In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.

Let ${\displaystyle X}$ be a projective complex algebraic variety. Because ${\displaystyle X}$ is a complex variety, its set of complex points ${\displaystyle X(\mathbb {C} )}$ can be given the structure of a compact complex analytic space. This analytic space is denoted ${\displaystyle X^{\mathrm {an} }}$. Similarly, if ${\displaystyle {\mathcal {F}}}$ is a sheaf on ${\displaystyle X}$, then there is a corresponding sheaf ${\displaystyle {\mathcal {F}}^{\text{an}}}$ on ${\displaystyle X^{\mathrm {an} }}$. This association of an analytic object to an algebraic one is a functor. The prototypical theorem relating ${\displaystyle X}$ and ${\displaystyle X^{\mathrm {an} }}$ says that for any two coherent sheaves ${\displaystyle {\mathcal {F}}}$ and ${\displaystyle {\mathcal {G}}}$ on ${\displaystyle X}$, the natural homomorphism

is an isomorphism. Here ${\displaystyle {\mathcal {O}}_{X}}$ is the structure sheaf of the algebraic variety ${\displaystyle X}$ and ${\displaystyle {\mathcal {O}}_{X}^{\text{an}}}$ is the structure sheaf of the analytic variety ${\displaystyle X^{\mathrm {an} }}$. More precisely, the category of coherent sheaves on the algebraic variety ${\displaystyle X}$ is equivalent to the category of analytic coherent sheaves on the analytic variety ${\displaystyle X^{\mathrm {an} }}$, and the equivalence is given on objects by mapping ${\displaystyle {\mathcal {F}}}$ to ${\displaystyle {\mathcal {F}}^{\text{an}}}$. In particular, ${\displaystyle {\mathcal {O}}_{X}^{\text{an}}}$ is itself coherent, a result known as the Oka coherence theorem, and also, it was proved in “Faisceaux Algebriques Coherents” that the structure sheaf of the algebraic variety ${\displaystyle {\mathcal {O}}_{X}}$ is coherent.

Another important statement is as follows: for any coherent sheaf ${\displaystyle {\mathcal {F}}}$ on an algebraic variety ${\displaystyle X}$ the homomorphisms

are isomorphisms for all ${\displaystyle q}$'s. This means that the ${\displaystyle q}$-th sheaf cohomology group on ${\displaystyle X}$ is isomorphic to the cohomology group on ${\displaystyle X^{\mathrm {an} }}$.

The theorem applies much more generally than stated above (see the formal statement below). It and its proof have many consequences, such as Chow's theorem, the Lefschetz principle and Kodaira vanishing theorem.

Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over ${\displaystyle \mathbb {C} }$ can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way.

For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either the rational functions or the identically infinity function (an extension of Liouville's theorem). For if such a function ${\displaystyle f}$ is nonconstant, then since the set of ${\displaystyle z}$ where ${\displaystyle f(z)}$ is infinity is isolated and the Riemann sphere is compact, there are finitely many ${\displaystyle z}$ with ${\displaystyle f(z)}$ equal to infinity. Consider the Laurent expansion at all such ${\displaystyle z}$ and subtract off the singular part: we are left with a function on the Riemann sphere with values in ${\displaystyle \mathbb {C} }$, which by Liouville's theorem is constant. Thus ${\displaystyle f}$ is a rational function. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere.