Some remarks on L-equivalence for cubic fourfolds and hyper-Kähler manifolds

1. Problem Statement and Context

The paper investigates the relationship between $L$-equivalence and $D$-equivalence (derived equivalence) for specific classes of algebraic varieties: cubic fourfolds and projective hyper-Kähler manifolds.

• $L$-equivalence: Two varieties $X$ and $Y$ are $L$-equivalent if there exists an integer $r > 0$ such that $\mathbb{L}^r ([X] - [Y]) = 0$ in the Grothendieck ring of varieties $K_0(\text{Var}_k)$, where $\mathbb{L} = [\mathbb{A}^1_k]$. This concept arises because the class of the affine line is a zero-divisor in this ring.
• $D$-equivalence: Two varieties are $D$-equivalent if their bounded derived categories of coherent sheaves are equivalent ($D^b(X) \cong D^b(Y)$).
• The Conjecture: A conjecture by Meinsma posits that for projective hyper-Kähler manifolds, $L$-equivalence implies $D$-equivalence. Similarly, for cubic fourfolds, the authors explore whether $L$-equivalence implies isomorphism (which, for cubic fourfolds, is equivalent to $D$-equivalence due to the Bondal–Orlov reconstruction theorem) or at least a weaker form of equivalence known as being Fourier–Mukai (FM) partners.

The central question is: Does $L$-equivalence force geometric or derived equivalence in these specific high-dimensional settings?

2. Methodology

The authors employ a combination of Hodge theory, lattice theory, and derived category techniques. The core methodology relies on the following logical chain:

1. Hodge-Theoretic Implications of $L$-equivalence:
   Using results by Efimov and Meinsma, the authors utilize the fact that $L$-equivalence implies equality of classes in the Grothendieck ring of Hodge structures. Specifically, if $X$ and $Y$ are $L$-equivalent, their transcendental Hodge structures $T(X)$ and $T(Y)$ are related.
2. Lattice-Theoretic Analysis:
   The authors analyze the transcendental lattices $T(X)$ and $T(Y)$ as Hodge lattices. They utilize:
   • Discriminant Groups: Showing that $L$-equivalence implies equality of discriminants for unimodular ambient lattices.
   • Endomorphism Rings: Assuming $\text{End}(T(X)) = \mathbb{Z}$ (which holds for very general varieties), they prove that equality of classes in the Grothendieck ring of Hodge structures implies the existence of a Hodge isometry between $T(X)$ and $T(Y)$ (possibly up to a twist by $\pm 1$).
   • Gluing Subgroups: Using Nikulin's theory of primitive embeddings to extend isometries from the transcendental part $T(X)$ to the full cohomology lattice $H^*(X, \mathbb{Z})$ or the Néron–Severi lattice.
3. Torelli Theorems:
   Once a Hodge isometry between the relevant cohomology lattices is established, the authors apply Torelli theorems:
   • For cubic fourfolds: The global Torelli theorem implies that a Hodge isometry preserving the polarization implies isomorphism.
   • For hyper-Kähler manifolds: The global Torelli theorem implies that a monodromy-preserving Hodge isometry implies birationality.
4. Derived Category Connections:
   For hyper-Kähler manifolds of K3$[n]$ type, birationality combined with specific lattice conditions implies $D$-equivalence (via results by Hoshiyama and others).

3. Key Contributions and Results

The paper provides a nuanced picture, proving strong results for "very general" cases while constructing counterexamples for "special" cases.

A. Cubic Fourfolds

Let $Y \subset \mathbb{P}^5$ be a smooth cubic fourfold.

1. Very General Case (Theorem 1.3 / Theorem 3.3):
   If $Y$ is a very general cubic fourfold (meaning it does not belong to any Hassett divisor $C_d$) and $Y'$ is $L$-equivalent to $Y$, then $Y \cong Y'$.

   • Significance: Since cubic fourfolds are determined by their derived categories (Bondal–Orlov), this implies $L$-equivalence implies $D$-equivalence in the very general case.

2. Special Case (Counterexample):
   The authors observe that for special cubic fourfolds (those containing specific surfaces, e.g., in the Hassett divisor $C_{20}$), $L$-equivalence does not imply isomorphism.

   • Example: They cite Fan and Lai's construction of two cubic fourfolds $Y, Y' \in C_{20}$ related by a Cremona transformation. These are $L$-equivalent and birational but not isomorphic.

3. Fourier–Mukai Partners (Theorem 1.5 / Theorem 3.6):
   For a very general cubic fourfold $Y$ in a specific Hassett divisor $C_d$ (where $d \equiv 0, 2 \pmod 6$, $d > 6$, and $9 \nmid d$), if $Y'$ is $L$-equivalent to $Y$, then $Y$ and $Y'$ are Fourier–Mukai partners.

   • Definition: They have equivalent Kuznetsov components ($A_Y \cong A_{Y'}$), even if the full derived categories are not equivalent.
   • Conjecture: The authors conjecture that $L$-equivalence always implies being FM partners for cubic fourfolds.

B. Hyper-Kähler Manifolds

The authors provide evidence supporting Meinsma's conjecture ($L$-equivalence $\implies$ $D$-equivalence) for projective hyper-Kähler manifolds under specific conditions.

1. General Framework (Theorem 1.7):
   Let $X, X'$ be deformation equivalent, $L$-equivalent projective hyper-Kähler manifolds with $\text{End}(T(X)) = \mathbb{Z}$.

   • LSV Manifolds: If $X, X'$ are birational to Laza–Saccà–Voisin (LSV) manifolds associated with very general cubic fourfolds, then $X$ and $X'$ are birational.
   • Moduli Spaces on Kuznetsov Components: If $X, X'$ are moduli spaces of objects in the Kuznetsov component of very general cubic fourfolds (with Mukai vectors of the same square), then $X$ and $X'$ are birational. If they are of K3$[n]$ type, they are $D$-equivalent.
   • OG10 Type: If $X$ is of OG10 type with specific lattice conditions ($NS(X) = U \oplus A_2(-1)$, $\text{disc } T(X) = 1$), then $X$ and $X'$ are birational and $D$-equivalent.
   • Hilbert Schemes: If $X$ is birational to $S[n]$ (Hilbert scheme of points on a K3 surface $S$) and satisfies lattice coprimality conditions, then $X$ and $X'$ are $D$-equivalent.

2. Mechanism:
   The proofs rely on showing that $L$-equivalence forces the transcendental lattices $T(X)$ and $T(X')$ to be Hodge isometric. Under the assumption $\text{End}(T(X)) = \mathbb{Z}$ and specific rank/discriminant constraints, this isometry extends to the full cohomology lattice, triggering the Torelli theorem to yield birationality. For K3$[n]$ types, birationality plus lattice constraints yields $D$-equivalence.

4. Significance

• Validation of Meinsma's Conjecture: The paper provides strong affirmative evidence for Meinsma's conjecture in the context of hyper-Kähler manifolds, particularly those arising from cubic fourfolds and K3 surfaces. It establishes that for "very general" members of these families, $L$-equivalence is a very strong condition, essentially forcing the varieties to be birational or isomorphic.
• Clarification of the "Special" Case: The paper highlights that the implication $L$-equivalence $\implies$ Isomorphism fails for special cubic fourfolds. This distinction is crucial for understanding the limits of motivic invariants.
• Refinement of Equivalence Notions: The work clarifies the hierarchy of equivalences. While $L$-equivalence does not always imply isomorphism for cubic fourfolds, it consistently implies the weaker notion of being Fourier–Mukai partners (equivalence of Kuznetsov components).
• Methodological Bridge: The paper successfully bridges the gap between the motivic world (Grothendieck rings) and the Hodge-theoretic world (transcendental lattices), demonstrating that motivic relations impose strict constraints on the underlying Hodge structures of these complex manifolds.

In summary, Billi and Li Bassi demonstrate that while $L$-equivalence is a subtle motivic invariant, for the specific and important classes of cubic fourfolds and hyper-Kähler manifolds, it acts as a powerful rigidity tool, often forcing the varieties to be isomorphic, birational, or derived equivalent, provided they are sufficiently general.