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The Chow-Kontsevich dilogarithm

This paper constructs a regulator in characteristic $p$ and derives an infinitesimal invariant for certain cycles by utilizing a variant of the Kontsevich $1\frac{1}{2}$ -logarithm function.

Original authors: Sinan Ünver

Published 2026-06-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sinan Ünver

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

The Big Picture: Measuring Shapes in a "Fuzzy" World

Imagine you are an architect trying to measure the volume of a building. In the real world (what mathematicians call "characteristic 0"), you have standard tools: rulers, protractors, and formulas like the one for the area of a triangle. You can measure things precisely.

However, this paper deals with a very strange, "fuzzy" version of geometry. Imagine the building is made of a material that is slightly squishy or has a tiny bit of "noise" in it. In math terms, this is working over a field of characteristic $p$ (a specific type of number system used in advanced algebra) and looking at dual numbers (numbers that have a tiny "error" or "infinitesimal" part attached to them, like $x + \epsilon$ , where $\epsilon$ is so small it's almost zero but not quite).

The author's goal is to build a new measuring tool (called a "regulator") that can take a complex shape (a "cycle") in this fuzzy world and assign it a single, meaningful number.

The Problem: Two Different Rulers

In the "normal" world, mathematicians already have a tool called the dilogarithm. Think of this as a special ruler that measures the "volume" of certain geometric shapes. It works beautifully when the world is crisp and clear.

But when the world gets "fuzzy" (characteristic $p$ ), the old ruler breaks. It's like trying to measure a liquid with a ruler made of wood; the wood absorbs the liquid, and the measurement gets messed up.

The paper points out that in this fuzzy world, there isn't just one way to measure. In fact, there seem to be two independent rulers needed to get the full picture:

The Old Ruler ( $\rho$ ): This is a modified version of the standard ruler, adapted to work in the fuzzy world.
The New Ruler ( $\rho_K$ ): This is the star of the show. It is a brand-new tool based on a function invented by the famous mathematician Kontsevich.

The New Tool: The Chow-Kontsevich Dilogarithm

The author constructs a new function called the Chow-Kontsevich dilogarithm (denoted as $\rho_K$ ).

What it does: It takes three functions (think of them as three different maps or coordinates) defined on a curve (a line or a loop) and combines them to produce a single number.
The "Kontsevich" Twist: The secret sauce of this new tool is a function Kontsevich called the "1 1/2-logarithm."
- Analogy: Imagine a standard logarithm is a straight line. A "1 1/2-logarithm" is like a line that has been bent or folded in a very specific, quirky way that only works in this fuzzy, characteristic $p$ world. It satisfies a special equation (like a puzzle piece fitting perfectly) that the standard logarithm doesn't.
Why it's special: The paper proves that this new ruler ( $\rho_K$ ) is independent of the old one ( $\rho$ ). If you change the "fuzziness" of your material, the old ruler changes its reading in one way, but the new ruler changes in a completely different way. You need both to understand the shape fully.

The Main Theorem: The Projective Line Test

To prove the tool works, the author tests it on the simplest possible shape: a projective line (which is like a circle or a straight line that wraps around).

The Result: When the author applies the new ruler to three specific points on this line, the result is a number calculated using that quirky "1 1/2-logarithm" function.
The Significance: This confirms that the new tool isn't just a random invention; it connects directly to the deep mathematical structures (K-theory) that mathematicians expect to exist in this fuzzy world.

The Second Part: Measuring "Cycles" (Moving Shapes)

The paper goes further. It doesn't just measure static points; it measures cycles.

Analogy: Imagine a cycle is a moving sculpture. It's a shape that exists in a 3D space but is defined by equations.
The Invariant: The author defines a way to measure these moving sculptures. The most important property of this measurement is stability.
- If you have two sculptures that look slightly different but are "equivalent" when you ignore the tiny fuzziness (specifically, if they are the same modulo $t^2$ ), the new ruler gives them the exact same number.
- This is like saying: "Even if the paint on the sculpture is slightly smudged, the underlying shape is identical, so my measurement should be identical."

Summary of Claims

Construction: The author built a new mathematical function ( $\rho_K$ ) that acts as a regulator (a measuring device) for curves in a specific type of fuzzy geometry (characteristic $p$ ).
Independence: This new function is mathematically distinct from the existing "infinitesimal" ruler. They provide different information.
Connection: The new function is directly linked to Kontsevich's "1 1/2-logarithm," a specific formula that behaves uniquely in this number system.
Application to Cycles: The author used this function to define an "invariant" for cycles (shapes) in 3D space. This invariant is robust: it doesn't change if the shape is slightly altered in a specific, controlled way.

What the paper does NOT claim:

It does not claim this has any immediate use in physics, engineering, or medicine.
It does not claim to solve the "scissors congruence" problem (cutting shapes into pieces) in the real world, though it draws inspiration from it.
It does not predict future technologies. It is purely a theoretical construction within the realm of algebraic geometry.

In short, the paper is about inventing a new, specialized ruler for a very abstract, "fuzzy" mathematical universe, proving that this ruler works, and showing that it measures things differently than any ruler we had before.

Technical Summary: The Chow-Kontsevich Dilogarithm

Problem Statement
The paper addresses the construction of regulator maps and infinitesimal invariants for algebraic cycles in characteristic $p \geq 5$ . While infinitesimal analogs of the Dehn invariant and dilogarithm maps have been well-established in characteristic 0 (via the work of Cathelineau, Bloch, Esnault, and Unver), the situation in characteristic $p$ presents distinct challenges. Specifically, the standard logarithm function is undefined due to denominators in its power series expansion. Furthermore, the structure of the indecomposable part of $K_3(k_2)$ (where $k_2 = k[t]/(t^2)$ ) differs in characteristic $p$ , suggesting the existence of two independent dilogarithm functions rather than one. The paper seeks to define a "Chow-Kontsevich dilogarithm" $\rho_K$ on curves over $k_2$ that serves as a characteristic $p$ analog to the infinitesimal Chow dilogarithm, utilizing Kontsevich's $1\frac{1}{2}$ -logarithm function.

Methodology
The construction proceeds through several technical stages:

Characteristic $p$ Logarithms and Dilogarithms:
The author utilizes Kontsevich's $1\frac{1}{2}$ -logarithm function, defined as $\mathcal{L}_1(s) = \sum_{1 \leq i \leq p-1} \frac{s^i}{i}$ . This function satisfies the fundamental four-term functional equation of information theory. Using this, a characteristic $p$ additive dilogarithm $\ell i^{(p)}_2: B_2(k_2) \to k$ is defined on the Bloch group. This map is expressed via the Bloch complex differential $\delta$ and a specific wedge product of logarithmic components $\ell^{(p)} = \frac{1}{2} \sum_{1 \leq i < p} i \cdot \ell_{p-i} \wedge \ell_i$ .
Construction of the Comparison 1-Form $\Omega^{(p)}$ :
To define a global regulator on a curve $C$ over $k_2$ , the author must compare local liftings of functions and the curve itself. In characteristic 0, this is done using a 1-form $\Omega$ derived from standard logarithms. In characteristic $p$ , the author constructs a unique 1-form $\Omega^{(p)}$ on $\Lambda^3 R^\times_\infty$ (where $R$ is a ring of characteristic $p$ ). This form is defined by its vanishing properties on specific elements and its explicit values on terms of the form $e^{\alpha t^a} \wedge e^{\beta t^b} \wedge e^{\gamma t^c}$ where $a+b+c=p$ .
Invariance under Reparametrization:
A critical technical hurdle is proving that the residue of $\Omega^{(p)}$ is independent of the choice of local coordinates (parametrization). The author proves that for a continuous automorphism $\sigma$ of $k((s))_\infty$ that is the identity modulo $(t)$ , the residue of $\Omega^{(p)}$ remains invariant. This relies on a detailed analysis of infinitesimal automorphisms and the expansion of $\sigma(e^{\alpha t^a})$ using a truncated exponential series.
Defining the Regulator $\rho_K$ :
For a smooth proper curve $C$ over $k_2$ and a set of smooth liftings $P$ , the regulator $\rho_K: \Lambda^3 k(C, P)^\times \to k$ is defined by summing local contributions. For a triple of functions $(f, g, h)$ , the value is computed by:
- Choosing local liftings of the curve and functions.
- Calculating the residue of the characteristic $p$ dilogarithm $\ell i^{(p)}_2$ on the local residues.
- Adding a "defect term" derived from the residue of the 1-form $\Omega^{(p)}$ comparing the local liftings to a generic lifting.
  The sum is shown to be finite and independent of all auxiliary choices (liftings and isomorphisms).
Infinitesimal Invariant of Cycles:
The regulator is extended to define an invariant for codimension 2 cycles in the space $\square^3_{k_\infty}$ (where $k_\infty = k[[t]]$ ). The author defines a subgroup $z^2_f(k_\infty, 3)$ of cycles with finite reduction. The map $\rho_K$ is shown to vanish on boundaries and to depend only on the reduction of the cycle modulo $(t^2)$ .

Key Contributions and Results

The Chow-Kontsevich Dilogarithm ( $\rho_K$ ): The paper constructs a regulator map $\rho_K: \Lambda^3 k(C, P)^\times \to k$ for a curve $C$ over $k_2$ . This map is distinct from the characteristic 0 infinitesimal Chow dilogarithm $\rho$ .
Linear Independence: The author demonstrates that $\rho$ and $\rho_K$ are linearly independent. Specifically, under the rescaling $t \mapsto a \cdot t$ , $\rho$ scales by $a^3$ , whereas $\rho_K$ scales by $a^p$ . This confirms the existence of two independent dilogarithm functions in characteristic $p \geq 5$ , consistent with the structure of $K_3(k_2)$ .
Explicit Computation: On the projective line $\mathbb{P}^1_{k_2}$ with coordinate $z$ , the regulator is explicitly computed. For a triple $(z-\alpha) \wedge (z-\beta) \wedge (z-\gamma)$ , the value is given by $a^p \cdot \mathcal{L}_1(s)$ , where $s$ is derived from the cross-ratio of the points and $a$ is a coefficient from the lifting.
Modulus Property: The paper proves that $\rho_K$ induces a well-defined map on cycles modulo $(t^2)$ . If two cycles are congruent modulo $(t^2)$ , they yield the same value under $\rho_K$ .
Relation to Goncharov's Conjecture: The construction is framed as an analog of Goncharov's conjecture on strong reciprocity laws. The map $\rho_K$ is expected to be the composition of the motivic complex map $\Lambda^3 k(C, P)^\times \to B_2(k_2)$ with the characteristic $p$ dilogarithm $\ell i^{(p)}_2$ .

Significance
The paper claims that the construction of $\rho_K$ provides a purely characteristic $p$ object that completes the picture of dilogarithm regulators in characteristic $p$ . While the additive dilogarithm $\ell i_2$ (derived from characteristic 0 methods) exists in characteristic $p$ , it is insufficient to capture the full structure of the $K$ -theory of $k_2$ . The addition of $\rho_K$ , based on Kontsevich's $1\frac{1}{2}$ -logarithm, allows for the construction of an isomorphism between the infinitesimal part of $K_3(k_2)$ and $k \oplus k$ .

Furthermore, the paper establishes an infinitesimal invariant for cycles in characteristic $p$ , analogous to results in characteristic 0. This suggests that the Chow-Kontsevich dilogarithm is the correct characteristic $p$ analog for studying motivic cohomology and reciprocity laws in this setting. The work does not propose new experimental applications but rather solidifies the theoretical framework for regulators and $K$ -theory in positive characteristic.