Monday, 28 August 2006

ag.algebraic geometry - A GAGA question

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 of
algebraic equations' fi=0fi=0
and inequalities gane0gane0, As is the case when Z is projective
space, the fifi may not be globally defined;
same for the gaga. In my situation,
the Zariski open set defined by intersecting the sets gane0gane0
is an affine set (in the usual schemy sense) and the fifi 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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