Li Yi

Li Yi

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

Birational Geometry and Deformation Theory

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Published:

The aim of this series of notes is to give a brief introduction to the basics of deformation theory, and then to focus on its applications to certain problems in birational geometry.

Part I. Invariance of plurigenera and Volume problems

Note-I.1 Levine’s deformation theoretical method to invariance of plurigenera problem [update 8.12]

Note-I.2 Levine–Fujiki’s approach to Deformation invariance of uniruled varieties [update 8.12]

Note-I.3 Nakayama’s invariance of plurigenera results [update 8.15]

Note-I.4 Algebraic proof of invariance of plurigenera with general type assumption

Note-I.5 Paun-Demailly’s analytic proof of invariance of plurigenera [update 8.25]

Note-I.6 Hacon-Mckernan-Xu’s log invariance of plurigenera and volumes

Note-I.7 Takayama’s proof of invariance of plurigenera

Note-I.8 Kollar’s MMP proof of invariance of plurigenera

Note-I.9 Deformation invariance of numerical Kodaira dimension of Nakayama

Note-I.10 Cao-Paun’s approach to invariance of plurigenera

Note-I.11 Siu’s analytic proof of invariance of plurigenera (without general type condition)

Note-I.11 Applications of invariance of plurigenera

Part II. Deformation of Positivities in Birational Geometry

Nite-II.1: Picard Group, Numerical Group in Families

Note-II.2: deFernex–Hacon variation of cones in Families

Note-II.3: Deformation bahavior of pseudo-effectiveness

Note-II.4: Deformation of nefness in family

Note-II.5: Deformation of numerical trivial condition

Part III. Deformation of minimal models, good minimal models

Note-III.1: Existence of good minimal on the closure

Note-III.2: Deformation openness of existence of good minimal model

Part I. Basic knowledge on deformation theory

Note-1: Kodaira-Spencer map, Kodaira-Spencer correspondence,

Note-2: A brief introduction to Obstruction Theory

Note-3: Maurer-Cartan equation, DGLA

Note-4: Bogomolov-Tian-Todorov Theorem, ddbar method and Ran’s T1 lifting theorem with applications

Part II. Deformation of (birational) morphism

Note-1: Blow up in deformation theory

Note-2: Deformation of (rational) curves in complex space

Note-3: Deformation of morphisms

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