Moduli Space: Numerical Equivalence Classes

• Moduli space of numerical equivalence classes is a framework that parametrizes algebraic structures by identifying objects with identical intersection-theoretic data.
• It utilizes numerical invariants such as Euler pairings and Chern characters to construct projective moduli spaces via quotient stacks and ample determinant line bundles.
• This approach refines classical moduli constructions through stratifications and intersection theory, classifying objects up to numerical rather than linear equivalence.

The moduli space of numerical equivalence classes provides a universal framework for parametrizing algebraic or geometric structures, such as vector bundles, sheaves, polarized varieties, or subvarieties, where objects are identified if they are indistinguishable under intersection-theoretic data. This concept appears in several contexts: as a quotient of the Grothendieck group by numerical equivalence, as loci in moduli of abelian varieties cut out by intersection conditions, and as moduli stacks for polarized models or log Calabi–Yau varieties modulo numerical data. The structure and projectivity of such moduli spaces are crucial for understanding coarse classification, ampleness of determinant line bundles, and the geometry of stratifications in moduli theory.

1. Numerical Equivalence and Associated Functors

Numerical equivalence is formally defined in several contexts. On polarized surfaces, the Grothendieck group $K(X)$ of coherent sheaves is considered modulo the kernel of the Euler pairing $\chi(E,F) = \sum_i (-1)^i \dim \mathrm{Ext}^i(E,F)$, yielding the numerical Grothendieck group $K_{\mathrm{num}}(X)$, which is a finite rank lattice. Numerical equivalence identifies objects whose Chern character or cycle class cannot be distinguished by integration or intersection numbers against fixed cycles or divisors.

For families of polarized varieties or fibrations, the equivalence relation is given by identifying tuples $(X,A)$ and $(Y,B)$ if there exists an isomorphism $\varphi:X\to Y$ such that $\varphi^*B-A$ is numerically trivial, i.e., it has vanishing intersection with all curves on $X$. This is formalized using functors such as $W\operatorname{Pic}^{\mathbb Q}$, the stack of $\mathbb Q$-Cartier, rank-1, reflexive sheaves modulo numerically trivial line bundles (Hattori, 26 Jan 2026).

On abelian varieties, numerical equivalence classes often correspond to symmetric endomorphisms under the principal polarization and are identified in the Néron–Severi group $\mathrm{NS}(A)$ of an abelian variety $A$ (Auffarth, 2015).

2. Construction of Moduli Spaces for Numerical Equivalence

The construction of moduli spaces of numerical equivalence classes is realized by passing from moduli functors that parametrize structures with fixed linear (Cartier, line bundle) data to their stacks or algebraic spaces where only numerical equivalence is preserved. In the context of klt good minimal models with Kodaira dimension one, the stack $N^{W,I}$ is built by forming the fiber product

$$N^{W,I}_{d,v,u,r,w} = \mathcal{M}_{d,v,u,r,w} \times_{\sigma,\operatorname{Pic}} W\operatorname{Pic}^{\mathbb Q}$$

with the universal family $(\mathcal U,\mathcal B)\to \mathcal M$ and quotienting by the Abelian sub-group-scheme $\operatorname{Pic}^0$ of numerically trivial sheaves. This yields a separated Deligne–Mumford stack of finite type whose coarse moduli space parametrizes numerical equivalence classes and, after seminormalization and normalization, yields a quasi-projective or projective scheme with ample CM line bundle (Hattori, 26 Jan 2026).

For vector bundles or sheaves on surfaces, the moduli stack $\mathcal{M}^\sigma(v)$ of $\sigma$-semistable objects of fixed numerical class $v\in K_{\mathrm{num}}(X)$ descends to a good moduli space $M^\sigma(v)$, identifying S-equivalence classes of semistable objects, which can be constructed projectively by using determinant line bundles (Tajakka, 2020).

In the context of abelian varieties, the locus in the Siegel modular variety $\mathcal{A}_n$ corresponding to principally polarized abelian varieties containing a subvariety of a given numerical class is cut out by explicit intersection-theoretic conditions in the period domain, yielding an irreducible closed subvariety $\mathcal{A}_n(\eta)$ (Auffarth, 2015).

3. Characterization and Classification via Intersection Theory

The numerical equivalence class of a subvariety or line bundle is typically characterized by intersection numbers. For abelian subvarieties $X \subset A$ of dimension $u$ and exponent $d$, the unique numerical class $\delta_X \in \mathrm{NS}(A)$ is determined by

$$(\delta_X^r \cdot \mathcal{L}^{n-r}) = \begin{cases} (n-r)! \, r! \, \binom{u}{r} \, d^{r} & 1 \leq r \leq u \ 0 & u+1 \leq r \leq n \end{cases}$$

where $\mathcal L$ is a principal polarization (Auffarth, 2015). This characterization provides a bijection between abelian subvarieties and primitive numerical classes satisfying these identities.

On smooth projective surfaces, $K_{\mathrm{num}}(X)$ is equipped with the Mukai pairing

$$(u,v)_M := -\chi(u^\vee\otimes v) = -\int_X \operatorname{ch}(u^\vee) \cdot \operatorname{ch}(v) \cdot \operatorname{td}_X$$

ensuring moduli spaces classify objects numerically indistinguishable under this pairing (Tajakka, 2020).

In the context of polarized klt fibrations, numerical classes are effectively the data retained after quotienting by the subgroup of numerically trivial classes, allowing moduli spaces to classify varieties up to this relation rather than up to linear equivalence (Hattori, 26 Jan 2026).

4. Properties and Structure of the Moduli Spaces

The moduli spaces of numerical equivalence classes inherit natural ampleness properties and stratifications from their construction:

• On polarized surfaces, the determinant line bundle $\lambda_\mathcal Q(w_Z)$ associated to the universal complex is ample on $M^\sigma(v)$, reflecting separation of numerical classes in the moduli space; this ampleness is key for projectivity (Tajakka, 2020).
• In the context of minimal models, the CM line bundle becomes ample on the normalized moduli space $N^\nu$, which is projective over the moduli of $\varepsilon$-stable quotients (Hattori, 26 Jan 2026).
• For abelian varieties, the strata in the Siegel modular variety corresponding to fixed numerical class have codimension $u(n-u)$, and are cut out by explicit linear equations in the Siegel upper half-space (Auffarth, 2015).

The moduli-theoretic consequences include the existence of well-behaved coarse moduli spaces parametrizing objects up to numerical equivalence and projective or quasi-projective structure, which is essential for intersection theory and positivity properties of ample determinants.

5. Relationship to Classical Moduli and Compactifications

Moduli spaces defined up to numerical equivalence refine or generalize classical constructions:

• On surfaces, the moduli space $M^\sigma(v)$, for $\sigma$ on the vertical wall in Bridgeland stability, is shown to be projective and to coincide with the Uhlenbeck compactification $M^{\mathrm{Uhl}}(v)$, thus linking semistable sheaf moduli and derived category stability to numerical data (Tajakka, 2020).
• In the moduli of abelian varieties, the strata defined by numerical equivalence recover classical loci of non-simple ppavs, such as Humbert surfaces and higher codimension analogues, described by explicit period and intersection conditions (Auffarth, 2015).
• The extension of Viehweg's construction to the klt and $\mathbb Q$-Cartier case allows projective moduli for good minimal models, repairing the quasi-finiteness issues in prior approaches based only on linear equivalence (Hattori, 26 Jan 2026).

The identification of moduli spaces via numerical equivalence supports the existence of natural bijections between different compactification schemes (e.g., Uhlenbeck, Bridgeland, and others).

6. Examples, Applications, and Further Context

Explicit examples include:

• Humbert surfaces in genus 2: loci of ppavs containing an elliptic curve of a fixed exponent $m$, corresponding to unique numerical class $\eta_m$, cut out by linear equations in Siegel space (Auffarth, 2015).
• Higher genus: the loci in $\mathcal{A}_n$ defined by $\mathcal{A}_n(\eta_{m,n})$ parametrize ppavs with specified elliptic subcurves.
• In the K-moduli setting, families of polarized varieties that are constant as maps to classical moduli spaces but exhibit strictly positive CM-degree highlight the necessity of constructing moduli up to finer equivalences (Hattori, 26 Jan 2026).

A plausible implication is that, in each context, the moduli space of numerical equivalence classes not only facilitates the existence of coarse moduli with desirable geometric properties (such as projectivity and ampleness), but also serves as a universal recipient for invariants and determinant bundles. This underpins the effective construction of "saturation" or "refinement" covers over classical moduli stacks, as needed in moduli problems with numerical equivalence as the fundamental invariant.