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 notes is to introduce fibrations in algebraic geometry (the classification theory of complex algebraic/analytic varieties). Fibrations are among the most powerful tools for the classification of varieties, and fibration structures are particularly well suited to inductive arguments. In these notes, we focus mainly on applications of fibrations to classification results. Fibrations are central in algebraic geometry.

In algebraic geometry (birational geometry), we understand varieties using fibrations.

Part I. Fibrations in Algebraic Geometry

This part of the notes is based on Fibrations in Algebraic Geometry and Applications by Professor Voisin. What is new is that we add more applications in birational geometry. The general idea is

We try to construct fibrations for which the fibers are 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, Shafarevich Map with Applications

Note-I.5 Core Fibration, Bogomolov Line Bundle, and Campana Special Variety with Applications

Note-I.6 Applications of the Leray Spectral Sequence in Fibration Problems

Part II. Singularities and Positivity of Fibrations

Part III. Foliation in Algebraic Geometry

Note-III.1 Campana–Păun’s Algebraic Criterion for Foliations and Cao–Păun’s Generalization

Part IV. Beauville–Bogomolov–Yau Decomposition

Note-IV.1 Local Triviality of the Albanese Fibration

Note-IV.2 Splitting of the Tangent Sheaf

Note-IV.3 Proof of the Beauville–Bogomolov–Yau Decomposition (Projective klt Pair)

Note-IV.4 Algebraic Approximation for Kähler Calabi–Yau Manifolds

Note-IV.5 Proof of the Beauville–Bogomolov–Yau Decomposition (Kähler klt Pair)

Part V. Structure Theorems for Projective varieties/Kähler varieties with nef anti-canonical bundle

In this part of the notes, I summarize classification results for projective/Kähler manifolds with positive tangent/cotangent bundles. The main references for this part are DPS 94, DPS 01, CCM21, Wang22, MW25, MW25, MWWZ25.

Note-IV.0 Overview [4.3]

Note-IV.1 Numerical Flatness Criteria

Note-IV.2 Positivities of the direct images [4.4]

Note-IV.3 Birational Geometry of the MRC/Albanese fibration

Note-IV.3 Criteria for Fibrations to Be Locally Trivial

Note-IV.4 Splitting of the Tangent Sheaf

Note-IV.5 Structure Theorem for klt Projective Varieties with Nef Anti-canonical Bundle

Note-IV.6 Structure Theorem for klt Kahler Varieties with Nef Anti-canonical Bundle

Part V. Hodge Theory and Fibrations

Note-V.1 Abelian Fibrations and BBDG Decomposition

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The purpose of this series of notes is to discuss the recent paper Moishezon morphism by Professor Kollár in detail. Let us first briefly answer the question WHY Moishezon?

First, Moishezon spaces have more functorial behavior (compared with projective varieties), as we will see in my notes. Secondly, from almost any projective variety we can construct a Moishezon space via bimeromorphic modification, making Moishezon spaces versatile in birational geometry. Thirdly, by Artin’s fundamental theorem, the category of Moishezon spaces appears naturally in moduli theory. Another compelling reason to consider the Moishezon category is that it allows cutand-paste operations similar to those we can perform in topology.

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