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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 periods are a class of numbers which includes, alongside the algebraic numbers, many well known mathematical constants such as the number π .
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 ...
(where B is a period ring, and V is a p-adic representation) which no longer have a G K-action, but are endowed with linear algebraic structures inherited from the ring B. In particular, they are vector spaces over the fixed field E := B G K {\displaystyle E:=B^{G_{K}}} . [ 4 ]
Real algebraic geometry is the study of real algebraic varieties. The fact that the field of the real numbers is an ordered field cannot be ignored in such a study. For example, the curve of equation x 2 + y 2 − a = 0 {\displaystyle x^{2}+y^{2}-a=0} is a circle if a > 0 {\displaystyle a>0} , but has no real points if a < 0 {\displaystyle a<0} .
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 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).
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry ...
Lefschetz theorem on (1,1)-classes (algebraic geometry) Lehmann–Scheffé theorem ; Leray's theorem (algebraic geometry) Leray–Hirsch theorem (algebraic topology) Lerner symmetry theorem ; Lester's theorem (Euclidean plane geometry) Levi's theorem ; Levitzky's theorem (ring theory) Lévy continuity theorem (probability)
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