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Non-liftable varieties via etale cohomology rings

The paper constructs a smooth projective variety in positive characteristic whose $\mathbb{Q}_{\ell}$ -coefficient étale cohomology ring cannot be realized as a scalar extension of any graded $\mathbb{Q}$ -algebra, thereby establishing a new obstruction to lifting varieties to characteristic zero.

Original authors: Runjie Hu, Siqing Zhang

Published 2026-07-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Runjie Hu, Siqing Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to build a house that exists in two different worlds at once: a world made of smooth, flowing water (characteristic zero) and a world made of rigid, pixelated blocks (positive characteristic). In mathematics, these "worlds" are different types of number systems used to describe shapes called varieties. For a long time, mathematicians have wondered if every beautiful shape built in the blocky world can be "lifted" or translated into the smooth world without breaking. It's like asking if a sculpture made of LEGO bricks can be perfectly melted down and reformed into a statue made of liquid glass.

To answer this, mathematicians use a special tool called "cohomology." Think of cohomology as a unique fingerprint or a DNA sequence for a shape. It records how the shape is connected, how many holes it has, and how its parts fit together. Usually, if a shape can be lifted from blocks to glass, its fingerprint in the blocky world should look like a simple, scaled-up version of a fingerprint from the smooth world. If the fingerprints don't match up, the shape is stuck in the blocky world and cannot be lifted. This paper dives deep into the structure of these fingerprints to see if there are shapes that are fundamentally "blocky" and cannot exist in the smooth world, no matter how hard you try.

The authors of this paper, Runjie Hu and Siqing Zhang, have discovered a brand new type of shape that refuses to lift. They constructed a specific, smooth, and connected 3D shape (a variety) using a prime number $p$ (where $p > 0$ ) as their building material. Their main finding is that the "fingerprint" of this shape, when analyzed with a specific set of mathematical tools (using coefficients from a field called $\mathbb{Q}_\ell$ ), is so strange that it cannot be the result of simply scaling up a fingerprint from the smooth world.

To understand why this is a big deal, imagine you have a set of instructions written in a secret code. If you could translate that code into English, you would have a "rational form" of the instructions. The authors proved that for their special shape, no such English translation exists. The code is too tangled; it requires a specific kind of mathematical "magic" (a division algebra) that simply doesn't exist in the smooth world. Because the fingerprint cannot be translated, the shape itself cannot be lifted to the smooth world. This is a new kind of proof that doesn't rely on the shape having obvious flaws or "pathologies" (like broken pieces or weird singularities). Instead, the shape is perfectly smooth and well-behaved, but its internal mathematical DNA is just too exotic for the smooth world to handle.

The construction of this shape is a bit like a complex game of "connect the dots" using a supersingular elliptic curve (a special kind of loop). The authors took this curve and created a giant, multi-layered structure by blowing up (expanding) specific points and lines. They carefully arranged these expansions so that the resulting shape's fingerprint would encode a "quaternion division algebra." You can think of this algebra as a set of four-dimensional directions that are impossible to flatten into a two-dimensional plane. The shape's fingerprint remembers these four-dimensional directions so clearly that if the shape could be lifted to the smooth world, it would force a four-dimensional object to fit into a two-dimensional space—a mathematical impossibility.

The authors then took this 3D shape and embedded it into a much larger 9-dimensional space, blowing it up one last time to create a final shape, $X$ . This final shape is "simply connected," meaning it has no holes or loops that can't be shrunk to a point, making it a very clean and simple object in terms of its connectivity. Despite being simple, its fingerprint still remembers the impossible four-dimensional directions from the earlier steps. The paper proves that because the fingerprint of $X$ cannot be defined over the rational numbers (the "smooth" base), the shape $X$ cannot be lifted to characteristic zero.

This result is significant because it answers a question posed by the famous mathematician Alexander Grothendieck. He wondered if the "homotopy types" (the fundamental shape structures) of these blocky shapes, once completed, would look like finite collections of building blocks (finite CW complexes). The authors' example suggests a "no" to this question, showing that these shapes can have a complexity that finite building blocks just can't capture.

In short, the paper proves the existence of a smooth, perfectly formed shape in a blocky mathematical world that is fundamentally incompatible with the smooth world. It's not broken, and it's not weird in the usual sense; it's just that its internal mathematical structure is so unique and rigid that it cannot be translated into the language of the smooth world. This provides a new, purely algebraic reason why some mathematical objects are stuck in their specific environment, forever unable to cross over to the other side.

Technical Summary: Non-Liftable Varieties via Étale Cohomology Rings

Problem Statement
The paper addresses the problem of lifting smooth projective varieties from positive characteristic $p$ to characteristic zero. While various obstructions to such liftings are known (e.g., those arising from finite group actions, failures of Hodge theory, or specific cohomological invariants like Betti numbers), the authors seek a new type of obstruction rooted intrinsically in the structure of $\ell$ -adic étale cohomology rings. Specifically, they investigate whether the étale cohomology ring $H^*(X, \mathbb{Q}_\ell)$ of a variety $X/\mathbb{F}_p$ can be defined over $\mathbb{Q}$ . A graded $\mathbb{Q}_\ell$ -algebra $R$ admits a $\mathbb{Q}$ -form if there exists a graded $\mathbb{Q}$ -algebra $R_\mathbb{Q}$ such that $R_\mathbb{Q} \otimes_\mathbb{Q} \mathbb{Q}_\ell \cong R$ . The authors aim to construct a variety where this condition fails, thereby providing a topological obstruction to lifting as suggested by Sullivan.

Methodology
The construction proceeds in two main stages, utilizing blow-ups to encode algebraic structures into the cohomology ring:

Construction of the Auxiliary Variety $Z$ :
- The authors start with a supersingular elliptic curve $E$ over $\mathbb{F}_p$ . Its rational endomorphism algebra $D = \text{End}(E) \otimes \mathbb{Q}$ is a four-dimensional quaternion division algebra over $\mathbb{Q}$ .
- They select generators $\alpha, \beta \in \text{End}(E)$ and a set of indices $I = \{\infty, 0, 1, \alpha, \beta\}$ .
- Inside the threefold $Y = (E \times E) \times \mathbb{P}^1$ , they consider graphs $\Gamma_\gamma$ of these endomorphisms (and coordinate copies).
- To ensure disjointness and distinct labeling, they choose finite, pairwise disjoint subsets $T_\gamma \subset \mathbb{P}^1$ with distinct cardinalities $m_\gamma$ .
- They define $Z$ as the blow-up of $Y$ along the disjoint union of curves $C_{\gamma, t} = \Gamma_\gamma \times \{t\}$ .
- Key Mechanism: The cohomology ring $H^*(Z)$ is shown to intrinsically recover a labeled configuration of subspaces in $H^1(Z) \cong V \oplus V$ (where $V = H^1(E)$ ). Specifically, the "rank-drop locus" of multiplication maps in the cohomology ring identifies exceptional divisors corresponding to each $\gamma$ , which in turn encode the kernels $K_\gamma$ . These kernels reconstruct the action of the division algebra $D$ on a two-dimensional vector space.
Construction of the Simply-Connected Variety $X$ :
- To satisfy the condition of being simply-connected (a requirement for Sullivan's topological obstruction), the authors embed $Z$ into $\mathbb{P}^9_k$ and define $X$ as the blow-up of $\mathbb{P}^9_k$ along $Z$ .
- They demonstrate that the cohomology ring $H^*(X)$ allows for the intrinsic recovery of $H^*(Z)$ . By identifying the hyperplane class and the exceptional divisor class through specific algebraic relations (e.g., annihilation properties and kernel structures of multiplication maps), they reconstruct the graded algebra of $Z$ from $X$ .
- This ensures that any obstruction present in $H^*(Z)$ persists in $H^*(X)$ .

Key Contributions and Results

Theorem 0.1: The paper constructs a simply-connected smooth projective variety $X$ over $\mathbb{F}_p$ such that its étale cohomology ring $H^*(X, \mathbb{Q}_\ell)$ admits no $\mathbb{Q}$ -form.
Non-Liftability: As a direct consequence (via Lemma 1.1), the variety $X$ does not admit a lifting to characteristic zero.
Intrinsic Recovery: The authors prove that the labeled configuration of subspaces encoding the quaternion division algebra action can be recovered solely from the abstract graded algebra structure of the cohomology ring, without reference to the geometric construction.
Contradiction via Division Algebras: The proof that no $\mathbb{Q}$ -form exists relies on the fact that if such a form existed, the recovered configuration would define an action of the 4-dimensional division algebra $D$ on a 2-dimensional $\mathbb{Q}$ -vector space, which is impossible.

Significance
The paper claims to provide the first example realizing Sullivan's topological obstruction to characteristic zero lifting, which posits that lifting is obstructed if the $\mathbb{Q}_\ell$ -homotopy data does not descend to $\mathbb{Q}$ .

Novelty of Obstruction: Unlike previous obstructions that rely on geometric pathologies (e.g., failure of Kodaira vanishing or specific Frobenius behaviors), this obstruction arises from the intrinsic structure of the $\ell$ -adic cohomology ring itself.
Relation to Grothendieck's Expectation: The result provides a negative answer to a question regarding Grothendieck's expectation that the $\ell$ -completion of the étale homotopy type of a smooth proper variety in characteristic $p$ should be homotopy equivalent to the $\ell$ -completion of a finite CW complex. The existence of a variety whose cohomology ring has no $\mathbb{Q}$ -form implies a fundamental difference between the homotopy types in characteristic $p$ and characteristic 0.
Comparison to Prior Work: The authors distinguish their work from other non-liftability results (e.g., Esnault-Srinivas-Stix, van Dobben de Bruyn) by noting that their example is simply-connected, whereas previous examples relying on $\ell$ -adic topology often depended on non-trivial étale fundamental groups.

The paper concludes that the $\ell$ -adic étale cohomology rings of varieties in characteristic $p$ and 0 possess an essential difference, demonstrating that the "nice" behavior expected of these rings can fail in a way that prevents lifting.

Technical Summary: Non-Liftable Varieties via Étale Cohomology Rings

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