Fourfold nonvanishing by minimal metrics and moving jets

1. Fix the minimal counterexample and its curve degrees

Proposition 2.1 replaces a counterexample by one fixed projective $\mathbb Q$-factorial terminal fourfold $Y$, with $K=K_Y$ nef, $\kappa(K)=-\infty$, $K^4=0$, $q(Y)=0$, and nef dimension $n(K)=4$. A fixed Cartier index $\iota$ gives $K\cdot C\geq1/\iota$ for every curve through a very general point. Nef dimension here is distinct from numerical dimension $\nu(K)$. Using abundance in dimensions at most three, Proposition 2.2 makes every proper positive-dimensional subvariety through that locus of general type on resolution.

Propositions 2.1–2.2 · pp. 5–7

2. Bound vanishing using bounded volume and diverging degree

For fixed small $\varepsilon>0$, take $L_t=r_t(K+tA)$ with $L_t^4\to\varepsilon^4$ and $r_t\to\infty$. Curve degrees satisfy $L_t\cdot C\geq r_t/\iota\to\infty$ although volume stays bounded. Moving high-order sections create base components. The moving-multiplicity input from LA controls ordinary powers of their ideals for the entire high-jet kernel. A general-type center with bounded degree and canonical intersection produces a curve of bounded degree, contradicting the diverging curve degrees. Proposition 3.5 therefore gives, for every allowed Cartier degree,

(pp. 10–12).

§§3.4–3.5; Proposition 3.5 · pp. 10–12

3. Exclude slices and correspondences to generate two-factor jets

For the two-factor argument, set $S_0=\iota K$ and

The analytic SNC hypothesis is not imposed directly on this bundle. Restrict moving base components to slices and exclude, in turn, a proper generically finite image, the entire bundle over a proper image, and a non-axis multisection over all of $Y$ (§6.4, pp. 26–29). In the last case, intersection with the axes gives

and Proposition 5.1 below controls its signed support. Remaining correspondences with both projections dominant and generically finite are excluded by Proposition 6.4. First rule out sweeping branch divisors; then LA, Lemma 7.5, p. 51 gives finitely many finite étale covers of a fixed open. A degree bound alone is not used to assert finiteness. Birational automorphism groups would then produce a dominant map from an abelian variety to $Y$, again contradicting positive canonical curve degree.

Propositions 6.3–6.4; Corollary 6.5 · pp. 23–30

The proof of Proposition 5.1 separates metric-based surjectivity of restriction from the nonzero boundary section supplied by three-dimensional semi-dlt abundance.

4. Fix the finite data before varying the characteristic

This gives $\epsilon(P;z)>10$ in Proposition 6.3 and one finite $10k_0$-jet surjection in Corollary 6.5. Section 7 now fixes $t,r_t$, a resolution $W$, an ample perturbation $H$, a complete-intersection flag with $\mu_{\min}(\Omega_W^1|_C)\geq0$, and a concrete strongly movable curve $\gamma$ on the blowup of a point. The numerical conditions are

They are supplemented by $J_x\gamma>0$ and

The curve is the pushforward of an ample complete intersection on one fixed birational model. This permits testing effective divisors after reduction without assuming specialization of an effective cone.

Theorem 7.1; §§7.1–7.2 · pp. 31–33

5. Compare two orders of the same determinant

Theorem 7.1 restates LA, Theorem 9.1, p. 62, which rules out coexistence of these fixed data. In §7.3 the characteristic-$p$ evaluation has generic rank $R=p^8$ but diagonal rank less than $R/4$. Testing the two actual determinant factors with $\gamma$ gives the conflicting inequalities

for a nonzero determinant $\sigma$. The jet point $z_*$ need not lie over the independently chosen point $x$. All finite data must be fixed over $\mathbb C$ before varying $p$.

§7.3 · pp. 33–34

1. Construct a nef boundary with nonempty floor

Assume it is not big. Positive Iitaka dimension, combined with general-type general fibers, would force bigness, so $\kappa(K_W+D_W)\leq0$. A dlt MMP and a small crepant perturbation preserving a Cartier index produce $M'\sim_{\mathbb Q}G'$, $\kappa(M')\leq0$, and a nef klt interval $M'-uD'$ for $0<u\leq\delta$. The signed divisor $G'$ is supported on $D'$. Its largest positive coefficient $a$ and small $s>0$ give

Here $0\leq B\leq D'$, $(Z',B)$ is lc, $N$ is nef, and $\mathcal J(Z',B)=\mathcal I_S$.

§5; (5.2)–(5.7) · pp. 15–17

2. Use endpoint metrics to obtain restriction onto the whole floor

Fix $0<b_0<1<b_2$ and construct effective SNC boundaries $C_i$ on a resolution. The endpoint residuals in (5.9) satisfy

MM, Theorem 1.1 supplies zero-Lelong metrics on the actual endpoint bundles. MM, Theorem 1.2 and local vanishing then give

Thus $H^0(Z',mN)\to H^0(S,mN|_S)$ is surjective for sufficiently large divisible $m$.

(5.9)–(5.10) · p. 18 · MM, Theorems 1.1–1.2 · pp. 1–2

3. Descend and lift a nonzero section of the whole floor

A nonzero boundary section is constructed separately. Apply adjunction and threefold semi-dlt abundance to the entire floor $T$ of a dlt blowup. Local vanishing in (5.12), giving $g_*\mathcal O_T=\mathcal O_S$, descends a whole-floor section, including conductor compatibility, to $S$ and then lifts it. Only now does $\kappa(N)=0$ follow.

(5.11)–(5.12) · p. 19

4. Apply klt semiampleness after obtaining nonvanishing

Finally apply GM, Corollary 5.3, p. 19 to $N/\alpha\sim_{\mathbb Q}K'+(1-s/\alpha)D'$. Semiampleness and Iitaka dimension zero give numerical triviality, contradicting its positive intersection with an ample cube because $K'$ is pseudo-effective, $D'\ne0$, and $1-s/\alpha>0$. The nonvanishing assumption of GM is obtained before its use.

§5, final klt adjoint · pp. 19–20 · GM, Corollary 5.3 · p. 19

This contradiction handles the signed-support branch. Return to the exclusion of slices and correspondences, then to two-factor jets and the Frobenius comparison that closes smooth nonvanishing.

1. Pass from smooth to lc nonvanishing

Hash, Theorem 1.4, p. 2 takes smooth fourfold canonical nonvanishing to nonvanishing for projective lc pairs of dimension at most four. Since nef $D$ is pseudo-effective, it gives $D\sim_{\mathbb R}G\geq0$ on the original normal variety.

§8 · pp. 34–35 · Hash, Theorem 1.4 · pp. 1–2

2. Rationalize with fixed support and clear the prescribed index

Lemma 8.1 writes this as a rational linear system involving finitely many prime and principal divisors. Fixing the zero coefficients and approximating the positive ones by a rational solution gives $D\sim_{\mathbb Q}G_{\mathbb Q}\geq0$ with the same support. Clearing denominators together with the prescribed $r$ produces a section of $mrD$ on $X$ itself, rather than only on a $\mathbb Q$-factorial model.

Lemma 8.1; Corollary 1.2 proof · pp. 34–35