Li Yi

Li Yi

I am a 3rd year PhD student majoring in birational geometry

My PhD Thesis Project: Kähler minimal model program

2 minute read

Published:

The aim of this note is to introduce the minimal model program for Kähler varieties. Compared with the well-known minimal model program for projective varieties, the theory of Kähler minimal model program will have the following three major difficulties: (1) Mori bend-and-break technique: Mori bend and break is not known for Kähler varieties, (2) The base point free theorem and cone theorem: Since Kähler manifold with a big line bundle is automatic projective, thus a big line bundle does not make sense on (non-projective) Kähler manifold, (3) Contraction theorem: In the classical MMP, we require the base point free theorem to produce some semi-ampleness divisor. This semi-ample divisor will induce the negative contraction morphism, and this approach is not very clear under Kaehler setting.

In general, we aim to study the following question: To what extent are natural geometric or sheaf-theoretic constructions on compact Kähler manifolds determined by analogous constructions in the projective setting? Further structural results could provide a general framework for reducing certain questions about Kähler manifolds to the algebraic setting.

Part I. My PhD Thesis on Kähler minimal model program

My PhD Dissertation Defense Slides: The Kähler Minimal Model Program and Its Applications to Deformation Problems

My PhD Thesis: The Kähler Minimal Model Program and Its Applications to Deformation Problems [working in progress]

My PhD Thesis (Chinese version): 凯勒极小模型纲领及其在变形理论中的应用 [working in progress]

Part-II. Reading Notes on the Kähler minimal model program

Note 0 An Overview of the Kähler Minimal Model Program [upd 10.9]

Note-1 Positivity in the Kähler Minimal Model Program [upd 10.10]

Note-2 Lelong Numbers and Quasi-psh Functions [TODO]

Note-3 Transcendental Volume Function [TODO]

Note-4 Boucksom’s Divisorial Zariski Decomposition [TODO]

Note-5 Generalized Kähler Pairs [TODO]

Note-6 Analytic Contractibility Theorem [upd 9.28]

Note-7 Divisorial Contractions for Kähler Threefolds [upd 3.30]

Note-8 Flipping Contractions for Kähler Threefolds [upd 4.9]

Note-9 On the Kähler 4-Fold MMP [TODO]

Note-10 Finite Generation Problem in the Kähler Setting [TODO]

Note-11 Termination Problems in the Kähler MMP

Note-12 Transcendental Base Point Free Conjecture

Note-13 Cao–Höring: Producing Rational Curves on Compact Kähler Manifolds [upd 5.23]

Note-14 Cone Theorem for the Kähler MMP [upd 12.28]

Note-15 Existence of Minimal Models and Good Minimal Models in the Kähler Setting

Note-16 Canonical Bundle Formulas and Subadjunction with Applications in the Kähler MMP

Note-17 Properties of Uniruled Kähler Manifolds

Note-18: BCHM for Projective Morphisms Between Complex Analytic Spaces

Note-19: Abundance for Kähler Threefolds

Supplement: Tools That Can Be Used to Reduce a Kähler Problem to a Projective Problem

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First, Moishezon spaces have more functorial behavior (compared with projective varieties), as we will see in my notes. Secondly, from almost any projective variety we can construct a Moishezon space via bimeromorphic modification, making Moishezon spaces versatile in birational geometry. Thirdly, by Artin’s fundamental theorem, the category of Moishezon spaces appears naturally in moduli theory. Another compelling reason to consider the Moishezon category is that it allows cutand-paste operations similar to those we can perform in topology.

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