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. The general idea is

We try to construct fibrations for which the fibers are instead simpler than the total space, reducing in principle the study to phenomena on the base.

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 [upd 10.19]

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. Applications of Leray Spectral sequence in fibration problem

Part II. Singularity and Positivities of Fibrations

Part III. Foliation in algebraic geometry

Note-III.1 Camapana-Paun’s algeraic criterion of foliation and Cao-Paun’s generalization

Part IV. Beauville-Bolgomolov-Yau Decomposition

Note-IV.1 Locally trivality of the Albanese fibration

Note-IV.2 Splitting of the tangent sheaf

Note-IV.3 Proof of Beauville-Bolgomolov-Yau decomposition (projective klt pair)

Note-IV.4 Algebraic approximation for Kahler Calabi-Yau

Note-IV.5 Proof of Beauville-Bolgomolov-Yau decomposition (Kahler klt pair)

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.

Note-IV.1 Criterion for Fibrations to be locally trivial

Note-IV.1 Structure theorem for projective manifolds with nef anti-canonical bundle

Note-IV.2 Structure theorem for klt projective varieties with nef anti-canonical bundle

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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