9 problems

Birational invariance conjecture for condition star. Then $X'/\mathbb{P}^1$ also satisfies condition $(\ast)$. This predicts that condition $(\ast)$ is preserved under square-birat…

Let $X/S$ be a threefold Mori fibre space, and let $\mathcal{P}(X/S)$ denote its pliability. Topological pliability conjecture. The pliability $\mathcal{P}(X/S)$ is, in a natural w…

Let $X/S$ and $Y/T$ be threefold Mori fibre spaces, and let rigidity mean birational rigidity of the Mori fibre space. Topological rigidity conjecture. Whether a threefold Mori fib…

Let $X/S$ and $Y/T$ be threefold Mori fibre spaces, with total spaces $X$ and $Y$ diffeomorphic. Topological rationality conjecture. If $X$ is rational, then $Y$ is also rational.…

Let $X\to S$ be a Mori fibre space, and let $\mathcal{P}(X/S)$ denote its pliability. Pliability algebraicity conjecture. The pliability $\mathcal{P}(X/S)$ is, in a natural way, an…

Let $X$ be a smooth uniruled variety. K-stable-fibre conjecture. The variety $X$ is birational to a Mori fibre space whose general fibre is K-stable. This conjecture links biration…

McKernan's conjecture. There is $\delta\in\mathbb R^{>0}$ such that if $f\colon X\to Z$ is a Mori fibre space, where $X$ is an $\epsilon$-lc $\mathbb Q$-factorial variety of dimens…

Cheltsov–Shramov conjecture. The varieties

Let $(Z,\Phi)$ be a Kawamata log terminal pair. Run the $(K_Z+\Phi)$-minimal model program via a birational map $f\colon Z\dashrightarrow X$, and write $\Delta=f_*\Phi$. Mori MMP c…