Lecture on the abundance

こんなストーリーは望んでいなかったけど、一応、科学者なので、真理に向き合わないといけないので、講義をします。OpenAIの公開証明に基づいてアバンダンス予想及び極小モデルの存在性を解説します。勉強しながらの講義になります。ご了承ください。正直パッとみた感じ60~70%はwellknown for expertsの議論で構成されていて、初めて見る議論でも局所的には慣れしたんだものばかり。教員の初めて見る議論を中心の講義をして、証明が本当にworkしてるかチェックします。
This is not how I had hoped the story would unfold. Nevertheless, as a scientist, I believe I have to face the truth, so I have decided to give a lecture series. In this series, I will explain the abundance conjecture and the existence of minimal models based on the proofs released by OpenAI. I will be learning the proofs myself as I prepare and deliver the lectures, so I ask for your understanding. To be frank, at first glance, about 60–70% of the proofs seem to consist of arguments that are well known to experts. Even the arguments I have never seen before look locally familiar. I plan to focus mainly on the arguments that are new to me, with the aim of carefully examining whether the proofs really work.

1. 10/9, Proof strategy and known steps. Set up for ACC for Lelong number
2. 10/16,
   ・ACC for Lelong number (verified): Unlike the usual ACC statements for lct or mld, where the underlying varieties are allowed to vary, this result involves a large amount of fixed data and therefore follows essentially as a consequence of the Izumi-type inequality of Demailly–Kollár.
   ・Reduction to no generic fibration admitting a positive closed current with null along a fiber : In a sense, this can be regarded as a current-theoretic analogue of the algebraic fiber-space strategy. I suspect that this approach had already been explored to some extent from an analytic perspective. However, the actual argument involves a clever combination of algebraic and analytic techniques. The argument does not essentially go as far as constructing a foliation, but rather serves merely as a reduction step. The crucial point here is the assumption \kappa=-\infty in the proof by contradiction. Normally, an argument of this kind would be expected to yield a contradiction to the assumption that K is not big, but the fact that this does not happen is precisely what makes the present situation different. As a consequence, just as in the three-dimensional case, the proof has to be divided into two parts: non-vanishing and extension.

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