7 problems

Let $(X/Z\ni o,D)$ be a log variety admitting at least one $\mathbb{Q}$-complement of $K_X+D$ near the fiber over $o$. It is exceptional over $o$ if every such complement has at mo…

A log surface is a log pair whose boundary has standard coefficients, and a minimal complement index is the least index of a complement in the relevant complement theory. In the ex…

Higher-dimensional boundedness conjecture. The conditions for boundedness formulated in the paper should be the most natural ones, and direct analogues of all four main theorems sh…

Finiteness conjecture for wlc klt models. The number of projective wlc klt models in a fixed 0-class is finite up to log isomorphism.

Let $S$ be a smooth $\mathbb{Q}$-acyclic surface, and let $(X,D)$ be a log smooth completion of $S$. Here, $K_X$ is the canonical divisor of $X$, $D$ is the boundary divisor, and…

Let $E\subseteq \mathbb{P}^{2}$ be a complex curve homeomorphic to the projective line, and let … be a minimal log resolution. Here, $K_X$ is the canonical divisor of $X$, $D$ is t…

Let $E\subseteq\mathbb{P}^2$ be a rational cuspidal curve, and let $(X,D)$ be a log resolution of $(\mathbb{P}^2,E)$. Write $\mathbb{C}^{**}=\mathbb{C}^1\setminus\{0,1\}$. Negativi…