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In algebraic geometry, a correspondence between algebraic varieties V and W is a subset R of V×W, that is closed in the Zariski topology.In set theory, a subset of a Cartesian product of two sets is called a binary relation or correspondence; thus, a correspondence here is a relation that is defined by algebraic equations.
Suppose that X is a smooth complex algebraic variety.. Riemann–Hilbert correspondence (for regular singular connections): there is a functor Sol called the local solutions functor, that is an equivalence from the category of flat connections on algebraic vector bundles on X with regular singularities to the category of local systems of finite-dimensional complex vector spaces on X.
In mathematics, the geometric Langlands correspondence relates algebraic geometry and representation theory.It is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from algebraic geometry. [1]
In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after Kevin Corlette and Carlos Simpson) is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.
In algebraic geometry, the relation between sets of polynomials and their zero sets is an antitone Galois connection. Fix a natural number n and a field K and let A be the set of all subsets of the polynomial ring K[X 1, ..., X n] ordered by inclusion ⊆, and let B be the set of all subsets of K n ordered by inclusion ⊆.
The Kobayashi–Hitchin correspondence has found a variety of important applications throughout algebraic geometry, differential geometry, and differential topology. By providing two alternative descriptions of the moduli space of stable holomorphic vector bundles over a complex manifold, one algebraic in nature and the other analytic, many ...
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).
Algebraic geometry is in many ways the mirror image of commutative algebra. This correspondence started with Hilbert's Nullstellensatz that establishes a one-to-one correspondence between the points of an algebraic variety, and the maximal ideals of its coordinate ring. This correspondence has been enlarged and systematized for translating (and ...
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