3 problems

Let $(X,\Delta)$ be a projective klt pair such that $K_X+\Delta$ is pseudo-effective, and let $A$ be an ample $\mathbb{Q}$-divisor on $X$. The finite generation conjecture. The adj…

Let $X$ be a normal projective variety, and let $\Delta_i$ be $\mathbb{Q}$-divisors on $X$ such that the pairs $(X,\Delta_i)$ are klt for $i=1,\dots,r$. The finite generation conje…

Adjoint-ring finite-generation conjecture. The adjoint ring $R(X;D_1,\dots,D_\ell)$ is finitely generated.