Conditional good minimal models for compact Kähler fourfolds

1. Fix the nef endpoint supplied by the MMP

Assumption 2.3 supplies an endpoint satisfying (i)–(iii) of Theorem 1.1. It assumes termination for arbitrary choices of negative rays; retain the actual adjoint line on this endpoint. Property (iv) still requires a first section, which termination alone does not provide.

Assumption 2.3 · p. 9; §9.4 · p. 84

2. Reduce nonvanishing to two cases on a smooth model

In the projective branch, Lemma 3.1 derives logarithmic Iitaka subadditivity from Assumption 2.2 and applies the manuscript's Theorem A.2, p.85. Appendices A–I prove that conditional projective abundance statement; it is not another independent premise. The uniruled branch uses the rational quotient, and the irregular branch the Albanese map, with nonvanishing in dimension at most three. Ou Theorem 1.1, p.1 relates non-uniruledness to canonical pseudo-effectivity for smooth compact Kähler manifolds. §3, pp.10–14

The remaining case is a smooth non-uniruled, non-Moishezon fourfold $W$ with $q(W)=0$. Proposition 4.1 handles $a(W)>0$ and Theorem 9.4 handles $a(W)=0$. The following proof sections trace these two branches.

Proposition 3.5 · pp. 13–14; Proposition 4.1 · p. 14; Theorem 9.4 · pp. 83–84

3. Return the section to the original adjoint and use the final premise

Push a canonical section on $W$ down to $Y$ and multiply by the section of the effective boundary. This yields nonvanishing for $J$ on the specified nef pair. Only now is the hypothesis $\kappa(Y,J)\geq0$ of Assumption 2.4 available. Applying it to that same actual $J$ gives global generation.

Proposition 3.5 · pp. 13–14; §9.4 · p. 84 · AN, Theorem 1.1 · p. 3

1. Construct an actual canonical pullback from twisted sections

For $0<a(W)=d<4$, resolve algebraic reduction and choose a very ample $H$ on its projective base $S_0$. Lower-dimensional fibre nonvanishing and ample twisting first give $\kappa(K_W+Nb^*H)\geq0$ for every sufficiently large integer $N$. At each stage choose $N>5k_i$, where $k_i$ is the current canonical Cartier index. Local rationality forces $H_i\cdot R_i=0$ for a ray negative for $K_{V_i}+NH_i$, so its step also belongs to a single empty-boundary $K$-program. Rechoosing $N$ does not change that program's boundary. Arbitrary termination and Assumption 2.4 make the twist semiample; Stein factorization yields actual line-bundle identities $$ K_V\sim_{\mathbb Q}g^*A_T,\qquad K_M\sim_{\mathbb Q}f^*A+R,\quad R\geq0. $$ Here $S\to T$ is a smooth projective resolution, $A$ is the pullback of $A_T$, $q(S)=0$, and $A$ is pseudo-effective. Proposition 4.3, pp.15–16

Proposition 4.3 · pp. 15–16

2. Obtain nonvanishing on bases of dimension one and two

For $d=1$, $S=\mathbb P^1$. For $d=2$, apply Assumption 2.2 to $r$-fold fibre powers over a curve. Fix the local form $t=x^e$ and the exceptional coefficient $h$ before varying $r$; the base coefficient has pole bound $mrh/e$. Letting $r\to\infty$ gives $$ \left(A-K_S+\sum_{P\ {\rm exceptional}}c_PP\right)\cdot H'\geq0 $$ (Lemma 4.4, pp.17–19). For the Zariski decomposition $A=P_0+N_0$, the remaining case $P_0^2=0$ gives $K_S\cdot P_0\leq0$; Riemann–Roch and $q(S)=0$ then produce sections.

Lemma 4.4; §4.3 · pp. 17–19

3. Produce a klt threefold adjoint from genus-one fibres

For $d=3$, the generic fibre has genus one. Identify its Hodge line by $\mathcal H^{\otimes m}\simeq\mathcal O_S(m(A-K_S))$, and use the modular sections $E_4^{m/4}$ and $\Delta_{\rm mod}^{m/12}$ with $12\mid m$. Their common divisorial order $\ell_P$ satisfies $\ell_P/m=1-t_P$, where $$ t_P=\min_{E\mapsto P}\frac{1+\operatorname{coeff}_E R} {\operatorname{ord}_E(f^*P)}. $$ The integrability calculation proves this on every required model, including exceptional base valuations. A general linear combination supplies an effective $\Xi$ on the original normal base $T$ such that $(T,\Xi)$ is klt and $K_T+\Xi\sim_{\mathbb Q}A_T$. No separate $\mathbb Q$-Cartier hypothesis on $K_T$ is imposed. Projective threefold nonvanishing supplies the section to pull back. Lemma 4.5 and §4.4, pp.19–23

Lemma 4.5; §4.4 · pp. 19–23

1. Obtain a nef interval and metric from the reduced boundary

Proposition 7.20, pp.62–65, takes a reduced-boundary pair $(T_0,G_0)$ to a nonextracting model $(T,G)$ with $$ A=K_T+G\ {\rm nef},\qquad K_T+tG\ {\rm nef\ and\ klt}\quad(t_0\leq t<1), $$ and a projective crepant dlt modification $h:(V,D)\to(T,G)$ with $J=K_V+D=h^*A$. It uses the special-termination arguments of §7 and reduction to ordinary klt programs; it does not add fourfold dlt termination to the premises.

Apply the zero-Lelong minimal-metric Lemma 5.1, p.23, inside this interval, add $(1-t)G$, and let $t\to1$ to obtain a zero-Lelong metric on the pullback of $J$.

Lemma 5.1 is proved analytically in §5. Volume-normalized capacity, differentiation of a Monge–Ampère equation and a Bochner estimate retaining its residual term are combined. Equation (89) cancels the residual measure on the same sublevel set; after taking the limit at fixed $t$, the Lelong upper bound (90) contradicts the positive lower bound (93). pp.25–33 This is not a direct application of the projective MM theorem to a nonprojective space.

Proposition 7.20 · pp. 62–65; Lemma 5.1; (89)–(93) · pp. 23–33

2. Match component sections within each cluster

Proposition 8.1, pp.65–81, gives $H^0(D,\mathcal O_D(mJ))\ne0$ when $D\ne0$. Full differents, including fractional coefficients, are retained on the normalization components. Das–Ou Corollary 1.3, p.4 supplies their semiample systems, but componentwise abundance alone does not glue sections. Choose components with maximal system dimension; match even residues on clusters of dominant conductor branches and arrange vanishing on nondominant branches. Lemma 8.4 multiplies all transported sections under a finite loop action, preserving nonzeroness and the required zeros.

Proposition 8.1; Lemmas 8.3–8.4 · pp. 65–72 · DO, Corollary 1.3 · p. 4

3. Control cycles using one ambient Kähler class

For a nonprojective torsion cluster, transport one ambient Kähler class on $V$ to the relevant surfaces. For the cycle automorphism $\varphi$ and a class $v$ of positive square, the construction gives $\varphi^*v-v\in\operatorname{NS}(Q)_{\mathbb R}$. Hodge index and integral lattice actions make the top-form multiplier a root of unity, permitting matching after a common power. This does not assert finite characters for arbitrary automorphisms of Kähler surfaces. Lemma 8.9, pp.73–74 Extend by zero on other clusters, and descend to the entire $D$ by codimension-one matching and $S_2$. Lemma 8.2 and §8.6, pp.66–67,81

Lemma 8.9 · pp. 73–74; §8.6 · p. 81

4. Kill the extension obstruction under the contradiction hypothesis

Work on smooth models with $a=q=0$. Lemma 9.1, p.81, gives finitely many prime divisors with linearly independent real cohomology classes. If $\kappa(M,L)=-\infty$, then every fixed vector bundle $\mathcal E$ satisfies $H^0(M,\mathcal E\otimes\mathcal O_M(mL))=0$ for all sufficiently large divisible $m$ (Lemma 9.2, pp.81–82). Take determinants of sections of maximal generic rank and compare their nonnegative integral coefficients on the finite set of primes. No global meromorphic frame is assumed at the start.

Proposition 9.3 assumes $\kappa(T,A)=-\infty$ to kill the extension obstruction. On $p:M\to V$, write $K_M+F\sim_{\mathbb Q}L=p^*J$, where $F$ has SNC support and coefficients at most one, allowing negative coefficients. The key comparisons are $$ p_*\mathcal O_M(mL-\lfloor F\rfloor) =\mathcal I_D\otimes\mathcal O_V(mJ), $$ $$ H^1(V,\mathcal I_D(mJ)) \hookrightarrow H^1(M,K_M+L_m),\qquad L_m\sim_{\mathbb Q}(m-1)L+\{F\}. $$ The injection is low-degree Leray and needs no higher-direct-image vanishing.

Zero Lelong numbers and the integrability margin of $\{F\}$ make the multiplier ideal trivial. DPS hard Lefschetz gives $H^0(M,\Omega_M^3\otimes L_m)\twoheadrightarrow H^1(M,K_M+L_m)$. Lemma 9.2 kills the source. High powers of the nonzero section on the whole reduced $D$ therefore extend to $V$, contradicting the assumption. Equations (176)–(179), pp.82–83

Lemmas 9.1–9.2; Proposition 9.3; (176)–(179) · pp. 81–83 · DPS, Theorem 2.1.1 · arXiv PDF p. 8

1. Reduce the absence of a frame to a model with no prime divisors

Theorem 9.4, pp.83–84, separates the existence of a nonzero meromorphic pluricanonical tensor from its absence. In the latter case, include every prime in $G_0$ and use Proposition 7.20. A nonzero $G$ would supply a section that, after dividing out divisor factors, gives the prohibited tensor. Thus $G=0$, leaving a model with no prime divisors and nef canonical line; Proposition 6.1 excludes it.

Theorem 9.4 · pp. 83–84

2. Build a foliation and spherical structure from differential forms

Proposition 6.1 starts with a zero-Lelong metric and DPS to obtain $\chi(\mathcal O_M)=0$, $h^{2,0}\geq1$ and $h^{3,0}\geq2$. Two- and three-forms produce an integrable saturated conormal line. Hodge index for positive currents gives $\alpha^2=0$ and, under the contradiction hypothesis, zero Lelong numbers for every positive current in $\alpha$. This permits a continuous self-map of the full compact convex set of positive currents. Its fixed point supplies a transverse spherical structure. §§6.1–6.4, pp.34–43

Proposition 6.1; §§6.1–6.4 · pp. 34–43

3. Use holonomy to contradict algebraic dimension zero

Index charts give finite holonomy for boundary meridians. A Selberg cover and compactification produce a torsion-free $\operatorname{PU}(2)$ representation. In the Zariski-dense case, the projective Shafarevich base from Campana–Claudon–Eyssidieux Theorem 1, p.2 contradicts $a=0$. A proper closure is virtually solvable; $q=0$ on every finite étale cover makes the image finite and hence trivial. Extending the developing map meromorphically by local charts and trace produces a nonconstant function, again contradicting $a=0$. §§6.5–6.6, pp.43–47

§§6.5–6.6 · pp. 43–47 · CCE, Theorem 1 · p. 2

4. Remove the negative part of a frame and return to a smooth model

If a meromorphic frame exists, write $K_{T_0}\sim_{\mathbb Q}P-N$, with $P,N\geq0$ and disjoint supports, and use $G_0=(\operatorname{Supp}P)_{\rm red}$. If $G=0$, pseudo-effectivity forces $N_T=0$. Otherwise the boundary-extension section and uniqueness of its meromorphic divisor when $a(T)=0$ give the same conclusion. Run the MMP on $(T,0)$ and use Assumption 2.4 to make the canonical line actually torsion. On a resolution, the signed exceptional divisor is effective by the support of a positive current and independence of prime classes. This returns canonical nonvanishing to the smooth model. Equations (180)–(181), p.84

Theorem 9.4; (180)–(181) · p. 84