Log adjoint and nef big vanishing on smooth projective varieties

Dimension zero is immediate, and degrees above the dimension vanish by the local Dolbeault resolution. In positive dimension we prove the analytic form that implies the first assertion. Suppose a holomorphic line bundle B has on X minus an SNC divisor D a smooth Hermitian metric h of curvature $i{\mathrm{\Theta }}_h\ge b\omega$ for a fixed $b>0$, and locally along D has the form $h=h0{\prod }_j\mathrm{\mid }{z}_j{\mathrm{\mid }}^{-2a_j}$, with h0 smooth positive and $0\le a_j<1$. Then $H^q(X,K_X\otimes B)=0$ for $q>0$. A smooth closed B-valued (n,q)-form f is square integrable for h, because every local integral of product $\mathrm{\mid }{z}_j{\mathrm{\mid }}^{-2a_j}$ is finite. All estimates below concern these top-holomorphic-degree forms; their metric-volume cancellation is important.

On X minus D construct a complete Kähler metric. Scale defining-section norms so $\log \mathrm{\mid }{s}_j{\mathrm{\mid }}^2<-1$, and add to a sufficiently large multiple of omega the forms $d{d}^c[-\log (-\log \mathrm{\mid }{s}_j{\mathrm{\mid }}^2)]$. Their leading normal term is a positive constant times $\mathrm{\mid }d{z}_j{\mathrm{\mid }}^2\mathrm{/}(\mathrm{\mid }{z}_j{\mathrm{\mid }}^2{\log }^2\mathrm{\mid }{z}_j\mathrm{\mid })$, while their tangential negative terms are bounded by a fixed multiple of omega. The resulting metric ${\omega }_P$ is Kähler and complete, since the radial length integral int dr/(r|log r|) diverges. For each $ϵ>0$ use the complete metric ${\omega }_{ϵ}=\omega +ϵ{\omega }_P$. Local metric coefficients and h are smooth on this open manifold. Distance cutoffs have compact support, tend to one, and have gradient bounded by 2/R; Lipschitz cutoffs suffice and may be smoothed in coordinate patches. Cutoff multiplication followed by local translation convolution approximates the joint graph domains of bar-partial and its formal adjoint. The error is the cutoff gradient term tending to zero and the local Friedrichs commutator: expand the smooth coefficients at the convolution center, subtract their constant part, and use the first derivative remainder to pass to zero in L2. Thus compactly supported smooth forms are a graph core.

Here is the integration identity on that core. In Kähler normal coordinates and a line frame, expand bar-partial as alternating covariant bar derivatives and its adjoint as the negative contraction of covariant holomorphic derivatives. In the sum $\mathrm{\parallel }\bar{\mathrm{\partial }}v{\mathrm{\parallel }}^2+\mathrm{\parallel }{\bar{\mathrm{\partial }}}_h^{\ast }v{\mathrm{\parallel }}^2$, integrate mixed derivative terms once. Their commutator is line curvature plus tangent curvature; the tangent-curvature contractions cancel with the same contractions in the holomorphic adjoint square because torsion is zero. The remaining line-curvature operator is $[i{\mathrm{\Theta }}_h,\mathrm{\Lambda }]$. Thus $\mathrm{\parallel }\bar{\mathrm{\partial }}v{\mathrm{\parallel }}^2+\mathrm{\parallel }{\bar{\mathrm{\partial }}}_h^{\ast }v{\mathrm{\parallel }}^2=\mathrm{\parallel }{D}_h^{\mathrm{\prime }}{}^{\ast }v{\mathrm{\parallel }}^2+\int ⟨[i{\mathrm{\Theta }}_h,\mathrm{\Lambda }]v,v⟩$. If curvature is diagonal with eigenvalues ${\lambda }_j$ in a unitary frame, its action on the coefficient $d{z}_1$ wedge ... wedge $d{z}_n$ wedge $d{\bar{z}}_J$ is multiplication by ${\sum }_{j\in J}{\lambda }_j$: the wedge/contraction commutator has one contribution precisely for each antiholomorphic index present. The formula therefore has the claimed sign and is positive for $q>0$. Graph-core approximation extends the inequality to the maximal Hilbert domains.

Denote that positive curvature operator by $B_{ϵ}$. The Hilbert-complex argument gives, for closed f with finite integral $<B_{ϵ}^{-1}f,f>$, a solution $\bar{\mathrm{\partial }}u=f$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{h,{\omega }_{ϵ}}^2\le \int ⟨B_{ϵ}^{-1}f,f{⟩}_{h,{\omega }_{ϵ}}$. Indeed let $T=\bar{\mathrm{\partial }}$ in degrees q−1 to q and $S=\bar{\mathrm{\partial }}$ in degrees q to q+1, closed on their maximal domains. Project a test vector v orthogonally to ker S. This projection remains in Dom T*, with the same T*, because Ran T is contained in ker S. On that kernel, the identity in step3.1 and weighted Cauchy–Schwarz bound the functional Tv $\mapsto <f,v>$ by the square root of the displayed energy times ||Tv||. It is well-defined by the same estimate. Extend this bounded functional and apply Hilbert representation; the representing u satisfies $Tu=f$ against every test vector. This is an actual construction, including the closed-range-independent case.

The estimate is uniform as epsilon tends to zero despite completeness blowing up near D. Diagonalize ${\omega }_{ϵ}$ relative to omega, with eigenvalues $t_j\ge 1$. The (n,q) coefficient norm times the volume factor is $\mathrm{\mid }{f}_J{\mathrm{\mid }}^2\mathrm{/}{\prod }_{j\in J}{t}_j$. Since $\text{curvature}\ge b\omega$, the inverse-curvature energy is bounded by ${\sum }_J\mathrm{\mid }{f}_J{\mathrm{\mid }}^2\mathrm{/}[b({\sum }_{j\in J}1\mathrm{/}{t}_j){\prod }_{j\in J}{t}_j]\le b^{-1}{\sum }_J\mathrm{\mid }{f}_J{\mathrm{\mid }}^2$. The last inequality uses $q\ge 1$ and $t_j\ge 1$. It also follows for nondiagonal curvature by inverse-operator monotonicity. On every compact subset of X minus D the metrics tend smoothly to omega. Weak subsequences of the solutions, taken on a countable compact exhaustion and diagonally, give $\bar{\mathrm{\partial }}u=f$ and $\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }{\mathrm{\mid }}_{h,\omega }^2\le b^{-1}\mathrm{\mid }\mathrm{\mid }f\mathrm{\mid }{\mathrm{\mid }}_{h,\omega }^2$. Fatou's inequality gives the global bound.

This equation extends across D as a distributional equation on X. The singular metric dominates a fixed smooth line metric, so u is ordinary locally L2. Around each normal-crossing coordinate hyperplane take a logarithmic cutoff with transition |z| in $(r^2,r)$; its derivative is bounded by C/(|z||log r|), whose squared area integral is O(1/|log r|). Product cutoffs handle crossings. The boundary term in the distributional equation tends to zero by Cauchy–Schwarz with u in L2 and this vanishing cutoff-gradient norm. Hence $\bar{\mathrm{\partial }}u=f$ globally. Smooth Dolbeault Hodge theory then implies vanishing of the class of f: pair the equation with its smooth harmonic representative and integrate by parts, or use the smooth Green resolution of the resulting distributional exact form. This proves the analytic assertion in step1.1.

For the first public assertion put $B=O_X(M-K_X)$. An ample rational divisor $A=M-K_X-\mathrm{\Gamma }$ has a positive metric obtained by taking a root of the Fubini–Study metric on a sufficiently divisible very ample multiple. Tensor that rational metric with the canonical divisor metrics of Gamma. The result is a genuine metric on B, with strictly positive curvature on X minus Supp Gamma and precisely the singular exponents $a_j$ in [0,1). Steps1.1–6.1 apply.

We give the big decomposition used for the second assertion. Choose a very ample divisor H. Bigness supplies a cubic, or in dimension n an n-th degree, lower bound for h0(rN) along sufficiently divisible r. On the fixed hyperplane $H,h0(H,rN\mathrm{\mid }H)=O(r^{n-1})$: choose a large ample twist B on H with B−N|H globally generated by F3, and choose a section of B−N|H which is nonzero on every component and associated point of H. Its r-th power embeds $O_H(rN)$ into $O_H(rB)$. The ample Hilbert-polynomial bound and Serre vanishing in F3 give $h^0(H,rB)=O(r^{n-1})$, hence the required bound. The finitely many associated-point avoidance conditions can be satisfied over C. The restriction sequence therefore gives a section of rN−H for some r. Write $rN=H+G$ with G effective. Resolve G by smooth-center blowups $\pi :Y\to X$. A positive exceptional divisor F with −F relatively ample exists for this composition: at each blowup its negative exceptional divisor is relatively ample; multiply the pulled-back earlier choices by sufficiently large integers before adding the next one. For a sufficiently small positive rational $t,H_Y=\pi \ast H-tF$ is ample, because a relatively ample line bundle tensored with a sufficiently high pullback of an ample line bundle is ample by the relative projective embedding and Serre's criterion. Thus r pi*N $=H_Y+G_Y,G_Y=\pi \ast G+tF$ has an ample rational first term and an effective SNC second term.

Let ${\omega }_Y$ be a fixed projective Kähler metric. By the attached curve-nef/ample bridge, piN+delta $A_Y$ is ample for every rational $\delta >0$ and fixed ample $A_Y$. Positive metrics on these rational ample bundles, tensored with the negative delta-th power of the fixed $A_Y$ metric, give smooth metrics on piN with $\text{curvature}\ge -C\delta {\omega }_Y$. The big decomposition in step8.1 gives a metric on the same bundle with $\text{curvature}\ge b0{\omega }_Y$ off $G_Y$ and logarithmic exponents equal to the coefficients of $G_Y\mathrm{/}r$. Take a convex combination of the two metric potentials, of weights 1−s and s. First fix $s>0$ so small that every exponent $s\mathrm{coeff}(G_Y)\mathrm{/}r$ is $<1$, then delta small enough that $sb0-(1-s)C\delta >0$. The combined metric satisfies exactly step1.1. Therefore $H^q(Y,K_Y+\pi \ast N)=0$ for $q>0$. This uses curve-nef, not an unmentioned metric-nef convention.

Finally $H^q(Y,K_Y+\pi \ast N)=H^q(X,K_X+N)$. Here is the required smooth-blowup calculation rather than a rational-singularity citation. For a blowup along a smooth center of codimension $r,K_Y=\pi \ast K_X+(r-1)E$: in a regular-center chart (z1,z1u2,...,z1ur), the Jacobian factor is $z{1}^{r-1}$. The standard affine charts of the regular-center Rees algebra have the projective-simplex monomial Cech contraction, giving $\pi \ast O_Y=O_X$ and $R^q\pi \ast O_Y=0$ for $q>0$. For $1\le a\le r-1$ use $0\to O_Y((a-1)E)\to O_Y(aE)\to O_E(-a)\to 0$. On the projective (r−1)-space fibres every cohomology of O(−a) vanishes in this range, by the standard homogeneous Laurent monomial computation. Relative local trivializations give the identical direct-image vanishing. Induction in a yields $\pi \ast O_Y((r-1)E)=O_X$ and all its higher direct images zero. Projection formula gives $\pi \ast {\omega }_Y={\omega }_X$ and $R^q\pi \ast {\omega }_Y=0$ for $q>0$. The same holds for the finite composition by Leray; tensoring N and Leray prove the asserted equality. Step9.1 therefore proves the second public assertion.