Let $\mathbb k$ be an algebraically closed field of characteristic zero.

I have two questions:

(1) Is an irreducible algebraic variety $X/\mathbb k$ of dimension at least 2 locally simply connected?

(2) Let $X/\mathbb k$ be an irreducible algebraic variety and $Y$ a closed subvariety of dimension at least 2. If $X$ is (locally) simply connected, is $X\setminus Y$ (locally) simply connected?

codimension$\geq 2$. Removing a closed subvariety of codimension at least 2 doesn't change $\pi_1$. $\endgroup$never), as soon as $\dim(X)>0$. For each point $x\in X$ you can find $f\in \mathscr{O}_{X,x}^\times$ which is not a square. Then (assuming $\mathrm{char}(\mathbb{k})\neq2$) $\mathrm{Spec}(\mathscr{O}_{X,x}[\sqrt{f}])$ extends to a nontrivial étale cover of some neighborhood $U$ of $x$, so $U$ is not simply connected. $\endgroup$1more comment