Low-dimensional nonvanishing from exact subadjoint resolution data

Let $E_1,\mathrm{.}\mathrm{.}\mathrm{.},E_r$ be the exceptional curves. Their intersection matrix Q is negative definite. Here is the needed construction from Hodge index, rather than an unmentioned contraction theorem. The pullback $B={\nu }^{\ast }A$ has $B^2=A^2>0$ and $B{E}_i=0$. Hodge index (equivalently its signature on numerical divisor classes) makes the quadratic form nonpositive on combinations of the $E_i$. Off-diagonal entries of Q are nonnegative, since distinct curves on the smooth surface have nonnegative local intersection multiplicities. If a nonzero coefficient vector z had $z^{T}Qz=0$, replacing z by |z| could only increase this quadratic form. It would remain zero by nonpositivity. Hodge index would then make the effective nonzero divisor $\sum \mathrm{\mid }{z}_i\mathrm{\mid }{E}_i$ numerically zero, impossible since it has positive intersection with an ample divisor on V. Thus Q is negative definite even as a matrix on the exceptional coefficient vectors.

Put $B_0=-Q$. Its off-diagonal entries are nonpositive and it is positive definite. The unique minimizer of 1/2 $x^T{B}_{0}x-\sum x_i$ has no negative coordinate: replacing x by |x| cannot increase the quadratic term and strictly reduces the linear term if a coordinate is negative. Its equation is $B_{0}x=1$; no coordinate can be zero, since that row would have nonpositive left side. Hence $x_i>0$ for every i, and x is rational since Q is an integral invertible matrix. Clearing denominators gives a positive integral exceptional divisor E with $E{E}_i<0$ for every i.

Here is a direct absolute ampleness construction, avoiding a relative numerical criterion. Choose an integer N larger than all coefficients of E, and let I be the ideal of the finite image of the exceptional locus. For k sufficiently large, the ample global-generation criterion gives a nonzero section of $I^N(kA)$, defining an effective F∈|kA|. Every exceptional valuation of the pullback of I is a positive integer, so ${\nu }^{\ast }F-E$ is effective. For every nonexceptional curve C not a component of ${\nu }^{\ast }F-E$ this implies $EC\le k{\nu }^{\ast }AC$. For the finitely many nonexceptional curve components of that divisor, the denominators ${\nu }^{\ast }AC$ are strictly positive; enlarge k to a real bound $k_0$ for all their ratios $EC\mathrm{/}({\nu }^{\ast }AC)$ as well. Thus ${\nu }^{\ast }A-\delta E$ has positive intersection with every nonexceptional curve when $\delta k_0<1$, and with each exceptional curve because $E{E}_i<0$. Its square is $A^2+{\delta }^2{E}^2>0$ for sufficiently small δ. The actual projective Nakai--Moishezon criterion on the smooth surface V makes ${\nu }^{\ast }A-\delta E$ ample. Scaling gives $a{\nu }^{\ast }A-\delta E$ ample for sufficiently small δ. If there are no exceptional curves, take $E=0$ and use that ν is an isomorphism.

Put $P=\mathrm{ceil}(-\mathrm{\Delta })$, with coefficientwise rounding. Because ${\nu }_{\ast }\mathrm{\Delta }\ge 0$ and all coefficients of Δ are $<1$, $P$ is an effective integral exceptional divisor. The coefficients of Δ+P lie in [0,1). Choose $\delta >0$ sufficiently small that $a{\nu }^{\ast }A-\delta E$ is ample and $\mathrm{\Gamma }=\mathrm{\Delta }+P+\delta E$ has every coefficient $<1$. It is effective and simple normal crossing by the resolution hypothesis.

For every integer n≥0, the Cartier divisor $T_n=P+n{\nu }^{\ast }A$ satisfies $T_n-K_V-\mathrm{\Gamma }{\sim }_Q(n+a){\nu }^{\ast }A+M-\delta E$.

The right side is ample: $a{\nu }^{\ast }A-\delta E$ is ample, and $n{\nu }^{\ast }A+M$ is curve-nef. Apply the actual SNC log-adjoint vanishing supplier on the SMOOTH V to obtain $H^i(V,O_V(T_n))=0$ for every $i>0$. Notice both the sign +δE in Γ and the rounding P; applying smooth vanishing directly to a possibly negative Δ would be invalid. No global klt replacement of M on W is used.

Normality gives ${\nu }_{\ast }{O}_V(P)=O_W$. Indeed a local rational function with divisor at least −P on V has no pole along any codimension-one point of W, because P is exceptional; the codimension-one characterization of regular functions on a normal variety makes it regular on W. Conversely regular functions pull back regularly and P is effective.

The coherent sheaf $R^1{\nu }_{\ast }{O}_V(P)$ is supported at finitely many points. For sufficiently large n, Serre vanishing gives $H^1(W,O_W(nA))=H^2(W,O_W(nA))=0$. Projection formula and the low-degree Leray sequence then identify $H^0(W,R^1{\nu }_{\ast }{O}_V(P)\otimes O_W(nA))$ with $H^1(V,O_V(P+n{\nu }^{\ast }A))=0$. A nonzero coherent sheaf of finite support has a nonzero global section after every line-bundle twist. Therefore $R^1{\nu }_{\ast }{O}_V(P)=0$.

For every n≥0, the same Leray sequence injects $H^1(W,O_W(nA))$ into $H^1(V,O_V(T_n))$, and, since R¹ has vanished, injects $H^2(W,O_W(nA))$ into $H^2(V,O_V(T_n))$. Both targets vanish by Section 2. This proves all the claimed cohomology vanishings. Since W is integral proper over $C$, $H^0(W,O_W)=C$, and therefore $\chi (O_W)=1$. This deduction proves exactly what is needed and does not label W rational or assert an unproved local rational-singularity theorem.

Use the ordinary smooth-surface Riemann–Roch formula on V for ${\nu }^{\ast }A$. The higher direct images $R^j{\nu }_{\ast }{O}_V$ are supported at finitely many points for every $j>0$. By projection formula, twisting by A does not change their lengths. The Leray Euler formula therefore cancels these finite-support corrections in the difference and gives $\chi (W,A)-\chi (W,O_W)=\chi (V,{\nu }^{\ast }A)-\chi (V,O_V)=1\mathrm{/}2{\nu }^{\ast }A({\nu }^{\ast }A-K_V)$.

This difference formula requires only that A be Cartier; it does not assume $K_W$ Q-Cartier or use a singular Riemann–Roch theorem. From the exact subadjoint relation, $K_V{\nu }^{\ast }A=-a{A}^2-\mathrm{\Delta }{\nu }^{\ast }A-M{\nu }^{\ast }A\le -a{A}^2$.

Indeed $\mathrm{\Delta }{\nu }^{\ast }A=D$ A≥0 because D is effective, while $M{\nu }^{\ast }A\ge 0$ because both are nef (or approximate ${\nu }^{\ast }A$ by ample divisors). Thus χ(W,A)≥1+(1+a)A²/2. Section 3 gives $h^0(W,A)=\chi (W,A)$, proving the positive section count and the desired nonvanishing.

Let W instead be a normal integral projective curve and A an ample Cartier divisor on W. A normal curve over C is smooth: its one-dimensional normal local rings are discrete valuation rings, hence regular, and C has characteristic zero. Put $g=h^1(W,{\mathcal{O}}_W)$ and choose a canonical divisor $K_W$ with ${\mathcal{O}}_W(K_W)={\omega }_W$. Since $W$ is integral projective, $h^0({\mathcal{O}}_W)=1$. F2 gives $h^0({\omega }_W)=g$ and $h^1({\omega }_W)=1$. Applying curve Euler-characteristic Riemann–Roch to $K_W$ yields $g-1=\chi ({\omega }_W)=\mathrm{deg}{K}_W+1-g$, so $\mathrm{deg}{K}_W=2g-2$. Suppose the actual subadjoint relation is $K_W+D+M\equiv -aA$ with D effective, M nef, and rational a satisfying $a\cdot \mathrm{deg}A>-2$. Taking degrees gives 2g(W)−2≤−$a\cdot \mathrm{deg}A<2$, hence g(W)≤1 because 2g(W)−2 is an even integer. Curve Riemann–Roch gives h⁰(W,A)≥deg $A+1-g(W)>0$ because deg A is a positive integer. This includes the index-one threefold endpoint $\gamma =1$, for which $a=1-\gamma =0$; strict log-Fano positivity must not be falsely claimed there. It also covers curve centres in the fourfold case and the slight negative-a perturbation allowed by $({\gamma }_1-1)\mathrm{deg}A<2$. A reduced point has $h^0(A\mathrm{\mid }W)=1$ directly.