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The rational numbers (), algebraic numbers (), algebraic periods and exponential periods as subsets of the complex numbers ().In mathematics, specifically algebraic geometry, a period or algebraic period [1] is a complex number that can be expressed as an integral of an algebraic function over an algebraic domain.
The global unpolarized period domain is the quotient of the local unpolarized period domain by the action of Γ (it is thus a collection of double cosets). In the polarized case, the elements of the monodromy group are required to also preserve the bilinear form Q , and the global polarized period domain is constructed as a quotient by Γ in ...
Algebraic geometry is a branch of mathematics which uses ... made many theorems in algebraic geometry simpler and sharper: For example, ... same period, Blaise Pascal ...
The simplest example of a period is , which arises as ∫ S 1 d z z = 2 π i . {\displaystyle \int _{S^{1}}{\frac {dz}{z}}=2\pi i.} Algebraic de Rham cohomology is used to construct periods as follows: [ 8 ] For an algebraic variety X defined over Q , {\displaystyle \mathbb {Q} ,} the above-mentioned compatibility with base-change yields a ...
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
For applications in algebraic geometry, namely, classification of complex projective varieties by their periods, the set of all Hodge structures of weight on is too big. Using the Riemann bilinear relations , in this case called Hodge Riemann bilinear relations , it can be substantially simplified.
Algebraic variety. Hypersurface; Quadric (algebraic geometry) Dimension of an algebraic variety; Hilbert's Nullstellensatz; Complete variety; Elimination theory; Gröbner basis; Projective variety; Quasiprojective variety; Canonical bundle; Complete intersection; Serre duality; Spaltenstein variety; Arithmetic genus, geometric genus, irregularity
In classical algebraic geometry, adjectives were often used as nouns: for example, "quartic" could also be short for "quartic curve" or "quartic surface". In classical algebraic geometry, all curves, surfaces, varieties, and so on came with fixed embeddings into projective space, whereas in scheme theory they are more often considered as ...
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