$E_6$-local systems from cubic threefolds

1. Problem Statement and Context

The paper addresses a fundamental question in arithmetic geometry and the theory of motives: Is every simple exceptional group realized as the algebraic monodromy group of a local system of geometric origin on a smooth complex variety?

• Background: Serre previously raised the question of whether exceptional groups (like $G_2, E_8$) appear as Galois groups of motives. While constructions exist for $G_2, E_7, E_8,$ and $F_4$ (often via Shimura varieties or rigid local systems on $\mathbb{P}^1 \setminus \{0, 1, \infty\}$), the case for $E_6$ remained open in the context of geometric local systems (those arising from the cohomology of algebraic families).
• Specific Gap: Prior to this work, no explicit construction was known for a local system of geometric origin with algebraic monodromy group $E_6$. Existing candidates (e.g., tautological local systems on Shimura varieties) were not proven to be of geometric origin.
• Goal: The authors aim to construct infinitely many pairwise distinct local systems of geometric origin whose algebraic monodromy group is exactly $E_6$, specifically arising from the moduli space of smooth cubic threefolds.

2. Methodology and Construction

The authors construct these local systems using the geometry of cubic threefolds and their associated Fano surfaces of lines.

2.1. Geometric Setup

• Family: Let $S$ be a smooth complex variety and $\mathcal{Y} \to S$ a versal family of smooth cubic threefolds (hypersurfaces of degree 3 in $\mathbb{P}^4$).
• Fano Surface: Associated with each cubic threefold $Y_s$ is a surface $F_s$ (the Fano surface) parametrizing lines contained in $Y_s$. This yields a smooth projective morphism $\pi: \mathcal{F} \to S$.
• Local System Construction:
  1. Take a unitary rank-one local system $L$ of finite order $n$ on the total space $\mathcal{F}$ (or extending from the fibers).
  2. Consider the middle cohomology of the fibers twisted by $L$: $V := R^2\pi_* L$.
  3. By Ehresmann's theorem, $V$ is a local system on $S$ of rank 27 (since $\dim H^2(F_s, \mathbb{Q}) = 10$, but the twist and cohomological shift yield a 27-dimensional representation in the specific context of the Fano surface's Albanese map).
  4. The algebraic monodromy group $M_s$ is defined as the Zariski closure of the image of the monodromy representation $\rho_s: \pi_1(S, s) \to GL(V_s)$.

2.2. The Reconstruction Technique (Section 2)

A core methodological innovation is a reconstruction technique derived from the derivative of the period map.

• Higgs Bundles: The authors analyze the infinitesimal Higgs bundle associated with the variation of Hodge structures (VHS) of $V$.
• Recovery of $L$: They prove that under certain conditions (specifically regarding the global generation of $L \otimes \omega_F$), the local system $L$ on the fiber $F$ can be functorially reconstructed from the $d$-th iterate of the Higgs field (where $d=\dim F = 2$).
• Key Insight: The reconstruction relies on the cup product map $\mu: \text{Sym}^2 H^1(F, T_F) \to H^2(F, \wedge^2 T_F)$. The base locus $B$ of the kernel of the dual map $\mu^\vee$ corresponds to the divisor of "lines of the second type" on the cubic threefold.
• Implication: If the monodromy group were smaller than $E_6$, the reconstruction would force the local system $L$ to be self-dual or decompose in a way that contradicts the specific branching rules of $E_6$ representations.

3. Key Technical Results

3.1. Global Generation and Rank Bounds (Section 3)

To apply the reconstruction technique, the authors establish specific geometric properties of the Fano surface $F$ of a generic cubic threefold:

• Theorem 3.1: For a generic line bundle $L \in \text{Pic}^0(F)$, the bundle $L \otimes \omega_F$ is globally generated. This ensures the necessary surjectivity conditions for the reconstruction maps.
• Theorem 3.2: The rank of the cup product map $\mu$ is exactly 50.
  • The space $H^0(F, \omega_F^{\otimes 2})$ has dimension 51.
  • The kernel of the dual map $\mu^\vee$ is 1-dimensional, generated by a section $\sigma$ cutting out the divisor of lines of the second type (lines where the normal bundle splits as $\mathcal{O}(1) \oplus \mathcal{O}(-1)$).
  • This divisor $B$ is ample, which is crucial for the reconstruction arguments.

3.2. Monodromy Analysis (Section 4)

The proof that the monodromy group is exactly $E_6$ proceeds in two steps:

1. Inclusion ($M^\circ \subseteq E_6$):
   • Using results from Lawrence and Sawin and Krämer, the monodromy group normalizes the Tannaka group $G_F$ associated with the convolution of perverse sheaves on the Albanese variety of $F$.
   • It is known that the derived group of $G_F$ is $E_6$ acting on the 27-dimensional representation. Thus, $M^\circ \subseteq E_6$.
2. Strictness ($M^\circ = E_6$):
   • The authors assume $M^\circ$ is a proper subgroup and derive a contradiction.
   • They analyze the branching of the 27-dimensional representation of $E_6$ restricted to maximal semisimple subgroups (e.g., $D_5, C_4, F_4, A_2 \times G_2$, etc.).
   • Non-Self-Duality: They prove $V$ is not self-dual (Lemma 4.4). This eliminates subgroups where the restriction of the 27-dim rep is self-dual (like $C_4, F_4, A_2, G_2$).
   • Rank Constraints: Using the reconstruction technique, they show $V$ contains a unique simple sub-local system $W$ of rank $\ge 13$.
   • Elimination: By checking the branching rules (Table 1), they show that for all non-self-dual maximal subgroups (e.g., $D_5, A_1 \times A_5, A_2 \times G_2$), the representation either splits into summands too small to contain $W$, or forces the monodromy group to be trivial, which is a contradiction.

4. Main Theorem and Results

Main Theorem:
There exists an integer $n_0 > 2$ such that for any versal family of smooth cubic threefolds $\mathcal{Y} \to S$ and any rank-one local system $L$ of finite order $n \ge n_0$ on the fibers:

1. Monodromy Group: The connected algebraic monodromy group of $V = R^2\pi_* L$ is isomorphic to $E_6$.
2. Representation: The action on the fiber $V_s$ is isomorphic to one of the two 27-dimensional irreducible representations of $E_6$.
3. Invariant Trace Field: The invariant trace field is $\mathbb{Q}(\zeta_n)$, where $\zeta_n = e^{2\pi i/n}$.
4. Distinctness: Local systems arising from different orders $n$ (with distinct cyclotomic fields) are non-equivalent. Furthermore, if two such systems are isomorphic, their underlying rank-one twists $L_1, L_2$ must be isomorphic.

5. Significance and Impact

• Resolution of the $E_6$ Case: This paper provides the first explicit construction of geometric local systems with monodromy group $E_6$, completing the list of simple exceptional groups ($G_2, F_4, E_6, E_7, E_8$) known to arise from geometry in this specific sense.
• New Geometric Source: Unlike previous constructions for other groups which often relied on Shimura varieties or rigid local systems on $\mathbb{P}^1$, this construction uses the rich geometry of cubic threefolds and their Fano surfaces.
• Motivic Implications: By specializing these local systems, the authors produce infinitely many motives with Galois group $E_6$, answering a version of Serre's question for this group.
• Methodological Advance: The "reconstruction technique" linking the Higgs field of the variation of Hodge structures to the underlying local system on the base variety is a powerful new tool. It allows for the recovery of geometric data (the line bundle $L$) from the monodromy representation, which is essential for proving the monodromy group is the full exceptional group and not a proper subgroup.
• Connection to Exceptional Geometry: The work highlights a deep connection between the exceptional group $E_6$ and the geometry of lines on cubic threefolds (specifically the divisor of lines of the second type), reflecting the fact that the Weyl group of $E_6$ acts on the lines of a cubic surface.