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| toric varieties | |
|---|---|
| Name | Toric varieties |
| Field | Algebraic geometry |
| Introduced | 1970s |
| Notable | David Cox; William Fulton; Victor Kac |
toric varieties
Toric varieties are algebraic varieties built from combinatorial data of lattices and fans, forming a bridge between algebraic geometry and combinatorics. They provide explicit models where tools from convex geometry, representation theory, and number theory interact, and they serve as accessible test cases in the study of birational geometry, mirror symmetry, and moduli problems.
Origins of toric varieties trace to work connecting algebraic tori with polyhedral geometry studied by David Mumford, Igor Dolgachev, and Michel Demazure; further development involved William Fulton, David Cox, and Tadao Oda. Major influences include classical algebraic geometry exemplified by the Italian school, the work of Alexander Grothendieck on schemes, and combinatorial geometry associated with Branko Grünbaum and Hermann Minkowski. Related figures and institutions that advanced the subject include the Institute for Advanced Study, Massachusetts Institute of Technology, and the European Mathematical Society. Applications have linked the topic to string theory through Edward Witten, mirror symmetry research via Maxim Kontsevich and Cumrun Vafa, and computational algebra through work at Symbolic Computation groups and the Max Planck Institute.
A fan consists of strongly convex rational polyhedral cones in a lattice; foundational combinatorial contributors include Gustave Choquet, Hermann Weyl, and John Nash. Key constructions use cones studied by Augustin-Louis Cauchy and Hermann Minkowski, while lattice theory builds on work of Richard Dedekind and Emmy Noether. The study of polytopes feeding into fans invokes results of George B. Dantzig and Garrett Birkhoff; duality notions connect to Sophus Lie and Élie Cartan in representation-theoretic contexts. Classical convexity theorems from Carathéodory and Helly underpin combinatorial lemmas used by Igor Shafarevich and Jean-Pierre Serre in algebraic settings. Important algorithmic developments are associated with David Avis and Komei Fukuda in computational polyhedral theory.
Affine toric varieties arise from semigroup algebras associated to cones; constructions are analogous to techniques used by Emmy Noether and Jean Leray in algebraic contexts. Projective embeddings rely on lattice polytopes and convex hulls, with combinatorial precedents in works by Pafnuty Chebyshev and Simon Newcomb on discrete structures. Classical projective geometry methods from Alexander Grothendieck and André Weil inform the study of ample line bundles and projective normality. Connections to the theory of torus actions recall research by Sophus Lie, Wilhelm Killing, and Évariste Galois on symmetry and group actions. Seminal classifications of toric Fano varieties build on methods used by Shigefumi Mori and Rei Kawamata in birational geometry.
Morphisms between toric varieties correspond to maps of fans; the categorical perspective links to Grothendieck's work and to developments by Alexander Beilinson and Vladimir Drinfeld in derived functors. Very ample and ample line bundles on toric varieties are governed by polytope data, reflecting classical projective embeddings considered by Joseph-Louis Lagrange and Bernhard Riemann. The study of equivariant embeddings connects to representation-theoretic frameworks of William Fulton and Roger Howe. Geometric invariant theory inputs from David Mumford and Frances Kirwan clarify quotient constructions and Cox ring constructions parallel to work by Igor Krichever and Ernest Vinberg. Notions of basepoint freeness and global generation are used in analogues of Mori theory explored by Shigefumi Mori and Yujiro Kawamata.
Cohomological calculations on toric varieties exploit combinatorial vanishing theorems reminiscent of Kodaira vanishing studied by Kunihiko Kodaira and Jean-Pierre Serre. Sheaf-theoretic methods trace back to Godement and Alexander Grothendieck; local cohomology techniques connect to works of Melvin Hochster and Robin Hartshorne. Cartier and Weil divisor theory on these varieties relates to foundational divisor theory from Heinrich Weber and Oscar Zariski. Intersection theory on toric varieties follows approaches from William Fulton and gives explicit Chow ring descriptions analogous to enumerative techniques in the work of Hermann Schubert and Frank S. Macaulay. Cohomology ring structures and equivariant cohomology tie into research by Atiyah and Bott, as well as Victor Guillemin and Shlomo Sternberg in symplectic contexts.
Resolution of singularities for toric varieties is achieved combinatorially via star subdivisions and refinements of fans, building on Hironaka's resolution program and algorithmic refinements by Heisuke Hironaka and Jan Denef. Factorization and flip phenomena mirror developments in the minimal model program by Shigefumi Mori and Claire Voisin, while explicit crepant resolutions appear in work related to Reid's recipe and Miles Reid. Toric birational maps are studied using techniques employed by Alessio Corti and Valery Alexeev in moduli compactifications. The explicit nature of toric modifications makes them tools in desingularization algorithms used by computing groups led by Bruno Buchberger and Grigory Gandolfi.
Notable examples include projective spaces associated with Bernhard Riemann and Grassmannian degenerations studied by David Eisenbud and Joe Harris. The connection to mirror symmetry involves Kontsevich, Maxim Kontsevich, Cumrun Vafa, and Philip Candelas, with Batyrev’s reflexive polytope construction bridging toric geometry and Calabi–Yau hypersurfaces relevant to string compactifications explored by Edward Witten and Andrew Strominger. Combinatorial mirror constructions relate to works by Victor Batyrev and Lev Borisov, while applications in gauge theory and supersymmetry tie to Edward Witten and Nathan Seiberg. Computational implementations and databases leverage contributions from the Simons Foundation and research groups at the Mathematical Sciences Research Institute.