CATALOGUE 034 · PAPER 02 / 14 Overview
Uniform indices for semi-log-canonical log Calabi–Yau pairs
Descend normalization trivializations by controlling residue cycles
A common index on normalization components does not by itself glue across the conductor. The argument bounds klt characters using a Hodge rank, then kills residue cycles with a uniform exponent independent of the number of components.
Read the manuscript on GitHub1. Main results
Theorem 1.1 — Cartier trivialization independent of component count
Fix $d\geq4$ and a finite set $\Phi\subset[0,1]\cap\mathbb Q$. Theorem 1.1, p.2 asserts that an integer $a=a(d,\Phi)>0$ works for every connected, projective, equidimensional slc pair $(X,B)$ of dimension $d$ over any algebraically closed field of characteristic zero satisfying $$ K_X+B\sim_{\mathbb Q}0,\qquad \{\text{nonzero coefficients of }B\}\subset\Phi: $$ the divisor $a(K_X+B)$ is Cartier and $$ \mathcal O_X\!\left(a(K_X+B)\right)\simeq\mathcal O_X. $$ The theorem's stated range $d\geq4$ is distinct from the consequence obtained by adding established lower-dimensional results. The exponent is independent of the numbers of normalization components and lc strata.
Theorem 5.1 — Top-form Hodge rank (input to the character bound)
The character bound uses a particular Hodge rank, not all Betti numbers. For a smooth projective $n$-fold $U$, let $T(U)\subset H^n(U,\mathbb Q)$ be the smallest rational Hodge substructure containing $H^{n,0}(U)$, and let $r(U)=\dim_{\mathbb Q}T(U)$. In Theorem 5.1, p.16, $n\geq2$ and $T$ is a complex projective $\mathbb Q$-factorial terminal $n$-fold with $K_T\sim0$. A smooth resolution $U$ satisfies $h^1(U,\mathcal O_U)=0$, and every rational image $Y$ with $0<\dim Y<n$ has rationally connected smooth projective resolution. The asserted bound is $r(T):=r(U)\leq R_n$.
Theorem 6.1 — A uniform klt pluricanonical character exponent
For $n\geq0,m\geq1$, Theorem 6.1, p.29 asserts an integer $L(n,m)>0$ for integral complex projective klt pairs $(V,\Delta)$ of dimension $n$, with $\Delta\geq0$ and $m(K_V+\Delta)$ an integral principal divisor. If a rational $m$-canonical form satisfies $\operatorname{div}(\theta)=-m\Delta$, then every $B$-birational self-map satisfies $$ f^*\theta=c\theta\quad\Longrightarrow\quad c^{L(n,m)}=1. $$ The map $f$ need not have finite order.
References on the arrows
Unprefixed theorem, lemma, section and equation numbers refer to this manuscript. The principal external inputs use the following keys.
- UI Uniform Pluricanonical Iitaka Fibrations · October 3, 2026 · PDF.
- LA Log abundance in characteristic zero · September 24, 2026 · PDF.
- ULI Uniform log Iitaka fibrations and bounded moduli denominators · October 4, 2026 · PDF.
- HJ Han–Jiang · Theorem 1.2 · arXiv v2.
- JL Jiang–Liu · Theorem 3.2 · arXiv.
- K Kollár, Sources of log canonical centers · arXiv v3.
2. Proof of Theorem 1.1
Proof route. The Hodge-rank bound in Theorem 5.1, together with the factorwise arguments, gives the character exponent in Theorem 6.1. That exponent kills the residue cycles here; normal lc indices and conductor descent are additional inputs to Theorem 1.1.
First trivialize the normalization components over the complex numbers. The remaining tasks are to kill all conductor residue ratios simultaneously and show that the descended form is an actual Cartier generator.
1. Trivialize normalization components in a common even degree
For the normalization $\nu:\coprod X_i\to X$, write $K_{X_i}+\Delta_i=\nu^*(K_X+B)|_{X_i}$. The conductor enters $\Delta_i$ with coefficient one. The normal lc index theorem, UI Theorem 1.2, p.2, supplies an even common integer $m$, depending only on $d,\Phi$, and rational $m$-canonical forms with $\operatorname{div}(\theta_i)=-m\Delta_i$. This trivializes the normalization components; descent remains to be proved.
2. Use the existing nonuniform trivialization to make ratios constant
Form the finite graph whose vertices are normalization components and whose edges are generic conductor nodes. For an edge $e:i\to j$, the branch correspondence $\tau$ compares the residues by $r_e=\tau^*\theta_T/\theta_S$. The original rational linear triviality supplies a possibly nonuniform principal Cartier degree $M$, divisible by $m$. Comparing its trivialization with the $\theta_i$ gives $$ r_e^{M/m}=b_i/b_j, $$ which proves that $r_e$ is constant. This step does not bound $M$. §7.3, pp.37–38
3. Transport residue ratios to minimal klt strata
Uniformity comes from transporting these constants to minimal klt strata. Within one dlt component, the $\mathbb P^1$-links of Kollár Theorem 10 and Proposition 14, pp.6,8–9 preserve residues: evenness removes the sign. Across the conductor, Lemma 7.2 constructs a $B$-birational map of minimal strata carrying the same multiplier $r_e$. It does not simply restrict an arbitrary birational map to a stratum. On a common SNC model, it compares residue fields and proves that the relevant local monomial matrix has determinant $\pm1$. Lemmas 7.1–7.2, pp.35–37
4. Kill every cycle character with one exponent
A closed walk therefore gives a $B$-birational self-map of one minimal klt stratum with character $\prod_e r_e$. Theorem 6.1 below gives $$ L=\operatorname{lcm}_{0\leq q\leq d-1}L(q,m),\qquad a=mL, $$ so every closed walk satisfies $\prod_e r_e^L=1$. Constants chosen along a spanning tree then satisfy $c_i=c_jr_e^L$ on every edge, and the forms $c_i\theta_i^L$ glue. Loops and multiple edges are included. No factor equal to the length of a cycle enters the exponent. Equations (7.10)–(7.11), p.38
The uniform character exponent is constructed in the proof of Theorem 6.1. It allows the following node-gluing step without introducing the cycle length into the exponent.
5. Match generators at nodes and extend by S2
One must obtain a Cartier generator, not merely a descended section. At the split node $F[[x,y]]/(xy)$, the canonical generator has branches $(dx/x,-dy/y)$; matching even-degree residues matches generators. Nonsplit nodes are checked after a quadratic extension and descended. The $S_2$ extension of these codimension-one generators identifies the degree-$a$ reflexive extension itself with $\mathcal O_X$. Finally, descent to a finitely generated field and faithfully flat base change transfer the conclusion from $\mathbb C$ to any algebraically closed characteristic-zero field. §§7.2–7.3, pp.37–39
3. Proof of Theorem 5.1
Assume $r(T)$ is unbounded. Organize the small-order covering families from the diagonal estimate by a chain quotient, then contradict volume asymptotics with an upper bound for sections.
1. Turn large rank into covering families with small orders
Lemma 4.1 bounds $r(T)$ from a Seshadri lower bound for a normalized ample class on a canonical model with $K\sim0$. Its diagonal argument uses the quantitative Hodge norm gap of Lemma 3.2. Unbounded rank would force small local positivity. Tracking, generic chains, and the small-volume Iitaka-fibre estimate of UI Lemmas 6.2–6.4, pp.25–26 then produce covering families of $\kappa=0$ subvarieties with small vanishing order for their complete section systems. Lemma 5.5, pp.18–20
2. Realize the chain field as a morphism
The chain field is realized as a morphism on a small terminal model. Applying LA Theorem 11.1, p.73 to $(T,\epsilon J)$ for a movable $J$ supplies termination and a semiample endpoint. This is an actual use of LA abundance. Lemma 5.6, pp.20–21
3. Calibrate volume and degree over a maximal base
The proof maximizes the dimension of a proper chain quotient and bounds its base using the rational-image hypothesis and UI's moduli-denominator and boundedness inputs. It then removes divisors mapping into codimension at least two in the base. Lemma 5.7 calibrates a big divisor $D$ and a linear functional $\ell_X$. With $v=\operatorname{vol}(D)^{1/n}$, $b=\dim Y$, and $s=n-b$, the scale is $$ c\delta\leq v\leq C\delta,\qquad \ell_X(D)\leq C\delta. $$ pp.21–25
4. Compare vertical jets and horizontal sections with volume
Comparison with a base hyperplane makes the small-order families vertical; maximality makes their chain leaves fill the generic fibre. Vertical jets bound the generic rank by $C(m\eta v)^s$. Determinants and cotangent slopes on the bounded base give $$ h^0(X,mD)\leq C(m\eta v)^s(m\delta)^b. $$ For sufficiently small $\eta$, this contradicts the volume leading term $m^nv^n/n!$. The section estimate on $\mathbb P^b$ in Lemma 5.8 must be linear in the rank for this comparison. §5.5, pp.25–28
4. Proof of Theorem 6.1
Estimate the multiplier on a volume form rather than the order of the self-map. Control the exponent lost in lifting by dimension alone, then bound the factor characters using their respective inputs.
1. Lift to a product cover and fix its factors
After crepant terminalization, ULI Proposition 5.1 and Lemmas 5.2–5.3, pp.12–13 provide a quasi-étale product cover. It has a rationally connected factor carrying all the boundary, an abelian factor, and boundary-free Calabi–Yau or symplectic factors. Lemma 6.4 uses a characteristic subgroup of the smooth-locus fundamental group to lift the map, and its $n!$-th power fixes the factor directions. The cover degree need not be bounded. For factor multipliers $d_i$ and form degrees $p_i$, the comparison is $$ c^{n!}=\prod_i d_i^{\,m/p_i}; $$ the cover degree does not enter this exponent. pp.30–32
Lemma 6.4 · pp. 30–32 · ULI, Proposition 5.1; Lemmas 5.2–5.3 · pp. 12–13
2. Bound the RC character in a bounded family
For the RC factor, fixed-index boundedness from Han–Jiang Theorem 1.2, pp.1–2 leads to the bounded-family representation theorem Jiang–Liu Theorem 3.2, p.8.
Lemma 6.5 · pp. 32–33 · HJ, Theorem 1.2 · pp. 1–2 · JL, Theorem 3.2 · p. 8
3. Bound the remaining character orders by Hodge rank
For the nonabelian boundary-free factors, UI Lemma 5.2, pp.21–22 supplies the rational-image property needed in Theorem 5.1. Abelian factors have the cohomology rank bound $\binom{2r}{r}$. The action on an integral polarized Hodge structure constrains a character of order $e$ by $\varphi(e)\leq b(r)$; take the least common multiple of the possible orders. Combining this with the RC bound yields $L(n,m)=n!E(n,m)$. Lemmas 6.5–6.6, pp.32–34
Lemma 6.6; Theorem 6.1 proof · pp. 33–34
The uniform rank bound comes from the proof of Theorem 5.1. Return the resulting character exponent to the cycle step of Theorem 1.1.
5. Which papers supply which steps
| Input | Role here | Check performed |
|---|---|---|
| UI Thm.1.2, p.2 | Thm.2.2 and §7.3: common normal lc index | Input statement and application compared |
| UI Thm.1.1, Props.4.1,4.5 | Prop.2.4: bounded chain-quotient bases | Target's input description read; proofs outside scope |
| UI Lems.6.2–6.4, pp.25–26 | Inputs 5.2–5.4 and Lemma 5.5: chains and small volumes | Input statements and uses compared |
| LA Thm.11.1, p.73 | Thm.2.3, Lemma 5.6 and §5.4: good models | Statement and rational-boundary applications compared; proof outside scope |
| ULI Prop.5.1, Lems.5.2–5.3, pp.12–13 | Input 6.3 and Lemma 6.4: product structure stable under further covers | Original statements and uses compared |
| HJ Thm.1.2 and JL Thm.3.2 | Lemma 6.5: RC character bound | Original statements compared; promotion from bounded varieties to log bounded pairs read only here |
| UI Lem.5.2, pp.21–22 | Lemma 6.6: applying Thm.5.1 to nonabelian factors | Input and application compared; also an indirect dependence on LA nonvanishing |
| K Thm.10, Prop.14 | Lemma 7.1: residue-preserving maps within a component | Original statements and even-degree sign calculation compared |
U4 is also cited in Inputs 5.2–5.4 as the source of geometric lemmas used through UI. Its original lemmas were not rechecked for this draft. A bibliography entry alone is not a dependency edge.
6. Return to the sources
Theorem 5.1; §5 · pp. 16–28 · Theorem 6.1; §6 · pp. 29–34 · §7 · pp. 35–39
Editorial reading covered the main statements, §§2–3, the chain quotient, calibration statement and vertical–horizontal jet comparison in §5, factor characters in §6, and residues, cycles, nodes and $S_2$ descent in §7. It did not check all diagonal calculations in Lemma 4.1, every calibration estimate in Lemma 5.7, or independently validate the fundamental-group lift and local valuation argument. Original slope inputs such as Campana–Păun, the underlying singular Beauville–Bogomolov literature, and the proofs of the UI and LA input theorems remain unchecked. This draft traces citations and applications; it does not certify the theorem.
The revision rechecked the statements and descent in §7, and organized proof introductions, stages and citations. The recorded scope for external inputs is retained from the initial draft. Source: October 5, 2026 version, PDF pagination, snapshot adc7f1241b42e322a6451854ab7e4b4c146bf78a.