Anticanonical nonvanishing from smooth semipositivity

1. The cover retains monodromy

The structure theorem gives a finite étale cover $S\to X$ with universal cover $\mathbf C^a\times F\times Z$. The Ricci-flat factor $F$ has a parallel nowhere-zero canonical frame; $Z$ has no positive-degree holomorphic forms and smoothly semipositive $-K_Z$. The manuscript passes to a finite-index deck subgroup, kills its action on $F$, and makes the closure of its action on $Z$ a connected compact torus $T$. It does not assert that $S$ itself is a product.

Theorem 2.1 · pp. 5–8 [DPS96, Structure Theorem, p. 1] [CDP, Theorem 1.4, p. 3]

2. From finite volume to a naturally invariant section

On the no-forms factor, $D=0,J=L=-K_Z$ satisfies the finite-volume and bounded-metric assumptions. An ordinary section, however, need not descend under the deck action. Theorem 7.2 takes a rational torus quotient and uses rank one of the invariant adjoint direct image to transfer two metrics to the base. Theorem 1.2 applies on that base, and its section lifts invariantly.

Theorem 1.2 · p. 3 Theorem 7.2 · pp. 24–25

3. Descend using invariance of the natural action

Multiply $s_Z\in H^0(Z,-mK_Z)^T\setminus\{0\}$ by the inverse $m$th powers of the invariant canonical frames on $\mathbf C^a$ and $F$. The product is deck-invariant, hence descends to a section of $-mK_S$. Its norm, for $e=\deg(S/X)$, gives $H^0(X,-meK_X)\ne0$. If $Z$ is a point, take $m=1,s_Z=1$. The argument requires some positive exponent, without uniform estimates for the cover degree or exponent.

Lemma 2.2; §7.3 · pp. 8, 25–26

1. Finite volume controls the tangent slope on a supporting face

Since $L+D$ is pseudoeffective but not big, cone duality supplies a nonzero movable class $\alpha$ with $L\cdot\alpha=D\cdot\alpha=0$. Finite volume enters the relative Bergman metric in Lemma 4.1. Near a multiple fiber $s_1=z_1^e$, the relative-form coefficient contains $z_1^{-(e-1)}$. A local upper bound after subtracting $(e-1)\log|z_1|^2$ from the Bergman weight extends the metric to the determinant of the saturated conormal sheaf. The positive HN part is a foliation, and positive-slope algebraicity then yields $\mu_{\min,\alpha}(T_Z)\ge-D\cdot\alpha=0$.

Lemma 4.1; Theorem 4.2; Corollary 4.3 · pp. 11–15 [BP08, Theorem 0.1, p. 2] [CP19, Theorem 1.1, p. 2] [GKP, Corollary 2.27; Theorem 4.2, pp. 8, 12] [BDPP, Theorems 2.2, 2.4, p. 6]

2. One determinant and a base controlling every section ratio

The no-forms assumption gives $\chi(\mathcal O_Z)=1$. The Riemann–Roch polynomial $\chi(K_Z+mL)$ has constant term $(-1)^d$, hence is nonzero for large $m$. Bounded weights imply $\mathcal I(h_L^m)=\mathcal O_Z$, so Hard Lefschetz produces infinitely many maps $-mL\to\Omega_Z^p$ for one fixed $p$. Stabilize their generic tail span and take its determinant $M$. Then $M\cdot\alpha\le0$ and there are $s_i\in H^0(Z,M+m_iL)$ with $m_i\to\infty$.

Every complete system $|uL+vM+wD|$, for $u,v,w\ge0$, has nonpositive degree against $\alpha$ and cannot be generically finite. Choose one with maximal image dimension, and take the relative algebraic closure of its image field in $\mathbf C(Z)$ to obtain $\dim S<\dim Z$. Products of sections and maximality put every ratio from these systems in $\mathbf C(S)$. The interpolation identity (6.5), together with $m_i\to\infty$, shows that $\rho=s_2/s_1$ has nonnegative order at horizontal divisors. The same ratio property, after clearing common vertical denominators, gives adjoint rank one for every $j\ge0$, including $j=0$.

Lemma 6.1 · pp. 19–20 [DPS01, Theorem 2.1.1, p. 8] Lemma 6.2 · pp. 20–22

3. Transfer two metrics and lift only permitted poles

Proposition 5.1 supplies a bounded semipositive line $A$ and a finite-volume metric on $-K_S+D_S$. Pullback of holomorphic forms is injective, so $S$ also has no positive-degree forms. Induction gives a section of $aA+bD_S$, which lifts to a nonzero section of $akL+cD$, contradicting the assumption. If the base is a point, the absence of horizontal poles already makes $\rho$ a section. The dimension-zero starting case is $a=1,b=0$.

Proposition 5.1 · pp. 15–19 §6.3 · pp. 22–23

1. Normalize vertical orders on the equidimensional normal model

Clear denominators in $c_T=\min_{\bar f(Q)=T}\operatorname{ord}_Q(\rho)/\operatorname{ord}_Q(\bar f^*T)$ and put $A=\mathcal O_S(\sum c_TT)$. Dividing $\rho$ by the pullback of its tautological rational section gives $e$ with nonnegative order at every prime divisor. Normality makes it regular. Above each $T$, at least one component has zero order in $e$. Equidimensionality is required of the normal model, not of the smooth resolution used for integration.

Proposition 5.1 · pp. 15–19

2. Zeroth moments carry volume; high moments carry boundedness

Using $\eta_Y=\pi^*s_D$, including its Jacobian, and local frames $ds,v$, set $$V_j(s)=\int_{Y_s}|(\eta_Y/f^*ds)e_v^j|^2_{\pi^*(h_Jh_L^{jk})},\qquad\Phi_j=-\log V_j.$$ Rank one identifies a Bergman weight and gives psh representatives of $\Phi_j$. For $j=0$, a boundary correction $D_S$ yields an adjoint measure equal to $f_*|\eta_Y|^2$, preserving finite volume. In contrast, $$\Psi=\lim_{\ell\to\infty}\left(\sup_{j\ge\ell}\frac{\Phi_j}{j}\right)^*$$ is a weight on $A$. Boundedness of $h_L$ gives $V_j\le C^jV_0$ and hence a lower bound for $\Psi$. On a component where $e$ does not vanish, $V_j\ge C_1C_2^j|s_1|^{2b_T}$ has $b_T$ independent of $j$. Division by $j$ and passage to the limit remove the boundary term. These two estimates produce bounds on both sides.

§5, (5.3)–(5.10) · pp. 16–19 [BP08, Theorem 0.1, p. 2]

3. The support condition permits lifting

Pull back a base section and multiply by $e^a$. Poles occur only over bad primes, hence are exceptional over $X$ or supported above $D$. Exceptional poles disappear under normal pushdown, and multiplication by $s_D^c$ clears the remaining ones. No new forbidden polar divisor survives on $X$.

Proposition 5.1 · pp. 15–19

1. A rational quotient gives rank one only in the invariant part

The vanishing $H^1(\mathcal O_Z)=0$ extends $T$ to an algebraic torus $G$ and identifies $T$-fixed and $G$-fixed sections for the natural action. A general fiber of the Rosenlicht quotient has a dense orbit. Commutativity makes the fundamental vector fields invariant; their wedge, combined with a base anticanonical frame, gives an invariant $\rho$ without horizontal poles. Ratios of invariant adjoint sections belong to $\mathbf C(Z)^G=\mathbf C(S)$, so the invariant subspace has dimension one for every $j\ge0$. The full adjoint space need not have rank one.

Theorem 7.2 · pp. 24–25

2. Positivity comes from an orthogonal quotient of the smooth direct image

Average the smooth semipositive metrics to make them $T$-invariant. Berndtsson's theorem gives the adjoint direct image its semipositive $L^2$ metric on the smooth locus. Compact averaging is a holomorphic orthogonal projection onto the invariant line, whose metric is therefore a semipositive quotient metric. The argument does not assume positivity of an arbitrary subbundle. This use of smooth coefficients distinguishes Proposition 7.1 from ordinary Proposition 5.1, which allows singular coefficients.

Proposition 7.1 · pp. 23–24 [Ber09, Theorem 1.2, p. 3]

3. Apply the ordinary theorem downstairs and lift invariantly

Proposition 7.1 uses the boundary estimates of Proposition 5.1 to construct both base metrics. The no-forms condition descends, and Theorem 1.2 gives a section of $aA+bD_S$. The base action is trivial and $e$ is invariant, so the lift remains invariant. Since $D=0$, every permitted pole is exceptional and disappears on pushdown. If the general orbit has dimension zero, the action is trivial and Theorem 1.2 applies directly; the point case is immediate as well.

Proposition 7.1 · pp. 23–24 Theorem 7.2 · pp. 24–25