Minimal metrics and interior injectivity for nef adjoints

1. Choose extremal sections with one fixed twist

Write $N=K_V+B$. Every coefficient of $B$ is less than one, while $B^-$ is $\pi$-exceptional; thus $e^{-b}$ is integrable for $b=[B]$. Fix very ample $H_0$ and put $A=\pi^*((n+1)H_0)$. Klt vanishing and Castelnuovo–Mumford regularity make $mN+A$ globally generated whenever $mD_H$ is Cartier. If a minimal weight $\varphi$ has positive Lelong number at $p$, choose $s_m$ maximizing $|s(p)|$ subject to

Global generation gives $s_m(p)\ne0$. A smaller-mass section with the same value will contradict this choice.

§2.2 · pp. 5–6 · Fuj, Theorem 3.2 · PDF p. 8

2. Localize on a fixed sublevel and solve the error

A subsequential limit of $\tau_m=m^{-1}\log|s_m|^2$ and minimality $\tau_\infty\leq\varphi+C$ allow one fixed $R$. For $g_m=\varphi+a/m-\tau_m$,

Positive Lelong number also supplies fixed $c$ for which $e^{-c\varphi-b}$ is nonintegrable at $p$, retaining the negative coefficients of $B$. Fix a cutoff $\chi$ and a convex function $f$ with $c\leq f'\leq d$. On a smooth affine open avoiding $B$ and the zeros of $s_m$, Lemma 2.3, with $S_m=(m-1)\tau_m+a/m+b$, gives

Here $C$ depends only on $\chi,f$, not on $m$ or the open set. The external input is Dem's smooth inverse-curvature estimate. The manuscript passes to singular weights by decreasing approximation and weak limits in spaces with fixed earlier weights.

Claim 2.2; (2.5)–(2.12); Lemmas 2.1, 2.3 · pp. 5–9 · Dem, Theorem 5.1 · p. 33

3. Extend past negative exceptional components and preserve evaluation

The section $w_m=\chi(g_m)s_m-u_m$ is holomorphic on the open set. An upper bound for $G_m=S_m+d\max(0,g_m)$ and its $L^2$ estimate first extend it to $V\setminus\operatorname{Supp}B^-$. Instead of assuming that bound on $B^-$, descend to the complement of a codimension-two set in $H$. The actual Cartier bundle $mD_H+A_H$ and normality of $H$ permit Hartogs extension; pulling back gives $w_m\in H^0(V,mN+A)$.

Near $p$, the section $s_m$ is nonvanishing, $\chi(g_m)=1$, and $S_m+f(g_m)=b+c\varphi+O_m(1)$. A nonzero value of $s_m-w_m$ at $p$ would contradict the finite correction norm and the nonintegrability fixed above. Hence $w_m(p)=s_m(p)$. This $O_m(1)$ is a local comparison for each $m$, distinct from the estimates requiring uniform constants.

§2.4; (2.13)–(2.15) · pp. 9–10

4. Contradict extremality with a uniform Hölder bound

Take $m>d+1$ and set $I=\int_Ve^{\varphi-b}<\infty$. The exponent $d/(m-1)$ in Hölder's inequality cancels the final power $m-1$, giving

All of $R,c,d,\chi,f,C',I$ are independent of $m$. For large $m$, $0<Q_m(w_m)<1$, so $Q_m(w_m)^{-m/2}w_m$ has mass one and larger evaluation at $p$ than $s_m$. This contradiction proves $\nu(\varphi,p)=0$. Corollary 2.4 then follows from Skoda integrability.

(2.16)–(2.17); Corollary 2.4 · pp. 10–11

1. Regularize the endpoints and align contraction with curvature

In Reg, zero Lelong numbers make the singular set of the decreasing smooth approximants empty. Dini’s theorem makes the curvature loss uniform, while monotonicity and Skoda integrability bound the integrals at each fixed exponent. The log-sum modification in (3.1)–(3.3) makes multiplication $s:L_1\to L_0$ by the canonical section of $E=\lfloor C_1\rfloor-\lfloor C_0\rfloor\geq0$ a contraction. On the complement of the SNC support it gives

The strict inequality $\lambda<1$ keeps the Bochner comparison constant $C_\lambda=(1-\lambda)^{-1}$ finite. The statement does not extend the argument to the endpoint $\lambda=1$.

Lemma 3.1; (3.1)–(3.6) · pp. 11–13 · Reg, Main Theorem 1.1 · p. 2

2. Construct the ordinary class map and uniform primitives

Lemma 3.2 supplies a fixed complete Kähler metric $\omega_c$ on the SNC complement $U$. Its bounded local potentials can be added to the weights on a fixed finite cover to make curvature positive with norm-comparison constants independent of $k$. Local primitives from Dem have holomorphic differences. Uniform upper bounds on the weights allow ordinary $L^2$ extension of these differences, producing a Čech cocycle in ordinary $H^1(V,K_V\otimes F)$.

The continuous map $\kappa_k$ of Proposition 3.3 kills the closure of exact forms. To split a cocycle in its kernel, apply one fixed Fréchet open mapping theorem with only a sup norm on the smaller cover prescribed. The cocycle family is uniformly bounded in the finitely many compact seminorms needed by that theorem. This gives splittings bounded in the prescribed sup norm; uniform integrability of the weights then gives $\bar\partial v=f$ and $\|v\|_k\leq C\|f\|_k$. No lift bounded in every seminorm, or linear choice of global primitive, is claimed.

Lemmas 2.1, 3.2; Proposition 3.3; (3.7)–(3.8) · pp. 5, 13–16

This step supplies primitives bounded uniformly for the varying weights. The Bochner estimate kills the image norm; only the comparison with a fixed Hilbert space kills the original ordinary cohomology class.

3. Combine Bochner estimates with the uniform primitive

Project a smooth representative $\gamma$ of an ordinary kernel class onto $u_k$, orthogonal to the closure of exact forms. Complete-metric cutoffs and the curvature comparison give $\|\bar\partial^*_{0,k}(su_k)\|_{0,k}=O(\sqrt{\varepsilon_k})$. The manuscript also uses cutoffs to place the form in the Hilbert-adjoint domain. Since the class map kills the closure, $\kappa_{0,k}(su_k)=0$; the preceding step gives $\bar\partial v_k=su_k$ with $\sup_k\|v_k\|_{0,k}<\infty$. Hence

This is an estimate in varying weights. Eliminating the ordinary class still requires the next comparison.

§§3.4–3.5; (3.9)–(3.14) · pp. 16–17

4. Return to a fixed Hilbert space and kill the original class

For a fixed smooth weight $h_*$, the bound $h_{1,k}\leq h_*+C_*$ gives bounded inclusions into fixed Hilbert spaces in both degrees zero and one. On compact subsets of $U$, where $s$ is nonvanishing, the preceding estimate gives strong convergence $u_k\to0$. Together with global boundedness in the fixed space, this implies $u_k\rightharpoonup0$.

Exact forms approximating $\gamma-u_k$ also pass to the fixed space. Applying Proposition 3.3 to the fixed weight gives a continuous class map with $\kappa_*(u_k)=[\gamma]$ for every $k$. Weak continuity into finite-dimensional ordinary cohomology forces $[\gamma]=0$, proving Theorem 1.2. The exact sequence (1.1) then gives surjectivity of $H^0(V,K_V+L_0)\to H^0(E,(K_V+L_0)|_E)$ when $E\ne0$. This concerns the entire divisor scheme, including nonreduced structure.

(3.15), §3.5 · pp. 17–18; (1.1) · p. 2

1. Obtain nonvanishing from the metric and Euler characteristic

Take a tensor power of Theorem 1.1’s metric on the actual bundle $\pi^*\mathcal O_X(rD)$. It satisfies LP’s generalized algebraic singularities condition with zero divisorial part. Set the auxiliary nef divisor and parameter to zero; $\chi(X,\mathcal O_X)\ne0$ then gives $\kappa(D)\geq0$.

Theorem 4.2; Corollary 4.1 proof · p. 19 · LP, Corollary D · p. 4

2. Apply the semiampleness criterion to the same metric

For a nef adjoint on a klt fourfold, GM gives semiampleness from a zero-Lelong metric on a positive Cartier multiple and nonvanishing. Supply the latter from the preceding step and use the same metric. Theorem 1.2 is not used in this corollary.

Theorem 4.3; Corollary 4.1 proof · p. 19 · GM, Corollary 5.3 · printed p. 499 / PDF p. 23