[Paper Review] An elementary semi-ampleness result for log canonical divisors

Shigetaka Fukuda|arXiv (Cornell University)|Mar 6, 2010

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper establishes that if the log canonical divisor $K_X + \Delta$ on a projective variety with Kawamata log terminal singularities is numerically equivalent to a semi-ample $\mathbf{Q}$-divisor, then $K_X + \Delta$ itself is semi-ample. The proof uses relative semi-ampleness theorems and numerical equivalence to reduce the problem to a birational geometry argument, ultimately concluding that the divisor is semi-ample via Zariski’s Main Theorem and ampleness preservation under numerical equivalence.

ABSTRACT

If the log canonical divisor on a projective variety with only Kawamata log terminal singularities is numerically equivalent to some semi-ample $\mathbf{Q}$-divisor, then it is semi-ample.

Motivation & Objective

• To establish a semi-ampleness criterion for log canonical divisors under numerical equivalence to semi-ample Q-divisors.
• To provide a direct, elementary proof of a special case of the log abundance conjecture for Kawamata log terminal pairs.
• To extend known results on numerically trivial log canonical divisors to the broader setting of numerically equivalent semi-ample divisors.
• To propose a subconjecture for log canonical singularities, generalizing the main result to a wider class of pairs.
• To offer a self-contained argument using relative semi-ampleness and numerical equivalence, avoiding advanced minimal model program techniques.

Proposed method

• Construct a morphism $f: X \to Y$ via the linear system $|lD|$ for a sufficiently large and divisible $l$, where $D$ is a semi-ample Q-Cartier Q-divisor numerically equivalent to $K_X + \Delta$.
• Show that $K_X + \Delta$ is $f$-nef by construction, as it is numerically equivalent to $f^*A$ for an ample divisor $A$ on $Y$.
• Apply the $\mathbf{Q}$-linear triviality result (Proposition 0.4) to the general fiber $F$ of $f$, proving that $(K_X + \Delta)|_F$ is $\mathbf{Q}$-linearly trivial.
• Use the relative semi-ampleness theorem (Proposition 0.7) to deduce that $K_X + \Delta$ is $f$-semi-ample, hence $m(K_X + \Delta) = g^*B$ for some $h$-ample divisor $B$ on $Z$.
• Establish that the composition $h: Z \to Y$ is birational and finite via numerical triviality of $f^*(mA - lB)$, and apply Zariski’s Main Theorem to conclude $h$ is an isomorphism.
• Conclude that $lB$ is ample on $Y$, hence $K_X + \Delta$ is semi-ample by numerical equivalence and ampleness preservation.

Experimental results

Research questions

• RQ1 Under what conditions is a log canonical divisor $K_X + \Delta$ semi-ample when it is numerically equivalent to a semi-ample $\mathbf{Q}$-divisor?
• RQ2 Can the semi-ampleness of $K_X + \Delta$ be deduced from relative semi-ampleness theorems under numerical equivalence?
• RQ3 Does the $\mathbf{Q}$-linear triviality of the restriction of $K_X + \Delta$ to general fibers imply its global semi-ampleness?
• RQ4 Can the log abundance conjecture be approached via numerical equivalence to semi-ample divisors in the Kawamata log terminal setting?
• RQ5 What is the minimal singularities condition under which the main result extends, particularly to log canonical pairs?

Key findings

• If $K_X + \Delta$ is numerically equivalent to a semi-ample $\mathbf{Q}$-Cartier $\mathbf{Q}$-divisor on a projective variety with Kawamata log terminal singularities, then $K_X + \Delta$ is semi-ample.
• The proof relies on the relative semi-ampleness theorem (Proposition 0.7) applied to a morphism $f: X \to Y$ induced by the numerically equivalent semi-ample divisor.
• The restriction of $K_X + \Delta$ to a general fiber $F$ is $\mathbf{Q}$-linearly trivial, as guaranteed by Proposition 0.4.
• The morphism $h: Z \to Y$ induced by the relative semi-ampleness is shown to be an isomorphism via Zariski’s Main Theorem, implying $K_X + \Delta$ is semi-ample.
• The ampleness of $lB$ on $Y$ follows from the ampleness of $mA$ and numerical equivalence, confirming that $K_X + \Delta$ is semi-ample.
• The result supports a subconjecture (Conjecture 0.8) that extends the claim to log canonical singularities, which was later partially confirmed by Gongyo in dimension ≤ 4.