Torsion cycles on Fermat varieties

1. Problem Statement

The paper addresses the behavior of algebraic cycles on Fermat varieties (hypersurfaces defined by $X_0^d + \dots + X_{n+1}^d = 0$). Specifically, it investigates whether cycles supported on "trivial" loci (linear subvarieties defined by coordinate hyperplanes $X_i=0$) are torsion in the Jacobian or intermediate Jacobian.

• Background: The Manin-Drinfeld theorem states that any degree 0 divisor supported on the cusps of a modular curve is torsion in the Jacobian. Rohrlich (1977) generalized this to Fermat curves ($n=1$), proving that degree 0 divisors supported on the "Fermat points" (intersections of coordinate hyperplanes) are torsion.
• The Gap: While Rohrlich's result was proven explicitly for curves, a unified proof using Mixed Hodge Theory (following Elkik's approach for modular curves) was lacking for higher dimensions. Furthermore, it was unclear if this torsion property extends to:
  1. Higher codimensional cycles (null-homologous cycles) on Fermat hypersurfaces of dimension $n > 1$.
  2. Higher Chow cycles (motivic cycles) on these varieties.

2. Methodology

The author employs Mixed Hodge Theory (MHS) and the theory of extensions of Hodge structures to reprove and generalize Rohrlich's results. The core methodology involves:

• Abel-Jacobi via Extensions: The paper utilizes the fact that the Abel-Jacobi map (and its higher analogues) can be described via extensions of Mixed Hodge Structures. A cycle $Z$ is torsion in the intermediate Jacobian if and only if the associated extension of MHS splits over $\mathbb{Q}$.
• Cohomological Splitting: The central technical task is to prove that a specific short exact sequence of cohomology groups splits as a sequence of Hodge structures. The sequence relates the cohomology of the variety $X$, the cohomology of the complement $X \setminus |D|$, and the cohomology with support on the divisor $|D|$.
• Group Action Analysis: The author analyzes the action of the group of roots of unity, $G_d^n \cong \mu_d^{n+1}$, on the cohomology of Fermat varieties. By decomposing cohomology into eigenspaces (characters) under this group action, the author constructs a projection operator to demonstrate the splitting of the exact sequence.
• Inductive Structure: The paper also utilizes the rational map structure between products of Fermat varieties to discuss the existence of non-torsion cycles, contrasting them with the "trivial" cycles studied in the main theorems.

3. Key Contributions and Results

A. Generalization of Rohrlich's Theorem (Theorem 1.2 & 3.1)

The paper proves that for a Fermat hypersurface $F_d^n$ of dimension $n$, let $S_d^p$ be the union of all linear subvarieties of codimension $p$ (intersections of $p$ coordinate hyperplanes).

• Result: The exact sequence of Hodge structures:
  $$0 \to H^{2p-1}(F_d^n, \mathbb{Q}) \to H^{2p-1}(F_d^n - S_d^p, \mathbb{Q}) \to H^{2p}_{S_d^p}(F_d^n, \mathbb{Q})_0 \to 0$$
  is split. Here, $H^{2p}_{S_d^p}(\dots)_0$ is the kernel of the cycle class map (generated by null-homologous linear combinations).
• Implication: Any null-homologous cycle supported on the linear subvarieties $S_d^p$ is torsion in the $p$-th intermediate Jacobian.
• Significance: This provides a "pure thought" proof of Rohrlich's theorem for curves ($n=1, p=1$) and extends it to higher dimensions ($n > 1$).

B. Higher Chow Cycles and Regulators (Theorem 1.3 & 4.2)

The author extends the logic to Higher Chow groups (motivic cohomology), specifically cycles of the form $y = \sum (Z_I, f_{I1}, \dots, f_{Iq})$, where $Z_I$ are linear subvarieties and $f_{Ij}$ are functions with divisors supported on linear subvarieties.

• Result: The regulator map $\text{reg}(y)$ to Deligne cohomology $H^{2p-q}_D(F_d^n, \mathbb{Q}(p))$ is zero (or torsion).
• Condition: This holds for $n > 1$. The proof relies on showing that the relevant exact sequence for simplicial schemes splits due to the trivial action of the group $G_d^n$ on the classes of these specific cycles.
• Conjecture Connection: Under the Bloch-Beilinson conjectures (which assert the injectivity of the regulator map for varieties over number fields), these cycles would be torsion in the Chow group itself.

C. Distinction from Non-Torsion Cycles (Section 5)

The paper clarifies that while "natural" cycles built from coordinate hyperplanes are torsion, Fermat varieties do possess non-torsion cycles.

• Source: These arise from the inductive structure of Fermat varieties (rational maps from products of lower-dimensional Fermat varieties).
• Examples:
  • The Gross-Schoen modified diagonal cycle on $(F_d^1)^3$, which maps to a likely non-torsion cycle on $F_d^3$.
  • Cycles constructed by Ross and Sarkar using functions on cusps, which are non-trivial in specific higher Chow groups (e.g., $CH_2(F_d^1, 2)$).
• Conclusion: The torsion results apply specifically to cycles supported on the "trivial" linear subvarieties, not to all cycles on the variety.

4. Significance

• Unification: The paper unifies the proof of the Manin-Drinfeld theorem and Rohrlich's theorem under the framework of Mixed Hodge Theory, moving away from explicit arithmetic calculations to structural Hodge-theoretic arguments.
• Dimensional Generalization: It successfully generalizes the torsion property from curves ($n=1$) to higher-dimensional Fermat hypersurfaces ($n > 1$) for both classical divisors and higher Chow cycles.
• Clarification of "Trivial" vs. "Interesting": It delineates the boundary between cycles that are structurally forced to be torsion (those supported on coordinate hyperplanes) and those that carry deep arithmetic information (those arising from product structures like the Gross-Schoen cycle).
• Methodological Utility: The use of group actions ($G_d^n$) to split exact sequences of Hodge structures provides a robust template for analyzing cycles on other varieties with high symmetry.

Summary of Main Theorems

Note: The paper acknowledges the use of AI (Gemini 3) for arguments and stylistic comments, but the mathematical core relies on the author's application of Deligne's and Shioda's work on Hodge structures and Fermat varieties.