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On characterization of toric varieties

This paper investigates Shokurov's conjecture and a generalization regarding the characterization of toric varieties, demonstrating that these characterizations fail in dimensions three and higher while verifying some weaker versions.

Original authors: Ilya Karzhemanov

Published 2026-02-12

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Ilya Karzhemanov

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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: The "Toric" Mystery

Imagine you are a detective trying to identify a specific type of building called a Toric Variety.

In the world of math, a "Toric Variety" is a special kind of shape that has a very high degree of symmetry. Think of it like a perfectly symmetrical crystal or a kaleidoscope pattern. These shapes are built using a grid system (like graph paper) and are very easy to understand because their symmetries are so predictable.

For a long time, mathematicians had a hypothesis (a guess) proposed by a brilliant researcher named V. V. Shokurov. The hypothesis was essentially a "litmus test" to identify these special buildings.

The Hypothesis (The Test):
"If you look at a building and count its 'walls' (divisors) and check how many different directions you can move without hitting a wall, and the numbers add up to a specific formula, then 100% of the time, that building must be a Toric Variety."

It was like saying: "If a car has 4 wheels, a steering wheel, and an engine, it must be a Toyota." (Obviously, that's not true in real life, but in math, they hoped this specific formula was a perfect identifier).

The Plot Twist: The "Fake" Toric Varieties

The author of this paper, Ilya Karzhemanov, decided to test this hypothesis. He asked: "Is this formula actually a perfect test, or are there 'imposters'?"

He discovered that for small, simple shapes (2D), the test works perfectly. But as soon as you get to 3D or higher (like a cube or a hypercube), the test fails.

He found a way to build what he calls "Fake Toric" (f-toric) varieties.

The Analogy:
Imagine you have a perfect, symmetrical crystal (the real Toric variety). Now, imagine you take that crystal, cut it up, rearrange the pieces, and glue them back together in a slightly weird way.

The Result: It still looks almost like the crystal. It still passes the "wall count" test (the formula works). It still has the same "energy" properties.
The Catch: If you look closely at the structure, it's actually a mess. It's not a crystal anymore. It's a "fake" crystal.

Karzhemanov proved that in dimensions 3 and higher, you can always build these "fake" crystals. They trick the formula, but they aren't actually Toric varieties. This means Shokurov's conjecture, as originally written, is false.

How Did He Build the "Fake"? (The Construction)

To prove his point, Karzhemanov didn't just guess; he built a specific example.

The Base: He started with a standard shape (a quadric surface in 4D space).
The Twist: He applied two different "flips" (like flipping a coin twice) to this shape. Imagine taking a piece of paper, folding it, cutting it, and flipping it over.
The Glue: He glued the shape back together based on these flips.
The Result: The resulting shape passes all the numerical checks. It has the right number of walls and the right "complexity" score. But, because of the way he glued it, it lost the perfect symmetry required to be a true Toric variety.

It's like building a house that has the exact same number of windows, doors, and rooms as a famous architect's design, but the floor plan is slightly off, so it doesn't feel like the original design.

The Appendix: Calling Out a "Fake Proof"

At the end of the paper, Karzhemanov adds a section called the Appendix. This is like a "Fact Check" section.

Recently, another group of mathematicians published a paper claiming they had proved Shokurov's conjecture was true. Karzhemanov read their paper and found a massive hole in their logic.

The Analogy:
Imagine someone claims to have proved that "All swans are white." They write a long proof, but in the middle, they make a huge assumption: "Assume there are no black swans."
Karzhemanov points out: "Wait, you can't just assume that! That's the whole point of the argument! And here is a specific example of a black swan (the fake variety I built) that breaks your logic."

He shows that their proof relied on a step that simply doesn't work for the complex shapes they were studying.

The Silver Lining (What Does Work?)

Even though the main conjecture is false, Karzhemanov doesn't throw the whole idea away. He shows that while the "perfect test" doesn't exist, there are weaker versions that still hold true.

The "Good Enough" Test: If the numbers match up perfectly in a specific way, the shape might not be a Toric variety, but it will definitely be a "Rational Variety."
What's a Rational Variety? Think of it as a shape that can be smoothly stretched and squashed into a simple ball or a sphere without tearing. It's not as fancy as a Toric variety, but it's still a very nice, well-behaved shape.

Summary for the Non-Mathematician

The Goal: Mathematicians wanted a simple formula to identify special, symmetrical shapes called Toric varieties.
The Discovery: The author found that in 3D and higher, this formula is a "trap." You can build shapes that pass the formula but aren't actually Toric. They are "Fakes."
The Proof: He constructed a specific 3D "Fake" shape to prove the formula fails.
The Correction: He also debunked a recent paper that claimed to have proved the formula works, showing their logic was flawed.
The Takeaway: While we can't use this specific formula to find Toric varieties anymore, we can use it to find a slightly less special (but still very cool) type of shape called a Rational Variety.

In short: The "Toric Detector" is broken for complex shapes, but it still works as a "Rational Detector."

1. Problem Statement

The paper addresses a conjecture proposed by V. V. Shokurov regarding the characterization of toric varieties via numerical invariants of log pairs.

The Conjecture (Conjecture 1.3):
Let $X$ be a normal, projective variety over $\mathbb{C}$ with $\mathbb{Q}$ -factorial singularities, and let $D = \sum d_i D_i$ be a $\mathbb{Q}$ -boundary (effective Weil divisors with $0 \le d_i \le 1$ ) such that:

The pair $(X, D)$ is log canonical (lc).
The divisor $-(K_X + D)$ is nef.

Let $r(X, D)$ be the dimension of the $\mathbb{Q}$ -vector subspace in $N_1(X) \otimes \mathbb{Q}$ spanned by the components $D_i$ . The conjecture states:
$\sum d_i \le r(X, D) + \dim X$
Furthermore, equality holds if and only if the pair $(X, D')$ is formally toric for some boundary $D' \ge \lfloor D \rfloor$ (i.e., $X$ is a toric variety and $D'$ is its toric boundary).

Context:

The conjecture was known to be true for $\dim X = 2$ and for specific cases in dimension 3 (e.g., plt pairs with $K_X+D \equiv 0$ ).
A related conjecture by J. McKernan (Conjecture 1.7) introduced the concept of complexity $c(X, D) = r(X, D) + \dim X - \sum d_i$ . It posited that if $c(X, D) < 1$ , the variety is toric.
Previous proofs relied on two main assertions: (1) varieties isomorphic to toric varieties in codimension 1 are toric, and (2) the existence of integral $\mathbb{Q}$ -complements.

2. Methodology

Karzhemanov employs a combination of counter-example construction and inductive proofs for weaker versions of the conjecture.

A. Disproving the General Conjecture (Dimensions $\ge 3$ )

The author constructs a specific class of varieties called "fake toric" (f-toric) varieties.

Definition: A variety $X$ is f-toric if there exists a boundary $D$ satisfying the conjecture's constraints (lc pair, $-(K_X+D)$ nef) such that the complexity $c(X, D) < 1$ , yet $X$ is not a toric variety.
Construction Strategy:
1. Start with a quadric hypersurface $Q \subset \mathbb{P}^4$ .
2. Define a group $G \cong \mathbb{Z}/2 \times \mathbb{Z}/2$ acting on $\mathbb{P}^4$ via specific automorphisms ( $g_1, g_2$ ).
3. Blow up a $G$ -invariant line $\ell \subset Q$ to obtain $W$ .
4. Take the quotient $X = W/G$ .
5. Construct a specific boundary $D$ on $X$ derived from the ramification divisors of the quotient map and the exceptional divisors of the blowup.
6. Verify that $K_X + D \equiv 0$ and calculate the absolute complexity $ac(X, D) = 3/4$ (which is $< 1$ ).
7. Prove $X$ is not toric by analyzing its singularities and the structure of its fan, showing it cannot be isomorphic to a toric variety even in codimension 1 due to the specific nature of its singular points (involving non-toric quotients).

B. Proving Weaker Versions

While the main conjecture fails, the author proves positive results under stricter conditions:

Proposition 1.14 (Prokhorov): If $K_X + D \equiv 0$ and $d_i = 1$ (integral boundary), then $c(X, D) \ge 0$ . If $c(X, D) = 0$ , $X$ is rationally connected.
Theorem 1.15: If $ac(X, D) = 0$, then $X$ is rational. If $ac(X, D) = 1$ and $X$ is rationally connected, $X$ is rational.
Method: These proofs utilize the Minimal Model Program (MMP), induction on dimension and Picard number, and the construction of "Morse functions" (pencils) to reduce the problem to lower-dimensional fibers.

C. Critique of Recent Literature

The Appendix analyzes a recent paper [6] claiming a proof of Conjecture 1.3. Karzhemanov identifies critical logical gaps:

Unproven Grading: The proof assumes the Cox ring is positively graded without justification.
False Algebraic Claim: It incorrectly claims that a ring with $d$ invariant divisors forming an lc pair must be a polynomial ring. A counter-example is provided using a surface $xy^3 = z^2 + 1$ with a $\mathbb{C}^*$ action.
Invalid Rationality Criterion: The proof applies a rationality criterion for conic bundles to a singular case where the criterion is not known to hold.

3. Key Contributions and Results

Disproof of Shokurov's Conjecture:
- Theorem 1.11: The class of f-toric varieties $T_{f,n}$ is non-empty for all $n \ge 3$ .
- Specifically, there exists a 3-fold $X$ with a boundary $D$ such that $K_X + D \equiv 0$ , $ac(X, D) = 3/4$, and $X$ is not toric.
- This demonstrates that the condition $c(X, D) < 1$ (or $ac(X, D) < 1$) is insufficient to guarantee that a variety is toric in dimensions $\ge 3$ .
Clarification of "Toric in Codimension 1":
- The paper highlights a gap in previous literature (specifically [25]) regarding the assertion that a variety isomorphic to a toric variety in codimension 1 is itself toric. The author shows this relies on the variety being a "Mori dream space" and that the indeterminacy locus must be torus-invariant, which is not always guaranteed for general boundaries.
Positive Results for Rationality:
- While the toric characterization fails, the paper establishes that under the condition $ac(X, D) = 0$ (or $1$ with rational connectedness), the variety $X$ is rational. This salvages a weaker form of the conjecture: low complexity implies rationality, though not necessarily the toric structure.
Critique of [6]:
- The paper provides a rigorous refutation of a claimed proof of Conjecture 1.3, exposing fundamental errors in the algebraic and geometric arguments used in that work.

4. Significance

Refining the Theory of Toric Varieties: The paper delineates the precise boundary where numerical criteria for toric varieties break down. It shows that while complexity $c(X, D)$ is a powerful invariant for rationality, it does not uniquely characterize toric varieties in higher dimensions without additional structural assumptions.
Counter-Examples in Birational Geometry: The construction of the "fake toric" variety $X$ (a quotient of a blowup of a quadric) provides a concrete, non-trivial counter-example in the Minimal Model Program, illustrating the complexity of singularities in quotients of smooth varieties.
Correction of Literature: By identifying the flaws in the recent "proof" [6], the paper prevents the propagation of incorrect results in the field of algebraic geometry and clarifies the necessary conditions for rationality and toric structures.
Open Questions: The paper concludes by posing questions about the nature of f-toric varieties (e.g., are they always quotients of projective bundles? Do they admit torus actions?), suggesting new directions for research into the classification of varieties with low complexity.

In summary, Karzhemanov's work serves as a critical correction to the field, proving that the elegant numerical characterization of toric varieties proposed by Shokurov holds only in low dimensions or under very specific constraints, and fails generally in dimension 3 and above.