Relative denominators and effective systems for log Calabi-Yau fibrations

1. Descend in the same degree and fix the exact presentation

The geometric generic fibre has dimension at most three. The low-dimensional slc index theorem principalizes its adjoint in a uniform degree. Ratios of conjugate principalizing functions are constants, so Hilbert 90 descends the principalization to the fibre over $\mathbb C(Z)$ in the same degree $p_0$. Comparison with the original pullback relation produces the exact equality (1.1), since a function with vertical divisor is constant on the generic fibre. The effective lc boundary gives the rank-one condition for an lc-trivial fibration. The qualitative canonical bundle formula then descends $\mathbf M$ to a nef divisor on a smooth model $W$ (§3, pp. 6–7).

§3 · pp. 6–7 · JL, Corollary 1.6 · FG, Theorem 3.6

2. Slice to detect the denominator and remove the error

For $P\subset W$, put $\alpha=\operatorname{coeff}_P D_W$ and let $t_P$ be the threshold for the crepant sub-pair. The coefficient of $M_W$ differs from $\alpha+t_P$ by an integer. It suffices to show $p(\alpha+t_P)\in\mathbb Z$. A transverse curve slice retains $\alpha$ by specifying canonical representatives through residues. After a relative dlt MMP, the error $E_N$ is vertical by generic-fibre negativity, and nefness plus fibre connectedness makes it proportional to the fibre. The original threshold-attaining component has zero error coefficient, forcing that proportionality constant to vanish. Locally one obtains

Pseudo-effectivity is used in the four-dimensional termination input (§4, pp. 7–10).

§4, (4.1)–(4.6) · pp. 7–10 · CT, Theorems 1.1–1.2

3. Obtain a form on the entire special fibre

Lemma 5.1 (pp. 10–15) clears the denominator in this identity. A depth theorem and conductor adjunction make the reduced fibre $T$ slc. Global ACC puts the different coefficients in a finite set. The JL index theorem supplies a nowhere-vanishing form $u$ of uniform even degree $p$ on the log pluricanonical line of all of $T$.

Lemma 5.1, (5.1)–(5.4) · pp. 10–13 · JL, Corollary 1.6

4. Make each residue ratio constant on the cyclic cover

Write $p_0\beta=a/m$ in lowest terms and normalize the base change $w^m=z$, obtaining $\pi:Y\to N$. Coefficient comparison makes every fibre multiplicity divisible by $m$, so $\pi$ is étale in codimension one. The form $s=(\pi^*\theta)^{\otimes p_0}\pi^*\psi_Cw^{-a}$ carries character $\zeta^{-a}$. On each normalized component, compare $\operatorname{res}(s^{p/p_0})$ with $\pi^*u$. Adjunction in a sufficiently divisible auxiliary degree shows their ratio is constant. This auxiliary degree may depend on the pair.

Lemma 5.1, (5.9)–(5.11) · p. 14

5. Match constants at double crossings and kill the character

Back in the uniform degree $p$, the next residues at a double crossing agree because $p$ is even, forcing the constants to agree across components. Connectedness handles the group action even when it permutes components. Thus $\zeta^{-ap/p_0}=1$, giving $m\mid p/p_0$ and $p\beta\in\mathbb Z$.

Lemma 5.1, (5.12), conclusion · p. 15

1. Transfer the complete section space to the base

Set $L=K_X+B$ and assume $p_0\mid l$. The exact presentation yields an equality inside $\mathbb C(X)$:

For the reverse inclusion, divide a source section by $\psi^{l/p_0}$. It is regular on the generic fibre and hence belongs to $\mathbb C(Z)$. Effectivity descends by testing a pullback component of positive multiplicity dominating each prime of the base. This valuation argument applies to reflexive sheaves ((6.1)–(6.2), p. 16).

(6.1)–(6.2) · p. 16

2. Construct birational systems on a base model with effective boundary

On a resolution $q:W\to Z$, the potentially negative exceptional coefficients of the crepant boundary are replaced as follows. Let $A$ be the strict transform of $B_Z$ plus the reduced exceptional divisor. Then

The left side is big, the boundary coefficients lie in a fixed DCC set, and $pM_W$ is nef Cartier. Apply [BZ, Theorem 1.3].

(6.3) · p. 16 · BZ, Theorem 1.3

3. Recover the embedded base field from all section ratios

Push its birational sections to $Z$, and transfer them to $X$. The common factor $\psi^{l/p_0}$ cancels in ratios, which generate the entire base field. Taking $m=\operatorname{lcm}(p_0,b_1,\ldots,b_d)$ works for every positive multiple (pp. 16–17).

Proposition 6.1, conclusion · pp. 16–17

1. Use extraction and ACC to reach boundedness

Construct ordinary klt boundaries to obtain small $\mathbb Q$-factorializations and extractions of specified valuations for the generalized pair. Applying global ACC also after extraction gives a finite boundary coefficient set and a uniform positive lower bound on generalized log discrepancies. A numerically trivial nef part is treated separately by ordinary global ACC. [Bir, Theorem 1.7] then bounds the bases up to isomorphism in codimension one (pp. 17–19).

Proposition 7.1 · pp. 17–19 · Bir, Theorem 1.7

2. Bound the torsion exponent through bounded topology

For $U=Z_{\mathrm{reg}}$, the group $H_1(U(\mathbb C),\mathbb Z)$ is unchanged upon removing codimension-two subsets. Semialgebraic triviality in a bounded family gives only finitely many such groups. A rationally connected resolution $Y$ has $\operatorname{Pic}^0(Y)=0$, so $\operatorname{Cl}(Z)$ is finitely generated. Kummer theory gives

For fixed $Z$, the orders on the left stay bounded as $n$ varies. Hence $H_1$ has no free summand. The finite list of finite groups supplies a uniform torsion exponent $T$.

Proposition 7.1, (7.1) · pp. 20–21

3. Clear actual coefficients before principalizing

Separately, the finite boundary coefficient set and integrality of $pM_Z$ give an integral actual divisor $qD$. Its class is torsion, so $\ell=Tq$ makes it principal (pp. 20–21).

Proposition 7.1, conclusion · p. 21

4. Pull the principal divisor back to the total space

Corollary 7.2 applies Theorem 1.1 with $D=0$ and pulls back the resulting principalization, choosing $r$ divisible by $\ell,p$:

Corollary 7.2, (7.2) · p. 21