[Paper Review] On Finiteness of B-representation and Semi-log Canonical Abundance

Christopher D. Hacon, Chenyang Xu|arXiv (Cornell University)|Jul 21, 2011

Advanced Algebra and Geometry14 references15 citations

TL;DR

This paper presents a new proof of the finiteness of B-representations for dlt pairs with semi-ample canonical divisors, leveraging Hodge theory and Kollár’s gluing theory on log canonical centers. As a key application, it establishes that the semi-log canonical abundance conjecture follows from the log canonical abundance conjecture, resolving relative cases and confirming conjectures by Birkar and Kollár on rational equivalence of numerically trivial divisors.

ABSTRACT

We give a new proof of the finiteness of B-representations. As a consequence of the finiteness of B-representations and Kollár's gluing theory on lc centers, we prove that the (relative) abundance conjecture for slc pairs is implied by the abundance conjecture for log canonical pairs.

Motivation & Objective

• To provide a new, Hodge-theoretic proof of the finiteness of B-representations for dlt pairs with semi-ample canonical divisors.
• To establish that the relative semi-log canonical abundance conjecture follows from the log canonical abundance conjecture via Kollár’s gluing theory.
• To resolve a long-standing open problem regarding the rational equivalence of numerically trivial divisors in the relative setting.
• To extend known results on semi-ample divisors in the log canonical case to the semi-log canonical setting, particularly in the relative projective case.
• To confirm conjectures by Birkar and Kollár on the equivalence of numerical and rational triviality for semi-log canonical pairs under relative semi-ampleness.

Proposed method

• Use Hodge-theoretic construction of the B-representation via log resolutions and cohomology of line bundles on the total space of a morphism.
• Apply the finiteness of B-representations in the case of Kodaira dimension zero, building on Gongyo’s result and Fujino’s framework.
• Utilize Kollár’s gluing theory for log canonical centers to relate the normalization of a semi-log canonical pair to its normal counterpart.
• Construct a quotient of a double point scheme via birational automorphisms and analyze the graph of B-representations over generic fibers.
• Apply Kollár’s injectivity theorem and base-point-free theorems to deduce global generation and semi-ampleness of line bundles over the base.
• Use induction on dimension and run a relative MMP with scaling to reduce to the klt case, where known results on good models apply.

Experimental results

Research questions

• RQ1 Does the finiteness of B-representations for dlt pairs imply the relative abundance conjecture for semi-log canonical pairs?
• RQ2 Can the semi-log canonical abundance conjecture be reduced to the log canonical abundance conjecture via gluing techniques?
• RQ3 Is a relative semi-log canonical divisor numerically trivial over a base if it is numerically trivial in the relative sense?
• RQ4 Does the relative semi-ampleness of the canonical divisor on the normalization of a semi-log canonical pair imply semi-ampleness on the original pair?
• RQ5 Can the relative version of the Birkar–Borisov–Borisov–Cheltenham conjecture be established in the absence of projectivity on the base?

Key findings

• The B-representation of a dlt pair with semi-ample canonical divisor has finite image for sufficiently divisible multiples, providing a new Hodge-theoretic proof of this finiteness.
• The relative semi-log canonical abundance conjecture holds if the log canonical pair obtained via normalization satisfies the abundance conjecture.
• The canonical divisor $K_X + abla$ is semi-ample over $S$ whenever its pullback to the normalization is semi-ample over $S$, under the given decomposition.
• The conjecture of Birkar on the semi-ampleness of nef divisors under certain base locus and positivity conditions is confirmed in the relative setting.
• For a semi-log canonical pair with $K_X + abla χ_{\mathbb{Q},S} 0$, it holds that $K_X + abla \sim_{\mathbb{Q},S} 0$, resolving a question posed by J. Kollár in 2009.
• The proof establishes the relative case of the abundance conjecture for semi-log canonical pairs in full generality, extending prior results that required projective base or absolute setting.