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algebraic variety
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algebraïsche variëteit
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object of study in algebraic geometry
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Smooth complex algebraic varieties are complex manifolds, including: Complex vector spaces.
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In another direction, toric varieties are algebraic varieties acted on by a torus.
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In terms of category theory, an algebraic group is a group object in the category of algebraic varieties.
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In mathematics, an algebraic surface is an algebraic variety of dimension two.
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Suppose that X is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements.
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The emphasis on algebraic surfaces—algebraic varieties of dimension two—followed on from an essentially complete geometric theory of algebraic curves (dimension 1).
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The conjectures concern the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields.
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In English, "manifold" refers to spaces with a differentiable or topological structure, while "variety" refers to spaces with an algebraic structure, as in algebraic varieties.
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The fundamental groups considered in algebraic geometry are also profinite groups, roughly speaking because the algebra can only 'see' finite coverings of an algebraic variety.
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Complex manifolds that can be embedded in Cn are called Stein manifolds and form a very special class of manifolds including, for example, smooth complex affine algebraic varieties.
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Alternatively, they may be described by polynomial equations, in which case they are called algebraic varieties, and if they additionally carry a group structure, they are called algebraic groups.
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In mathematics, the Hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
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In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
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In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry, in particular as the Zariski topology on an algebraic variety or the spectrum of a ring.
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An abstract algebraic variety is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings.
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The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a moduli space, the structure of which remains an area of active research.
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In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result contributing to the Riemann–Roch problem for complex algebraic varieties of all dimensions.
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Since holomorphic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds.
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They relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of 'Tate cycles' (the fixed cycles for a suitably Tate-twisted action) that conjecturally picks out the algebraic cycles.
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In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions.
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Conjecturally there is just one essential type of global L-function, with two descriptions (coming from an algebraic variety, coming from an automorphic representation); this would be a vast generalisation of the Taniyama–Shimura conjecture, itself a very deep and recent result (as of 2009) in number theory.
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Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry.
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Taking the case of K the rational number field Q, and V a non-singular projective variety, we can for almost all prime numbers p consider the reduction of V modulo p, an algebraic variety Vp over the finite field Fp with p elements, just by reducing equations for V. Again for almost all p it will be non-singular.
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They describe properties of analytic invariants, called local zeta functions, of the number of points on an algebraic curve or variety of higher dimension.
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A single equation defines a hypersurface, and simultaneous Diophantine equations give rise to a general algebraic variety V over K; the typical question is about the nature of the set V(K) of points on V with co-ordinates in K, and by means of height functions quantitative questions about the "size" of these solutions may be posed, as well as the qualitative issues of whether any points exist, and if so whether there are an infinite number.
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