Set of Chow

In mathematics , Chow's theorem is an example of the relationship between analytic geometry and algebraic geometry.

The sentence says that a closed analytical subspace of complex-projective space must be a sub-variety of . An analytical subspace that is closed in the standard topology is therefore also closed in the Zariski topology . ${\displaystyle \mathbb {C} P^{n}}$ ${\displaystyle \mathbb {C} P^{n}}$

The theorem makes it possible to use methods of classical algebraic geometry to study any analytic subspaces.

The theorem was proven by Chow in 1949 , the proof simplified by Remmert and Stein in 1953 , before Serre received it in 1956 as a consequence of his GAGA principle ( Géométrie Algébrique et Géométrie Analytique ).

Some uses:

• By applying it to the function graph , one obtains that every holomorphic mapping of a compact projective variety into an algebraic variety is an algebraic morphism .
• Every meromorphic function is rational . ${\displaystyle f\colon \mathbb {C} P^{n}\to \mathbb {C} P^{n}}$
• With Chow's theorem one can deduce from the analytical embeddability in one that every compact Riemann surface is an algebraic curve . ${\displaystyle \mathbb {C} P^{n}}$

literature

• Chow, W.-L .: On Compact Complex Analytic Varieties, American Journal of Mathematics, Vol. 71, No. 4, pp. 893-914
• Gunning, RC and H. Rossi: Analytic functions of several complex variables, AMS Chelsea, Providence
• Serre, J.-P .: Géométrie algébrique et géométrie analytique, Annales de l'institut Fourier, Vol. 6, pp. 1-42