Existence of minimal models for varieties of log general type

2009/11/13 by Caucher Birkar, Paolo Cascini, Christopher D. Hacon +1 · 25 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Meromorphic and Entire Functions

paper · pdf · doi:10.1090/s0894-0347-09-00649-3

openalex publication_date 2009/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25

Abstract

We prove that the canonical ring of a smooth projective variety is finitely generated.

Citations

• A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring
• Log canonical models for the moduli space of curves: First divisorial contraction
• Pluricanonical systems of projective varieties of general type

Cited by

• Fano Varieties in Mori Fibre Spaces
• On Lagrangianity of p-supports of holonomic D-modules and q-D-modules
• On isolated log canonical singularities with index one
• A klt generalized pair with infinitely generated canonical ring
• Towards boundedness of minimal log discrepancies by Riemann--Roch theorem
• Complexity one varieties are cluster type
• On K-polystability for log del Pezzo pairs of Maeda type
• On semipositivity, injectivity and vanishing theorems
• Discreteness of 𝐹-jumping numbers at isolated non-ℚ-Gorenstein points
• Boundedness and volume of generalised pairs
• Manin’s b-constant in families
• Algebraic surfaces and hyperbolic geometry
• Perfectoid C_i transfer
• Cone theorem and Mori hyperbolicity
• On birational involutions of \mathbb P³
• Geometry of linear determinantal quartic 3-folds via their intermediate Jacobian
• On the anti-canonical geometry of weak ℚ -Fano threefolds II
• Semi-terminal modifications of demi-normal pairs
• Injectivity Theorems with Multiplier Ideal Sheaves and Their Applications
• Existence of flips for generalized lc pairs
• Moduli of polarized Calabi-Yau pairs
• On Eckl's pseudo-effective reduction map
• Nonexistence of asymptotic GIT compactification
• Birational geometry [wikipedia]
• Canonical ring [wikipedia]
• Flip (algebraic geometry) [wikipedia]

Related

• Existence of minimal models for varieties of log general type
• Existence of minimal models for varieties of log general type II
• On the non-vanishing conjecture and existence of log minimal models
• Quasi-log varieties
• On the anti-canonical ring and varieties of Fano type