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| Toric geometry | |
|---|---|
| Name | Toric geometry |
| Caption | Fan and polytope illustration |
| Field | Algebraic geometry |
| Introduced | 1970s |
| Notable people | David Mumford; William Fulton; Victor Kac; Tadao Oda; G. Kempf |
Toric geometry is a branch of algebraic geometry that studies algebraic varieties containing algebraic tori as dense open subsets and equipped with combinatorial structures that encode geometric data. It links concepts from algebraic geometry, convex geometry, combinatorics, and representation theory, enabling explicit constructions and calculations. Toric methods are central in the work of many mathematicians and have strong connections to mirror symmetry, symplectic geometry, and number theory.
Toric geometry grew from interactions among researchers such as David Mumford, William Fulton, Tadao Oda, and G. Kempf, and developed alongside subjects like convex polytopes studied by Branko Grünbaum and Peter McMullen. Key institutions that nurtured the field include Princeton University, Harvard University, the University of Tokyo, and the Institute for Advanced Study. Influential works include Fulton’s textbook and Oda’s monograph, while conferences at Oberwolfach and MSRI helped disseminate advances linked to mirror symmetry and the homological conjectures of Alexander Grothendieck and Pierre Deligne.
Constructions in toric geometry rely on convex geometry results due to Hermann Minkowski and applications of polyhedral theory advanced by Egon Balas and David Ziegler. Lattice polytopes, Newton polytopes associated to works of Olga Taussky-Todd and Igor Shafarevich, and support functions studied by Michel Kervaire connect to Ehrhart theory developed by Eugène Ehrhart. The correspondence between polytopes and projective toric varieties draws on combinatorial tools used by Richard Stanley and Gian-Carlo Rota. Methods for resolving singularities reference the desingularization techniques of Heisuke Hironaka and the subdivision strategies related to Jean-Pierre Serre’s local analysis.
A toric variety is built from a fan, a collection of strongly convex rational polyhedral cones, reflecting ideas from John Conway’s and Branko Grünbaum’s polyhedral studies. The fan construction is central in the work of Tadao Oda, and it interfaces with representation-theoretic perspectives exemplified by Victor Kac and James E. Humphreys. Compactifications such as the Deligne–Mumford compactification and toroidal embeddings studied by Mumford and Faltings illustrate broader moduli problems. Techniques for studying orbit decompositions relate to methods used by André Weil, Claude Chevalley, and Jean-Pierre Serre.
Algebraic properties of toric varieties, including normality, Cohen–Macaulayness, and Gorenstein conditions, connect to commutative algebra results by David Eisenbud, Daniel Buchsbaum, and Alexander Grothendieck. Geometric notions such as singularity types are analyzed using resolution methods influenced by Heisuke Hironaka and Shigefumi Mori’s minimal model program. Intersection theory on toric varieties connects to work by William Fulton and Robert MacPherson, while deformation theory draws on ideas from Pierre Deligne and Michael Artin. Criteria for ampleness and projectivity reflect contributions from Kodaira and Kunihiko Kodaira’s theorems on embeddings.
Cohomological computations for line bundles on toric varieties exploit combinatorial vanishing theorems related to Serre duality and Kodaira vanishing, informed by Jean-Pierre Serre and Kunihiko Kodaira. Cartier and Weil divisors are treated using lattice data reminiscent of Claude Chevalley’s arithmetic methods. The study of Picard groups and divisor class groups reflects techniques developed by Oscar Zariski and André Weil, while vanishing and Riemann–Roch formulas use invents from Friedrich Hirzebruch and John Milnor. Equivariant cohomology methods bring in ideas from Raoul Bott and Robert Bott’s work with Atiyah, and localization techniques echo contributions by Michael Atiyah and Isadore Singer.
Toric varieties provide explicit examples for mirror symmetry studied by Maxim Kontsevich and Cumrun Vafa, and they feature in enumerative geometry problems associated with Gromov–Witten theory advanced by Edward Witten and Alexander Givental. Classic examples include projective spaces, Hirzebruch surfaces, and toric Fano varieties, with classification efforts influenced by K. Sato and Victor Batyrev. Applications to symplectic geometry reflect the moment map techniques of Michèle Vergne and Atiyah, while combinatorial optimization links back to Leonid Khachiyan and George Dantzig. Computational aspects employ software traditions from David J. Eisenbud’s groups and packages developed in SageMath and Macaulay2 communities.
Generalizations of toric geometry include spherical varieties investigated by Ernest Vinberg and Dmitry Timashev, tropical geometry inspired by Imre Simon and Jean-Jacques Moreau, and log geometry developed by Arthur Ogus and Kazuya Kato. Connections to cluster algebras trace to Sergey Fomin and Andrei Zelevinsky, while relations with noncommutative algebraic geometry involve Maxim Kontsevich and Yuri Manin. Recent trends link toric techniques to arithmetic geometry in the tradition of John Tate and Jean-Pierre Serre, and to string-theoretic model building influenced by Edward Witten and Cumrun Vafa.