[Paper Review] On the moduli part of the Kawamata-Kodaira canonical bundle formula

Caucher Birkar, Yifei Chen|arXiv (Cornell University)|Jul 17, 2012

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper proves that an arbitrarily small perturbation of the moduli b-divisor in the Kawamata-Kodaira canonical bundle formula is semi-ample, by constructing a resolution and a suitable effective divisor that makes the perturbed divisor semi-ample. This result resolves a conjecture of Fujino and Gongyo, showing that if $ f: X \to Z $ is a smooth morphism of smooth projective varieties with $ -K_X $ semi-ample, then $ -K_Z $ is also semi-ample.

ABSTRACT

It is conjectured that the moduli b-divisor of the Kawamata-Kodaira canonical bundle formula associated to a klt-trivial fibration $(X,B) o Z$ is semi-ample. In this paper, we show the semi-ampleness of an arbitrarily small perturbation of the moduli b-divisor by a fixed appropriate divisor which roughly speaking comes from a section of $K_X+B$. We apply the above result to settle a conjecture of Fujino and Gongyo: if $f\colon X o Z$ is a smooth surjective morphism of smooth projective varieties with $-K_X$ semi-ample, then $-K_Z$ is also semi-ample. We list several counter-examples to show that this fails without the smoothness assumption on $f$.

Motivation & Objective

• To prove the semi-ampleness of a perturbation of the moduli b-divisor in the Kawamata-Kodaira canonical bundle formula for klt-trivial fibrations.
• To resolve a conjecture by Fujino and Gongyo on the semi-ampleness of $ -K_Z $ for smooth fibrations $ f: X \to Z $ when $ -K_X $ is semi-ample.
• To establish that the moduli b-divisor, while not necessarily semi-ample itself, becomes semi-ample after a small perturbation via a fixed divisor from a section of $ K_X + B $.
• To provide counterexamples showing the smoothness assumption is essential for the conjecture to hold.

Proposed method

• Use the minimal model program to construct a resolution $ \phi: W \to Y $ of an Ambro model $ Y \to Z $, where $ Y $ is a resolution of the base $ Z $ of a klt-trivial fibration.
• Define a $ \mathbb{Q} $-divisor $ \Delta_Y = B_Y + E $ on $ Y $, where $ E \geq 0 $ is an exceptional $ /Z $ $ \mathbb{Q} $-divisor such that $ (Y, \Delta_Y) $ is klt.
• Apply Theorem 1.2 to show that for $ \alpha \gg 0 $, the divisor $ \alpha M_W + G $ is semi-ample on $ W $, where $ G \leq \phi^*D_Y $ and $ D_Y \sim_{\mathbb{Q}} K_Y + \Delta_Y + bM_Y $.
• Use the negativity lemma to show that $ G $ is zero over a neighborhood of a general point $ z \in Z $, implying $ G $ is exceptional and does not dominate $ z $.
• Construct a $ \mathbb{Q} $-divisor $ P_W \geq 0 $ such that $ \alpha M_W + G \sim_{\mathbb{Q}} P_W $, with $ P_W $ disjoint from the fiber over $ z $, so that $ P_Z = \sigma_*\phi_*P_W \geq 0 $ is a $ \mathbb{Q} $-divisor with $ z \notin \operatorname{Supp} P_Z $.
• Conclude that the stable base locus of $ M_Z $ does not contain any point $ z \in Z $, so $ -K_Z $ is semi-ample.

Experimental results

Research questions

• RQ1 Is the moduli b-divisor in the Kawamata-Kodaira canonical bundle formula semi-ample for klt-trivial fibrations?
• RQ2 Can the semi-ampleness of the moduli b-divisor be established via perturbation by a fixed divisor from a section of $ K_X + B $?
• RQ3 Does the semi-ampleness of $ -K_X $ imply the semi-ampleness of $ -K_Z $ for smooth fibrations $ f: X \to Z $?
• RQ4 What happens to the semi-ampleness of $ -K_Z $ when the smoothness assumption on $ f $ is dropped?
• RQ5 Can the canonical bundle formula be used to prove semi-ampleness of $ -K_Z $ without relying on known results about nef or big anticanonical divisors?

Key findings

• The moduli b-divisor $ M_Y $ in the canonical bundle formula is not necessarily semi-ample, but an arbitrarily small perturbation $ \alpha M_W + G $ is semi-ample for sufficiently large $ \alpha \gg 0 $, where $ G $ is an effective $ \mathbb{Q} $-divisor on a resolution $ W \to Y $.
• The perturbation $ \alpha M_W + G $ is constructed so that $ G \leq \phi^*D_Y $ and $ R(W, lG) \simeq R(Y, lD_Y) $ for some $ l > 0 $, ensuring compatibility with section rings.
• The result implies that $ -K_Z $ is semi-ample whenever $ f: X \to Z $ is a smooth morphism of smooth projective varieties and $ -K_X $ is semi-ample, thus resolving a conjecture of Fujino and Gongyo.
• The proof relies on the negativity lemma to show that the perturbing divisor $ G $ is zero over a neighborhood of any point $ z \in Z $, so that $ \alpha M_Z \sim_{\mathbb{Q}} P_Z \geq 0 $ with $ z \notin \operatorname{Supp} P_Z $, proving $ -K_Z $ is base-point-free in the stable base locus sense.
• Counterexamples show that the smoothness assumption on $ f $ is essential: without it, $ -K_X $ semi-ample does not imply $ -K_Z $ semi-ample.
• The construction works even in the special case where $ K_X + B \sim_{\mathbb{Q}} 0 $, taking $ D_Y = E $, and still yields semi-ampleness of the perturbed moduli divisor.