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.
`quasi-projective'' I mean X is defined by
a finite number ofalgebraic equations' $f_i = 0$
and inequalities $g_a \ne 0$, As is the case when Z is projective
space, the $f_i$ may not be globally defined;
same for the $g_a$. In my situation,
the Zariski open set defined by intersecting the sets $g_a \ne 0$
is an affine set (in the usual schemy sense) and the $f_i$ 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. )