In this question $\mathbb F$ is a field and $P({\mathbb F}^{n+1})$ is the projective space of dimension $n$ over $\mathbb F$. The term algebraic variety means a subset of $P({\mathbb F}^{n+1})$ which is the set of zeroes of finitely many homogeneous polynomials with coefficients in $\mathbb F$. The field $\mathbb K$ is the field of non-standard complex numbers (real and imaginary parts are hyperreal) and $\mathbb C$ is the subfield of complex numbers. If V is a variety in $P({\mathbb K}^{n+1})$, is the shadow of V a variety in $P({\mathbb C}^{n+1})$ ? A point is in the shadow of V means it has some projective coordinates in ${\mathbb C}^{n+1}$ infinitesimally close to some projective coordinates in ${\mathbb K}^{n+1}$ of some point in V.