[Paper Review] A geometric characterization of toric singularities

Joaquí­n Moraga, Roberto Svaldi|arXiv (Cornell University)|Aug 3, 2021

Algebraic Geometry and Number Theory27 references4 citations

TL;DR

This paper provides a complete solution to a conjecture by Shokurov on the geometric characterization of toric singularities by introducing a complexity invariant for log canonical pairs over a base. It proves that when this complexity is zero, the morphism is formally isomorphic to a toric morphism, and the boundary components map to toric invariant divisors, thus giving a precise criterion for formal toricness in terms of birational geometry and class group rank.

ABSTRACT

Given a projective contraction $π\colon X ightarrow Z$ and a log canonical pair $(X, B)$ such that $-(K_X+B)$ is nef over a neighborhood of a closed point $z\in Z$, one can define an invariant, the complexity of $(X, B)$ over $z \in Z$, comparing the dimension of $X$ and the relative Picard number of $X/Z$ with the sum of the coefficients of those components of $B$ intersecting the fibre over $z$. We prove that the complexity of $(X,B)$ over $z\in Z$ is non-negative and that when it is zero then $(X,\lfloor B floor) ightarrow Z$ is formally isomorphic to a morphism of toric varieties around $z\in Z$. In particular, considering the case when $π$ is the identity morphism, we get a geometric characterization of singularities that are formally isomorphic to toric singularities. This gives a positive answer to a conjecture due to Shokurov.

Motivation & Objective

• To resolve Shokurov's conjecture on the geometric characterization of toric singularities using a complexity invariant.
• To establish a criterion for when a morphism is formally toric based on the non-negativity and vanishing of a geometric complexity invariant.
• To generalize previous results on Q-factorial and log canonical singularities to the non-Q-factorial case.
• To prove that complexity zero implies formal toricness and descent of moduli divisors in generalized pairs.

Proposed method

• Define the complexity invariant $ c_z(X/Z,B) = \dim X + \dim_{\mathbb{Q}} \operatorname{Cl}_{\mathbb{Q}}(X/Z) - \sum_{B_i \cap \pi^{-1}(z) \neq \emptyset} b_i $ for a log canonical pair over a base.
• Use the relative version of the generalized log minimal model program to analyze the behavior of the complexity invariant under blow-ups and modifications.
• Construct a relative $ \mathbb{A}^1 $-bundle $ C_{\infty} \to X $ to lift the pair to a higher-dimensional setting where the complexity can be controlled.
• Apply formal toricness criteria via the structure of lc places and the descent of moduli divisors in generalized pairs.
• Use the fact that $ K_X + B \sim_{\mathbb{Q},Z} 0 $ when complexity is zero to deduce formal toricness via deformation and completion techniques.
• Leverage the commutative diagram of morphisms to transfer formal toricness from the total space to the base, proving the main theorem.

Experimental results

Research questions

• RQ1 When is a morphism $ \pi: X \to Z $ formally isomorphic to a toric morphism at a closed point $ z \in Z $?
• RQ2 What is the precise geometric meaning of the complexity invariant $ c_z(X/Z,B) $ in the context of log canonical pairs?
• RQ3 Under what conditions does the vanishing of complexity imply that $ X \to Z $ is formally toric and $ K_X + B \sim_{\mathbb{Q},Z} 0 $?
• RQ4 Can the complexity invariant be used to characterize singularities formally isomorphic to toric singularities without assuming $ \mathbb{Q} $-factoriality?
• RQ5 Does the moduli divisor $ M $ descend over a neighborhood of $ z \in Z $ when $ c_z(X/Z,B+M) = 0 $?

Key findings

• The complexity invariant $ c_z(X/Z,B) $ is always non-negative when $ -(K_X + B) $ is nef over a neighborhood of $ z \in Z $.
• If $ c_z(X/Z,B) = 0 $, then $ K_X + B \sim_{\mathbb{Q},Z} 0 $, and $ X \to Z $ is formally toric at $ z $.
• Under the formal isomorphism, the components of $ \lfloor B \rfloor $ are mapped to the completion of toric invariant divisors.
• The result generalizes previous work by Kollár, Mc Kernan–Keel, Prokhorov, and Yao to the non-Q-factorial case.
• The moduli divisor $ M $ descends over a neighborhood of $ z $ when $ c_z(X/Z,B+M) = 0 $, under the generalized pair setting.
• The proof uses a relative $ \mathbb{A}^1 $-bundle construction to reduce the problem to a higher-dimensional generalized pair where formal toricness is detectable.