[Paper Review] Positivity criteria for log canonical divisors and hyperbolicity

Steven Lu, De‐Qi Zhang|arXiv (Cornell University)|Jul 31, 2012

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper establishes geometric criteria for the nefness and ampleness of log canonical divisors $K_X + D$ on complex projective varieties with simple normal crossing divisors, using the minimal model program and adjunction. It proves that under Mori hyperbolicity or Brody hyperbolicity assumptions, $K_X + D$ is nef (and ample under mild component conditions), offering a geometric sharpening of the Nakai-Moishezon criterion and verifying it without relying on deep conjectures in low dimensions.

ABSTRACT

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. By adjunction and running the log minimal model program, natural to our setting, we obtain a geometric criterion for K_X+D to be numerically effective as well as a geometric version of the cone theorem, generalizing to the context of log pairs these results of Mori. A criterion for K_X+D to be pseudo-effective with mild hypothesis on D follows. We also obtain, assuming the abundance conjecture and the existence of rational curves on Calabi-Yau manifolds, an optimal geometric sharpening of the Nakai-Moishezon criterion for the ampleness of a divisor of the form K_X+D, a criterion verified under a canonical hyperbolicity assumption on (X,D). Without these conjectures, we verify this ampleness criterion with assumptions on the number of ample and non ample components of D.

Motivation & Objective

• To establish geometric criteria guaranteeing positivity (nef, pseudo-effective, ample) of log canonical divisors $K_X + D$ on complex projective varieties with reduced divisors $D$.
• To provide a geometric version of the cone theorem and a sharpened Nakai-Moishezon criterion for ampleness of $K_X + D$ under hyperbolicity assumptions.
• To verify the ampleness criterion without relying on the abundance conjecture or existence of rational curves on Calabi-Yau manifolds in dimensions two and three.
• To characterize extremal rational curves in the closure of the Mori cone that violate $K_X + D$-non-negativity, particularly in relation to strata of $D$.

Proposed method

• Uses the minimal model program (MMP) via induction on dimension, applying adjunction to reduce positivity questions to lower-dimensional cases.
• Defines and analyzes $\mathcal{D}$-rational curves, $\mathcal{D}$-compact varieties, and $\mathcal{D}$-rational curves in strata of $D$ to classify extremal rays in $\overline{\operatorname{NE}}(X)$.
• Applies the canonical bundle formula and log canonical singularities theory, particularly using results from [12] on log canonical singularities and their resolution.
• Imposes geometric constraints on rational curves $\ell$ in strata of $D$, showing that $\ell \cdot (K_X + D) \in \{1, 2, \dots, 2n\}$, with equality to 1 if $\ell$ is not $\mathcal{D}$-compact.
• Uses the minimal resolution $\eta: X' \to X$ to analyze the behavior of $K_{X'} + D' + \Gamma' + E$ and derive genus bounds via adjunction and intersection theory.
• Applies the dlt (divisorially log terminal) condition and log canonical singularities to control the geometry of $\Gamma$ and $D$ near singular points, especially in dimension ≤3.

Experimental results

Research questions

• RQ1 Under what geometric conditions is $K_X + D$ nef for a smooth projective variety $X$ and reduced divisor $D$ with simple normal crossings?
• RQ2 Can the Nakai-Moishezon criterion for ampleness of $K_X + D$ be sharpened geometrically, especially under hyperbolicity assumptions?
• RQ3 What is the structure of the Mori cone $\overline{\operatorname{NE}}(X)$ when $K_X + D$ is not nef, and which rational curves generate the extremal rays?
• RQ4 How do the singularities of $D$ and the geometry of strata influence the positivity of $K_X + D$?
• RQ5 Can the ampleness of $K_X + D$ be verified without assuming the abundance conjecture or rational curves on Calabi-Yau manifolds in low dimensions?

Key findings

• The closure of the Mori cone $\overline{\operatorname{NE}}(X)$ is generated by the $(K_X + D)$-non-negative part and at most countably many extremal $\mathcal{D}$-rational curves $\ell_i$, with $-\ell_i \cdot (K_X + D) \in \{1, 2, \dots, 2n\}$.
• If $(X,D)$ is Mori hyperbolic, then $K_X + D$ is nef, providing a geometric criterion for nefness.
• For $\dim X = n$, if $D$ has at least $n-3$ ample components and $n-2$ irreducible components, then $K_X + D$ is ample, verifying the ampleness part of Conjecture 1.1 in low dimensions.
• In dimension ≤3, if $(X,D)$ is Brody hyperbolic and $D + \Gamma$ is dlt with $\Gamma \geq 0$, then $K_X + D + \Gamma$ restricts to an ample divisor on every irreducible component of $D + \lfloor \Gamma \rfloor$.
• The extremal $\mathcal{D}$-rational curves $\ell_i$ satisfy $-\ell_i \cdot (K_X + D) = 1$ if $\ell_i$ is not a $\mathcal{D}$-compact rational curve.
• The paper verifies the geometric ampleness criterion without assuming the abundance conjecture or existence of rational curves on Calabi-Yau manifolds in dimensions two and three.