Can we think of any complex disc around origin(say $\Delta$) as a projective variety?
I am asking this question because I have encountered the product $\mathbb{P}^{n} \times \Delta$ in an algebro-geometric context,which I think may be interpreted in $2$ ways:
$(1)$ product of $2$ complex manifolds one of dimension $n$ and other is of dimension $1$ (please correct me if I am wrong)
$(2)$ fiber product of projective space of dimension $n$ and some projective variety in $\mathbb{P}^{n}$.(for my purpose this kind of interpretation will be more relevant,but I don't see how is that the case)
Also can we say that any smooth $r$ dimensional complex manifold is essentially a projective scheme? If so how?
Any help from anyone is welcome