Mori theory

Mori theory
Name	Mori theory
Field	Algebraic geometry
Developed by	Shigefumi Mori
Notable ideas	Minimal Model Program, extremal rays, flips, abundance conjecture
Related	Birational geometry, Fano varieties, Calabi–Yau manifolds

Mori theory is a central framework in Algebraic geometry that reshaped the classification of higher-dimensional Complex manifolds and projective varieties. Originating in work by Shigefumi Mori and influenced by ideas from Kunihiko Kodaira, Federigo Enriques, and the Italian school associated with Guido Castelnuovo, it connects techniques from the study of Fano varietys, Calabi–Yau manifolds, and the birational methods used in the classification of surfaces and Threefolds.

Introduction

Mori theory builds on foundational results from Italian school investigations and the classification of surfaces by Federigo Enriques and Guido Castelnuovo, bringing tools from the work of Alexander Grothendieck on schemes, Jean-Pierre Serre on cohomology, and Maxim Kontsevich on enumerative geometry into the birational program. The subject crystallized after breakthroughs by Shigefumi Mori in the study of extremal contractions on threefolds and influenced subsequent work of Yujiro Kawamata, Vladimir V. Shokurov, Paul Hacking, and James McKernan. It aims to reduce classification problems to the study of well-understood building blocks such as Fano varietys, minimal models appearing in the Calabi–Yau regime, and varieties admitting Mori fiber spaces akin to structures studied by Enriques and Kodaira.

Minimal Model Program

The Minimal Model Program (MMP) formalizes steps to produce minimal models or Mori fiber spaces using techniques developed by Shigefumi Mori, extended by V. V. Shokurov, Yujiro Kawamata, Christopher Hacon, and James McKernan. Starting with a projective variety similar to examples in Fano and K3 surface studies, one analyzes the canonical divisor as in the work of Kunihiko Kodaira to decide birational modifications inspired by Castelnuovo’s contraction theorem and Enriques classification. The MMP employs cone theorems from results related to Mori and Sakai and uses tools developed in the context of Grothendieck’s cohomological machinery and Hodge theory as in the work of Phillip Griffiths and Wilfried Schmid.

Extremal Rays and Contraction Theorems

Extremal rays arise in the structure theorem for the cone of curves, a concept advanced by Shigefumi Mori and connected to earlier ideas of Mumford and Zariski. The cone theorem, refined by Mori and later by Kawamata and Shokurov, identifies countably many extremal rays analogous to descriptions in Mori’s threefold results and constructs associated contraction morphisms reminiscent of contraction techniques used by Castelnuovo. Contraction theorems classify possible contractions into divisorial, flipping, and fiber type cases, paralleling classification approaches seen in work by Enriques and modern developments by Hacon and McKernan.

Flips, Flops, and Termination

The introduction of flips resolved obstacles in higher-dimensional birational transformations, a leap achieved by Shigefumi Mori for threefolds and extended in higher dimensions by V. V. Shokurov, Christopher Hacon, James McKernan, and Caucher Birkar. Flops, studied in the context of Calabi–Yau manifolds and influenced by examples from Miles Reid and Mark Gross, provide crepant birational maps preserving canonical class; flips change the birational model while improving the canonical divisor as in approaches by Kawamata and Shokurov. Termination of flips, a subtle issue addressed by work of Hacon, McKernan, Birkar, and conjectures formulated by Shokurov, remains central to proving full existence of minimal models and is tied to abundance-type statements historically related to conjectures considered by Kodaira.

Applications and Examples

Mori theory has been applied to classify threefolds as in Shigefumi Mori’s landmark results and to resolve parts of the Yamabe-like classification problems linked to Fano varietys and Calabi–Yau families studied by Candelas and Strominger. It underpins proofs about rational connectedness influenced by Frédéric Campana and János Kollár, informs moduli problems connected to work by David Mumford and Mikhail Gromov, and intersects mirror symmetry themes explored by Maxim Kontsevich and Lev Borisov. Concrete examples include classifications of del Pezzo surfaces extended to threefold analogues treated by Iskovskikh and Yuri Manin, and explicit flop transitions analyzed by Reid in local models.

Extensions and Higher-Dimensional Results

Extensions of Mori theory to higher dimensions build on methods by V. V. Shokurov and the birational boundedness results of Caucher Birkar, Paolo Cascini, Christopher Hacon, and James McKernan. Recent progress toward the abundance conjecture and existence of minimal models in arbitrary dimension draws on collaborative advances by Hacon, McKernan, Kawamata, Shokurov, and Birkar, and connects to classification programs influenced by Grothendieck’s stacks formalism and deformation techniques developed by Kodaira and Masatake Kuranishi. These higher-dimensional results also interact with topics studied by Maxim Kontsevich in derived categories and by Paul Seidel in symplectic geometry, broadening the impact of Mori-theoretic ideas across Algebraic geometry.

Category:Algebraic geometry