Iitaka's subadditivity via Hodge-theoretic techniques

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Iitaka's subadditivity via Hodge-theoretic techniques

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Recursos

Take any fibration f: X → Z and you can now read the paper's proof of κ(X) ≥ κ(F) + κ(Z) as one chain of inequalities, and say which tool powers each link. You can also state the bound the theorem gives for a concrete family.

The claim

For a surjective projective morphism with connected fibres between smooth projective varieties over an algebraically closed field of characteristic zero, κ(X) ≥ κ(F) + κ(Z), where F is the geometric generic fibre [0:58]. Since (−∞) + a = −∞, the only real work is the case where both terms are non-negative. Here is a quick check [1:22]: take an elliptic surface over a genus-2 curve. The fibres give 0 and the base gives 1, so κ(X) ≥ 1.

The chain

The proof [1:45] shows κ(X) ≥ κ(Y, D) ≥ d + κ(Z), with d = κ(F).

First link: the canonical bundle formula [2:04]. The relative Iitaka fibration X̃ → Y has fibres of Kodaira dimension zero, and K_X̃ ∼ g*(K_Y + B + M) + R. Comparing sections gives κ(X) ≥ κ(Y, D) with D = K_Y + B + M, and D is big on the generic fibre over Z.
The Hodge line [2:47]. The least-index root cover of each fibre has geometric genus one, so its top form is the entire highest Hodge line. M is a multiple of that line, and the period map writes it as M = p*L.
Adjoint positivity [3:22]. The goal is K_S + jL big for large j. Log general type only gives K plus the boundary, so the boundary and the ramification divisors must be removed.
Rank loss [3:43]. On every such divisor E, ν(J|_E) < r = ν(J). Sections grow like c·j^r, and each removal costs only O(j^(r−1)), so for large j the line's own growth pays for the boundary.
Last link [4:42]. Fujino's twisted weak positivity, then klt interpolation, then the addition theorem for fibres of log general type together give d + κ(Z). Extension of scalars then covers any algebraically closed field of characteristic zero.

Rule to take away [5:08]: find the line, check that it loses rank on every divisor you want to remove, and let the j^r growth do the rest.

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