Exact orders and invariant anticanonical linear systems

1. The curvature-null condition forces full rank one

The set $k=\{0\}\cup\{h\in\mathbb C(X)^\times:\ell((h)_\infty)=0\}$ is a field. The poles of coefficients in a monic equation bound the poles of an algebraic element, proving relative algebraic closedness and finite generation. Take $X\xleftarrow{\pi}Z\xrightarrow{g}Y$ with $\mathbb C(Y)=k$. Bounding the poles of ratios of a very ample base system by one null divisor gives $\ell(\pi_*g^*H)=0$. Dividing any generic-fiber section of $E=K_{Z/X}$ by its Jacobian section again gives a function with null poles, hence an element of $k$. Thus the full adjoint direct image at exponent zero has rank one.

Lemmas 3.1–3.2 · pp. 9–10

2. Turn a horizontal strictness patch into integrability and finite generation

For $\nu<n$, the intersection $\int_Z(\pi^*\alpha)^\nu\wedge g^*\omega_H\wedge\pi^*\omega_0^{n-\nu-1}$ vanishes. Semipositivity makes the top-degree form itself zero and gives $\ker\pi^*\alpha\subseteq\ker dg$ on the maximum-rank locus. For $\nu=n$, the inclusion is automatic. A smaller patch then satisfies $\pi^*\alpha\geq cg^*\omega_H$. If $\nu=0$, a positive-dimensional base cannot occur.

Prepare an equidimensional toroidal model $X\xleftarrow pW\xrightarrow fY$ by birational modifications without extending the field. Metric descent, Theorem 2.1, uses full rank one and this patch to give an effective rational correction $A_0$ with $p_*f^*A_0=0$ and an unweighted integrable metric on $-K_Y+A_0$. Its strictness on an ordinary open set allows Controlled boundary, Theorem 4.3, to supply common $\epsilon,k_0$ and a finitely generated ring in finitely many directions. Patch strictness is not itself described as a global lower bound.

§3.2, (3.4)–(3.6) · pp. 10–11

[OpenAI26e, Theorem 2.1, p. 7]

[OpenAI26f, Theorem 4.3, pp. 9–10]

3. Reduce an arbitrary fixed auxiliary line to two directions

Choose input divisors $N_1,N_2$ with distinct exponents $u_1,u_2$. Solve $u_1t_j^1+u_2t_j^2=1$ and $t_j^1+t_j^2=1/m_j$. For $D(\mathbf t)=t^1N_1+t^2N_2$, this gives

Here $h_j\in k^\times\otimes_{\mathbb Z}\mathbb Q$ denotes formal rational powers of rational functions, not an extension obtained by adjoining roots. Bounding the principal corrections' poles by $N_1,N_2$ places them in this field. If $k=\mathbb C$, the fixed-support limit already gives the desired divisor. Otherwise, vertical minima are minima of finitely many rational affine functions, and become affine on a short rational segment at the endpoint. Apply finite generation in the two endpoint directions and Proposition 2.2 to obtain $N\sim_{\mathbb Q}L$ with $v(N)=0$.

§4, (4.1) · pp. 11–12

1. Select among finitely many actual generators

Write $t_j=a_j/d_j$, $d_j=a_j+b_j$, and clear denominators so that $s_j=h_j^{\epsilon kd_j}$ is a section in bidegree $(a_j,b_j)$. Express each section as a sum of generator monomials. At least one monomial has valuation no greater than that of the sum. Every off-ray generator consumes at least one unit of first degree, so their total contribution is $O(a_j)$. Since $a_j/d_j\to0$, some degree-$(0,b)$ generator $r$ satisfies $v(r)/(\epsilon kb)\leq-v(D_0)$. A finitely generated numerical cone alone would not provide this rational function.

Lemma 2.1 · pp. 5–6

2. Do not assume the correction has valuation zero

Set $h_*=r^{1/(\epsilon kb)}$. The generator's section inequality and the vertical minima give $p^*D_0+f^*(\operatorname{div}(h_*)+A/\epsilon)\geq0$; horizontal coefficients are nonnegative by passage to the limit. Push forward by $p$ to remove the correction, obtaining $N=D_0+\operatorname{div}_X(h_*)\geq0$. Only now use nonnegativity of a centered valuation to conclude $0\leq v(N)\leq0$. The exceptional correction may have positive valuation on a higher model.

Proposition 2.2 · pp. 7–8

1. Lift an extremal source weight to forms

Use the left action $(g\cdot s)(x)=g(s(g^{-1}x))$. If $c$ is the natural weight of $L$ on the source $S$ of a cocharacter $\lambda$, then $c>0$ for a nonzero effective cocharacter. Concavity of the orbit logarithmic norm for the semipositive metric makes $c$ the largest fixed-component weight. The highest-weight coefficient in localization is $\chi(S,(l-1)L|_S)$, with value one at zero. For large $l$, cohomology of weight $(l-1)c$ is therefore nonzero. Hard Lefschetz with smooth coefficient $lL$ lifts it to eigensections of $\Omega_X^a\otimes lL$ with fixed $a$.

§7.1 · p. 14

[DPS01, Theorem 2.1.1; Corollary 2.1.2, p. 8]

2. Retain the determinant-rank shift

Let the generic span of the images have rank $e$, and saturate its determinant in $\bigwedge^e\Omega_X^a$ to fix a line $M$. The manuscript uses Calabi–Yau metrics and the subbundle curvature inequality to obtain $\ell(M)\leq0$. Fixing $e-1$ basis vectors and varying the last produces eigensections of $M+m_jL$ of weight $(m_j-e)c$. If $c_M$ is the weight of $M$ on $S$, then

is bounded. Effectivity and the intersection inequality give $\ell(N_j)=0$. This construction uses Theorem 1.1 for arbitrary fixed $M$, without requiring $-M$ pseudoeffective. Replacing the weight by $m_jc$ would lose the necessary rank shift.

§7.2 · pp. 14–15

3. Turn exact order into a finite product of characters

The generic orbit formula $v_\lambda(\operatorname{div}s)=\mu-w$ shows that a zero-order section has normalized character $\chi/m$ with $\langle\chi/m,\lambda\rangle=c$. Varying the cocharacter identifies the support function of the closed convex hull of normalized section weights with the source-weight function $c$. Positivity in every nonzero direction puts zero in the interior, hence in the convex hull of finitely many actual rational weights. Clear a rational convex combination's denominators and multiply the corresponding eigensections to obtain a nonzero invariant section in a positive anticanonical multiple. The trivial torus and point cases are immediate. For an ineffective action, pass to the effective quotient: its entire kernel, including its finite part, acts trivially on the natural anticanonical line, so invariance lifts to $T$.

Lemmas 6.1–6.2 · p. 13

§7.3 · p. 15

1. Verify the null condition and the center

For $\nu<n$, $\ell$ is nonnegative on the pseudoeffective cone and $\ell(L)=0$, so $0\leq\ell(N_j)=\ell(M)\leq0$. For $\nu=n$, $\ell=0$ by definition. The generic arc gives $\mathbb C(X)\hookrightarrow F((t))$, and the image of $\operatorname{ord}_t$ is zero or $d\mathbb Z$. At the closed point's image $\xi$, the local ring maps into $F[[t]]$ and units have order zero, giving centeredness.

Corollary 5.1 · p. 12

2. Clear denominators while preserving zero order

Divide a nonzero valuation by $d$, and retain the zero valuation as permitted in Theorem 1.1. This fixed normalization preserves the $o(m_j)$ condition. Clear the coefficients of the resulting $N\sim_{\mathbb Q}L$ and its rational linear equivalence by a common $r>0$. Then $D=rN\sim rL$ is integral and effective, and its arc order remains exactly zero. For a point, use the zero divisor.

Corollary 5.1 · p. 12

1. Retain the nonzero Euler number at the source

Let $P$ be the convex hull of fiber characters of $A=-K_W$ on the fixed components. For a nonzero cocharacter $\lambda$, nefness and the attracting decomposition identify the source weight $\mu_+$ with the support function of $P$. Faithfulness and the natural tangent-determinant character give $\mu_+>0$, so $0$ lies in the interior of $P$.

Expand the fixed-point formula at infinity. The coefficient of the maximal cohomological weight $m\mu_+$ is $Q(m)=\chi(S_+,mA|_{S_+})$; a negative normal weight at every other component removes its contribution to this coefficient. The connected torus acts trivially on $H^j(W,\mathcal O_W)$, giving $Q(0)=\chi(W,\mathcal O_W)\ne0$. Riemann–Roch and hard Lefschetz with coefficient $(m+1)A$ produce eigenforms of one degree at unbounded exponents.

§10, (10.1)–(10.2) · pp. 25–27 [DPS01, Theorem 2.1.1; Corollary 2.1.2, p. 8]

2. The determinant-rank shift becomes a fixed arc order

Let $s$ be the rank of the generic span of these forms and $M$ its saturated determinant. The cotangent-tensor inclusion and nef anticanonical bundle make $-M$ pseudoeffective; for $p=0$ the underlying line is trivial. Completing each varying form with a fixed finite choice of basis forms gives

For the generic orbit arc, order is fiber weight minus section weight. Hence $v(N_i)=\mu_M+s\mu_+$ is constant. The rank shift $s$ is essential to this computation.

§10, (10.1)–(10.2) · pp. 25–27 [LMPTX23, Theorem 4.1, print p. 1409 / PDF p. 17] Lemmas 6.1–6.2 · p. 13

3. Multiply actual characters close to each vertex

Theorem 9.1 supplies sections of $rA$ with arbitrarily small $v(N)/r$. One eigencomponent has no greater order, and its normalized character approaches $\mu_+$ in the chosen direction. Nonnegative orders for all cocharacters put actual normalized characters in $P$. An exposing direction for each vertex then supplies characters approaching that vertex. Choose finitely many close enough that their convex hull still contains zero. Clear a rational convex combination and multiply the corresponding nonzero sections to obtain a naturally invariant section. For the trivial torus, approximate transfer already supplies the section.

The approximate-transfer input removes the fixed divisor from the arc order, then extends first jets after nef reduction to prove bigness. Its full analytic extension proof is not reproduced here.

Theorem 9.1 · p. 21 §9, (9.3)–(9.7) · pp. 22–24 Lemmas 6.1–6.2 · p. 13

1. The invariant field forces rank one at every positive exponent

Let $F$ consist of zero and invariant rational functions $u$ for which $bA-(u)_\infty$ is pseudoeffective for some $b>0$. Pole inequalities and triviality of finite orbits of a connected torus make $F$ relatively algebraically closed and finitely generated. Its base $Y$ has trivial action, and every ample base divisor $H$ satisfies a domination $bp^*A-f^*H$ pseudoeffective.

Three-index invariant ratios belong to $F$, forcing nonnegative horizontal orders of $\delta_s/\delta_r$. Subtract vertical minima on a normal equidimensional model to obtain a base line $B$ and $\tau\in H^0(ap^*A-f^*B)^T$, with order zero on a component above each base prime. Globalize invariant adjoint sections by a sufficiently ample base twist. Their ratios again have anticanonically controlled poles and belong to $F$, so the invariant adjoint rank is one for every $l\ge1$. Ordinary rank one is not assumed.

§11, (11.1)–(11.3) · pp. 27–29

2. A bounded metric, nef reduction, and an isolated pole

Lemma 8.2 uses smooth coefficient metrics and the orthogonal quotient given by compact averaging. The limit of high moments gives a bounded semipositive metric on $B$. The condition $q(X)=0$ implies $q(Y)=0$. In Lemma 8.3, flattening the nef reduction ensures numerical triviality on every contracted curve. Numerical pullback and removal of torsion using $\operatorname{Pic}^0=0$ then descend $B$ up to rational linear equivalence to a nef line $L$ on a smaller base.

If $\dim Z=b>0$, Lemma 8.4 concentrates curvature of $B_k=kL+\epsilon H$ at one point. For sufficiently large $m=ap_0$, decomposition (11.4) gives an invariant metric $h_J$ on $J=(m+1)A_S$ and a function $\Psi$ with neat analytic singularities satisfying

The ideal equality combines sufficiency from integrability of the remaining singular factor with necessity from nonintegrability of uncancelled integral divisorial poles.

Lemmas 8.2–8.4 · pp. 17–20 [Leh15, Theorem 1.2, p. 1] [EKL95, Theorem 3.1, p. 14] §11, (11.4)–(11.5) · pp. 29–30

3. Average a lift of a nonzero class and remove the exceptional factor

The Cao–Demailly–Matsumura theorem lifts global classes in $\mathcal I(h_J)/\mathcal I(h_Je^{-\Psi})$ with coefficient $K_S+J$. Near the chosen fiber, $s_E(\tau g^*e_L)^{p_0}$ belongs to the larger ideal and glues with zero outside the fiber. At a submersion point its coefficient is a unit, while the transverse pole exponent $3b/2\ge b$ forces nonintegrability. Its quotient class is therefore nonzero.

The base and frame actions are trivial, so this class is invariant. Compact averaging of a lift preserves the same nonzero class. The resulting invariant section of $mA_S+E$ descends after removing the exceptional Jacobian factor, and extends across codimension two on smooth $X$. This gives a nonzero invariant section of $mA$. A point base is handled directly by pushing down $\tau$.

[CDM17, Theorem 1.1, p. 2] §11, (11.4)–(11.5) · pp. 29–30

1. Pull back all but finitely many primes from one base

Let $P$ be the field of functions with poles supported on controlled primes. Rational connectedness gives $\operatorname{Pic}^0(V)=0$. Choose finitely many controlled prime classes spanning the rest. A principal relation involving any further prime has its function in $P$, making that prime vertical over a model of $P$. A flat intermediate model ensures that no prime of $V$ maps to codimension at least two on the base.

The rational pullback $p^\diamond=\pi_*f^*$ kills divisors exceptional for base modifications, including their divisor linearizations. Apart from finitely many exceptions, a controlled prime $D$ equals $p^\diamond Q$ for a base prime $Q$. Lemma 12.2 supplies full adjoint rank one at zero and an invariant effective correction $J$ satisfying $p^\diamond[J]^T=0$.

§12.1; Lemma 12.2 · pp. 31–33

2. Weighted approximation gives a disappearing correction

Mix an anticanonically dominated ample base metric with the smooth metric by a small coefficient. This produces $\mathcal I(h)=\mathcal O_U$ and a strict base-direction lower bound. The multiplier-ideal condition supplies the generic isomorphism required by Păun–Takayama. Rank-one fiber integration gives a strict metric on $-K_Y+J$, but the integrable object is $|s_J|^2e^{-\varphi}$.

Theorem 2.2 of Controlled boundary makes every coefficient of $\Psi=G/r-\sigma^*J-K_{Y'/Y}$ less than one and supports $D_0=\Psi_-$ on $\sigma^*J$ and the exceptional locus. Thus $p^\diamond[D_0]^T=0$. Pushing down the canonical invariant rational section shows that the character component also vanishes. The ordinary rational linear equivalence in the approximation theorem is not automatically treated as equivariant.

[PT14, Setup 3.2.1; Theorem 3.3.5, pp. 20, 24] [OpenAI26f, Theorem 2.2, pp. 6–7] §12.2, (12.5)–(12.9) · pp. 33–34

3. Turn invariant finite generation into a cone of classes

Choose a rational polytope around $[D_0]^T$ in $\operatorname{Pic}^T(Y')_{\mathbb R}$. A small exceptional adjustment supplies an ample reserve, and each underlying vertex class becomes adjoint to a common ample part and an SNC klt boundary. Adjoint finite generation on a common divisible lattice makes the multigraded ring $R$ finitely generated. The actual linearized line bundles define its torus action; general boundary representatives need not be invariant.

Split generators into eigencomponents. Exact projection to weight zero and Gordan's lemma make $R^T$ finitely generated by multihomogeneous elements $u_i$. For any invariant effective base divisor $R_0$, express $[D_0]^T+t[R_0]^T$ as a rational convex combination of the vertices for sufficiently small $t>0$. Clear denominators and torsion, express its canonical section in the $u_i$, and choose a nonzero monomial of the same multidegree. Only equality of degrees is needed, not equality of zero divisors with the original section.

Apply $p^\diamond[D_0]^T=0$. Then $p^\diamond[R_0]^T$ belongs to the cone of $p^\diamond[\operatorname{div}u_i]^T$. Adjoin the finitely many invariant controlled exceptions identified earlier to obtain the required rational polyhedral cone. Its generators are effective but need not be controlled.

[CL12, Theorem A, p. 2] §12.3, (12.10) · pp. 34–35