Abundance after nonvanishing for compact Kähler fourfolds

1. Descend generation to the original space in positive Iitaka dimension

For $\kappa(X,D)\geq1$, take the crepant ordinary Q-factorial Kähler model of DHP, Theorem 6.1, p. 43. Since the original pair is klt, so is the output; HLL, Theorem 4.1, pp. 17–18 supplies semiampleness. Normality and the projection formula show that every section is pulled back from $X$, and evaluation surjectivity descends to $X$ (K4N, Proposition 5.2, p. 34). Generation on a birational model alone is not the endpoint.

Proposition 5.2 · p. 34

2. Match the zero divisor support with a dlt floor

In the remaining case, let $M=\operatorname{div}(s_0)/m_0$ be the normalized zero divisor of a nonzero section. Suppose $M\neq0$. On a log resolution, raise the coefficients of its strict transforms and the exceptional primes to one. With $G_Y$ the crepant boundary, the effective representative in Lemma 5.3 (pp. 34–35) is $P_Y=p^*M+(B_Y-G_Y)$. Its support is exactly the floor, and comparison with exceptional divisors preserves Iitaka dimension zero.

Lemma 5.3 · pp. 34–35

3. Ensure that contraction targets remain Kähler

Kählerness of the targets of projective MMP steps cannot be obtained from projectivity of the original space. Following the supported construction of DHP, Theorem 7.2, p. 46, the manuscript prepares threefold floor contractions (Proposition 3.1, p. 9) and their extension to the ambient space (Proposition 4.3, p. 24). The latter uses conormal vanishing and finite analytic thickenings.

Claim 5.4 (pp. 37–42) verifies nefness of the descended Bott–Chern class, a current dominating a positive Hermitian form, and positive top intersections on every positive-dimensional subspace, thereby obtaining a Kähler target. The full analytic contraction constructions are outside the checks completed for this draft.

Propositions 3.1, 4.3・Claim 5.4 · pp. 9, 24, 37–42

4. Preserve nonzero support using the original nefness

Proposition 5.1 (p. 33) produces an ordinary Q-factorial Kähler dlt fourfold $(V,B)$, a nef $A=K_V+B$, nonzero $P\sim_{\mathbb Q}A$, $\operatorname{Supp}P=\operatorname{Supp}S$, and $\kappa(V,A)=0$. Survival of $P$ uses nefness of the original $D$: if it disappeared, the pullback of $M$ on a common resolution would be nonzero, effective, nef and exceptional, contradicting negativity (pp. 45–46).

Proposition 5.1・§5.3 · pp. 33, 45–46

5. Combine whole-floor generation with supported lifting

Lemma 5.6 also gives a special projective resolution: its strict boundary and exceptional support have globally smooth distinct SNC components, exceptional crepant coefficients are below one, and it is generically an isomorphism on the image of each intersection component of distinct strict floor primes. Under this resolution condition, Theorem 6.1 (p. 46) generates $A|_S$. Theorem 1.2 then yields $\kappa(V,A)\geq1$, a contradiction. Thus $M=0$; normality makes $s_0$ nowhere vanishing, and it trivializes the original holomorphic line (§13, pp. 88–89).

Lemma 5.6・Theorem 6.1・§13 · pp. 43–46, 88–89

1. Identify stratum adjoint lines with ambient restrictions

Proposition 6.3 (pp. 48–50) identifies adjoints on normal Kähler dlt strata indexed by intersections with restrictions of one ambient holomorphic line. Sufficiently divisible even degrees remove iterated-residue signs. Sections agreeing on every subordinate stratum descend uniquely to the entire reduced floor. Apply DO, Corollary 1.3, p. 4 to three-dimensional strata through small models; lower strata inherit generation by restriction (K4N §7.1, pp. 50–51). Componentwise generation does not supply agreement on intersections.

Proposition 6.3・§7.1 · pp. 48–51

2. Reduce extension to the residue comparison of two markings

Proposition 7.3 (pp. 56–58) reduces extension from the floor of a stratum to at most one residue comparison. Torsion-freeness of $R^1f_*\mathcal O_T(-\lfloor G\rfloor)$ in Lemma 7.1 (pp. 51–52) extends generic-fibre matching over every parameter. When a comparison is necessary, a general $\mathbb P^1$ fibre of a perturbed Mori contraction has two coefficient-one markings. When both come from one prime, use the degree-two exchange on the normal finite Stein space of its normalization. On a common graph, the invertible adjoint subsheaves inside meromorphic pluriforms agree; abstract Q-linear equivalence would not suffice.

Lemma 7.1・Proposition 7.3 · pp. 51–52, 56–58

3. Use the common Kähler class for fixed-degree finite images

On nonprojective surfaces, the action of all birational self-maps on pluriforms need not have finite image. Section 8 restricts to the groupoid generated by links of three-dimensional strata and their inverses. Restrictions of a single ambient Kähler form give classes $c_Z^2>0$ on smooth minimal surfaces. Lemma 8.2 (p. 60) shows that links preserve these classes modulo the real Néron–Severi space. In algebraic dimension zero, Lemma 8.3 (p. 61) bounds volume-character orders by $\varphi(n)\leq b_2$; algebraic dimension one is treated separately through the elliptic fibration. Proposition 8.1 (p. 59) asserts finite image on section spaces in each fixed degree, not finiteness of the groupoid.

Proposition 8.1・Lemmas 8.2–8.3 · pp. 59–61

4. Build compatible sections by norm products and descend to the floor

Proposition 9.2 (pp. 64–65) proceeds from points to three-dimensional strata. Extend an already compatible lower collection, then in the curve and surface stages take norm products of finitely many transports to enforce invariance. Lemma 9.1 preserves lower residues even when a surface link contracts a floor curve to a point. Avoiding finitely many hyperplanes chooses a section satisfying all required nonvanishing conditions, preserving generation. Proposition 6.3 finally descends the tuples to the floor. No new ambient section on $V$ has yet been constructed.

Lemma 9.1・Proposition 9.2 · pp. 63–65

Generation on the whole floor enters step 5 of Theorem 1.1 and supplies the hypothesis of the following lifting argument.

1. Formulate the finite-neighborhood lifting target precisely

The generated restriction $\mathcal O_V(G)|_S$, with $G=qP$, gives $f:S\to T=\mathbb P^b$ and an actual identity $\mathcal O_V(G)|_S\simeq f^*\mathcal O_T(1)$. Near each compact fibre, a root pair and a full normalized cyclic cover produce a reduced Cartier divisor $E$, with $I=\mathcal O_Z(-E)$ and $g:E\to U$. Proposition 10.2 (p. 68) seeks

These are sheaves of complex vector spaces on the underlying support of $E$. No holomorphic map from the thickening to $U$, or $\mathcal O_U$-coherence of its pushforward, is assumed. Neighborhoods for lifting a section germ may depend on the order.

Proposition 10.2 · p. 68

2. Retain special-support obstructions through split residues

A second canonical root and residue give a split injection from the obstruction sheaf $R^1g_*A_{-a}$ into $R^1h_*\omega_H$ for an SNC support (Proposition 10.4, pp. 71–72). The derived retraction retains classes supported at special parameters. Proposition 11.1 (p. 75) identifies the latter with the lowest step $F_0\mathcal M^1$ of a right filtered $\mathcal D$-module. Apply Sai, Theorem 1, p. 1 to constant real Hodge modules on smooth SNC strata and assemble a finite strict filtration with pure quotients. The proof does not assume a blanket Kähler direct-image theorem for arbitrary mixed Hodge modules.

Propositions 10.4, 11.1 · pp. 71–72, 75–80

3. Exclude maps from ample lines on the projective base

Lemma 11.2 (pp. 80–81) supplies the filtered comparison needed to transfer vanishing from SS, Theorem 16.3.10, printed p. 734 / chapter PDF p. 16.

Corollary 11.3 (pp. 81–82) then gives, on the projective parameter space,

for every ample $N$, including torsion in the kernel.

Lemma 11.2・Corollary 11.3 · pp. 80–82

4. Kill the obstruction at an arbitrary prescribed point

Assuming shorter lifting orders, the current connecting map becomes a degree-$k$ derivation $\delta_k$ on the Laurent graded algebra (Lemma 12.1, pp. 82–83). The adjugate identity in Lemma 12.2 (pp. 84–85) turns its values on base coordinates into a map to the symbol kernel without inverting the Jacobian of the parameter change.

Proposition 10.5 (pp. 73–74) supplies a projective coordinate-power cover, unbranched over an arbitrarily prescribed $t_*$, and one compact SNC graph. Section 12.3 (pp. 86–87) kills the resulting ample-line map by vanishing and descends the result at $t_*$. Since that point was arbitrary, special-support obstructions disappear as well.

Proposition 10.5・Lemmas 12.1–12.2・§12.3 · pp. 73–74, 82–87

5. Turn lifting in all integer degrees into section growth

The order-zero part of the same local identity and the split injection then kill negative-degree obstructions. The derivation rule for an invertible Laurent frame extends this to every integer degree, closing finite lifting induction. Finally cyclic invariants yield coherent layer quotients of $J_j=\mathcal O_V(-\lceil jG/r\rceil)$. Their direct images $F_j=f_*(J_j/J_{j+1})$ satisfy $F_{j-r}=F_j\otimes\mathcal O_T(1)$. Serre vanishing is applied to these coherent layer quotients. Euler characteristics of finite filtrations give at least $N-C$ quotient sections, and the final loss in passing to ambient sections is bounded by the fixed $h^1(V,\mathcal O_V)$. Some degree therefore has two independent sections, proving $\kappa(V,A)\geq1$ (pp. 87–88). Convergence of an infinite thickening is unnecessary.

§§12.3–12.4 · pp. 87–88

Return the resulting growth of sections to step 5 of the main proof, contradicting the zero-Kodaira counterexample.