Level structures on cyclic covers of $\mathbb{P}^n$ and the homology of Fermat hypersurfaces

1. Problem Statement

The paper addresses the relationship between the topology of a smooth projective hypersurface $Z' \subset \mathbb{P}^n$ of degree $d > 1$ and the topology of its associated $\mu_d$-cover $Z \to \mathbb{P}^n$, which is totally ramified along $Z'$.

Specifically, the author seeks to establish a precise connection between:

• The primitive cohomology of the branch locus $Z'$ (denoted $H_P(Z')$).
• The primitive homology of the covering space $Z$ (denoted $H_P(Z)$), specifically its $\mu_d$-coinvariants.

A primary motivation for this work is to resolve a question posed by Beauville regarding the moduli space of smooth cubic surfaces ($n=3, d=3$). Beauville asked whether a "marking" (a specific basis choice) of a cubic surface determines a level 3-structure on the associated cubic threefold. This paper aims to provide a general framework for such relationships, particularly in the context of ball quotient descriptions of moduli spaces.

2. Methodology

The author employs a combination of classical algebraic topology, representation theory of group rings, and explicit computations on Fermat hypersurfaces.

A. Algebraic Framework (Group Rings)

The core technical tool is the analysis of modules over the group ring $\mathbb{Z}G$ for a finite group $G$ (specifically the cyclic group $\mu_d$).

• Augmentation and Co-augmentation: The paper defines the augmentation map $\epsilon_G: \mathbb{Z}G \to \mathbb{Z}$ and the co-augmentation $\epsilon_G: \mathbb{Z} \to \mathbb{Z}G$.
• Invariants and Coinvariants: For a $\mathbb{Z}G$-module $M$, the paper studies the relationship between the invariants $M^G$ and the coinvariants $M_G = M/I_G M$ (where $I_G$ is the augmentation ideal).
• Key Maps: Two specific maps are constructed for any module $M$:
  1. $r_M: (M/M^G)_G \to \mathbb{Z}/d \otimes M^G$ (derived from the co-augmentation).
  2. $s_M: (\mathbb{Z}/d) \otimes M^G \to (M_G)^G$ (derived from the dual structure).
     These maps relate the reduction modulo $d$ of invariants to the coinvariants.

B. Geometric Setup

• The Cover: Let $Z' \subset \mathbb{P}^n$ be defined by a homogeneous polynomial $F$ of degree $d$. The cover $Z \subset \mathbb{P}^{n+1}$ is defined by $F + z_{n+1}^d = 0$.
• Primitive Cohomology: The paper utilizes the Lefschetz hyperplane theorem and the Gysin sequence to relate the homology of the affine complement $\mathring{Z} = Z \setminus Z'$ to the primitive homology of $Z$ and $Z'$.
• Fermat Hypersurfaces: To prove the main theorems, the author reduces the problem to the case where $Z'$ is a Fermat hypersurface ($F = \sum z_i^d$). This allows for an explicit description of the $\mu_d$-action on the homology using tensor products of augmentation ideals.

C. Monodromy and Vanishing Cycles

The paper incorporates the theory of vanishing cycles and Picard-Lefschetz transformations. It relates the monodromy representation of the moduli space of hypersurfaces to the action of the group $\mu_d$ on the homology of the cover, utilizing generalized Picard-Lefschetz transformations (as defined by Pham).

3. Key Contributions and Results

Main Theorem (Theorem 3.3)

The central result establishes a natural surjection between the reduction modulo $d$ of the primitive cohomology of $Z'$ and the $\mu_d$-coinvariants of the primitive homology of $Z$.

• Case $n$ is even: The map $r_{Z'}: H_P(Z')^{\mu_d} \to \mathbb{Z}/d \otimes H_P(Z')$ is surjective. The diagram involving the intersection pairing commutes, identifying the image of the primitive homology of $Z$ with the non-degenerate quotient of $H_P(Z')$ modulo $d$.
• Case $n$ is odd: The map $s_{Z'}: \mathbb{Z}/d \otimes H_P(Z') \to H_P(Z)^{\mu_d}$ is surjective.
• Kernel: In both cases, the kernel of the surjection is a subgroup of $\mathbb{Z}/d$.

Corollary 3.4 (Prime Degree Case)

If $d = p$ is a prime, the situation becomes more precise. The ring of integers of the cyclotomic field $\mathbb{Q}(\mu_p)$ is a Dedekind domain.

• The map becomes an isomorphism between the image of the intersection pairing on $Z$ and the primitive cohomology of $Z'$ reduced modulo $p$.
• This provides a canonical identification of the level structure on the cover with the marking of the branch locus.

Application to Cubic Surfaces and Threefolds (Example 3.5)

• $n=3$ (Cubic Surface): For a smooth cubic surface $Z' \subset \mathbb{P}^3$, the primitive homology is isomorphic to the dual of the root lattice $Q(E_6)$. The paper proves that a level 3-structure on the associated cubic threefold $Z$ is equivalent to a Weyl group $W(E_6)$-marking of the cubic surface $Z'$.
  • Significance: This answers Beauville's question, confirming that the moduli space of marked cubic surfaces is a ball quotient (a result previously established by Allcock-Carlson-Toledo via different methods).
• $n=4$ (Cubic Threefold): The result implies that the primitive cohomology of a cubic threefold inherits a unitary structure from the cubic surface.

Generalizations (Remark 3.6)

The author extends the applicability of these results to other geometric settings:

• Del Pezzo surfaces of degree 2: Viewed as double covers of $\mathbb{P}^2$ ramified along a quartic curve. The result links the level 2 structure on the genus 3 curve to the geometry of the del Pezzo surface.
• K3 Surfaces:
  • Degree 4 K3 surfaces (double covers of degree 2 del Pezzo surfaces) relate to genus 3 curves.
  • Degree 6 K3 surfaces ($\mu_3$-covers of $\mathbb{P}^1 \times \mathbb{P}^1$) relate to genus 4 curves.
  • Significance: This validates Kondō's ball quotient descriptions for the moduli spaces of genus 3 and 4 curves, showing they persist when imposing level structures.

Correction of Previous Work

The paper corrects an error in the author's previous work [7] (and related work by Gvirtz and Skorobogatov) regarding the description of the (co)primitive cohomology of smooth Fermat hypersurfaces as modules over their symmetry groups. The explicit module structure is derived in Section 4 (Corollary 4.2).

4. Significance

1. Resolution of Open Questions: It provides a direct and rigorous answer to Beauville's question about the relationship between markings of cubic surfaces and level structures on cubic threefolds.
2. Unification of Moduli Spaces: It unifies the understanding of moduli spaces of curves (genus 3 and 4) and surfaces (cubic, del Pezzo, K3) through the lens of cyclic covers and level structures. It demonstrates that ball quotient descriptions are robust under the imposition of level structures.
3. Algebraic Topology of Covers: The paper offers a general method for relating the homology of a branched cover to the homology of the branch locus using group ring techniques, which is applicable beyond just hypersurfaces in projective space.
4. Explicit Computations: By utilizing the Fermat hypersurface case, the paper provides explicit algebraic descriptions of these cohomology groups, correcting previous misconceptions about their structure.

In summary, Looijenga establishes a bridge between the discrete geometry of markings on algebraic varieties and the continuous geometry of their moduli spaces as ball quotients, using the machinery of cyclic covers and level structures.