5 problems

Let $(X,B)$ be a log variety, let $x\in X\setminus LCS(X,B)$ be a closed point, and let $H\in\operatorname{Div}(X)\otimes\mathbb{Q}$ be ample and normalized at $x$. Effective-build…

Let $X$ be a normal variety, let $B$ be a log divisor on $X$, and let $W\subset X$ be a closed subvariety with $a(\eta_W;X,B)=0$. Let $W^\nu$ be the normalization of $W$, and let…

The Base Change Conjecture. Then $K_Y+B_Y$ is $\mathbb{Q}$-Cartier and, if $f':(X',B^{X'})\to Y'$ is induced by a birational base change $\sigma:Y'\to Y$, then

The precise inverse of adjunction conjecture.

Let $(X,B)$ be a log variety and let $j:W\to X$ be the normalization of an irreducible component of $LCS(X,B)$ such that $K_X+B$ is log canonical at the generic point of $j(W)$. Th…