A GAGA question.
Say I have a ``quasi-projective'' (*) subvariety X over the complex
numbers within a smooth complex algebraic variety Z.
True or False: The analytic and algebraic closure of X (within Z) coincide.
I guess the answer must be `True' and is contained somewhere within Serre's GAGA,
or some elucidation thereof. If I'm right
could someone point me to a precise reference, either within GAGA or
elsewhere? If I'm wrong I'd love to hear about it.
Elucidation:
(*) By `quasi-projective'' I mean X is defined by
algebraic equations'
a finite number of
and inequalities , As is the case when Z is projective
space, the may not be globally defined;
same for the . In my situation,
the Zariski open set defined by intersecting the sets
is an affine set (in the usual schemy sense) and the are polynomials
on this affine set.
My Z is probably projective -- I'm not positive here, just pretty sure.
(My Z is obtained by iterating the construction of taking the
bundle over a smooth projective variety whose
fiber is the Grassmannian of d-planes within said variety's tangent space. )
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