Li Yi

Li Yi

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

Fibration and Foliation in Algebraic Geometry

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The aim of this series of note is to introduce the fibration in Algebraic geometry (classification theory of complex algebraic/analytic varieties). Fibration is the most powerful tool for the classification of varieties and fibration structure is very suitable for the induction process. In this notes we will major focus on the possible applications of the fibrations in the classification results. Fibration is central in Algebraic Geometry.

In algebraic geometry (birational geometry), we understand variety using fibration.

Part I. Fibrations in Algebraic Geometry

This part of notes is based on the Fibrations in algebraic geometry and applications by Professor Voisin, what’s new is we will add more applications in birational geometry.

Note-I.0 General Machine to construct Fibrations

Note-I.1. Iitaka Fibrations with applications

Note-I.2. Albanese map with applications [upd 10.10]

Note-I.3. MRC Fibrations with applications

Note-I.4. Gamma reduction, Shafarovich map with applications

Note-I.5. Core fibration, Bogomolov line bundle and Campana special variety with applications

Note-I.6. How to use Harder-Narasimhan filtration

Note-I.7. Applications of Leray Spectral sequence in fibration problem

Part II. Singularity of Fibrations

Part III. Positivities of Fibrations

Note-III.1. The Iitaka conjecture

Note-III.2. Nef reduction

Note-III.3. Pseudo-effective reduction

Part IV. Structure theorems for projective manifold/Kahler manifold with positive tangent/cotangent bundles

In this part of notes I will summarize the classification results for projective/Kahler manifolds with positive tangent/cotangent bundles. The major references of this part of notes is DPS 94, DPS 01.

Part V. Hodge Theory and Fibrations

Note-V.1: Abelian fibrations and BBDG decomposition

Part VI. Fibration with specific varieties

Note-VI.1. Toric Fibrations

Note-VI.2 Elliptic fibrations

Note-VI.3 Calabi-Yau Fibrations

Note-VI.4 Abelian Fibrations

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