On the distribution of mixed Hodge loci

Technical Summary: On the distribution of mixed Hodge loci

Problem Statement
This paper investigates the distribution of the Hodge locus $HL(S, V^\otimes)$ associated with an admissible, graded-polarized integral variation of mixed Hodge structures (VMHS) $V$ over a smooth, irreducible complex algebraic variety $S$. The Hodge locus is defined as the set of points $s \in S^{an}$ where the fiber $V_s$ admits additional Hodge tensors (rational tensors in $T^{a,b}(V_s, \mathbb{Q})$ that are Hodge classes) compared to the generic fiber.

While the work of Brosnan-Pearlstein-Schnell established that the Hodge locus is a countable union of strict algebraic subvarieties, the distribution of these subvarieties remains a central problem. The paper focuses on the Zilber-Pink conjecture for mixed variations, specifically the "all-or-nothing" dichotomy: if the "typical" part of the Hodge locus is non-empty, is it Zariski-dense in $S$? Furthermore, the paper seeks to understand the conditions under which the Hodge locus is empty or strictly closed, particularly in relation to the "level" of the variation.

Methodology
The author employs a synthesis of intersection theory, o-minimal geometry, and the theory of Mumford-Tate groups. The core methodological framework involves:

1. Intersection-Theoretic Description: Building on Klingler's realization, the Hodge locus is viewed as a union of intersection loci. The paper refines the decomposition of the Hodge locus into typical and atypical parts, and introduces a crucial intermediate notion: the transverse Hodge locus ($HL(S, V^\otimes)_{\text{trans}}$). A special subvariety is transverse if it arises from a "transverse intersection" of the period map image with a Hodge subvariety in the sense of differential geometry (satisfying a dimension equality).
2. V-Likelihood: The paper introduces the notion of a $V$-likely mixed Hodge subclass. A strict mixed Hodge subclass $(M, DM) \subset (P, DP)$ is $V$-likely if it satisfies a specific dimension inequality involving the period map and quotient maps by normal subgroups with unipotent radicals. This condition generalizes "factorwise $V$-admissibility" from the pure case.
3. O-minimal and Ax-Schanuel Tools: The proofs rely heavily on the Mixed Ax-Schanuel theorem (Bakker-Brunebarbe-Tsimerman, Gao-Klingler) and the Geometric Zilber-Pink theorem (Baldi-Urbanik). These tools allow the author to control the algebraicity of the period map and the distribution of weakly special subvarieties.
4. Lie Algebraic Analysis: The paper utilizes the structure of the Lie algebra of the generic Mumford-Tate group, specifically analyzing the weight filtration and Hodge numbers of $\text{Lie}(P)$ to derive constraints on the existence of transverse intersections.

Key Contributions and Results

• The All-or-Nothing Theorem for Transverse Loci: The primary result (Theorem 1.8) establishes that for any strict mixed Hodge subclass $(M, DM)$, the transverse Hodge locus of type $M$, $HL(S, V^\otimes, M)_{\text{trans}}$, is either empty or Zariski-dense in $S$.

  • Corollary: If the typical Hodge locus $HL(S, V^\otimes)_{\text{typ}}$ is non-empty, then the full Hodge locus $HL(S, V^\otimes)$ is Zariski-dense in $S$ (Theorem 1.4). This implies that the Zilber-Pink conjecture part (a) (finiteness of the atypical locus) implies part (b) (density of the typical locus).

• Characterization of Density via Likelihood: The paper proves that the non-emptiness of the transverse Hodge locus is equivalent to the existence of a $V$-likely strict mixed Hodge subclass (Theorem 1.14).

  • Euclidean Density: Under the condition that $\text{Gr}^W_{-2}(\text{Lie}(P)) = 0$ (which implies the variation has no "purely unipotent" factors of a certain type), the existence of a $V$-likely subclass is equivalent to the transverse locus being dense in the Euclidean topology (Theorem 1.11).
  • Counterexamples: The paper provides explicit examples (Propositions 6.2 and 6.3) showing that in the general mixed case, a $V$-likely subclass may yield a transverse locus that is Zariski-dense but not Euclidean-dense (e.g., dense in $(\mathbb{C}^\times)^n$ but not in the Euclidean topology due to roots of unity).

• Emptiness Criteria and Level: The paper extends results from pure variations regarding the "level" of the variation. It proves that if the associated graded variation $\text{Gr}(V)$ has level at least 3 (in the sense of Baldi-Klingler-Ullmo) and satisfies specific monodromy and geometric conditions (e.g., $P_0^{\text{der}} = H_0$ and no dominant morphism to $\mathbb{G}_m$), then the transverse Hodge locus is empty (Theorem 1.19, Theorem 10.3).

  • This emptiness implies that the full Hodge locus consists only of atypical subvarieties.
  • Consequently, the "factorwise positive dimensional" part of the Hodge locus is a strict Zariski-closed subset (Corollary 1.21).

• Refinement of the Zilber-Pink Conjecture: The paper clarifies the relationship between the atypical and transverse loci. It shows that under the Zilber-Pink conjecture, the density of the typical locus is equivalent to the density of the transverse locus.

Significance and Claims
The paper claims to provide a comprehensive framework for understanding the distribution of Hodge loci in the mixed setting, which is significantly more complex than the pure case due to the presence of unipotent radicals and extensions.

• Resolution of the "All-or-Nothing" Property: The author establishes that the "all-or-nothing" principle, previously known for pure variations, holds for the transverse Hodge locus in the mixed case. This serves as a strong partial verification of the Zilber-Pink conjecture for mixed variations.
• New Criteria for Emptiness: By linking the emptiness of the Hodge locus to the "level" of the graded variation and the structure of the unipotent radical, the paper provides new, verifiable criteria to determine when a Hodge locus is trivial. This improves upon classical results by Brosnan-Pearlstein-Schnell in situations where the level is high.
• Distinction between Topologies: The work highlights a fundamental difference between pure and mixed variations: in the mixed case, Zariski-density does not automatically imply Euclidean-density for prescribed types, a phenomenon illustrated by families of extensions of $\mathbb{Z}(0)$ by $\mathbb{Z}(1)^n$.

The paper concludes that the study of the transverse Hodge locus is the correct setting for analyzing density phenomena in mixed Hodge theory, and that the interplay between the "likelihood" of a subdatum and the geometric constraints (level, monodromy) fully determines the distribution of the Hodge locus.