Hyper-representations (Hyper-Kähler varieties)

• Hyper-representations, for hyper-Kähler varieties, are Galois or motivic representations of their cohomology, uniquely characterized by the property that their structure is predominantly determined by the degree-two (second cohomology) part.
• A central theorem states that the full André motive and l-adic Galois representation of deformation-equivalent hyper-Kähler varieties with sufficient b 2 are isomorphic if their degree-two counterparts are, subject to technical conditions on odd cohomology.
• This control by degree two has profound implications, like guaranteeing that two varieties over a finite field with equivalent degree-two l-adic representations have the same zeta function, showcasing rare arithmetic rigidity.

Hyper-representations, in the context of Salvatore Floccari’s work on the motive and Galois-theoretic structure of hyper-Kähler varieties, refer to the entirety of Galois or motivic representations associated to the (co)homology of such varieties—revealing how intricate, higher-order cohomological or arithmetic features are fully governed, up to technical refinements, by their degree-two (second cohomology) part. This principle is formalized both at the level of André motives and ℓ-adic Galois representations, and has profound consequences for arithmetic geometry, the structure of derived categories, and the informed paper of algebraic cycles.

---

1. Galois Representations and Hyper-Kähler Varieties: The Notion of Hyper-Representation

A hyper-Kähler variety is a smooth, simply connected, projective algebraic variety over a field $K \subset \mathbb{C}$ whose space of holomorphic 2-forms is spanned by a unique nowhere degenerate form. From the arithmetic and motivic viewpoint, hyper-representations refer to the action of the absolute Galois group $\mathrm{Gal}(\bar{K}/K)$ on the total ℓ-adic étale cohomology: $$\sigma_X: \mathrm{Gal}(\bar{K}/K) \to \prod_j \mathrm{GL}\left( H^j_{\mathrm{\acute{e}t}}(X_{\bar{K}}, \mathbb{Q}_\ell) \right).$$ The central phenomenon is that, for hyper-Kähler varieties, the “total” motivic or Galois representation is essentially determined by its structure in degree two, i.e., $H^2$ or the corresponding piece of the André motive: the “second cohomology governs all.”

This conceptual reduction—the control of all representation-theoretic and motivic information by degree two—distinguishes hyper-Kähler hyper-representations from the typically more intricate situation in other classes of varieties.

---

2. Main Theorems: André Motive and Galois Representation Controlled by Degree 2

The primary result is as follows:

Theorem (André Motive Governed by Degree 2):

Let $X_1$ and $X_2$ be deformation equivalent hyper-Kähler varieties over a field $K\subset\mathbb{C}$ with $b_2(X_i) > 6$. If there exists a Hodge isometry

$$f \colon H^2(X_1, \mathbb{Q}) \to H^2(X_2, \mathbb{Q}),$$

then (up to a finite extension of the base field and under a technical condition for odd cohomology)

$\mathcal{H}^*(X_1) \cong \mathcal{H}^*(X_2)$

as André motives.

Consequence:

If $f$ is Galois-equivariant on ℓ-adic cohomology, then, after finite extension,

$$H^*_{\mathrm{\acute{e}t}}(X_1, \mathbb{Q}_\ell) \cong H^*_{\mathrm{\acute{e}t}}(X_2, \mathbb{Q}_\ell)$$

as Galois modules, including the ring structure.

For varieties with trivial odd cohomology, the motivic Galois group of the total motive $\mathcal{H}_X^*$ surjects onto that of $\mathcal{H}_X^2$ with kernel the defect group: $\operatorname{Ker}\left( \Gmot(\mathcal{H}_X^*) \to \Gmot(\mathcal{H}_X^2) \right) =: P(X)$ which measures the deviation from “governed by degree 2”.

If odd cohomology is nontrivial, as in generalized Kummer type, the role of $\mathcal{H}_X^2$ is replaced by the Kuga–Satake motive of an associated abelian variety, subject to a technical Tannakian-inclusion condition.

---

3. Technical Assumptions: The Role of Odd Cohomology and the Kuga–Satake Construction

A critical subtlety arises when $H^{2k+1}(X) \neq 0$. Let $A$ denote the Kuga–Satake abelian variety attached to $H^2(X)$; the induced abelian motive $\mathcal{H}^1_A$ must be contained in the tannakian category generated by $\mathcal{H}_X^*$. This technical assumption is verified for known examples such as generalized Kummer varieties.

Under this hypothesis, the motivic Galois group of the total motive decomposes as

$\Gmot(\mathcal{H}^*_X) = P(X) \times \MT(H^*_X)$

with $\MT(H^*_X)$ the Mumford–Tate group and $P(X)$ the defect group.

It is conjectured that $P(X)$ is always trivial (would follow from the Hodge conjecture), so all cohomological cycles are “motivated”.

---

4. Implications and Applications

A. Mumford–Tate Conjecture

For all currently known hyper-Kähler deformation types, the Mumford–Tate conjecture holds for their cohomology. This is foundational for deducing the precise structure of associated motives and Galois representations.

B. Varieties over Finite Fields

If two smooth projective varieties $Z_1, Z_2$ over a finite field admit liftings to characteristic zero hyper-Kähler varieties with equivalent (degree two) ℓ-adic representations, then (after finite extension)

$H^*_{Z_1, \ell} \cong H^*_{Z_2, \ell}$

as Galois modules—hence $Z_1$ and $Z_2$ share the same zeta function. This indicates that the total arithmetic nature of such a variety is determined by its geometry in degree two, a rare and powerful rigidity.

C. Motivic and Arithmetic Rigidity

Under variation in families, the “defect group” $P(X)$ is locally constant; this gives rise to rigidity phenomena akin to parallel transport for motives and Galois representations.

D. Structural Consequences

• The “hyper-representation” (total Galois action or motive) of any such variety is fully encoded by $H^2$, modulo controlled technicalities.
• LLV Lie algebra symmetries and automorphic forms interact deeply with these representations, suggesting deeper links with representation theory.

---

5. Future Directions and Open Questions

Key challenges and conjectures remain:

1. Triviality of the Defect Group: Proving $P(X) = 1$ would show that all cohomology is motivically determined and all Hodge cycles are motivated.
2. Generalization of Technical Assumptions: Fully justifying the inclusion of the associated Kuga–Satake motive for broader families of hyper-Kähler varieties.
3. Extension to Unknown Deformation Types: Discovery and classification of new hyper-Kähler types may expand the reach of these rigidity results.
4. Number-Theoretic Implications: Equal degree-two ℓ-adic representations guarantee equal zeta functions; understanding the family-level and rationality consequences remains a frontier.
5. Representation-Theoretic Symmetries: Investigating the interaction with the LLV algebra and automorphic forms may reveal new structures in Hodge theory and arithmetic geometry.

---

Summary Table: Hyper-Representation Control

Context	Main Statement	Technical Requirement
Motives	$\mathcal{H}^*_X$ is determined by $\mathcal{H}^2_X$, up to isomorphism	Odd cohomology controlled via Kuga–Satake assumption
Galois repr. ($\ell$)	$H^*_{X,\ell}$ as Galois module determined by $H^2_{X,\ell}$	As above
Finite fields	Equal Galois reps $\implies$ equal zeta functions	Lifting to characteristic 0, Mumford–Tate holds

---

Concluding Perspective

The structure of hyper-representations for hyper-Kähler varieties—a motif deeply linked to degree-two geometry—offers a rare example of “governance by cohomology” in complex algebraic and arithmetic geometry. This body of results not only consolidates the Torelli philosophy for hyper-Kähler varieties but anchors modern motivic and Galois theory in concrete, explicit isomorphism phenomena. The landscape remains open for breakthroughs tied to the defect group, the extension to new deformation types, and the integration of these ideas with deep conjectures in Hodge theory and arithmetic geometry.