Lifting sections for log abundance in fourfolds

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Lifting sections for log abundance in fourfolds

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Recursos

Suppose you have a nef log canonical adjoint and only one effective section. This class shows how that section is forced to multiply: you push it into the coefficient-one boundary, get generation there, and lift it order by order until the Iitaka dimension becomes positive.

The supported lifting theorem

Take a ℚ-factorial dlt pair (V, C) and set A = K_V + C. Let G be a nonzero effective Cartier divisor with G ∼ qA [0:44]. Two conditions [0:52]:

Supp G ⊆ Supp⌊C⌋, meaning G lives only on the coefficient-one boundary;
𝒪_V(G)|_{G_red} is semiample on the whole reduced scheme.

Then κ(V, A) > 0. A does not need to be nef, and an extra coefficient-one divisor H may cross S = G_red anywhere [1:08]. If a component of G has coefficient ½, the theorem does not apply [1:22].

Why it works

Semiampleness gives f: S → Pℓ, but f need not extend to any neighborhood of S [1:40]. The proof therefore moves to the frame bundle of 𝒪_V(G), which carries a 𝔾_m scaling action with weights. Two cyclic covers produce a reduced divisor E = (y = 0) [2:01].
Lifting from y^k to y^{k+1} works like adding one more Taylor term. The first failure is a cohomology class that obeys Leibniz, so the obstruction is a derivation δ_k [2:19]. A residue injection turns it into a tensor v of weight m = a + k > 0 [2:41].
Horizontal part: a nonzero horizontal part would give an ample line bundle mapping into the kernel of the first symbol of a Hodge module. Kodaira–Saito vanishing rules this out [3:15].
Vertical part: v = s₀ ⊗ ε, and the actual Euler action gives sε = −ms. Since m > 0, the scalar is nonzero, so s₀ = 0 [3:40].
Lifting therefore works at every finite order. The layers come back twisted by 𝒪(1), 𝒪(2), …, while the loss in passing back to V is bounded by one fixed cohomology group. So h⁰(nG) is unbounded [4:13].

Using it in fourfolds

Take D nef with κ = 0 and D ∼_ℚ M ≥ 0, and suppose M ≠ 0. Raise the coefficients along Supp M to one and pass to a dlt minimal model. The negativity lemma keeps M_* ≠ 0, and lower-dimensional abundance gives semiampleness on its reduced support. The lifting theorem then gives κ > 0, which is a contradiction. Hence D ∼_ℚ 0 [4:47]. Together with a separate fourfold nonvanishing result, this gives log abundance in dimension at most four. The case κ > 0 needs effective minimal models on the Iitaka fibers.

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