Algebraic geometry and analytic geometry

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.

Main statement

Let

X

{\displaystyle X}

be a projective complex algebraic variety. Because

X

{\displaystyle X}

is a complex variety, its set of complex points

X

(

C

)

{\displaystyle X(\mathbb {C} )}

can be given the structure of a compact complex analytic space. This analytic space is denoted

X

a

n

{\displaystyle X^{\mathrm {an} }}

. Similarly, if

F

{\displaystyle {\mathcal {F}}}

is a sheaf on

X

{\displaystyle X}

, then there is a corresponding sheaf

F

an

{\displaystyle {\mathcal {F}}^{\text{an}}}

on

X

a

n

{\displaystyle X^{\mathrm {an} }}

. This association of an analytic object to an algebraic one is a functor. The prototypical theorem relating

X

{\displaystyle X}

and

X

a

n

{\displaystyle X^{\mathrm {an} }}

says that for any two coherent sheaves

F

{\displaystyle {\mathcal {F}}}

and

G

{\displaystyle {\mathcal {G}}}

on

X

{\displaystyle X}

, the natural homomorphism

Hom

O

X

(

F

,

G

)

→

Hom

O

X

an

(

F

an

,

G

an

)

{\displaystyle {\text{Hom}}_{{\mathcal {O}}_{X}}({\mathcal {F}},{\mathcal {G}})\rightarrow {\text{Hom}}_{{\mathcal {O}}_{X}^{\text{an}}}({\mathcal {F}}^{\text{an}},{\mathcal {G}}^{\text{an}})}

is an isomorphism. Here

O

X

{\displaystyle {\mathcal {O}}_{X}}

is the structure sheaf of the algebraic variety

X

{\displaystyle X}

and

O

X

an

{\displaystyle {\mathcal {O}}_{X}^{\text{an}}}

is the structure sheaf of the analytic variety

X

a

n

{\displaystyle X^{\mathrm {an} }}

.