of poles with some ample twist depending on $m$, but that observation supplies neither a sublinear bound on the twist nor support in a canonical divisor. Also, replacing a fibration by a finite ramified cover changes the canonical divisor by ramification; the étale descent proof does not apply verbatim to such a cover.

Outcome.

Outcome: Conditional reduction.

The attempt proves a canonical-direction pole-cost criterion and explains exactly why the apparently weaker sublinear ample-pole criterion only works directly over a big-canonical base. It also gives a complete finite-étale-product special case. What is missing is geometric control of the pluricanonical direct images strong enough to remove poles on a general non-big base. Assuming all good minimal models or abundance would use another unresolved conjectural package, not complete this argument. No novelty claim is made for the special cases or standard descent facts.

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Sources

• S1 C. Birkar, Iitaka conjecture C_{n,m} in dimension six, arXiv:0806.4413 [math.AG], 2008. https://arxiv.org/abs/0806.4413. Catalogue formulation source.
• E1 Junyan Cao and Mihai Păun, Kodaira dimension of algebraic fiber spaces over abelian varieties, Invent. Math. 207 $2017$, 345--387; arXiv:1504.01095v3 $2016$, Theorem 1.1 and Sections 2--4. https://arxiv.org/html/1504.01095v3.
• E2 Houari Benammar Ammar, Note on Iitaka Conjecture $C_{n,m}$, arXiv:2510.06412v2 (12 December 2025), Theorem 1.2. Recent manuscript result, not used in any proof in this notebook. https://arxiv.org/html/2510.06412v2.