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A counterexample to the PIA conjecture for minimal log discrepancies

This paper presents counterexamples that disprove both the PIA conjecture for minimal log discrepancies and the LSC conjecture for families.

Original authors: Yusuke Nakamura, Kohsuke Shibata

Published 2026-05-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yusuke Nakamura, Kohsuke Shibata

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 understand the structural integrity of a building. In the world of mathematics, specifically a field called birational geometry, the "building" is a shape (called a variety), and the "structural integrity" is measured by something called a Minimal Log Discrepancy (MLD). Think of the MLD as a "safety score" for a specific point on the shape. A higher score means the point is smoother and safer; a lower score means it's more jagged or "singular."

For a long time, mathematicians believed in a very helpful rule of thumb called the PIA Conjecture (Precise Inversion of Adjunction). Here is the simple version of that rule:

The "Wall" Analogy: Imagine you have a 3D building (the shape $X$ ) and you paint a specific wall on it (the divisor $D$ ). The PIA conjecture said: "If you want to know how safe the wall is, you don't need to look at the whole building. You can just look at the wall itself, and the safety score will be exactly the same."

Mathematicians hoped this rule was always true because it would make solving complex problems much easier, like solving a puzzle by only looking at one piece.

The Big Discovery: The Rule Breaks

In this paper, authors Yusuke Nakamura and Kohsuke Shibata found a specific case where this rule fails. They built a mathematical "building" that looks safe from the outside, but when they tried to apply the PIA rule to a specific wall, the numbers didn't match.

The Counterexample:
They created a shape using a special kind of symmetry (like a spinning top that repeats its pattern every three turns).

The Whole Building ( $X$ ): They calculated the safety score of a specific point on the whole building.
The Wall ( $D$ ): They calculated the safety score of that same point, but only looking at the wall.
The Result: The scores were different. The wall was actually "safer" (had a higher MLD) than the rule predicted it should be based on the whole building.

This proves that the PIA conjecture is not always true. Specifically, it fails when the building has a dimension of 5 or higher (imagine a 5D or 6D object).

The "Ghost" in the Machine: Virtually Free Actions

Why did the rule break? The authors discovered a hidden concept they call "Virtually Free" actions.

Think of a group of people (a symmetry group) dancing around a room (the shape).

Free Action: Everyone is dancing, and no two people ever bump into each other or stand on the same spot. The dance is perfectly smooth.
Free in Codimension One: Everyone is dancing smoothly, except maybe in a tiny, invisible corner of the room.
Virtually Free (The New Concept): This is a tricky middle ground. The authors found that in their counterexample, the "dancers" (the symmetry group) seemed to be dancing smoothly enough that the safety score should have stayed the same, but because of a subtle "ghost" interaction (a specific type of singularity), the score changed.

They proved that if the "dancers" are Virtually Free, the safety score usually stays the same. However, their counterexample showed a case where the "dancers" were virtually free, but the wall was not smooth enough (it wasn't "klt," a technical term for "mildly singular") for the rule to hold.

The Ripple Effect: The LSC Conjecture

The paper also touches on a second idea called the LSC Conjecture (Lower Semi-Continuity).

The Analogy: Imagine you have a family of buildings that change slightly as you turn a dial (like a slider). The LSC conjecture said: "As you turn the dial, the safety score of the buildings should never suddenly jump up; it can only stay the same or go down."

The authors used their counterexample to show a scenario where, as you turn the dial, the safety score suddenly jumps up for a specific family of buildings. This breaks the LSC conjecture for families, but only when the buildings in the family have "rough" edges (non-klt singularities).

What Does This Mean?

The Rule Has Limits: The PIA conjecture is a powerful tool, but it's not a universal law. It works for smooth buildings or buildings with mild bumps, but it fails for complex, highly symmetric shapes with specific types of rough edges.
Dimension Matters: The rule holds true for 3D and 4D shapes (according to the paper's discussion), but it breaks down in 5D and higher.
Refining the Theory: The authors suggest that if we add a condition that the "wall" must be smooth (klt), the rule might be saved. This helps mathematicians refine their theories to know exactly when they can trust the rule and when they need to be careful.

In summary: The authors built a mathematical "trap" that looked like it should follow the standard rules of geometry, but it didn't. This discovery forces mathematicians to update their rulebook, adding new conditions and concepts (like "Virtually Free") to explain why the rule failed in this specific, high-dimensional case.

1. Problem Statement

The paper addresses the PIA (Precise Inversion of Adjunction) Conjecture, a fundamental hypothesis in birational geometry concerning Minimal Log Discrepancies (mld).

Context: The minimal log discrepancy is a numerical invariant measuring the severity of singularities. The PIA conjecture posits a precise equality between the mld of a pair $(X, D + \Delta)$ at a point $x$ on a normal Cartier divisor $D$ , and the mld of the restricted pair $(D, \Delta|_D)$ at the same point.
The Conjecture: Specifically, Conjecture 1.3 states that for a log pair $(X, \mathfrak{a})$ and a normal Cartier prime divisor $D$ not contained in the cosupport of $\mathfrak{a}$ :
$\text{mld}_x(X, \mathfrak{a} \mathcal{O}_X(-D)) = \text{mld}_x(D, \mathfrak{a} \mathcal{O}_D)$
Known Status: The inequality " $\leq$ $\leq$ " is known to always hold. The conjecture asserts the reverse inequality " $\geq$ $\geq$ ". It has been proven in several cases:
- $\dim X = 2$ (Shokurov).
- $X$ is smooth (Ein, Mustață, Yasuda).
- $X$ is a locally complete intersection (Ein, Mustață).
- $\dim X = 3$ (proven in this paper).
- $X$ has quotient singularities and $D$ is klt (Kawamata log terminal) at $x$ (Nakamura, Shibata).
The Gap: The conjecture was open for higher dimensions ( $\dim X \geq 4$ ) and, crucially, for cases where the divisor $D$ itself has non-klt singularities.

2. Methodology

The authors employ a combination of quotient singularity theory, group actions, and arc space theory to construct a counterexample.

A. Construction of the Counterexample

The authors define a specific family of varieties using cyclic group actions:

Setup: Let $k$ be an algebraically closed field of characteristic zero. Let $\xi$ be a primitive third root of unity.
Space: Consider $A^{n+3}_k = \text{Spec}(k[x_1, \dots, x_{n+3}])$ .
Group Action: Define a cyclic group $G = \langle \gamma \rangle \cong \mathbb{Z}/3\mathbb{Z}$ acting on the space via the diagonal matrix $\gamma = \text{diag}(1, \xi, \xi^2, \xi, \dots, \xi)$ .
Quotient Varieties:
- $A_n = A^{n+3}_k / G$ .
- $B_n$ is the hypersurface defined by $f = x_1^3 + x_2^3 + x_3^3 = 0$ inside $A^{n+3}_k$ , and $\bar{B}_n = B_n / G$ is its image in the quotient.
Target: The pair $(\bar{A}_n, \bar{B}_n)$ at the origin $x_0$ .

B. The Concept of "Virtually Free" Actions

A major theoretical contribution is the introduction of "virtually free" group actions to explain why the mld behaves unexpectedly in this construction.

Definition: A $G$ -action on a normal variety $X$ is virtually free if, for any $G$ -equivariant proper birational morphism $Y \to X$ from a normal variety $Y$ , the action on $Y$ is free in codimension one.
Properties:
- It lies strictly between "free" and "free in codimension one."
- Proposition 3.5: If a $G$ -action is virtually free, the minimal log discrepancy is preserved under the quotient: $\text{mld}_x(X, \mathfrak{a}) = \text{mld}_{x'}(X/G, \mathfrak{a})$ .
- Theorem 3.10: If $X$ is klt, "virtually free" is equivalent to "free." This explains why counterexamples cannot be constructed using klt varieties (as the action would have to be free, preserving standard behavior).

C. Calculation of mld

The authors calculate the mld for the constructed example:

For the ambient space: Using the age formula for quotient singularities, they determine $\text{mld}_{x_0}(\bar{A}_n) = 2 + n/3$ .
For the divisor: They prove the action on $B_n$ is virtually free (Lemma 4.1). Consequently, $\text{mld}_{x_0}(\bar{B}_n) = \text{mld}_{x_0}(B_n)$ . Since $B_n$ is a product of a surface singularity and affine space, $\text{mld}_{x_0}(B_n) = n$ .
The Discrepancy:
- $\text{mld}_{x_0}(\bar{A}_n, \bar{B}_n) \leq \text{mld}_{x_0}(\bar{A}_n) - 1 = 1 + n/3$ .
- $\text{mld}_{x_0}(\bar{B}_n) = n$ .
- For $n \geq 2$ , $n > 1 + n/3$ . Thus, $\text{mld}_{x_0}(\bar{A}_n, \bar{B}_n) < \text{mld}_{x_0}(\bar{B}_n)$ , violating the PIA conjecture.

3. Key Contributions

Disproof of the PIA Conjecture: The paper provides the first counterexample to the PIA conjecture for $\dim X \geq 5$ .
- Result: For $n \geq 2$ , the equality fails for the pair $(\bar{A}_n, \bar{B}_n)$ .
- Significance: This demonstrates that the PIA conjecture does not hold generally for varieties with quotient singularities if the divisor $D$ is not klt.
Introduction of "Virtually Free" Actions:
- This new concept explains the phenomenon where $\text{mld}(X/G) = \text{mld}(X)$ even when the group action is not free.
- It bridges the gap between free actions (where mld is preserved) and non-free actions (where mld usually drops).
Counterexample to the LSC Conjecture for Families:
- The authors use a similar construction to disprove the LSC (Lower Semi-Continuity) Conjecture for Families (Conjecture 5.3).
- They construct a flat family where the mld of the special fiber ( $t=0$ ) is strictly lower than the mld of the general fibers ( $t \neq 0$ ), violating lower semi-continuity.
- Crucial Condition: This failure occurs specifically when the fibers are not klt.
Proof of PIA in Dimension 3:
- The authors provide a proof that the PIA conjecture holds for $\dim X = 3$ (Proposition 6.1), filling a gap in the literature.

4. Results Summary

Dimension	Status of PIA Conjecture	Condition
$\dim X = 2$	True	General (Shokurov)
$\dim X = 3$	True	General (Proven in this paper)
$\dim X = 4$	Open	Question 6.2
$\dim X \geq 5$	False	Counterexample provided (Theorem 1.5)
General	True	If $D$ is klt at $x$ (Nakamura-Shibata)
General	False	If $D$ is not klt (Counterexample)

Specific Counterexample Values (for $n \geq 2$ ):

$\text{mld}_{x_0}(\bar{A}_n, \bar{B}_n) \leq 1 + \frac{n}{3}$
$\text{mld}_{x_0}(\bar{B}_n) = n$
Since $n > 1 + n/3$ , the conjecture fails.

5. Significance

Refining Conjectures: The paper forces a re-evaluation of the PIA conjecture. It suggests that the conjecture requires the additional assumption that the divisor $D$ is klt at the point of interest. Without this assumption, the "precise" equality fails.
Singularities Theory: It highlights the pathological behavior of minimal log discrepancies in the presence of non-klt singularities and non-free group actions.
Families and Deformation: The counterexample to the LSC conjecture for families clarifies the limits of deformation invariance for singularities. It shows that while terminal/canonical singularities deform well, the behavior of mld in families with non-klt fibers is unstable.
Future Directions: The paper leaves the dimension 4 case open and poses questions about whether the conjecture can be salvaged by restricting to klt divisors (Question 6.4).

In conclusion, Nakamura and Shibata have fundamentally altered the understanding of the PIA conjecture by demonstrating its failure in higher dimensions for non-klt divisors, introducing the concept of "virtually free" actions to explain the underlying geometry, and providing a rigorous proof for the dimension 3 case.