Bounded anticanonical metrics on klt pairs and torus quotients

1. Produce unbounded twisted differentials

The polynomial $\chi(\widetilde W,K_{\widetilde W}+t\rho^*(r_0P))$ is nonzero at zero. Nonzero cohomology therefore occurs for infinitely many positive integers, and a subsequence fixes its degree. Positive powers of the bounded metric have trivial multiplier ideals, so hard Lefschetz produces nonzero sections of $\Omega^p_{\widetilde W}\otimes n_i\rho^*P$ for a fixed $p$. The case $p=0$ finishes immediately. Otherwise, saturating the determinant of their generic span gives effective divisors $N_i\sim_{\mathbb Q}M+m_iP$ with $m_i\to\infty$. The sums of positive twisting exponents are unbounded; numerical nontriviality of $P$ is not added as a premise. The cotangent tensor theorem applies using the klt pair, effective boundary and nef anticanonical class, giving pseudoeffectivity of $-M$.

Lemma 2.1 · pp. 5–6 [DPS01, Theorem 2.1.1, p. 8] [LMPTX23, Theorem 4.1, print p. 1409 / PDF p. 17]

2. Exclude new pencils and subtract minimal vertical ratios

Among diagrams $\pi:S\to W$ and $f:S\to Y$ with connected fibers and with $d\pi^*P-f^*A$ pseudoeffective for some $d>0$ and ample $A$, maximize $\dim Y$. A pencil dominated by $P$ and an exceptional divisor, nonconstant on the general fiber, would enlarge this base and is excluded. Applying this to ratios of the $N_i$ on a normal equidimensional model makes their horizontal parts affine in $m_i$. Taking a difference and subtracting, over each base prime, the minimum coefficient-to-multiplicity ratio yields $$r\pi^*P=f^*V+H,\qquad H\ge0.$$ Above each prime divisor of the base, at least one component is absent from $H$. This is stronger than an effective decomposition and is what controls the later metric. A zero-dimensional base already gives a section.

Lemmas 2.2–2.3 · pp. 7–8

3. Turn rank-one integrals into a bounded essential-supremum metric

Set $R=K_S-\pi^*(K_W+D)$, $E=\lceil R\rceil$, and $\Xi=E-R$. Then $E$ is effective and exceptional, $\Xi$ has SNC support and coefficients in $[0,1)$, and $K_S+\pi^*P+\Xi=E$. Maximality forces $f_*\mathcal O_S(E+ar\pi^*P)$ to have rank one for every $a\ge0$. The bundle $B_a=(ar+1)\pi^*P+\Xi$ is integral and has trivial multiplier ideal.

Positive curvature of the adjoint direct image gives, for a local generating section $s=f^*v\,s_H$, $$u_a(y)=-\frac1a\log\int_{S_y}|s|^{2a}_{h_P^r}\,d\mu_y\ \longrightarrow\ -\log\mathop{\rm ess\,sup}_{S_y}|s|^2_{h_P^r}.$$ The manuscript does not call the intersection of the exponent-dependent base-change opens Zariski open: it uses a common full-measure set and local $L^1$ compactness of psh functions. Near the component absent from $H$, the section $s$ is nonzero, bounding the essential supremum positively from below even toward a missing base divisor. The opposite estimate follows from properness. Psh extension gives a locally bounded semipositive metric on $V$, without a continuity assumption.

Lemma 2.4; Proposition 2.5 · pp. 9–11 [PT14, Theorems 3.3.4–3.3.5, p. 24]

4. Start at the zero twist and push the section down

Mix a small amount of a positive-current metric associated with $d\pi^*P-f^*A$ into the bounded metric. The coefficient margin in the klt boundary, Skoda integrability at small exponents and Hölder give a metric on $\pi^*P+\Xi$ with strict positivity in base directions and trivial multiplier ideal. Adding the metrics of $a f^*V+aH$ gives exactly $\mathcal O_S(-aH)$ as multiplier ideal for every $a\ge0$. The adjoint bundle becomes $E+a f^*V$, and Kollár–Nadel vanishing gives $$H^i(Y,\mathcal G\otimes\mathcal O_Y(aV))=0\quad(i>0,\ a\ge0),\qquad\mathcal G=f_*\mathcal O_S(E).$$ Since $E\ge0$, vanishing at $a=0$ yields $Q(0)=h^0(Y,\mathcal G)>0$. Thus $Q(a)=\chi(Y,\mathcal G(aV))$ is a nonzero polynomial and provides a section for some $a>0$. Multiply by $s_H^a$ and use $\pi_*\mathcal O_S(E)=\mathcal O_W$ to obtain $H^0(W,arP)\ne0$. The original Euler premise required nonzero value, not positive value.

Lemma 2.6; proof of Theorem 1.1 · pp. 11–12 [Fuj16, Theorem 1.3, p. 2]

1. Separate the auxiliary GIT linearization from the natural bundle

Pass to the effective image and average the smooth metric over the maximal compact subgroup. Positivity of the generic natural anticanonical weight pattern and finiteness of the patterns permit a small character perturbation of $L+\delta J$ with nonempty stable=semistable locus $U$. The quotient $W=U/T$ is projective. Finite stabilizer slices give a $\mathbb Q$-factorial klt pair and $D=\sum(1-1/e_B)B$. The torus adjoint representation is trivial, so the orbit-direction determinant introduces no character. Thus $L|_U=q^*P$ is an isomorphism of naturally equivariant $\mathbb Q$-line bundles. The auxiliary character must not be transferred to $L$.

Propositions 3.2–3.3 · pp. 13–14 Remark 3.6 · p. 18

2. Prove the orbit bound before applying the minimum principle

Translate the weak weight inequalities on $U$ into orders of moving lifts along algebraic arcs. Convexity of the orders and the initial weight calculation exclude poles. Compactifying the action graph and selecting a curve upgrades the arcwise conclusion to a locally uniform norm bound. The supremum along an orbit therefore defines a quotient norm. Its negative logarithm is the infimum of a psh function independent of imaginary exponential coordinates, hence is psh by Kiselman's principle. Slice bounds, the disc test for finite quotients and pullback to a resolution preserve local boundedness. Restriction to curves also establishes nefness of $P$.

Lemma 3.4 · pp. 15–16 Proposition 3.5 · pp. 16–17 [DWZZ19, Theorem 1.1, p. 1]

3. Supply the Euler premise and extend the invariant section

Rational connectedness of $Z$ gives $\chi(\mathcal O)=1$ on a smooth resolution of the quotient. Theorem 1.1 supplies a section of $mP$; taking a power clears the denominators in the equivariant identification. On projective $W$ the section has bounded norm for the bounded quotient metric. The orbit-supremum definition bounds its pullback throughout $U$. Relative to the smooth upstairs metric, removable singularities extend the bounded coefficients across all of $Z\setminus U$, including its divisors. Invariance persists by equality on the dense open set. The manuscript treats trivial actions and a point quotient as well.

§3.5; proof of Theorem 1.2 · pp. 17–18

1. Tensor invariant canonical frames with the section

The companion's structure theorem expresses the universal cover of a finite étale cover $X'\to X$ as $\mathbb C^q\times B\times Z$. It retains invariant canonical frames on the first two factors and residual monodromy on $Z$ inside an algebraic torus. Tensor the section from Theorem 1.2 with their anticanonical frames. The result is deck invariant and descends to $X'$. The finite cover itself need not be a product.

[OpenAI26a, Theorem 2.1, p. 5] Theorem 1.2 · p. 2

2. Multiply the exponent by the cover degree

For $d=\deg(X'/X)$, the finite étale norm sends the nonzero section to $H^0(X,-mdK_X)$. This proves nonvanishing at some positive exponent on the original $X$. The inputs from the companion are its independent structure theorem and norm lemma, not its nonvanishing theorem; the argument does not assume that conclusion circularly.

[OpenAI26a, Lemma 2.2, p. 8] Corollary 4.1 · p. 18