On the motive of O'Grady's six dimensional hyper-Kähler varieties

This paper establishes that the rational Chow motive of O'Grady's six-dimensional hyper-Kähler varieties, constructed as symplectic resolutions of singular moduli spaces on abelian surfaces, is generated by the motive of the underlying surface, thereby confirming the Hodge and Tate conjectures for these varieties.

The Gist

Imagine you are an architect trying to understand a massive, incredibly complex cathedral. This cathedral is a hyper-Kähler variety—a shape so intricate and high-dimensional that it's hard to visualize, let alone study.

For decades, mathematicians have known that these "cathedrals" are built using blueprints from a much simpler, flatter building: an Abelian surface (which is essentially a donut-shaped surface with two holes, or a "complex torus").

The big question was: Can we prove that the complex cathedral is truly just a fancy rearrangement of the simple donut? If we can, it means we can solve all the hard problems about the cathedral by just looking at the donut.

This paper by Salvatore Floccari answers "Yes" for a specific, very difficult type of cathedral known as an OG6 variety.

Here is the story of how he did it, broken down into simple concepts.

1. The Problem: A Broken Building

The OG6 variety starts as a "moduli space." Think of this as a giant catalog or a map of all possible ways to arrange certain mathematical objects (sheaves) on a donut.

• The Issue: This map is usually perfect and smooth. But for this specific type of arrangement, the map gets "cracked" and "broken" (singular). It has sharp corners and jagged edges where the math stops making sense.
• The Fix: Mathematicians O'Grady and others found a way to "smooth out" these cracks. They performed a surgical repair called a symplectic resolution. Imagine taking a crumpled piece of paper and carefully ironing it flat until it's a perfect, smooth sheet again. This new, smooth shape is the OG6 variety.

2. The Detective Work: Finding the Hidden Blueprint

Floccari's goal was to prove that this newly smoothed-out OG6 variety is "motivated" by the original donut (the Abelian surface). In math-speak, this means its "Chow motive" (a fancy way of saying its fundamental algebraic DNA) is built entirely out of the DNA of the donut.

To do this, he didn't just look at the OG6 variety directly. He used a clever trick discovered by other mathematicians (Mongardi, Rapagnetta, and Saccà):

• The Double-Decker Bus: They realized the OG6 variety is actually the result of taking a different, even stranger shape (let's call it Shape Y) and folding it in half.
• Shape Y is a "K3[3]-type" variety. It's like a different kind of cathedral, but one that mathematicians already knew how to build using a K3 surface (a different type of mathematical surface, often compared to a sphere with extra handles).

3. The "Aha!" Moment: Connecting the Dots

Floccari's breakthrough was connecting the dots between the three players:

1. The Donut (Abelian Surface A)
2. The K3 Surface (S)
3. The OG6 Variety

He proved a chain reaction:

• Step 1: The strange "Shape Y" is actually built from a K3 surface.
• Step 2: This specific K3 surface is "isogenous" to the Kummer surface (which is the Donut folded in half). In simple terms, the K3 surface and the Donut are "cousins"—they share the same genetic code, just arranged slightly differently.
• Step 3: Because the K3 surface is a cousin of the Donut, and Shape Y is built from the K3 surface, then Shape Y is ultimately built from the Donut.
• Step 4: Since the OG6 variety is just a folded version of Shape Y, the OG6 variety is also built from the Donut.

4. The Formula: The Recipe Book

The paper doesn't just say "it's related"; it gives the exact recipe. Floccari writes out a formula showing exactly how to construct the OG6 variety's "soul" (its motive) using pieces of the Donut.

It looks something like this:

OG6 Variety = (Donut parts) + (Two Donuts combined) + (A Donut folded in half) + (Lots of tiny mathematical Lego bricks).

This formula is powerful because it tells us that if we understand the Donut, we automatically understand the OG6 variety.

5. Why Does This Matter? (The Consequences)

Why should a general audience care? Because this proof solves two famous, century-old riddles for these shapes:

• The Hodge Conjecture: This is a rule about how shapes are made of smaller, invisible building blocks. Floccari proved that for OG6 varieties, these blocks behave exactly as the rule predicts.
• The Tate Conjecture: This is a similar rule about how these shapes behave when you change the number system (like switching from real numbers to finite fields). He proved this holds true too.

The Big Picture Analogy

Imagine you have a Master Key (the Abelian surface).
For a long time, mathematicians had a Locked Door (the OG6 variety) and didn't know if the Master Key opened it.
Floccari found a Middle Key (the K3 surface) that fits both. He proved that the Middle Key is just a copy of the Master Key. Therefore, the Master Key does open the Locked Door.

In summary: This paper proves that a very complex, 6-dimensional mathematical shape is actually just a sophisticated remix of a simple 2-dimensional donut. This allows mathematicians to use the simple donut to solve difficult problems about the complex shape, confirming that the universe of these shapes is more connected and orderly than we thought.

Technical Summary

1. Problem Statement

The paper addresses the structure of the rational Chow motive of a specific class of hyper-Kähler varieties known as OG6-type varieties. These varieties arise as symplectic resolutions of singular moduli spaces of semistable sheaves on a complex Abelian surface $A$.

While it was previously established that the motives of smooth moduli spaces on Abelian surfaces (generalized Kummer varieties) and O'Grady's 10-dimensional varieties ($\widetilde{M}_{2v_0}$) belong to the tensor category generated by the motive of the underlying surface, the case of the 6-dimensional OG6 varieties ($\widetilde{K}_{v}$) remained open. The central question is whether the rational Chow motive $h(\widetilde{K}_v)$ belongs to the pseudo-Abelian tensor subcategory $\langle h(A) \rangle_{\text{CHM}}$ generated by the motive of the Abelian surface $A$.

2. Methodology

The author employs a multi-step strategy combining geometric constructions, Hodge theory, and motivic decomposition:

• Geometric Construction (Mongardi–Rapagnetta–Saccà): The proof relies on the construction of $\widetilde{K}_v$ as a resolution of a quotient. Specifically, $\widetilde{K}_v$ is related to a smooth hyper-Kähler variety $Y$ of K3$[3]$-type via a degree 2 symplectic birational involution. The resolution process involves a sequence of blow-ups and contractions of singular loci (specifically, blowing up singular points and contracting $\mathbb{P}^2$-bundles to quadrics).
• Birational Geometry and Moduli Spaces: The author proves that the K3$[3]$-type variety $Y$ is birational to a smooth moduli space $M_w(S, L)$ of stable sheaves on a K3 surface $S$.
• Transcendental Lattice Analysis:
  • The author establishes a chain of Hodge isometries between the transcendental lattices of the varieties involved.
  • Using Lemma 3.4, a rational Hodge isometry is induced between $H^2_{\text{tr}}(Y, \mathbb{Q})$ and $H^2_{\text{tr}}(\widetilde{K}, \mathbb{Q})$ (scaled by a factor of 2).
  • Using Remark 3.5, $H^2_{\text{tr}}(\widetilde{K}, \mathbb{Q})$ is identified with $H^2_{\text{tr}}(A, \mathbb{Q})$.
  • Crucially, the author shows that the K3 surface $S$ (associated with $Y$) is isogenous to the Kummer K3 surface $T$ (the minimal resolution of $A/\pm 1$).
• Motivic Decomposition:
  • The author utilizes the blow-up formula for motives to decompose the motives of the intermediate varieties in the resolution diagram.
  • The author invokes Bülles' theorem (which states that birational hyper-Kähler varieties have isomorphic motives) and results by Huybrechts (isogenous K3 surfaces have isomorphic motives) to link $h(Y)$ back to $h(A)$.

3. Key Contributions

• Explicit Motive Formula: The paper provides an explicit decomposition of the rational Chow motive of the OG6 variety $\widetilde{K}_v$ in terms of the motive of the Abelian surface $A$ and its associated Kummer variety $Km(A)$.
• Proof of Motivic Generation: It confirms that $h(\widetilde{K}_v) \in \langle h(A) \rangle_{\text{CHM}}$, resolving the open question for this specific deformation type.
• Isogeny of Motives: The work provides strong evidence for the conjecture that isogenous hyper-Kähler varieties have isomorphic rational Chow motives.

4. Main Results

Theorem 1.1 (Main Theorem):
Let $A$ be a complex Abelian surface, $v = 2v_0$ a Mukai vector with $v_0$ primitive, effective, and of square 2, and $H$ a $v$-generic polarization. The rational Chow motive of the OG6 variety $\widetilde{K}_v(A,H)$ is given by:
$$h(\widetilde{K}_v) = h(Km(A)(3)) \oplus h(Km(A) \times Km(A))(-1) \oplus h(Km(A \times A))(-1) \oplus 137h(Km(A))(-2) \oplus 512\mathbb{Q}(-3)$$
Note: $Km(B)$ denotes the Kummer variety $B/\pm 1$, and the notation $M(n)$ refers to the Tate twist.

Corollary 1.2 (Consequences):

1. Abelian and Finite Dimensional: The Chow motive of $\widetilde{K}_v$ is Abelian and Kimura finite-dimensional.
2. Hodge and Tate Conjectures: The Hodge conjecture and the Tate conjecture hold for $\widetilde{K}_v$ and all of its self-products. This is derived from the fact that the motive is generated by $A$, for which these conjectures are known to hold for self-products.

5. Significance

• Completion of the Classification: This result completes the picture for the motives of hyper-Kähler varieties constructed from Abelian surfaces, covering the generalized Kummer type, the 10-dimensional O'Grady type, and now the 6-dimensional O'Grady type.
• Validation of Conjectures: By proving the motive is Abelian, the paper immediately validates the Hodge and Tate conjectures for these varieties, which are notoriously difficult to approach directly via cohomology.
• Support for the "Isogenous Motives" Conjecture: The paper supports Conjecture 5.1, which posits that isogenous hyper-Kähler varieties have isomorphic motives. The author demonstrates that the motive of the OG6 variety depends entirely on the isogeny class of the underlying Abelian surface.
• Methodological Bridge: The paper successfully bridges the gap between the geometry of singular moduli spaces on Abelian surfaces and the well-understood geometry of moduli spaces on K3 surfaces via the intermediate K3$[3]$-type variety.

In summary, Floccari proves that the complex geometry of 6-dimensional O'Grady varieties is entirely controlled by the motive of the underlying Abelian surface, providing a precise formula and confirming major conjectures regarding algebraic cycles on these manifolds.