Complex projective variety
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| Complex projective variety | |
|---|---|
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| Name | Complex projective variety |
| Field | Algebraic geometry |
| Introduced | 19th century |
Complex projective variety is a subset of complex projective space defined by the vanishing of homogeneous polynomials; it is a central object linking Bernhard Riemann, Oscar Zariski, André Weil, Alexander Grothendieck, and Jean-Pierre Serre through problems in classification, cohomology, and moduli. As an object it connects the work of Felix Klein, Bernhard Neumann, David Mumford, Shing-Tung Yau, and Simon Donaldson while appearing in the study of Calabi–Yau manifolds, K3 surfaces, and classical examples such as projective hypersurfaces and projective curves. Complex projective varieties serve as a testing ground for tools from Hodge theory, sheaf theory, intersection theory, and birational geometry developed across the 20th and 21st centuries by figures like Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, and Phillip Griffiths.
Definition and basic examples
A complex projective variety is defined inside Complex projective space P^n by a homogeneous ideal whose common zeros form a closed set in the Zariski topology; classic early examples include projective hypersurfaces studied by Rene Descartes, plane curves such as nonsingular conics and cubics studied by Isaac Newton and Gottfried Leibniz, and higher dimensional varieties like the Veronese and Segre embeddings associated to Giuseppe Veronese and Corrado Segre. Important explicit families include smooth plane curves classified by genus following Bernhard Riemann and canonical curves appearing in the work of Max Noether and Federigo Enriques. Other notable examples are del Pezzo surfaces connected to Pasquale del Pezzo and Fano varieties central to problems addressed by Iskovskikh and Vladimir Iskovskikh.
Projective embeddings and homogeneous coordinates
Embeddings into projective space use homogeneous coordinates introduced in the classical period and formalized in modern language by Karl Weierstrass and Bernhard Riemann; the projective embeddings of curves and surfaces are studied through very ample line bundles following results of Alexander Grothendieck and criteria by David Mumford and Jean-Pierre Serre. Constructions like the Veronese map of Giuseppe Veronese, the Segre map of Corrado Segre, and the Plücker embedding for Grassmannians studied by Julius Plücker give canonical projective models employed in invariant theory developed by David Hilbert and Emmy Noether. Techniques from geometric invariant theory of David Mumford and the ampleness criteria of Andrew J. Sommese are used to produce and compare projective embeddings in work influenced by Michael Artin and John Tate.
Algebraic and analytic perspectives
From the algebraic perspective a complex projective variety is a scheme of finite type over Complex numbers with projective structure formalized by Alexander Grothendieck; from the analytic perspective it admits a structure of a compact complex analytic space treated in the work of Kiyoshi Oka, Henri Cartan, and Hiroshima Kodaira. The dictionary between algebraic and analytic categories is exemplified by the GAGA results of Jean-Pierre Serre and extended by comparisons in Hodge theory by Phillip Griffiths and Pierre Deligne. Bridging approaches involve the use of coherent sheaves and analytic sheaf cohomology studied by Kōsaku Yosida and spectral sequence techniques refined by Jean Leray and H. Cartan.
Topological and geometric properties
As compact complex manifolds in the smooth case, projective varieties have topological invariants studied via singular cohomology and Hodge decompositions developed by W. V. D. Hodge and refined by Pierre Deligne and Phillip Griffiths. Betti numbers, Chern classes, and intersection numbers appear in the work of Hirzebruch and Raoul Bott, while Lefschetz theorems studied by Solomon Lefschetz and Picard–Lefschetz theory govern hyperplane section behavior. Examples connecting topology to arithmetic include the use of étale cohomology by Alexander Grothendieck and Pierre Deligne and comparisons to complex topology in results linked to Serre duality and the Hirzebruch–Riemann–Roch theorem of Friedrich Hirzebruch.
Line bundles, divisors, and cohomology
Line bundles and Cartier divisors on projective varieties are central through the Picard group and the Néron–Severi group studied by André Néron and Francois Severi, with the Riemann–Roch theorem and its generalizations by Hirzebruch and the Grothendieck–Riemann–Roch formalism of Alexander Grothendieck governing dimensions of global sections. Cohomology of coherent sheaves, developed in foundational form by Jean-Pierre Serre and Alexander Grothendieck, yields vanishing theorems such as Kodaira vanishing and Kawamata–Viehweg vanishing proved by Kunihiko Kodaira and Yujiro Kawamata and Eckart Viehweg. The study of ample and nef divisors connects to the Minimal Model Program advanced by Shigefumi Mori and consolidated by contributions from János Kollár and Vyacheslav Shokurov.
Singularities and resolution
Singularities of projective varieties are classified using techniques developed by Oscar Zariski and modern birational methods by Shigefumi Mori, Heisuke Hironaka who proved resolution of singularities in characteristic zero, and later developments by Markushevich and Miles Reid in surface and threefold theory. Notions such as rational singularities, log canonical and Kawamata log terminal singularities are central to the Minimal Model Program and appear in the work of János Kollár, Vyacheslav Shokurov, and Christopher Hacon. Resolution techniques employ blowups introduced by Max Noether and formalized in scheme language by Alexander Grothendieck and Heisuke Hironaka.
Moduli and classification theories
Moduli spaces parametrizing projective varieties, such as the moduli of curves constructed by David Mumford and compactified by Pierre Deligne and David Mumford (Deligne–Mumford compactification), and moduli of vector bundles developed by Maxwell Atiyah and Raoul Bott and later by Simon Donaldson, are central classification tools. Geometric Invariant Theory of David Mumford underpins many constructions, while Torelli theorems and period maps studied by Phillip Griffiths and Wilhelm Schmid relate Hodge structures to moduli. Classification programs for surfaces by Federigo Enriques and Francesco Severi and the higher-dimensional Minimal Model Program led by Shigefumi Mori and János Kollár aim to organize projective varieties into birational equivalence classes and to produce canonical models such as Fano varieties studied by Vladimir Iskovskikh and Yuri Manin.