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On the smoothability problem with rational coefficients

This paper demonstrates that the existence of smooth rational algebraic cycles up to homological equivalence on smooth projective complex varieties would contradict the Hartshorne conjecture, while simultaneously providing an unconditional solution to a symplectic variant of this smoothing problem.

Original authors: Olivier Benoist, Claire Voisin

Published 2026-07-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Olivier Benoist, Claire Voisin

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 perfect city out of smooth, polished marble. In the world of mathematics, specifically a field called algebraic geometry, the "city" is a complex shape called a variety, and the "buildings" are special sub-shapes called algebraic cycles. For a long time, mathematicians have asked a simple but tricky question: Can every building in this city be smoothed out until it's perfectly round and free of jagged edges, without changing its fundamental identity?

Think of "homological equivalence" as a way of saying two buildings are the same if they occupy the same amount of space and wrap around the city in the same way, even if one is a jagged rock and the other is a smooth sphere. The big question is: If you have a jagged rock that represents a specific type of space, can you always melt it down and reshape it into a smooth sphere that still counts as the same rock? When we use whole numbers (integers) to count these shapes, the answer is a hard "no"—sometimes the jagged rocks are too weird to ever become smooth. But what if we use fractions (rational numbers)? Maybe the jagged rocks are just a mix of smooth spheres that we haven't figured out how to separate yet. This paper dives into that "fraction" version of the problem, exploring whether the universe of shapes is secretly made of smooth pieces, even if we can't see them directly.

The authors, Olivier Benoist and Claire Voisin, tackle this by looking at two famous mathematical ideas that seem unrelated. The first is the "smoothing problem" mentioned above: can we always find smooth shapes to represent our fractional counts? The second is Hartshorne's Conjecture, a bold guess about how shapes fit inside a giant projective space. Hartshorne's idea suggests that if a shape is small enough compared to the space it lives in, it must be a "complete intersection"—basically, it's formed by the simple crossing of a few smooth surfaces, like how two planes cross to make a line.

Here is the twist: The paper proves that these two ideas cannot both be true at the same time. If Hartshorne's Conjecture is correct (meaning small shapes are always simple crossings), then the answer to the smoothing problem with fractions must be "no." In other words, even if we allow ourselves to use fractions, there are still some jagged algebraic rocks that cannot be built from smooth spheres. The authors show this by imagining a specific type of mathematical playground called a Grassmannian (a space that organizes all possible sub-planes in a larger space). They demonstrate that if the shapes in this playground follow Hartshorne's rules, the "smoothness" we hoped for with fractions simply vanishes. It's like discovering that if your city follows a specific zoning law, you can't build a smooth park in the middle of a jagged district, no matter how much you try to mix the materials.

However, the story doesn't end with a dead end. The authors also explore a different world: symplectic geometry, which is like the physics version of this math, dealing with smooth, flowing shapes in a "symplectic" universe. In this world, they prove a happy result: yes, you can always smooth things out! They show that in a symplectic manifold (a smooth, curved space with a special kind of flow), any fractional count of a shape can be built from smooth, symplectic sub-shapes. This is a huge deal because it tells us that the "jaggedness" we found in the algebraic world isn't caused by some fundamental topological flaw in the shape of the universe itself. If the universe were just a smooth, symplectic blob, we could always smooth everything out. The fact that we can't in the algebraic world means the problem is specific to the rigid, algebraic rules of the game, not a general impossibility.

So, what is the final verdict? The paper doesn't solve the smoothing problem for rational coefficients once and for all; instead, it sets a trap. It proves that if you believe Hartshorne's Conjecture is true, you must accept that the smoothing problem has a negative answer. Since Hartshorne's Conjecture is widely believed to be true, this paper strongly suggests that the answer to the smoothing problem is "no." It's a clever logical maneuver: rather than finding a jagged rock and proving it can't be smoothed, they prove that if the rocks could be smoothed, the whole city would break the rules of geometry. On the flip side, their symplectic result confirms that there are no hidden, universal topological barriers stopping us from smoothing things out; the barrier is purely algebraic. The paper leaves us with a clear picture: the universe of algebraic shapes is more stubborn and jagged than we hoped, but only because of the specific rules of algebra, not because of the shape of space itself.

Technical Summary: On the Smoothability Problem with Rational Coefficients

Problem Statement
The paper addresses the "smoothability problem" for algebraic cycles on smooth projective complex varieties. Specifically, it investigates whether the $\mathbb{Q}$ -vector space of homologically non-trivial algebraic cycles, $H^{2c}(X, \mathbb{Q})_{\text{alg}}$ , is generated by the fundamental classes of smooth subvarieties of codimension $c$ . This is the rational analogue of a question posed by Borel and Haefliger regarding integral coefficients, for which counterexamples are known (e.g., Hartshorne, Rees, and Thomas). While the integral case is known to fail in general, no counterexample has been found for the rational case (Question 0.1), and it remains an open problem whether the original integral counterexamples (such as the second Chern class of the tautological bundle on $G(3,6)$ ) are $\mathbb{Q}$ -linear combinations of smooth subvarieties.

Methodology
The authors employ two distinct methodological approaches to address the problem:

Algebraic Geometry and Vector Bundle Splitting: To analyze the rational smoothability problem, the authors focus on Grassmannians $G(k, n)$ . They utilize the relationship between smooth codimension 2 subvarieties and rank 2 vector bundles (via Serre's construction and Barth–Lefschetz-type theorems). The core strategy involves embedding projective spaces $\mathbb{P}^k$ and $\mathbb{P}^{n-k}$ into $G(k, n)$ in various ways. By assuming the validity of Hartshorne's Conjecture (which posits that smooth subvarieties of high codimension in projective space are complete intersections, or equivalently, that high-rank vector bundles on $\mathbb{P}^n$ split into line bundles), the authors deduce constraints on the splitting behavior of vector bundles on the Grassmannian. They prove that if the conjecture holds, any rank 2 vector bundle on $G(k, n)$ (for sufficiently large $k, n$ ) must split as a direct sum of line bundles. This splitting forces the second Chern class $c_2(E)$ to be proportional to $c_1(L)^2$ , thereby restricting the cycle classes available.
Symplectic Topology and Rational Homotopy Theory: To address the existence of topological obstructions, the authors construct a symplectic analogue of the problem. They utilize Sullivan's rational homotopy theory to construct topological complex vector bundles on a compact symplectic manifold $(M, \omega)$ with prescribed Chern classes (specifically, multiples of given rational cohomology classes). They then apply the Auroux–Donaldson theorem, which guarantees the existence of asymptotically holomorphic sections of high tensor powers of these bundles whose zero loci are symplectic submanifolds.

Key Results

Incompatibility with Hartshorne's Conjecture (Theorem 0.3): The primary algebraic result establishes that Question 0.1 (rational smoothability) and Conjecture 0.2 (Hartshorne's conjecture on complete intersections) cannot both be true. Specifically, if Hartshorne's conjecture holds for codimension 2 (i.e., if Conjecture 1.1 regarding the decomposability of rank 2 bundles on $\mathbb{P}^n$ for $n \ge n_0$ is true), then Question 0.1 fails for codimension 2 cycles on Grassmannians $G(k, n)$ when $k, n-k$ are sufficiently large.
- Mechanism: Under the assumption of Hartshorne's conjecture, any smooth codimension 2 subvariety $Z \subset G(k, n)$ is a complete intersection, implying its cohomology class $[Z]$ is proportional to $c_1(L)^2$ . However, the space $H^4(G(k, n), \mathbb{Q})$ is not generated solely by such classes; thus, not all algebraic cycles can be smoothed.
- Extension: The authors extend this logic to unstable vector bundles (Corollary 1.9) and higher-rank bundles (Proposition 1.11), showing that positive answers to generalized splitting questions would impose severe linear relations on Chern classes, contradicting the generation of cohomology by smooth cycles.
Positive Result in the Symplectic Category (Theorem 0.4 / Theorem 2.5): The authors prove that for any compact symplectic $C^\infty$ manifold $(M, \omega)$ , the $\mathbb{Q}$ -vector space $H^{2c}(M, \mathbb{Q})$ is generated by the fundamental classes of symplectic $C^\infty$ submanifolds of codimension $2c$ .
- Mechanism: By constructing topological vector bundles with specific Chern classes via rational homotopy theory and applying the Auroux–Donaldson theorem, they demonstrate that any rational cohomology class can be realized as a linear combination of symplectic submanifolds.

Significance and Claims
The paper claims that Theorem 0.4 demonstrates the absence of "purely topological obstructions" to the rational smoothability problem in a very strong sense. Since the symplectic category (which is topological in nature regarding the existence of submanifolds) allows for the generation of all rational cohomology classes by smooth submanifolds, any potential counterexample to Question 0.1 in the algebraic category must arise from specific algebraic constraints rather than general topology.

Consequently, the authors argue that the incompatibility result (Theorem 0.3) serves as a "negative indication" concerning Hartshorne's conjecture. If Hartshorne's conjecture were true, it would force algebraic cycles to be "too rigid" to generate the full rational cohomology, contradicting the expectation that rational coefficients should allow for smoothing. The paper concludes that the failure of rational smoothability (if it occurs) is deeply tied to the rigidity predicted by Hartshorne's conjecture, rather than a failure of topological realizability.

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