TITLE: A GAGA question
QUESTION [11 upvotes]: 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 
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.  )

REPLY [14 votes]: Yes, it is true: the analytic and algebraic closures of $X$ in $Z$ coincide (and you don't need at all to assume that $Z$ is smooth).
You may suppose that X is open in $Z$: if it isn't, just replace $Z$ by the Zariski (=algebraic) closure of $X$ in $Z$.  Then Serre's Proposition 5, page 11 of his famous GAGA article "Géométrie Algébrique et Géométrie Analytique" says exactly that the analytic closure of $X$
in $Z$ is $Z$, i.e. coincides with its Zariski closure. [ Caution:  his "X" takes the place of your "Z" , the big ambient variety]
You can also find the result (in English !) in the fine book by Joseph L. Taylor  "Several Complex Variables with Connections to Algebraic Geometry and Lie Groups", published by the AMS in its series Graduate Studies in Mathematics ( Volume 46).There the result you need is Proposition 13.4.6, page 344.