CM Line Bundles in Moduli Theory

• CM line bundles are algebro-geometric objects associated with families of polarized varieties that encode stability and intersection-theoretic data.
• They are constructed using determinants of cohomology and Deligne pairings, and their positivity properties ensure the ampleness and projectivity of moduli spaces.
• The degree of the CM bundle governs invariants such as the Donaldson–Futaki invariant, linking algebraic stability with differential and non-Archimedean geometric properties.

A CM (Chow–Mumford) line bundle is an algebro-geometric object naturally associated to a family of polarized varieties, encapsulating stability-theoretic and intersection-theoretic information. It arises in the context of moduli theory, particularly in the study of K-stability, moduli of canonically polarized varieties, and K-moduli of Fano or Calabi–Yau type varieties. Constructed via determinants of cohomology or Deligne pairings, the CM line bundle’s positivity properties (nefness, bigness, ampleness) have deep consequences for the projectivity and structure of moduli spaces, and its degree governs Donaldson–Futaki invariants in non-Archimedean Kähler geometry.

1. Construction and Definition

Let $f: X \to S$ be a flat, projective morphism with $n$-dimensional, $S_2$ geometrically connected fibers, and let $L$ be a relatively ample $\mathbb{Q}$-line bundle on $X/S$. The Knudsen–Mumford determinant expansion expresses

$$\det f_* \mathcal{O}_X(mL) = \bigotimes_{i=0}^{n+1} \lambda_i^{\otimes \binom{m}{i}},$$

where the Hilbert polynomial of the fiber $X_s$ is

$$\chi(X_s,\mathcal{O}_{X_s}(mL_s)) = a_0 m^n + a_1 m^{n-1} + O(m^{n-2}).$$

After setting $\mu = (2 a_1)/((n+1)! a_0^2)$, the CM line bundle is defined as

$$\lambda_{\mathrm{CM},f} := \lambda_{n+1}^{\otimes (\mu + 1/((n-1)! a_0))} \otimes \lambda_n^{\otimes (-2/(n! a_0))}.$$

Alternatively, in terms of intersection theory, when $S$ is a smooth curve,

$$\deg \lambda_{\mathrm{CM},f} = \mu L^{n+1} + \frac{1}{n! a_0} (K_{X/S}\cdot L^n).$$

For Q-Gorenstein families with klt or slc fibers and auxiliary boundary divisor $A$, the log version generated via Deligne pairings is

$$\lambda_{\mathrm{CM},T} = T_*\big((n+1) L^n \cdot (K_{X/S} + A) - n L^{n+1}\big) \in \mathrm{Pic}_\mathbb{Q}(S)$$

(Hattori, 26 Jan 2026, Hattori, 2023, Patakfalvi et al., 2015, Grieve, 22 Sep 2025).

2. Positivity Properties and Main Theorems

The last decade has clarified that the positivity properties of the CM bundle directly control the ampleness and projectivity of moduli spaces. For KSBA (Kollár–Shepherd-Barron–Alexeev) stable families, the CM line bundle $\lambda_{\mathrm{CM}}$ is ample on the proper KSBA moduli space $M^{\rm ksba}$, and hence $M^{\rm ksba}$ is projective (Patakfalvi et al., 2015). For families of klt good minimal models with Kodaira dimension $\kappa=1$, the seminormalization of the K-moduli is quasi-projective, and the CM line bundle pulled back to the normalization becomes ample after a suitable base change (Hattori, 26 Jan 2026).

A representative result for specially K-stable, maximal variation families states:

Given $(X, A, L) \to S$ a projective flat $\mathbb{Q}$-Gorenstein family with klt, specially K-stable fibers and maximal variation, the (log-)CM line bundle $\lambda_{\mathrm{CM},T}$ is ample on $S$ (Hattori, 2023).

Tables below summarize the main positivity results:

Setting	Base Object	CM Bundle Property
KSBA moduli, canonically polarized (Patakfalvi et al., 2015)	$M^{\rm ksba}$	Ample
K-moduli with $\kappa=1$, klt good minimal models (Hattori, 26 Jan 2026)	Normalization $M^\nu$	Ample
Special K-stability, maximal variation (Hattori, 2023)	Projective $S$	Ample

In all cases, ampleness is deduced from variations of the Nakai–Moishezon criterion and the analysis of the CM bundle’s intersection-theoretic degrees.

3. Moduli Spaces and Applications

The construction and ampleness of the CM line bundle enable the projectivity of moduli spaces in various birational settings.

• For canonically polarized varieties, the coarse moduli space $M^{\rm ksba}$ is projective (Patakfalvi et al., 2015).
• For moduli of uniformly adiabatically K-stable klt-trivial fibrations, projectivity of any proper subspace follows via the positivity of the CM bundle (Hattori, 2023).
• In the context of log Calabi–Yau fibrations or moduli with $\kappa=1$, quasi-projectivity and eventual ampleness after base change and normalization are established (Hattori, 26 Jan 2026).
• For K-semistable quasimaps or ε-stable quotients, projectivity and the ampleness of the associated CM bundle on their coarse moduli are also proven (Hattori, 26 Jan 2026).

This ampleness underlies the general principle that the CM bundle is a canonical polarization on moduli spaces of K-stable or KSBA-stable objects.

4. Intersection-Theoretic Formulas and Slope Theory

For big and nef line bundles (including the ample case), the CM bundle can be expressed explicitly in terms of intersection numbers: $$\lambda_{\rm CM}(X/B, L) = \frac{1}{a_0^2}\left(a_0\,\langle K_{X/B}, L^n\rangle - a_1\,\langle L^{n+1}\rangle\right)$$ with $a_0 = L^n / n!$ and $a_1 = -K_X \cdot L^{n-1}/2(n-1)!$ (Grieve, 22 Sep 2025).

The Donaldson–Futaki invariant of any test configuration is recovered as the $G_m$-weight of the CM line bundle along the central fiber. The first variation of the CM bundle for deformation to the normal cone yields the slope theory of Ross–Thomas, now generalized to big and nef line bundles. The continuity of the CM bundle under approximations by ample classes is essential for applications to non-Archimedean K-stability and the analysis of degenerations.

The following explicit formula connects the slope invariants: $$\mathrm{DF}(X, L; Z) \propto \mu(X; L) - \mu_c(L; I_Z)$$ where $Z \subset X$ is a proper subscheme and $L$ is big and nef (Grieve, 22 Sep 2025).

5. Ampleness After Normalization, Numerical Equivalence, and Quasimaps

The case $\kappa=1$ introduces unique challenges: moduli spaces are typically non-proper, and maps to K-moduli of quasimaps lack quasi-finiteness. The remedy involves introducing a third moduli space classifying numerical equivalence (rather than linear equivalence) classes, bridging the original moduli to the proper K-moduli of quasimaps (Hattori, 26 Jan 2026). The central diagram is

$$\begin{array}{ccc} M^{\mathrm{sn}} & \xrightarrow{\beta} & M_{\mathrm{qmaps}} \ \downarrow \xi & & \uparrow \alpha \ N^{\mathrm{sn}} & \xrightarrow{\alpha} & M_{\mathrm{qmaps}} \end{array}$$

Here, $\xi$ is projective, $\alpha$ is quasi-finite. Pulling back the ample CM bundle from $M_{\mathrm{qmaps}}$ via this diagram recovers ampleness on $M^\nu$ and quasi-projectivity on $M^{\mathrm{sn}}$ (Hattori, 26 Jan 2026).

6. Relation to Differential and Non-Archimedean Geometry

For KSBA-stable (or K-polystable) varieties, the CM bundle is the algebro-geometric manifestation of the Weil–Petersson metric, connecting the Yau–Tian–Donaldson conjecture (existence of Kähler–Einstein metrics) to the projectivity of the moduli spaces (Patakfalvi et al., 2015). The ampleness of the CM line is both an algebro-geometric and a differential-geometric phenomenon, encoding the positivity of the Mabuchi functional's Hessian and determining the K-stability of fibers.

Generalizations to non-proper or non-ample settings (big and nef, or log settings) are crucial for connecting higher-dimensional moduli with recent advances in non-Archimedean geometry and stability theory (Grieve, 22 Sep 2025). The continuity property of the CM bundle across the ample cone is fundamental for these developments.

7. Contemporary Significance and Open Questions

The projectivity and positivity results for CM line bundles have completed several major conjectures, such as Odaka’s prediction for positivity in families of maximal variation with special K-stable fibers (Hattori, 2023). These results unify approaches to K-stability for Fano, Calabi–Yau, and more singular (klt/slc) situations.

Open directions include the pursuit of purely algebraic proofs of CM-positivity in the Fano/K-polystable cases (currently reliant on analytic methods) and further extensions to settings with only semipositivity, as well as the study of the CM line bundle in degenerating and non-reduced families.

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Key Citations:

• (Hattori, 26 Jan 2026) On positivity of CM line bundles on the moduli space of klt good minimal models with $\kappa=1$
• (Hattori, 2023) Special K-stability and positivity of CM line bundles
• (Grieve, 22 Sep 2025) CM-line bundles and slope $\K$-semistability for big and nef line bundles along subschemes
• (Patakfalvi et al., 2015) Ampleness of the CM line bundle on the moduli space of canonically polarized varieties