Fibration and Foliation in Algebraic Geometry

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.

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

Part I. Canonical 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 The General Machine for fibration

Note-I.1 Iitaka Fibrations with Applications [upd 7.5]

Note-I.2 Albanese Map with Applications [upd 10.10]

Note-I.3 MRC Fibrations with Applications [upd 10.19]

Note-I.4 Nef Reduction with Applications

Note-I.5 Algebraic Reduction with Applications

Part II. Fibrations from the Minimal Model Program

Note-II.1 Fano type morphism as Outputs of the MMP

Note-II.2 Fano Fibrations (Mori Fiber Spaces)

Note-II.3 Calabi–Yau Fibrations

Note-II.4 Canonical bundle formulas

Part III. Foliation in Algebraic Geometry

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

Note-IV.2 Algebraically Integrable Foliations: From Foliations to Fibrations

Part V. Beauville–Bogomolov–Yau Decomposition

In this part of the notes, I summarize recent developments on the Beauville–Bogomolov decomposition for singular (klt) Calabi–Yau varieties, in both the projective and the Kähler settings. The major reference of my note is Beauville-Bogomolov decomposition for klt varieties by Henri Guenancia.

Note-V.1 Local Triviality of the Albanese Fibration

Note-V.2 Splitting of the Tangent Sheaf

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

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

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

Part VI. Structure Theorems for Projective/Kähler Varieties with Nef Anti-canonical Bundle

In this part of the notes, I summarize classification results for projective/Kähler varieties with nef anti-canonical bundle. This is where the whole machinery of the series comes together: the MRC and Albanese fibrations (Part I), positivity of direct images (Part III), splitting of the tangent sheaf, and the BBY decomposition (Part V). The main references for this part are DPS 94, DPS 01, CCM21, Wang22, MW25, MW25, MWWZ25.

Note-VI.0 Overview [4.3]

Note-VI.1 Numerical Flatness Criteria

Note-VI.2 Positivities of the Direct Images [4.4]

Note-VI.3 Birational Geometry of the MRC/Albanese Fibration

Note-VI.4 Criteria for Fibrations to Be Locally Trivial

Note-VI.5 Splitting of the Tangent Sheaf

Note-VI.6 Structure Theorem for klt Projective Varieties with Nef Anti-canonical Bundle

Note-VI.7 Structure Theorem for klt Kähler Varieties with Nef Anti-canonical Bundle

Note-VI.8 On the Hacon–McKernan Question