Li Yi

Li Yi

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

Birational Geometry Note: Kahler minimal model program

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The aim of this note is to introduce the minimal model program for Kahler varieties. Compared with the well-known minimal model program for projective varieties, the theory of Kahler minimal model program will have the following three major difficulties: (1) Mori bend-and-break technique: Mori bend and break is not known for Kahler varieties, (2) The base point free theorem and cone theorem: Since Kahler manifold with a big line bundle is automatic projective, thus a big line bundle does not make sense on (non-projective) Kahler 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.

Note-0 Overview of Kahler minimal model program [upd 10.9]

Note-1 Transendental Volume (for big cohomology class)

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

Note-3 Analytic Contractibility Theorem [Update 9.28]

Note-4 Divisorial constraction for Kahler 3-fold [Update 3.30]

Note-5 Flipping contraction for Kahler 3-fold MMP [Update 4.9]

Note-6 Kaehler 4-fold MMP

Note-7 Das-Hacon’s transcendental MMP for projective varieties

Note-7 Finite generation problem in Kahler MMP

Note-8 Termination problem in Kahler MMP

Note-9 Transcendental Base point free Conjecture,

Note-10 Rational curves on Kaehler varieties [update 5.23]

Note-11 Cone theorem for Kahler MMP [Update 5.23]

Note-12 Projectivity for Kaehler morphism

Note-13 Canonical bundle formulas and subadjunction with applications in Kahler MMP

Note-14 How to use divisorial Zariski decomposition?

Note-15 How to use generalized pair in the Kahler setting?

Note-16 Properties of uniruled Kahler manifold

Note-17: BCHM for projective morphism between complex analytic spaces by Fujino and Das-Hacon-Paun

Note-18: Abundance for Kahler varieties

Supplement: Tools can be used to reduce the Kahler problem to projective problem [upd 9.30]

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