Li Yi

Li Yi

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

My PhD Thesis Project: Kähler minimal model program

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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. Reading Notes on the Kähler minimal model program

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

Note-1 The Bott-Chern classes

Note-2 Positivities in the Kähler minimal model program [upd 10.10]

Note-3 Lelong number and quasi-psh functions

Note-4 Transcendental volume function

Note-5 Boucksom’s divisorial Zariski decomposition

Note-6 Generalized Kähler pairs

Note-7 Analytic contractibility Theorem [upd 9.28]

Note-8 Divisorial contractions for Kähler threefolds [upd 3.30]

Note-9 Flipping contractions for Kähler threefolds [upd 4.9]

Note-10 On the Kähler 4-fold MMP

Note-11 Das-Hacon’s transcendental MMP for projective varieties

Note-12 Finite generation problem under the Kähler setting

Note-13 Termination problems in Kähler MMP

Note-14 Transcendental base point free conjecture,

Note-15 Cao–Horing’s Producing rational curves on compact Kähler manifolds [upd 5.23]

Note-16 Cone theorem for the Kähler MMP [upd 12.28]

Note-17 Existence of minimal model, good minimal models in the Kähler setting

Note-18 Canonical bundle formulas and subadjunction with applications in the Kähler MMP

Note-19 Properties of uniruled Kähler manifolds

Note-20: BCHM for projective morphism between complex analytic spaces

Note-21: Abundance for Kähler threefolds

Supplement: Tools can be used to reduce a Kähler problem to a projective problem

Part II. 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]

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