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In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985), after Eugenio Calabi who first conjectured that such surfaces might exist, and Shing-Tung Yau (1978) who proved the Calabi conjecture.
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics.
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative". The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective space. In view of the correspondence between line bundles and divisors, there is an equivalent notion of an ample divisor.
In algebraic geometry, the Kodaira dimensionκ(X) measures the size of the canonical model of a projective variety X.
In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials.
In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero. The implications for the group with index q = 0 is usually that its dimension — the number of independent global sections — coincides with a holomorphic Euler characteristic that can be computed using the Hirzebruch–Riemann–Roch theorem.
In algebraic geometry, a Fano variety, introduced by Gino Fano in, is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties.
In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and divisors, there is an equivalent notion of a nef divisor.
In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories.
In algebraic geometry, the Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space determined by L. This is 1 less than the dimension of the section ring of L
In commutative algebra, the multiplier ideal associated to a sheaf of ideals over a complex variety and a real number c consists (locally) of the functions h such that
In differential geometry, algebraic geometry, and gauge theory, the Kobayashi–Hitchin correspondence relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence is named after Shoshichi Kobayashi and Nigel Hitchin, who independently conjectured in the 1980s that the moduli spaces of stable vector bundles and Einstein–Hermitian vector bundles over a complex manifold were essentially the same.
In several complex variables, the Ohsawa–Takegoshi L2 extension theorem is a fundamental result concerning the holomorphic extension of an -holomorphic function defined on a bounded Stein manifold to a domain of higher dimension, with a bound on the growth. It was discovered by Takeo Ohsawa and Kensho Takegoshi in 1987, using what have been described as ad hoc methods involving twisted Laplace–Beltrami operators, but simpler proofs have since been discovered. Many generalizations and similar results exist, and are known as theorems of Ohsawa–Takegoshi type.
Jean-Pierre Demailly was a French mathematician who worked in complex geometry. He was a professor at Université Grenoble Alpes and a permanent member of the French Academy of Sciences.
Lawrence Man Hou Ein is a mathematician who works in algebraic geometry.
Mihnea Popa is a Romanian-American mathematician at Harvard University, specializing in algebraic geometry. He is known for his work on complex birational geometry, Hodge theory, abelian varieties, and vector bundles.
In the mathematical fields of differential geometry and algebraic geometry, the Frankel conjecture was a problem posed by Theodore Frankel in 1961. It was resolved in 1979 by Shigefumi Mori, and by Yum-Tong Siu and Shing-Tung Yau.
In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds. K-stability is of particular importance for the case of Fano varieties, where it is the correct stability condition to allow the formation of moduli spaces, and where it precisely characterises the existence of Kähler–Einstein metrics.
References
- Ein, Lawrence; Lazarsfeld, Robert (1993), "Global generation of pluricanonical and adjoint linear series on smooth projective threefolds.", J. Amer. Math. Soc., 6 (4): 875–903, doi: 10.1090/S0894-0347-1993-1207013-5 , MR 1207013 .
- Fujita, Takao (1987), "On polarized manifolds whose adjoint bundles are not semipositive", Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, pp. 167–178, doi:10.2969/aspm/01010167, ISBN 978-4-86497-068-6, MR 0946238 .
- Helmke, Stefan (1997), "On Fujita's conjecture", Duke Mathematical Journal, 88 (2): 201–216, doi:10.1215/S0012-7094-97-08807-4, MR 1455517 .
- Siu, Yum-Tong (1996), "The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi", Geometric complex analysis (Hayama, 1995), World Sci. Publ., River Edge, NJ, pp. 577–592, MR 1453639, Zbl 0941.32021 .
- Smith, Karen E. (2000), "A tight closure proof of Fujita's freeness conjecture for very ample line bundles" (PDF), Mathematische Annalen, 317 (2): 285–293, doi:10.1007/s002080000094, hdl: 2027.42/41935 , MR 1764238, S2CID 55051810 .