Metric method in Birational Geometry

In this series of notes, we summarize several standard metric methods that are useful in birational geometry.

I. Ohsawa–Takegoshi Extension Theorem

I.1. The Ohsawa–Takegoshi Extension Theorem and Why It Is Useful in Birational Geometry [upd 10.8]

II. Invariance of Plurigenera and extension problem

II.1. Păun’s Analytic Proof of Invariance of Plurigenera [upd 5.11]

II.2. Demailly-Hacon-Paun’s Proof of DLT Extension Theorem

III. Positivity of Direct Images with applications

III.1. Păun–Takayama’s Construction of Singular Hermitian Metrics on Direct Images of Relative (Pluri)canonical Sheaves

III.2. Hacon–Popa–Schnell’s Construction of Singular Hermitian Metrics on Direct Images of Relative (Pluri)canonical Sheaves [upd 10.24]

III.3. Applications of positivity of direct images

IV. Metric methods in the canonical bundle formulas

IV.1. Hacon–Păun’s Metric Method for the Canonical Bundle Formula

V. Metric Methods in Boundedness Problems

V.1. The Theorem of Angehrn and Siu

V.2. Birational Boundedness for Varieties of General Type

VI. Metric Methods in the Abundance Conjecture

VI.1. Supercanonical Metrics and the Abundance Conjecture

VI.2. Metrics with Minimal Singularities

VI.3. Siu and Păun’s Analytic Proof of Finite Generation of the Canonical Ring and Shokurov’s Non-vanishing