Arithmetic Period Map for Cubic Fourfolds

• Arithmetic period maps connect the geometry of cubic fourfolds with the arithmetic of Shimura varieties by encoding polarized Hodge structure variations.
• The framework utilizes lattice theory, level structures, and Heegner divisor analysis to classify special cubic fourfolds and explain modularity phenomena.
• This construction underpins Torelli theorems and facilitates explicit moduli compactifications via GIT, bridging algebraic geometry with automorphic forms.

The arithmetic period map for cubic fourfolds is a morphism that connects the geometry of cubic fourfolds to the arithmetic of orthogonal Shimura varieties, encoding the variation of polarized Hodge structures in a way that allows powerful interactions between algebraic geometry, automorphic forms, and arithmetic. This framework plays a central role in the classification and moduli theory of cubic fourfolds, informs the theory of special divisors (Noether–Lefschetz loci), and facilitates applications such as complex multiplication theory and modularity for transcendental motives.

1. Polarized Hodge Structures and the Period Domain

Let $X \subset \mathbb{P}^5$ be a smooth cubic fourfold. Its middle cohomology $H^4(X, \mathbb{Z})$ is equipped with the intersection pairing

$\phi_X(\alpha, \beta) = \int_X \alpha \cup \beta,$

forming an odd unimodular lattice of signature $(21,2)$ (Yu et al., 2018). The square of the hyperplane class $\eta_X = c_1(\mathcal{O}_X(1))^2$ defines the primitive cohomology lattice

$$H^4_0(X, \mathbb{Z}) = \eta_X^\perp \subset H^4(X, \mathbb{Z}),$$

which is even of discriminant 3 and signature $(20,2)$. Abstractly, $(H^4(X, \mathbb{Z}),\, \eta_X,\, \phi_X)$ can be identified with $(\Lambda,\, \eta,\, \phi)$, where $$\Lambda \simeq E_8^{\oplus 2} \oplus U^{\oplus 2} \oplus A_2(-1)$$ and $\eta^2 = 3$.

The Hodge decomposition on $H^4(X, \mathbb{C})$ is of weight 4 with

$h^{3,1} = h^{1,3} = 1, \quad h^{2,2} = 20.$

Thus, $H^{3,1}(X)$ is a line in the primitive cohomology $\Lambda_0 \otimes \mathbb{C}$ and is isotropic for $\phi$. The period domain is then defined as

$$\widetilde{D} = \{ [\omega] \in \mathbb{P}(\Lambda_0 \otimes \mathbb{C}) \mid \phi(\omega, \omega) = 0,\; \phi(\omega, \bar{\omega}) < 0 \},$$

which has two connected components; one is fixed and denoted $\widehat{D}$ (Yu et al., 2018, Looijenga, 2010, Li et al., 2012).

2. The Arithmetic Monodromy Group

The relevant arithmetic group is

$G = \mathrm{Aut}(\Lambda, \phi, \eta),$

the group of automorphisms preserving both the lattice and $\eta$. Its index-2 subgroup

$$\widehat{\Gamma} = \{ g \in G \mid g \text{ preserves the component } \widehat{D} \}$$

is an arithmetic subgroup of $O(\Lambda, \phi)$. Generators can be realized as reflections in roots of $\Lambda_0$ (i.e., $r$ with $\phi(r, r) = -2$) and elements stabilizing $\eta$, though a finite presentation remains elusive (Yu et al., 2018).

The monodromy group acts properly discontinuously on the period domain. It is generated by reflections associated to:

• Nodal vectors ($\phi(r,r) = -2$),
• "Special" vectors (e.g., for Heegner divisors or Noether–Lefschetz loci), leading to locally finite hyperplane arrangements in the period domain (Looijenga, 2010).

3. The Arithmetic Period Map and Torelli Theorems

Given a marking $\Phi: H^4(X, \mathbb{Z}) \to \Lambda$ with $\Phi(\eta_X) = \eta$, the local period map is

$$\widetilde{P}: \{ \text{marked cubic fourfolds} \} \to \widehat{D},\ \ (X, \Phi) \mapsto [\Phi(H^{3,1}(X))].$$

This map is infinitesimally injective (infinitesimal Torelli) and, upon passage to the moduli space $M$ of smooth cubic fourfolds, yields the arithmetic period map

$$P: M \to \widehat{\Gamma} \backslash \widehat{D},\ \ [X] \mapsto [\Phi(H^{3,1}(X))].$$

Voisin's global Torelli theorem proves that $P$ is an open embedding; Laza and Looijenga determine its image as the complement of two $\Gamma$-invariant hyperplane arrangements: $H_\Delta$ (nodal cubics) and $H_\infty$ (sextic degenerations). Explicitly,

$$P(M) = \widehat{\Gamma} \bigl( \widehat{D} \setminus H_{\Delta} \setminus H_{\infty} \bigr)$$

and on a larger moduli $M_1$ with ADE singularities,

$$P(M_1) = \widehat{\Gamma} \bigl( \widehat{D} \setminus H_{\infty} \bigr) [1806.04873].$$

4. Moduli, Level Structures, and Shimura Varieties

For integral and arithmetic refinements, a Deligne–Mumford stack $\widetilde{\mathcal{C}}^{[N]}$ of cubic fourfolds with level-$N$ structure is constructed, for $N \geq 1$ coprime to 2310 (to trivialize residual automorphisms on $H^4(X,\mathbb{Z}/N)$) (Ito, 12 Dec 2025). The stack parametrizes tuples $(X, \lambda, \alpha)$ where $\alpha$ is an isometry

$$\alpha:\ R^4\pi_{*}\mathbb{Z}/N\mathbb{Z}(2)\ \to (L_0 \otimes \mathbb{Z}/N\mathbb{Z})_S$$

taking $h^2$ to a fixed vector $v$.

The arithmetic period map

$$j_N : \widetilde{\mathcal{C}}^{[N]}_{\mathbb{C}} \to \mathrm{Sh}_{K_N}(L)_{\mathbb{C}}$$

maps to a (connected component of a) Shimura variety of type $SO(2,20)$, where $K_N$ is the principal level-$N$ arithmetic subgroup of $SO(L)$ fixing $v$ and acting trivially modulo $N$. The map is algebraic, étale, and descends to $\mathbb{Q}$ when $N$ is coprime to $2310$. This construction is directly analogous to the period map for polarized K3 surfaces with level structure (Ito, 12 Dec 2025). The image of $j_N$ is a locally closed subset, and the Torelli-type result implies injectivity.

The period map structure can be summarized as follows:

Structure	Description	Reference
Domain $D$	$$\{ [\omega] \mid Q(\omega, \omega)=0,\ Q(\omega, \bar{\omega}) > 0 \}$$ in $\mathbb{P}(V)$	(Yu et al., 2018)
Arithmetic group $\Gamma$	$O(\Lambda_0)$ or its stabilizer subgroup, acting on $D$	(Looijenga, 2010)
Arithmetic period map $P$	$$M \to \Gamma \backslash D, [X] \mapsto [H^{3,1}(X)]$$	(Ito, 12 Dec 2025)
Image locus	$$P(M) = \Gamma \backslash (D \setminus H_\Delta \setminus H_\infty)$$	(Yu et al., 2018)

5. Special Cubic Fourfolds, Heegner Divisors, and Modularity

A cubic fourfold is called special of discriminant $d$ if it contains an algebraic surface whose cohomology class with $h^2$ spans a rank-2 sublattice of discriminant $d$. The loci of such fourfolds correspond to irreducible divisors $C_d$ in the moduli, and are nonempty for $d \equiv 0,2 \pmod{6}$ and $d > 6$ (Li et al., 2012).

Arithmetic theory realizes these as Heegner divisors in the period domain: for each $d$, a Heegner divisor $D_d$ is cut out by vectors in the primitive lattice with fixed norm and class in $\Lambda_0^\vee/\Lambda_0$. Geometric properties correspond to these divisors:

• $d=8$ iff $X$ contains a plane,
• $d=14$ iff $X$ is Pfaffian, and so on (Li et al., 2012).

One can compute the degrees $N_d$ of these divisors via intersection theory. The generating series

$\Theta(q) = -2 + \sum_{d > 2} N_d q^{d/6}$

is a modular form of weight 11 and level 3, encoding enumerative invariants of special cubic fourfolds in its Fourier coefficients. Vector-valued extensions realize the relations in the Picard group of the moduli, and congruence and growth properties of $N_d$ are controlled by modular and automorphic theory (Li et al., 2012).

6. Arithmetic Compactification and Functoriality

Geometric Invariant Theory (GIT) yields compactifications of the moduli of cubic fourfolds. These compactifications admit explicit descriptions as (semi-)toroidal or Looijenga type: the GIT compactification $\overline M$ coincides with $\overline{\Gamma \backslash D}^{H_\infty}$, where one blows up the Baily–Borel boundary along the arrangement $H_\infty$ and contracts to obtain normal crossings (Yu et al., 2018). The functoriality of Looijenga compactifications extends to situations with additional symmetries (automorphism refinements), inheriting the modular and period-theoretic properties from the cubic case.

The construction is compatible with moduli of K3 surfaces and provides a conceptual framework for understanding compactifications, boundary components, and degenerations in relation to period maps.

7. Complex Multiplication, Field of Definition, and Modularity

A cubic fourfold $X/\mathbb{C}$ is of CM type if its Mumford–Tate group is abelian; equivalently, the endomorphism algebra of its transcendental motive is a CM field. The arithmetic period map realizes the following properties (Ito, 12 Dec 2025):

• The point $j_N(X,\alpha)$ is a special point on the Shimura variety, rational over the maximal abelian extension $E^{\mathrm{ab}}$ of the reflex field $E$ of the CM type.
• $X$ admits a model over $E^{\mathrm{ab}}$.
• For rank-21 ("singular") cubic fourfolds, modularity for the $l$-adic Galois representation attached to $T(X)$ follows: there exists a weight-3 CM newform $f$ such that $L(\rho_l, s) = L(f, s-1)$.

This close relationship between period maps, Shimura varieties, Galois theory, and automorphic forms is a hallmark of the arithmetic theory of cubic fourfolds (Ito, 12 Dec 2025, Li et al., 2012).

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Special cubic fourfolds thus serve as a geometric and arithmetic bridge between Hodge theory, lattice-theoretic methods, and modular/automorphic representation theory. The arithmetic period map provides the organizing structure, encompassing Torelli-type results, explicit modularity of enumerative invariants, and deep field-of-definition statements for motives of cubic fourfolds.