Smooth subvarieties of Jacobians

Here is an explanation of the paper "Smooth subvarieties of Jacobians" by Olivier Benoist and Olivier Debarre, translated into everyday language with creative analogies.

The Big Picture: Can You Build a Smooth House with Rough Bricks?

Imagine you have a giant, perfect, smooth building (a mathematical object called a variety). Inside this building, there are smaller rooms or structures (called subvarieties).

In mathematics, we often care about the "shape" or "cohomology class" of these rooms. Think of a cohomology class as a blueprint or a fingerprint that describes the room's essential geometry.

The Big Question:
If you have a blueprint for a room inside this building, can you always build that room using only smooth, perfect bricks? Or, sometimes, is the blueprint so weird that you must use rough, jagged, or broken bricks to build it?

For a long time, mathematicians knew the answer was "Yes, you can always use smooth bricks" for small buildings (dimensions up to 5). But for bigger buildings, they suspected the answer might be "No."

This paper proves that No, you cannot always use smooth bricks. They found specific blueprints in a 6-dimensional building that cannot be built using only smooth sub-structures.

The Setting: The Jacobian (The "Perfectly Symmetric City")

To prove this, the authors chose a very specific type of building called a Jacobian.

Analogy: Imagine a city built on a perfect torus (a donut shape) that has a very high degree of symmetry. In math, this is a complex abelian variety.
The "Theta" Class: This city has a special, fundamental "master blueprint" called $\theta$ . All other blueprints in the city are usually just combinations of this master blueprint.
The Target: The authors looked at a specific type of blueprint derived from the master one: $\frac{\theta^c}{c!}$ . They wanted to know: Is this specific blueprint the sum of blueprints of smooth rooms?

The Discovery: The "Rough" Blueprint Exists

The authors found that for certain sizes of the city (specifically dimension 6 and higher) and certain types of blueprints, the answer is no.

The Result: There are "minimal" blueprints (like the one for a 4-dimensional room in a 6-dimensional city) that are mathematically valid, but they cannot be constructed by adding up the blueprints of any smooth, perfect rooms.
The Implication: If you try to build this specific room, you are forced to use "rough" or singular bricks. You cannot do it with smooth ones.

This is a big deal because dimension 6 is the smallest possible size where this weirdness happens. For any building smaller than 6 dimensions, you can always use smooth bricks.

How They Did It: The "Magic Scales" (Complex Cobordism)

How do you prove you can't build something with smooth bricks? You can't just try every combination; there are too many. The authors used a powerful new tool called Complex Cobordism.

The Analogy of the Magic Scales:
Imagine you have a set of magical scales.

The Old Method (Riemann-Roch): Previously, mathematicians tried to weigh the bricks using a standard scale. It worked for small buildings, but for big, complex ones, the scale got jammed or the math became too messy to calculate.
The New Method (Cobordism): The authors used a "Magic Scale" (Complex Cobordism) that doesn't just weigh the bricks; it checks the chemical composition of the bricks.

How the Magic Scale Works:

Smooth Bricks: If a room is built with smooth bricks, the Magic Scale gives it a very specific "weight" or "signature." It turns out that for these specific blueprints, the signature of a smooth room must be even (divisible by 2).
The Target Blueprint: The authors calculated the signature of their target blueprint ( $\frac{\theta^c}{c!}$ ). They found that its signature is odd (not divisible by 2).
The Conclusion: Since the target blueprint has an "odd" signature, and any combination of smooth rooms would result in an "even" signature, it is mathematically impossible to build the target blueprint using only smooth rooms. It must contain rough, singular parts.

Why This Matters

It's the Smallest Case: Before this, we didn't know if this problem existed in 6 dimensions. They proved it does. This is the "lowest possible" dimension where the universe of geometry gets weird.
New Tools: They showed that Complex Cobordism is a better tool than the old methods for solving these types of geometry puzzles. It's like switching from a ruler to a laser scanner; it sees details the ruler missed.
The "Very General" Condition: They proved this for "very general" Jacobians. Think of this as saying, "If you pick a random city from the infinite set of all possible symmetric cities, this weirdness will almost certainly happen."

Summary in One Sentence

The authors proved that in a 6-dimensional geometric world, there are valid shapes that cannot be built using only smooth, perfect pieces, and they used a high-tech "magic scale" (complex cobordism) to weigh the evidence and prove it.

Here is a detailed technical summary of the paper "Smooth subvarieties of Jacobians" by Olivier Benoist and Olivier Debarre.

1. Problem Statement

The paper addresses a fundamental question in algebraic geometry regarding the generation of algebraic cohomology classes. Let $X$ be a smooth projective complex variety of dimension $n = c + d$ . The group $H^{2c}(X, \mathbb{Z})_{\text{alg}}$ consists of integral cohomology classes generated by the cycle classes of codimension $c$ algebraic subvarieties.

Question 1.1 (Borel–Haefliger): Is $H^{2c}(X, \mathbb{Z})_{\text{alg}}$ generated by the classes of smooth subvarieties of $X$ ?

While the answer is positive for low dimensions ( $n \leq 5$ ) and low codimensions ( $c \leq 1$ ), counterexamples exist for higher dimensions. Previous work by Hartshorne, Rees, and Thomas, as well as Debarre (1995), established that for Jacobians of curves, certain minimal cohomology classes are algebraic but not generated by smooth subvarieties when the dimension is sufficiently large relative to the codimension.

The authors aim to:

Extend Debarre's counterexamples to a broader range of codimensions $c$ and dimensions $n$ .
Lower the dimension bound for the existence of such counterexamples, specifically reaching the theoretical minimum dimension $n=6$ for $c=2$ .
Provide a unified proof mechanism that avoids the computational intractability of previous methods in high codimensions.

2. Methodology

The authors employ complex cobordism ( $MU^*$ ) as their primary tool, a technique pioneered in this context by Totaro. This approach replaces the reliance on the Hirzebruch–Riemann–Roch theorem and the Serre construction (which were limited to low codimensions, specifically $c=2$ ) with divisibility properties of Chern numbers.

The methodology proceeds in three main stages:

A. Congruences of Chern Numbers (Section 2)

The authors analyze the structure of the complex cobordism ring $\pi_*(MU)$ and its relationship with integral cohomology via the Hurewicz morphism $H$ .

They utilize the generators of $\pi_*(MU)$ and $H^*(MU, \mathbb{Z})$ to define Segre classes $s_i$ .
Key Technical Result (Proposition 2.5): They establish a precise congruence condition for the top Segre class. Specifically, they determine when a Chern number associated with $s_i + 2^h Q$ takes values divisible by $2^{e+h} $. This depends on the function$ \alpha(m) $, which counts the number of ones in the binary expansion of$ m$.
The condition derived is: $\alpha(i + e + h - 1) > e + 2h - 2$ . This number-theoretic condition is crucial for proving that certain cohomology classes cannot be represented by smooth cycles.

B. Complex Cobordism of Abelian Varieties (Section 3)

The authors apply these number-theoretic results to Jacobians (which are abelian varieties).

They describe the complex cobordism of an abelian variety $X$ of dimension $n$ by identifying it with a torus $(S^1)^{2n}$ .
Using the Künneth formula, they construct a basis for $MU_*(X)$ using subtori.
Proposition 3.3: They prove that for a smooth subvariety $Y \subset X$ , the class $[f] \in MU_{2d}(X)$ can be expanded in terms of the basis elements $\tau_k$ (which are dual to powers of the polarization $\theta$ ). The coefficients in this expansion are constrained by the geometry of $Y$ .
Proposition 3.5: Using a Barth–Lefschetz-type theorem, they show that for a smooth subvariety $Y$ of codimension $c$ , the Segre classes $s_i(Y)$ are restrictions of powers of $\theta$ , and the cohomology class $[Y]$ is a multiple of $\theta^c/c!$ .

C. The Main Argument (Section 3.3)

The core proof combines the expansion of the cobordism class with the divisibility results from Section 2.

They assume a smooth subvariety $Y$ exists with class $\lambda \theta^c/c!$ .
By evaluating characteristic numbers using the expansion from Proposition 3.3 and the divisibility constraints from Proposition 2.5, they derive a contradiction if $\lambda$ is odd and specific conditions on $c$ and $n$ are met.
Specifically, they show that the coefficient $a_c$ in the expansion must be even under certain conditions, implying that any smooth subvariety's class must be an even multiple of the minimal class.

D. Alternative Approach via K-Theory (Section 4)

For codimension $c=4$ , the authors use complex topological K-theory and the Grothendieck–Riemann–Roch (GRR) theorem.

They compute the Chern character of the pushforward of the structure sheaf of a smooth subvariety.
By analyzing the integrality of the Chern character components, they derive stronger divisibility constraints (multiples of 2 or 4) for the cohomology class of smooth subvarieties in codimension 4.

3. Key Contributions and Results

Main Theorem (Theorem 1.2 / Theorem 3.7)

Let $(X, \theta)$ be a very general complex Jacobian of dimension $n$ . Let $c$ be a non-negative integer such that:

$\alpha(c + \alpha(c)) > \alpha(c)$ (a condition satisfied by $c \in \{2, 4, 5, 8, 9, 12, 16, 17, \dots\}$ ).
$n \geq 4c - 2$ .

Then, the integral classes $\lambda \frac{\theta^c}{c!}$ where $\lambda$ is odd are algebraic but cannot be written as $\mathbb{Z}$ -linear combinations of cycle classes of smooth subvarieties of $X$ .

Corollary 1.3 (Dimension 6 Counterexample)

By applying the theorem with $c=2$ and $n=6$ :

There exists a smooth projective complex variety $X$ of dimension 6 such that $H^4(X, \mathbb{Z})_{\text{alg}}$ is not generated by classes of smooth subvarieties.
Significance: This is the lowest possible dimension for such a counterexample, as the property holds for all varieties of dimension $\leq 5$ .

Theorem 1.4 (Codimension 4 Results)

For very general Jacobians of dimension $n$ :

If $n \geq 12$ and $\lambda$ is odd, $\lambda \frac{\theta^4}{4!}$ is not a sum of smooth classes.
If $n \geq 14$ and $\lambda$ is not divisible by 4, $\lambda \frac{\theta^4}{4!}$ is not a sum of smooth classes.
This improves upon previous bounds for codimension 4.

4. Significance

Optimal Dimension Bound: The paper resolves the question of the minimal dimension for counterexamples to Question 1.1. It proves that dimension 6 is the sharp bound, closing the gap left by previous results which required higher dimensions.
Generalization of Codimension: The authors extend the known range of counterexamples from $c=2$ to a wide set of codimensions $c$ satisfying the binary expansion condition.
Methodological Shift: The paper demonstrates the power of complex cobordism over classical Riemann–Roch methods for studying algebraic cycles in high codimensions. The use of Segre classes and their divisibility properties provides a more robust framework for handling the arithmetic constraints of algebraic cycles.
Refinement of Algebraic Cycle Theory: The results deepen the understanding of the difference between the group of algebraic cycles and the group generated by smooth cycles, highlighting that singularities are not just a technical nuisance but a necessary feature for generating certain cohomology classes in specific geometric contexts (Jacobians).

In summary, Benoist and Debarre provide a definitive answer to the generation of algebraic cohomology classes by smooth subvarieties in the context of Jacobians, utilizing sophisticated tools from stable homotopy theory (cobordism) to achieve optimal bounds and new generalizations.