[Paper Review] On characterization of toric varieties

Ilya Karzhemanov|arXiv (Cornell University)|Jun 18, 2013

Algebraic Geometry and Number Theory20 references3 citations

TL;DR

This paper investigates V. V. Shokurov's conjecture on characterizing toric varieties via log canonical pairs with nef anti-canonical divisors and bounded boundary divisor coefficients. Using geometric and birational geometry techniques, the author proves that the conjecture fails in dimensions ≥3, though weaker versions hold. A critical flaw is identified in a recent claimed proof of the conjecture, undermining its validity.

ABSTRACT

We study the conjecture due to V.\,V. Shokurov on characterization of toric varieties. We also consider one generalization of this conjecture. It is shown that none of the characterizations holds true in dimension $\ge 3$. Some weaker versions of the conjecture(s) are verified.

Motivation & Objective

• To investigate V. V. Shokurov's conjecture characterizing toric varieties via log canonical pairs with nef anti-canonical divisors.
• To test whether the conjecture holds in higher dimensions, particularly ≥3.
• To analyze and identify a critical error in a recent claimed proof of the conjecture (Paper [5]).
• To explore weaker forms of the conjecture, such as torification or formal toric structure.
• To examine whether varieties satisfying the conjecture's conditions are compactifications of the torus $(\mathbb{C}^*)^n$ or Mori dream spaces.

Proposed method

• Analyzes log canonical pairs $(X, D)$ with $-(K_X + D)$ nef and $X$ $\mathbb{Q}$-factorial, focusing on the invariant $r(X,D)$, the rank of the Néron-Severi group spanned by the $D_i$.
• Applies dlt modifications to reduce to the $\mathbb{Q}$-factorial case, preserving the key invariants and numerical conditions.
• Uses the structure of toric varieties and their fans to construct counterexamples in dimension $n \geq 4$ via products with $\mathbb{P}^1$.
• Employs the Cox ring and torus actions to analyze the failure of the conjecture, particularly in the case of singularities arising from group quotients.
• Identifies a critical error in the proof of [5]: the claim that $d$ invariant divisors on a $d$-dimensional $\mathbb{C}^*$-variety imply the ring is a polynomial ring is false.
• Constructs a counterexample using $Y \subset \mathbb{C}^3$ defined by $xy = z^3 + z^2 + z$, with $\mathbb{C}^*$-action, showing $\pi_1(Y) = \mathbb{Z}/3$, so $Y \not\simeq \mathbb{C}^2$.

Experimental results

Research questions

• RQ1 Does Shokurov's conjecture that $\sum d_i \leq r(X,D) + \dim X$ with equality iff $X$ is toric hold in dimensions $\geq 3$?
• RQ2 Can a variety satisfying the conjecture's conditions be formally toric, even if not regularly or analytically isomorphic to a toric pair?
• RQ3 Is the claimed proof of the conjecture in [5] valid, particularly its key step involving $\mathbb{C}^*$-invariant divisors and polynomial rings?
• RQ4 Do varieties satisfying the conjecture's conditions arise as quotients of toric varieties by finite groups or admit regular $(\mathbb{C}^*)^k$-actions?
• RQ5 Is $X$ a compactification of $(\mathbb{C}^*)^n$ or a Mori dream space under the conjecture's conditions?

Key findings

• The conjecture fails in dimension $n \geq 3$: a counterexample is constructed via $\mathfrak{X} = X \times (\mathbb{P}^1)^{n-3}$ with $ac(\mathfrak{X}, \mathfrak{D}) = \frac{3}{4}$, $K_{\mathfrak{X}} + \mathfrak{D} \equiv 0$, but $\mathfrak{X}$ is not toric.
• The proof in [5] is inconsistent, primarily due to the false claim that $d$ $\mathbb{C}^*$-invariant divisors on a $d$-dimensional variety imply the ring is a polynomial ring.
• A counterexample is given: $Y \subset \mathbb{C}^3$ defined by $xy = z^3 + z^2 + z$ with a $\mathbb{C}^*$-action has two invariant divisors but $\pi_1(Y) = \mathbb{Z}/3$, so $Y \not\simeq \mathbb{C}^2$, disproving the key step in [5].
• The local version of the conjecture (with $Z = X$) is true: if $K_X + D \equiv 0$ and $D_i$ are $\mathbb{Q}$-Cartier, then $X \simeq (\mathbb{C}^n \ni 0)/\mathfrak{A}$ with $\mathfrak{A}$ finite abelian diagonal, so $(X, \llcorner D \lrcorner)$ is toric.
• The conjecture's equality case implies $X$ is toric only under strong assumptions; in general, the condition $\sum d_i = r(X,D) + \dim X$ does not imply toric structure in $\dim \geq 3$.
• Weaker versions of the conjecture—such as $X$ admitting a torification or being formally toric—are not ruled out, but the original conjecture fails.