Algebraic cycles of some Fano varieties with Hodge structure of level one

1. Problem Statement and Context

The paper addresses fundamental questions in algebraic geometry concerning algebraic cycles on smooth projective varieties, specifically focusing on Fano varieties whose cohomology carries a Hodge structure of level one.

• The Integral Hodge Conjecture (IHC): While the rational Hodge conjecture remains a major open problem, the integral version is known to fail in general (e.g., counterexamples by Atiyah-Hirzebruch and Kollár). The paper investigates whether the IHC holds for specific classes of Fano varieties where the Hodge structure is "simple" (level one).
• The Gap in Codimension 3: For many varieties, the IHC is known for codimensions 1 and 2, but the case of codimension 3 (and higher) in higher-dimensional varieties (specifically fivefolds) is difficult.
• Motivic Decomposition: The authors aim to understand the structure of Chow groups and their motivic decompositions for these varieties, specifically testing the finite dimensionality of motives in the sense of Kimura.

2. Methodology

The authors employ a multi-faceted approach combining classical algebraic geometry, Hodge theory, and modern motivic techniques:

• Étale Motivic Cohomology: A core methodological shift is the use of étale Chow groups ($CH^i_{\text{ét}}(X)$), also known as Lichtenbaum cohomology. These groups are defined via the hypercohomology of Bloch's cycle complex in the étale topology.
  • The authors utilize the comparison map $\kappa_i: CH^i(X) \to CH^i_{\text{ét}}(X)$.
  • They leverage the fact that for torsion classes, the étale cycle class map is surjective onto the torsion part of the Hodge classes, whereas the classical map is not.
• Conservative Families of Functors: They use the framework of triangulated categories of motives ($DM_{\text{ét}}$). By showing that a morphism is an isomorphism if it is an isomorphism after tensoring with $\mathbb{Q}$ and $\mathbb{Z}/\ell$, they reduce integral problems to rational and torsion problems.
• Decomposition of the Diagonal: The paper relies heavily on the decomposition of the diagonal (introduced by Bloch and Srinivas). If the diagonal $\Delta_X$ can be decomposed into cycles supported on lower-dimensional subvarieties, it implies vanishing results for unramified cohomology and representability of Chow groups.
• Intermediate Jacobians: For varieties with Hodge level one, the middle cohomology $H^{2m+1}(X, \mathbb{Z})$ supports a weight-one Hodge structure, allowing the construction of an Intermediate Jacobian $J(X)$, which is a principally polarized abelian variety. The authors relate the Chow groups to the torsion and rational points of these Jacobians.
• Geometric Constructions:
  • Analysis of linear subspaces (lines, planes) contained in the varieties.
  • Use of incidence varieties and Fano schemes of $r$-planes in double covers (detailed in Appendix A).
  • Construction of specific rational maps and blow-ups to analyze unramified cohomology.

3. Key Contributions and Results

A. Classification and Étale Chow Groups of Level One Varieties

The authors revisit the classification of smooth complete intersections with Hodge level one (Deligne-Rapoport classification), including:

• Fano threefolds (cubic, quartic, (2,3) intersections).
• Intersections of two or three quadrics.
• Cubic fivefolds.

Result: They prove that for these varieties, the comparison map $\kappa_i: CH^i(X) \to CH^i_{\text{ét}}(X)$ is an isomorphism for all $i$. This implies that the integral Hodge conjecture holds for these varieties if and only if the étale version holds.

B. Motivic Decomposition

The paper establishes a motivic decomposition for these varieties in the category of étale motives.

• Theorem: For a smooth projective variety $X$ of dimension $2m+1$ with Hodge level one (e.g., cubic fivefolds, intersections of quadrics), the motive decomposes as:
  $$h_{\text{ét}}(X) \cong \bigoplus \mathbb{L}^k \oplus h^1_{\text{ét}}(J(X)) \otimes \mathbb{L}^m$$
  where $\mathbb{L}$ is the Lefschetz motive and $J(X)$ is the intermediate Jacobian.
• Significance: This confirms that these motives are finite-dimensional in the sense of Kimura, as they are built from finite-dimensional Lefschetz motives and the motive of an abelian variety.

C. The Main Theorem: Quartic Double Fivefolds

The primary contribution is the resolution of the Integral Hodge Conjecture for smooth quartic double fivefolds.

• Definition: A variety $X$ which is a double cover of $\mathbb{P}^5$ branched along a smooth quartic hypersurface.
• Theorem 7.17: The Integral Hodge Conjecture holds for $X$. Specifically, the cycle class map $c_3: CH^3(X) \to Hdg^6(X, \mathbb{Z})$ is surjective.
• Proof Strategy:
  1. Unramified Cohomology: They prove the vanishing of specific unramified cohomology groups ($H^4_{\text{nr}}(X, \mathbb{Q}/\mathbb{Z}(4)) = 0$ and $H^5_{\text{nr}}(X, \mathbb{Q}/\mathbb{Z}(5)) = 0$) using geometric arguments involving blow-ups along lines and fibrations over weighted projective spaces.
  2. Griffiths Group: They show that the Griffiths group $Griff^3(X)$ is trivial (or torsion-free in a way that vanishes in the relevant context).
  3. Kernel/Cokernel Analysis: By analyzing the kernel and cokernel of the comparison map $\kappa_3$, they demonstrate that $\kappa_3$ is an isomorphism. Since the rational Hodge conjecture holds (due to the double cover structure lifting classes from $\mathbb{P}^5$), the isomorphism $\kappa_3$ forces the integral Hodge conjecture to hold.

D. Gushel-Mukai Varieties

The paper briefly extends these results to Gushel-Mukai varieties (odd dimensions), confirming that the comparison maps are isomorphisms for these cases as well.

4. Significance and Impact

1. Resolution of a Specific IHC Case: The proof of the Integral Hodge Conjecture for quartic double fivefolds is a significant advancement. While the IHC is known to fail for general varieties, identifying a broad class of high-dimensional Fano varieties where it holds is crucial for understanding the boundary between rational and non-rational behavior.
2. Validation of the Étale Approach: The paper successfully demonstrates that working with étale motivic cohomology provides a more robust framework for studying integral cycles. The isomorphism $CH^i(X) \cong CH^i_{\text{ét}}(X)$ for these varieties suggests that the "obstructions" to the IHC (often torsion-related) are captured and resolved by the étale topology.
3. Motivic Finite Dimensionality: By explicitly constructing the motivic decomposition involving the intermediate Jacobian, the authors provide concrete examples of finite-dimensional motives in higher dimensions, supporting the broader conjecture that many geometrically "simple" varieties have finite-dimensional motives.
4. Refined Unramified Cohomology: The results contribute to the understanding of refined unramified cohomology (a concept developed by Schreieder), linking the vanishing of these groups directly to the validity of the IHC and the triviality of Griffiths groups.

In summary, the paper bridges classical Hodge theory, the geometry of Fano varieties, and modern motivic homotopy theory to prove that for a specific, important class of fivefolds, the algebraic cycles are fully characterized by their topological and Hodge-theoretic invariants.