Abundance for Minimal 3-Folds

Abstract

In the first section, we give a quick introduction to the birational classification theory due to Enriques-Kodaira-Shafarevich-Iitaka, which divides the n-dimensional algebraic varieties into n + 2 disjoint classes according to a single invariant called the “Kodaira dimension” k. A nice thing of the theory is that a variety of intermediate Kodaira dimension has a canonical structure of a fibre space unique up to birational equivalence (“Iitaka fibration”).

This classification would be incorporated to the Minimal Model Program through the “abundance conjecture”, which asserts that the canonical divisor of a minimal variety should be semiample, meaning in particular that a numerical invariant v(X) computes the Kodaira dimension k(X). Some easy cases where the abundance conjecture is actually verified are discussed in Section 2. Along with the minimal model theory, the existence of Iitaka fibration is also essential in the argument.

In Section 3, the non-negativity of the Kodaira dimension of a minimal threefold is proved. The key to the proof is the pseudo-effectivity of c₂ proved in Lecture III. We are exceptionally lucky in this case, because the Todd classes involve only c₁ and c₂ in dimension three.

By the result in Section 3, we can find an effective divisor S in a pluricanonical linear system on a minimal threefold. The proof of the abundance conjecture relies on the analysis of linear systems on S. Principal difficulties are caused from the fact that S may be highly reducible and non-reduced. The complete proof of the three-dimensional abundance conjecture requires fairly technical materials (log-minimal models etc.), which would be beyond the scope of these lectures intended for non-specialists. Instead, Section 4 gives a proof in easier cases where S is smooth. Still the discussion involves the heart of the idea and, hopefully, will illustrate guiding principles toward the proof in general cases.