6 problems

Let $X$ be a complex analytic space that is $S_2$ and has pure dimension $\geq 3$, let $x\in X$ be a point, and let $\pi:\bar X\to X$ be the normalization with corresponding point…

Let $(s,S)$ be an $S_2$ surface. Finite-dimensionality conjecture. There is a finite-dimensional local Picard group $\operatorname{\mathbf{Pic}}^{\rm loc}(s,S)$ if and only if $S$…

Let $X$ be an excellent scheme that is $S_2$ and has pure dimension $\geq 4$. Let $x\in X$ be a closed point and let $x\in D\subset X$ be an effective Cartier divisor. Kollár's res…

Let $X$ be an excellent scheme that is $S_2$ and has pure dimension $\geq 3$, let $x\in X$ be a closed point, and let $\pi:\bar X\to X$ be the normalization, with $\bar x$ denoting…

Let $X$ be a scheme, let $x e in Dsubset X$ be an effective Cartier divisor, and let … be the restriction map on local Picard groups. Kollár's restriction conjecture. The map…

Kollár's local Lefschetz conjecture. Grothendieck's local formulation remains true under weaker hypotheses than those imposed in SGA 2.