Minimal metrics and interior injectivity for nef adjoints

1 Introduction

A nef line bundle admits smooth metrics whose curvatures approach semipositivity. A limit of those metrics may acquire singularities. For an adjoint bundle, the question is whether the canonical and boundary terms prevent logarithmic poles in a least singular semipositive metric. We prove that they do for projective klt pairs, and use zero-Lelong endpoint metrics to establish an injectivity theorem in ordinary coherent cohomology.

We use the convention that a local metric weight \phi gives squared norm |\xi|^2e^{-\phi}. Semipositivity means that its curvature current i\partial\bar\partial\phi is nonnegative. Such a metric has minimal singularities if every other semipositive weight \psi on the same line bundle satisfies \psi\le\phi+C_\psi for a constant C_\psi. Weights on rational line bundles are defined by clearing denominators and dividing by the same positive integer. Lelong numbers measure the logarithmic singularities of these weights; in particular, zero Lelong numbers imply local integrability of e^{-t\phi} for every fixed t>0.

For a normal variety H and an effective rational divisor \Theta, the klt condition means that K_H+\Theta is \Q-Cartier and, on a log resolution \pi:V\to H, the crossing divisor B in \pi^*(K_H+\Theta)=K_V+B has every coefficient less than one. The actual rational line bundle, rather than its numerical class, is the object in the following theorem.

Consequently, for every minimal weight \phi on \pi^*(K_H+\Theta),

where \mathcal I(t\phi) consists of holomorphic germs f for which |f|^2e^{-t\phi} is locally integrable. We record the precise independence of the metric and the Cartier multiple in Corollary 2.4.

Demailly–Peternell–Schneider constructed minimal-singularity metrics on pseudoeffective line bundles and proved zero Lelong numbers in the big and nef case [2, Theorem 1.5 and Proposition 1.7]. Nefness alone does not give this conclusion. For example, on a ruled surface over an elliptic curve, a nef divisor bundle can have its divisor metric as a minimal metric, with a positive Lelong number along the divisor [14, Corollary 1.2 and Example 3.5]. Gongyo–Matsumura asked how to construct a zero-Lelong semipositive metric on a nef log canonical bundle in their study of injectivity and abundance [11, Question 5.6]. Theorem 1.1 gives a positive answer in the projective klt setting on every projective log resolution, without bigness or nonvanishing. Once one zero-Lelong semipositive metric exists, every minimal metric is no more singular and therefore also has zero Lelong numbers. Thus the universal assertion about minimal metrics expresses the same existence conclusion on each pullback bundle.

1.1 Interior boundaries and restriction of sections

Our second result concerns a family of effective rational divisors between two fixed endpoints. Their coefficients may cross integers. The resulting changes in the integral part of the divisor give natural inclusions of line bundles. Zero-Lelong metrics at the endpoints control the induced map on ordinary H^1.

The strict inequality \lambda<1 provides a positive coefficient in the curvature comparison. Effectivity and the common crossing support provide integrability for the fractional boundary weights. The theorem concerns the displayed integral line bundles and the natural sheaf map; its endpoint assumptions are on the actual rational bundles L-C_i.

One direct consequence explains the connection with restriction of sections. Put E=\lfloor C_1\rfloor-\lfloor C_0\rfloor, an effective integral divisor, which may be nonreduced. If E\ne0, then

Indeed, the connecting homomorphism for 0\to\cO_V(K_V+L_1)\to\cO_V(K_V+L_0) \to\cO_V(K_V+L_0)|_E\to0 has image equal to the kernel of the injective H^1 map. Thus the comparison lifts sections from the entire divisor scheme E at once.

Classical root-cover injectivity, including a reduced boundary, appears in Esnault–Viehweg [7, Theorem 5.1]. Fujino’s transcendental approach develops complete metrics on an analytic complement and comparison with coherent cohomology [10, Section 3, Lemmas 3.1–3.2 and Claim 1]. Matsumura extends analytic injectivity to metrics with transcendental singularities and obtains uniform estimates for primitives [19, Theorem 1.3 and Sections 5.2–5.3]. Our proof uses the same harmonic-form and local-primitive methods, with a comparison of two endpoint weights. We prove the required uniform primitive estimate and the return to ordinary cohomology explicitly.

1.2 The two analytic arguments

For the metric theorem, write N=\pi^*(K_H+\Theta)=K_V+B. The adjoint hypothesis enters twice. The klt coefficients make the adjoint density integrable, even where B has negative exceptional coefficients. Klt vanishing and nefness then give one ample bundle A_H on H whose pullback A=\pi^*A_H makes mN+A globally generated for every positive integer m with m(K_H+\Theta) Cartier. Normalize a section s by its integral

where a is a smooth weight of A and b the divisor weight of B. Choose a normalized section maximizing its value at a proposed positive-Lelong point.

The L^{2/m} normalization has its origin in Narasimhan–Simha’s work on ample canonical bundles [20]; its role in the later envelope construction is explained in [1, Section 5]. Tsuji constructs canonical singular metrics using normalized sections of A+mK_X with a fixed sufficiently ample twist [22, Sections 1.2 and 2]. Berman–Demailly develop integral-normalized adjoint envelopes and their algebraic approximation [1, Definition 5.3 and Proposition 5.19]. Their general pseudoeffective approximation allows twists p_mA with p_m\to\infty and p_m/m\to0 [1, Equation (5.24)]. Here the fixed twist and the klt density support an extremal argument: a localization estimate produces a better normalized section whenever the minimal weight has a positive Lelong number.

The normalized extremal sections have a semipositive limit after the twist is divided by m. Minimality bounds that limit by the chosen minimal weight. Consequently a fixed sublevel set of the difference of the weights has adjoint mass tending to zero. A localized \bar\partial equation, solved on an affine open set, produces a holomorphic competitor whose weighted norm is bounded by that mass. The exact localization constant is independent of m and of the affine open set. Positive Lelong number forces this competitor to have the same value at the chosen point. A Hölder estimate whose final bound is independent of m makes its normalization strictly smaller, contradicting extremality.

Two details keep the construction uniform and valid on a log resolution. The weak limits in the localization lemma are taken in fixed weighted Hilbert spaces. Extension across negative exceptional coefficients is performed on the normal base H in the actual Cartier bundle m(K_H+\Theta)+A_H, and then pulled back. The complete Kähler L^2 estimate supplies the local analytic solution [4, Theorem 5.1]; the localization and extension arguments are given in Section 2.

For injectivity, regularize both endpoint metrics, retaining arbitrarily small curvature loss and uniform integrability at every fixed exponent. A logarithm of a sum of endpoint weights makes the inclusion of line bundles a contraction. On the complement of the crossing divisor, a complete Kähler metric with bounded local potentials allows local L^2 solutions. Their holomorphic differences extend across the divisor and define ordinary Čech classes. An open-mapping argument on one fixed finite cover gives a uniform bound for primitives of forms whose ordinary class vanishes.

A Bochner comparison then makes the image of a harmonic representative small. All approximating representatives embed into a single fixed Hilbert space; weak convergence there preserves the original ordinary cohomology class. This is the final step from weighted estimates on the complement to the coherent-cohomology injection. The argument is independent of the adjoint metric theorem: it uses only the endpoint metrics stated in Theorem 1.2.

1.3 A fourfold application and organization

The metric theorem also gives a short abundance argument for a projective complex klt fourfold (X,\Delta) with nef rational adjoint and \chi(X,\cO_X)\ne0. Lazić–Peternell’s nonvanishing theorem uses generalized algebraic singularities of a semipositive metric [17, Corollary D]; zero Lelong numbers satisfy that condition with no divisorial part. With a first section obtained, Gongyo–Matsumura’s criterion gives semiampleness [11, Corollary 5.3]. The precise application is Corollary 4.1; it has no numerical-dimension restriction. For nef adjoints of numerical dimension at most one, Liu–Xu already obtain good minimal models for projective klt pairs of dimension at most four with nonzero Euler characteristic [18, Corollary 5.2]; in the nef case this gives semiampleness. Lazić’s work supplies a semiampleness criterion using uniform multiplier-ideal comparisons, and an algebraic approximation theorem for supercanonical currents, under their respective hypotheses [16, Theorems A–B].

In the compact Kähler setting, Höring–Lazić–Lehn obtain nonvanishing for non-uniruled \Q-factorial klt pairs with nef adjoint of numerical dimension one and nonzero Euler characteristic [13, Theorem A]. Under these hypotheses in dimension four, they obtain semiampleness when the minimal pullback current has zero Lelong numbers [13, Corollary B].

Section 2 proves the metric theorem, the arbitrary-data localization estimate and the multiplier-ideal consequence. Section 3 proves the independent interior injectivity theorem. Section 4 states the two published fourfold criteria and applies them to the same minimal metric.

2 Zero Lelong numbers for nef klt adjoints

We prove Theorem 1.1. The construction uses sections only after one fixed twist pulled back from an ample bundle on the normal base. Its central estimate must reduce a normalized section’s mass without changing its value at a point of positive Lelong number. We first recall the top-degree L^2 estimate, then construct the normalized extremal sections. The affine localization lemma turns their small sublevel mass into the required competitor.

2.1 Weights and the complete L^2 estimate

A local weight \varphi of a Hermitian metric gives the squared norm |\xi|^2e^{-\varphi} in its local frame. Semipositive curvature means that the local weights are plurisubharmonic. A metric on a rational line bundle is defined by taking a positive integral tensor power and dividing its weights by that integer. All identifications below are identifications of rational line bundles, rather than of numerical classes.

If a weight \eta is a weight on K_V, the expression e^\eta denotes a density: in a canonical coordinate frame it is multiplied by the corresponding coordinate volume. Likewise, for an F-valued top form v, the expression |v|^2e^{-\psi}, with \psi a weight on F, denotes a density. These densities are invariant under changes of coordinates and frames. In particular, the integral of the latter does not depend on an auxiliary Kähler metric.

We shall use the following form of the complete L^2 theorem. The matrix observation in its statement will be responsible for the uniform constants.

2.2 The fixed twist and normalization

The zero-dimensional case is immediate, so put n=\dim V>0 and fix a Kähler form \omega on V. Choose compatible canonical divisors and write

The support of B is simple normal crossing, every coefficient is strictly less than one, and the negative part B^- is exceptional over H. The last assertion follows because the nonexceptional coefficients are those of the effective boundary \Theta.

Let b=[B] be the divisor weight, a rational linear combination of logarithms of squared absolute values of local divisor equations. In crossing coordinates it has the form

Thus e^{-b} is locally integrable. Negative coefficients cause no failure of this integrability statement.

Fix a very ample Cartier divisor H_0 on H, and set A_H=(n+1)H_0 and A=\pi^*A_H. Give A a smooth semipositive metric with weight a. For every positive integer m for which mD_H is Cartier, the line bundle mD_H+A_H is globally generated. Indeed, for 1\le j\le n,

is ample. Klt Kawamata–Viehweg vanishing, in the form of [9, Theorem 3.2], gives the cohomology vanishings for Castelnuovo–Mumford regularity. The global-generation conclusion of [15, Theorem 1.8.5(i)] then applies. Pullback proves that mN+A is globally generated as well.

Since N is nef, it is pseudo-effective. Choose a semipositive metric \phi on N with minimal singularities. Such a metric is obtained as the upper envelope, relative to a smooth reference weight, of the normalized plurisubharmonic weights; see [2, Definition 1.4 and Theorem 1.5]. Its defining property is that every other semipositive weight \psi on N satisfies

for a constant C_\psi. We shall show that \phi has zero Lelong numbers.

Suppose, to the contrary, that its Lelong number at p\in V is positive. For each allowed m define

The integrand is a canonical density by Equation (2.1). The integral is finite, continuous in s, positive for s\ne0, and homogeneous of degree 2/m. These facts also show that Q_m(s)=1 is a compact subset of the finite-dimensional section space: restrict Q_m first to the unit sphere of any vector-space norm and use its positive minimum. Choose s_m maximizing the absolute evaluation at p on this compact set, using any fixed norm on the one-dimensional fiber for this m. Global generation implies

Put \tau_m=m^{-1}\log|s_m|^2, a weight on N+A/m.

Define the globally scalar difference, almost everywhere,

Its values on the zero divisor of s_m may be assigned arbitrarily when taking integrals; that divisor has measure zero. By Equation (2.2), \tau_\infty\le\phi+C_\infty. Choose once and for all R>C_\infty+1. The preceding convergence and domination imply

Indeed, the indicators tend to zero almost everywhere, because \phi-\tau_\infty\ge-C_\infty where both weights are finite; the densities have one integrable majorant on a fixed finite coordinate cover.

It is now enough to construct sections w_m\in H^0(V,mN+A) such that

where C_0 is independent of m. For large m this gives Q_m(w_m)<1, so scalar normalization would produce a section of gauge one with larger absolute evaluation at p. We next construct these competitors and prove the two displayed properties.

2.3 A singular weighted equation with a fixed constant

We will cut off s_m on \{g_m<-R\} and correct the resulting \dbar error. The weight for this equation must serve two purposes. Its singularity near p must force the correction to vanish at p after the holomorphic competitor has been extended. Its growth where g_m is positive must also allow a uniform conversion from the quadratic estimate to Q_m. The first requirement determines its slope c near -\infty; a fixed upper slope bound d will give the second.

Positive Lelong number gives local coordinates at p and a sufficiently small fixed \alpha>0 with \phi(z)\le\alpha\log|z|^2+O(1). On a cone of positive angular measure on which all crossing coordinates have modulus comparable to |z|, the weight e^{-c\phi-b} is bounded below by a positive constant times |z|^{-2(c\alpha+\sum\beta_j)}. Choose a fixed c>0 large enough that c\alpha+\sum\beta_j\ge n. Then

This choice is valid even if some \beta_j are negative.

Choose a smooth function 0\le\chi\le1 equal to one on (-\infty,-R-1] and to zero on [-R,\infty). Choose a smooth convex increasing function f which is affine of slope c near -\infty, has f''>0 on the closed transition interval [-R-1,-R], and satisfies c\le f'\le d for one fixed d. Such a function is obtained by prescribing a nonnegative compactly supported second derivative positive on that interval. In particular there is a fixed constant C_f with

All of R,c,\chi,f,d,C_f remain fixed as m varies.

For a fixed m, choose a dense smooth affine open U=U_m\subset V avoiding the supports of B and \operatorname{div}(s_m). This open set need not contain p. We will recover the value at p only after extending the resulting holomorphic section to all of V. On U the function g=g_m is psh, since the weight \tau_m is locally pluriharmonic there and the metrics \phi,a are semipositive. Put

This is a smooth semipositive weight on F_m|_U. Divisor weights and \tau_m are pluriharmonic on U, so its curvature is m^{-1}i\partial\bar\partial a. We regard s_m as an F_m-valued top form.

We apply the lemma to the adjoint data already fixed on U. The exact density identity is

so its sublevel mass is M=\varepsilon_m. Applying Lemma 2.3 with F=F_m, g=g_m, and s=s_m gives an F_m-valued top form u on U with

where C=C_{\chi,f} is independent of m and of U.

2.4 Extension, evaluation, and the extremal contradiction

For m>d+1 set

Equation (2.12) shows that w_m is holomorphic. Since G\ge S, the cutoff term has G-norm squared at most \varepsilon_m. Since S+f(g)\le G+C_f, the G-norm squared of u is at most e^{C_f}C\varepsilon_m. Therefore

This constant is independent of m.

To extend the section, rewrite its weight as

It is locally bounded above away from \Supp B^-: the coefficient m-1-d is positive, the psh weights in the last two terms are locally bounded above, and b has that property away from its negative components. Consequently Equation (2.13) gives ordinary local L^2 bounds there. Holomorphic L^2 removal across analytic sets extends w_m to V\setminus\Supp B^-.

There is an open H^\circ\subset H, with complement of codimension at least two, over which \pi is an isomorphism. It misses the image of B^-. The section just obtained descends on H^\circ to a section of the actual Cartier bundle \cO_H(mD_H+A_H). Normality extends it across H\setminus H^\circ: after trivializing that line bundle, this is the Hartogs extension of regular holomorphic functions on a normal space. Pullback now gives a global section, again denoted w_m, of mN+A. By projectivity and GAGA it is an algebraic global section as well. It agrees with the constructed section on U by the identity theorem. In particular Equation (2.13) continues to concern the same section. No estimate across the negative exceptional divisor was needed for this extension.

Near p, the weight \tau_m is smooth by Equation (2.4), whereas g\to-\infty. Thus \chi(g)=1 and f(g)=cg+\text{constant} near p on U, so

If s_m(p)-w_m(p) were nonzero, its holomorphic coefficient would have a positive lower bound on a smaller neighborhood. The finite weighted integral in Equation (2.12) would then contradict Equation (2.6). Removing the analytic null set V\setminus U does not change this divergence. This also applies when p lies on B^-, since both sections are now holomorphic at p. We conclude

The bounded factor denoted O_m(1) is used only to test integrability for a fixed m; it never enters Equation (2.13).

It remains to convert that quadratic estimate to the original gauge. Hölder’s inequality gives

There is no unbounded factor hidden in the second integral. Put \eta=d/(m-1)\in(0,1). Direct expansion yields

Both exponentials on the right are canonical densities. The first has integral one; the second has integral at most 1+I, where

Finiteness follows from local upper bounds for \phi and local integrability of e^{-b} on a fixed finite coordinate cover. Applying Hölder to Equation (2.17) gives

All integrals are unchanged by the omitted analytic null set. Together with Equations (2.5) and (2.16), this proves

For large m, therefore, 0<Q_m(w_m)<1. Multiply w_m by Q_m(w_m)^{-m/2}>1. The new section has gauge one and, by Equation (2.15), strictly larger absolute evaluation than s_m. This contradicts the choice of s_m, regardless of the rate at which s_m(p) or \varepsilon_m tends to zero. Hence \phi has no positive Lelong number at any point. This proves Theorem 1.1.

3 An interior injectivity theorem

We prove Theorem 1.2 from its two endpoint metrics, using the top-degree L^2 estimate in Lemma 2.1. The proof does not use Theorem 1.1 or its extremal-section construction. The output is the natural map on ordinary coherent cohomology. The proof therefore needs both a weighted estimate on the complement of the crossing divisor and a comparison that retains the original ordinary class. The fixed-cover construction below supplies this comparison together with a uniform bound for primitives.

For a class in the kernel, we will construct weighted harmonic representatives whose images have norms tending to zero. The endpoint metrics make multiplication a contraction and supply the curvature comparison for this estimate. Local solutions of the \dbar equation leave holomorphic differences on overlaps; controlling a splitting of these differences on one fixed cover gives uniformly bounded global primitives. The extended differences form a cocycle representing the ordinary cohomology class of the original form. Finally, comparison in one fixed Hilbert space lets the small images force the original ordinary class to vanish. Classical injectivity for roots of divisors, including an additional reduced boundary, is proved in [7, Theorem 5.1]. The local and uniform-primitive methods have precedents in [19, Sections 5.2–5.3]; the precise comparison needed for the present line bundles is proved here.

3.1 Endpoint regularization and the comparison weights

Fix a Kähler form \omega on V, and put n=\dim V. The zero-dimensional assertion is trivial, so assume n>0. We use the following consequence of regularization and exponential integrability.

Apply the lemma simultaneously to L-C_0 and L-C_2, obtaining weights \psi_{0,k},\psi_{2,k} with a common sequence 0<\epsilon_k\le1 tending to zero. Let

For a rational divisor C, write [C] for its divisor weight. The endpoint weights \psi_{i,k} live on the rational bundles L-C_i. Adding [C_i] gives weights on L. We first raise the weight at the first endpoint above that at the second, so that the natural inclusion of the eventual integral bundles will be a contraction. The symbols \Phi_{i,k} below are weights on L, whereas h_{i,k} are weights on L_i=L-\lfloor C_i\rfloor. On U set

The two summands in Equation (3.1) are weights on the same rational line bundle L-C_0, so the expression is intrinsically well-defined. The logarithm of a sum of exponentials preserves the common curvature lower bound: locally add a potential for \epsilon_k\omega to both summands and use the psh log-sum inequality. Moreover \psi'_{0,k}\ge\psi_{0,k}. Effectivity of C_2-C_0 gives a uniform local upper bound for its divisor weight, so \psi'_{0,k} retains both the upper bounds and all the fixed-exponent integrability bounds of Lemma 3.1.

The weights h_{i,k} are smooth metrics on L_i|_U. They have uniform local upper bounds in frames extending across D, and

on every relatively compact coordinate neighborhood B, where d\lambda_B is coordinate volume. To verify the last assertion, write

Every coefficient of either fractional divisor lies in [0,1). Choose one Hölder exponent a>1 so close to one that its product with each of these finitely many coefficients is still less than one. The exponential of the fractional divisor weight is in L^a by the crossing-coordinate integral. The conjugate Hölder exponent for the remaining weights is allowed by the fixed-exponent bounds; a second Hölder inequality handles the two summands in h_{1,k}. This proves Equation (3.4) uniformly in k.

Let E=\floor{C_1}-\floor{C_0}\ge0, and let s:L_1\to L_0 be multiplication by the canonical section of E. Since \Phi_{2,k}\le\Phi_{0,k}, we have \Phi_{1,k}\le\Phi_{0,k} and

Thus multiplication is a contraction on all form degrees for any one fixed background metric. On U the curvatures are \theta_{i,k}=i\partial\bar\partial\Phi_{i,k}; divisor weights contribute no curvature there.

3.2 A complete metric with bounded local potentials

The singular set must be removed to use smooth weighted harmonic forms. We choose the complete metric carefully so that local solvability still has a uniform comparison with the given weights. The bounded-potential construction follows the complete-metric method of [10, Lemma 3.1]; we include the calculation because both bounded potentials and completeness are needed in the uniform primitive estimate.

Fix this metric for the rest of the section. When D\ne\varnothing, the function r=\log\log(-t) is a smooth proper exhaustion of U with bounded gradient in \omega_c. If \zeta is a smooth function equal to one on (-\infty,1] and zero on [2,\infty), then \zeta(r/R), as R\to\infty, gives compactly supported smooth cutoffs tending to one and with gradient O(R^{-1}). When D is empty, the constant cutoff one suffices.

The curvature comparison following Equation (3.2) becomes

since \theta_{2,k}+\epsilon_k\omega_c\ge0 as well.

3.3 A fixed-cover primitive estimate and the ordinary class map

Fix a holomorphic line bundle F on V. The local L^2 theorem gives primitives on coordinate neighborhoods, but a global primitive requires their holomorphic differences on overlaps to split. We use one finite cover and its fixed spaces of holomorphic sections to make that splitting estimate uniform in the weights. The extended differences also define a class in the ordinary group H^1(V,K_V\otimes F). Thus the class map below has ordinary coherent cohomology as its target, and the primitive estimate shows that its kernel consists of actual weighted exact forms. This local-solutions and holomorphic-cocycle comparison follows the method of [10, Section 3, Claim 1 and Lemma 3.2]. For a smooth metric h on F|_U, write L^2_h(n,q) for the Hilbert space of F-valued (n,q) forms using h and \omega_c. The Dolbeault operator is its maximal distributional realization, and Z^1_h=\ker(\dbar:L^2_h(n,1)\to L^2_h(n,2)).

Both sequences h_{0,k} and h_{1,k} satisfy this proposition, by Equations (3.4) and (3.6). For each weight, parts (i) and (ii) give

Indeed, part (i) kills the closure of the exact forms, while part (ii) provides a primitive for every form in the kernel. We will keep track of the closures explicitly when comparing different weighted spaces. The proposition also applies to a fixed smooth line metric from V: its curvature is bounded below by -A_0\omega_c for some fixed A_0, and its local upper and integral bounds are automatic. This last observation will allow us to compare all approximation parameters in one Hilbert space.

3.4 Harmonic representatives and the Bochner comparison

Take a class in the kernel of the map in Theorem 1.2, and represent it by a smooth closed L_1-valued (n,1) form \gamma on V. All norms in this subsection use \omega_c; a subscript i,k indicates the line weight h_{i,k}, and integrals of pointwise inner products use dV_{\omega_c}. The norms \|\gamma\|_{1,k} are bounded uniformly. Indeed, enlarging \omega to \omega_c decreases the top-form-valued (0,1) density, and the coefficients of \gamma are bounded on a finite cover, where Equation (3.4) applies.

In the maximal L^2_{1,k} Dolbeault complex, the closure of the image of degree-zero forms lies in the closed kernel of the degree-one operator. Subtract its orthogonal projection from \gamma, obtaining u_k. Thus

The adjoint assertion follows from orthogonality to the actual range; it does not require that range to be closed. Local elliptic regularity for the smooth weighted Dolbeault Laplacian on U makes u_k smooth there.

Write \Lambda=\Lambda_{\omega_c} and let D'_{i,k} be the (1,0) Chern derivative. In bidegree (n,1) put

These are pointwise nonnegative operators, and A_{1,k}\ge(1-\lambda)A_{0,k}. The smooth compact-support Bochner–Kodaira identity is

Here D'_{i,k}v=0 in top holomorphic degree, and [\omega_c,\Lambda] is the identity in this bidegree. The same identity on complete manifolds, with the indicated domain interpretation and lower curvature bound, is discussed in [19, Proposition 2.4].

To justify its use without already knowing the extra terms are integrable, use the compactly supported smooth cutoffs constructed above, whose gradients tend uniformly to zero in \omega_c. Apply Equation (3.10) to the cutoff multiples of u_k and add \epsilon_k times their squared norms. The left-hand derivative terms tend to zero by Equation (3.9) and the gradient bound. The two terms retained on the right are nonnegative, so lower semicontinuity gives

The commutators with the cutoffs contain only their derivatives; they do not introduce constants from derivatives of the varying line metrics.

The Kähler identity D^{\prime *}=i[\Lambda,\dbar] shows that this formal operator commutes with holomorphic multiplication by s. Combining Equations (3.5) and (3.6) with Equation (3.11) therefore gives

For the curvature term one first uses A_{1,k}\ge(1-\lambda)A_{0,k} and then the line-norm contraction. One may take C_\lambda=(1-\lambda)^{-1}: this factor bounds both the derivative term and the curvature term in the source energy. The strict interior hypothesis keeps this comparison factor finite.

We also need the target \dbar adjoint. The form su_k is smooth, closed, and square integrable. Apply the compact-support identity to cutoff multiples of this form. On its right-hand side use Equation (3.12), replace the curvature commutator by the larger nonnegative A_{0,k}, and let the gradient errors tend to zero. It follows that the formal adjoint is square integrable and

This also verifies the Hilbert-adjoint domain condition. In detail, complete-metric cutoffs and local smooth approximation give density in the maximal graph norm, so integration by parts against compactly supported forms extends to that domain. The formal adjoint, now known to be L^2, is its Hilbert adjoint. Thus Equation (3.13) has not assumed the existence of a global primitive or an adjoint-domain statement in advance.

We have shown that the image of the harmonic representative has a small adjoint norm. The fixed-cover proposition supplies a uniformly bounded primitive for this image; pairing these two statements will make its entire target norm tend to zero.

3.5 From a small image to vanishing of the ordinary class

Multiplication by s is bounded in degrees zero and one and commutes with the distributional Dolbeault operator. Hence Equation (3.9) implies that s\gamma-su_k is in the target closure of the actual exact forms. Apply the target class map in Proposition 3.3. It annihilates that closure and agrees with ordinary cohomology on s\gamma. The latter class is zero by the choice of \gamma. Consequently \kappa_{0,k}(su_k)=0.

The norms of su_k are uniformly bounded by Equations (3.5) and (3.9). Part (ii) of Proposition 3.3 gives global forms v_k on U with

The adjoint-domain conclusion above now justifies

To retain the original class while passing to a limit, fix a smooth metric on L_1 over all of V, with local weight h_* and use the same complete background \omega_c on U. Uniform upper bounds in a finite framed cover give the global metric comparison

It gives one bounded inclusion from each varying weighted space into the fixed space, in both degrees zero and one. In particular the u_k have uniformly bounded fixed norms.

On any compact subset K\subset U the multiplier s is nowhere zero, and the target weights h_{0,k} have uniform upper bounds. Thus Equation (3.14) implies that u_k\to0 in the fixed L^2 norm on K. This local strong convergence and the global bound imply global weak convergence to zero. Indeed, approximate any fixed L^2 test form by one supported in a compact subset of U; its pairing tends to zero by local convergence, and the norm of the remaining tail controls its pairing uniformly in k. Concentration of norm near D does not affect this weak convergence.

For each k, choose actual exact forms \dbar z_{k,l} converging to \gamma-u_k in Equation (3.9). By Equation (3.15), their primitives belong to the fixed degree-zero space and their derivatives converge in the fixed degree-one norm. Therefore \gamma-u_k belongs to the fixed closure of actual exact forms. Apply Proposition 3.3 once more, now to the fixed smooth metric with weight h_*. Its continuous class map \kappa_* annihilates that fixed closure, so

The fixed closed-form space is a closed Hilbert subspace, and a continuous linear map from it to finite-dimensional ordinary cohomology is weakly continuous. Since u_k\rightharpoonup0, the last displayed identity forces [\gamma]=0.

This proves analytic Dolbeault injectivity. Dolbeault theory and GAGA identify it with the natural map in algebraic coherent cohomology, proving Theorem 1.2.

4 Abundance for fourfolds with nonzero Euler characteristic

The metric theorem gives abundance for klt fourfolds when the Euler characteristic is nonzero. A published nonvanishing criterion uses this Euler hypothesis to produce the first section; a second criterion then gives semiampleness. We state both inputs in terms of the actual adjoint bundle on a resolution, so that the same zero-Lelong metric supplies their analytic hypotheses.

This proof combines Theorem 1.1 with published nonvanishing and semiampleness criteria. It isolates the use of \chi(X,\cO_X)\ne0: that condition enters the nonvanishing criterion, whereas the metric theorem has no Euler-characteristic hypothesis. Related work on multiplier-ideal approximations and supercanonical currents studies this analytic setting [16, Theorems A–B].

The two algebraic inputs

For a \mathbb Q-Cartier divisor P on a normal projective variety, the metric condition in [17, Definition 2.12] requires a positive integer r, with rP Cartier, and a resolution f:Z\to X carrying a singular metric on f^*\mathcal O_X(rP) whose curvature has the form

The divisor sum is finite. This is called a metric with generalized algebraic singularities. The sum may be zero; in particular, every semipositive metric with zero Lelong numbers everywhere meets this definition.

This is [17, Corollary D], the unconditional four-dimensional consequence of their Theorem C(i). Its interval includes t=0. We will take N=0, so both metric assumptions concern D.

This is the form of [11, Corollary 5.3] used here. Positivity of the Cartier multiple is also explicit in their Theorem 5.1, from which that corollary follows. Neither of these fourfold results requires X to be \mathbb Q-factorial or non-uniruled. Their dimension-three inputs are already incorporated in the stated corollaries.

Application to the minimal metric

The argument imposes no restriction on the numerical dimension of D. The Euler-characteristic hypothesis is used only to obtain the first section; the interior-injectivity theorem of this article is not an input to this particular application.