Hodge Structures in Sextic Fourfolds Equipped with an Involution

Abstract: To each ternary sextic $f(X_0, X_1, X_2)$ whose associated plane curve is smooth, the Shioda construction attaches a smooth sextic fourfold $X \subset \mathbb{P}^5$ whose defining equation $f(X_0, X_1, X_2) - f(Y_0, Y_1, Y_2)$ is fixed under the involution $ι: (X_0, X_1, X_2, Y_0, Y_1, Y_2) \mapsto i \cdot (Y_0, Y_1, Y_2, -X_0, -X_1, -X_2)$. The induced action $ι^* : H^4(X, \mathbb{Q}) \to H^4(X, \mathbb{Q})$ fixes a Hodge substructure $H \subset H^4(X, \mathbb{Q})$ whose Hodge coniveau is 1. By the general Hodge conjecture, we expect that there should exist a divisor $Y \subset X$ for which $H \subset \ker\left( H^4(X, \mathbb{Q}) \to H^4(X \setminus Y, \mathbb{Q}) \right)$. We verify this prediction in case the Waring rank of $f(X_0, X_1, X_2)$ takes on its minimum possible value, partially answering a question of Voisin (J. Math. Sci. Univ. Tokyo '15).

• The paper confirms Voisin’s prediction by demonstrating that all invariant Hodge classes of coniveau 1 vanish outside divisors in sextic fourfolds with involution.
• It employs an explicit algebraic reduction to a partial differential equation, providing a constructive algorithm for finding suitable rational vector fields.
• The results validate geometric coniveau expectations and offer computational tools for advancing the generalized Hodge and Bloch conjectures.

Hodge Structures and Cycles in Sextic Fourfolds with Involution

Overview and Motivation

The paper "Hodge Structures in Sextic Fourfolds Equipped with an Involution" (2603.29157) investigates a specific case of the generalized Hodge conjecture regarding the structure of $H^4$ for a class of sextic fourfolds in $\mathbb{P}^5$ with symmetry under an explicit involutive automorphism. The author focuses on fourfolds arising from the Shioda construction: given a smooth ternary sextic $f(X_0, X_1, X_2)$, one considers $X \subset \mathbb{P}^5$ defined by $f(X_0, X_1, X_2) - f(Y_0, Y_1, Y_2) = 0$. This hypersurface enjoys a linear involution $\iota$ acting by cyclically rotating and conjugating coordinates, which induces an action on cohomology. The $\iota$-invariant part of middle cohomology defines a rich Hodge substructure, and its coniveau is central to the interplay between the generalized Hodge and Bloch conjectures.

The central claim is a confirmation of a prediction by Voisin regarding the generalized Hodge conjecture, for the fourfolds $X$ as above, in the setting where the plane sextic $f$ has minimal Waring rank (i.e., can be written as a sum of three sixth powers of linear forms). The author proves that all invariant Hodge classes of coniveau 1 indeed vanish outside divisors, confirming the geometric coniveau expectation in this case.

Hodge-theoretic Reformulation and Algebraic Criteria

A principal contribution is the effective translation of the Hodge-theoretic prediction into an explicit algebraic criterion. Specifically, for each $q \in \{0,1,2\}$, and for every $\mathbb{P}^5$0-anti-invariant polynomial $\mathbb{P}^5$1 of appropriate degree, the generalized Hodge conjecture for the $\mathbb{P}^5$2-invariant part is reduced to the solvability of a partial differential equation of the form

$\mathbb{P}^5$3

in the local ring, for a rational vector field $\mathbb{P}^5$4 with controlled denominators. This reduction exploits the explicit description of primitive cohomology of hypersurfaces in terms of residue forms (per Griffiths, Voisin, and Lewis) and a careful analysis of pole orders of primitives.

The proof leverages refined properties of the logarithmic de Rham complex and the degeneration properties of the Hodge filtration. The author presents a new, cohomologically elementary derivation (eschewing reliance on deep theorems such as Deligne’s degeneration statement), clarifying and controlling the possible pole order in the primitives and verifying the algebraic nature of any solution.

Explicit Solution in the Case of Minimal Waring Rank

The main technical result provides an explicit, constructive solution to the divisibility condition above in the case when $\mathbb{P}^5$5 is the Fermat sextic or is related via block-diagonal change of coordinates to a direct sum of three sixth powers of orthogonal linear forms (which translates, geometrically, to $\mathbb{P}^5$6 having Waring rank 3). The author develops an algorithm—resembling and extending Griffiths-Dwork reduction—that, for every anti-invariant input, produces a rational vector field primitive. A key combinatorial lemma ensures that the reduction survives at all stages, which relies on strong structural facts about the exponents of the monomials (namely, they remain $\mathbb{P}^5$7 due to repeated division by the Jacobian ideal).

This machinery transports to all fourfolds of the form $\mathbb{P}^5$8 where $\mathbb{P}^5$9 has rank 3, via block coordinate change; thus, the proof applies to an 8-dimensional family of plane sextics, and after further geometric considerations, to a 17-dimensional family in the relevant moduli.

Geometric and Cycle-Theoretic Interpretation

The divisors constructed in the proof, arising as vanishing loci of combinations of the partial derivatives of $f(X_0, X_1, X_2)$0 with respect to carefully chosen linear subspaces, are interpreted geometrically as ramification divisors for explicit rational fibrations of $f(X_0, X_1, X_2)$1 by $f(X_0, X_1, X_2)$2’s. This geometrization echoes broader heuristics regarding the effectivity of coniveau predictions for Hodge classes, and the algebraic cycles constructed reflect the expected depth-1 coniveau directly.

The author notes that the context is highly amenable to computational exploration because the anti-invariant Hodge classes are characterized explicitly and the divisibility test (the PDE above) is algorithmic. In this specific case, the difficulty that usually arises in higher cohomology—the lack of a purely algebraic detection of which classes must vanish—is circumvented.

Implications and Directions for Future Research

The work provides an unconditional positive answer to a version of Voisin’s question for the Shioda family with minimal Waring rank, confirming the geometric coniveau property for the $f(X_0, X_1, X_2)$3-invariant part of $f(X_0, X_1, X_2)$4. This case is of notable interest because it links to the Bloch generalization for products of K3 surfaces and to the structure of the Chow group of 0-cycles.

Practically, the algorithmic nature of the criterion suggests that further advances—possibly via symbolic or even numerical computation—might extend the result to larger families of sextics, although the combinatorial obstacles and the necessity for “non-strict” solutions (where the vanishing only occurs modulo higher powers of $f(X_0, X_1, X_2)$5) become apparent for more general $f(X_0, X_1, X_2)$6.

Theoretically, the explicit, elementary reduction provides a new point of departure for attempts at the generalized Hodge conjecture in similar contexts, and the approach opens the possibility of generalization if suitable combinatorial or algebraic reductions can be discovered for broader families.

Conclusion

This paper provides a rigorous, explicit, and effective confirmation of the geometric coniveau 1 property for the $f(X_0, X_1, X_2)$7-invariant part of middle cohomology for a significant family of sextic fourfolds. The reduction to an explicit algebraic partial differential equation and the construction of explicit primitives marks a technical advance in the subject, with implications for both the generalized Hodge conjecture and practical computations in algebraic geometry. The methods developed highlight a direction for future explorations of the interface between explicit cohomological tools, Hodge theory, and algebraic cycles.