5 problems

Openness conjecture. The set $X_{\mathrm{tame}}$ is open in $X$.

Closedness conjecture. In the situation of Corollary 44, the set $W$ is always closed in $X$; equivalently, the set

Generic invariance conjecture. The geometric generic fiber $X_{\overline{\eta}}$ is of Fano type.

Let $f:X\to Y$ be a smooth projective morphism with connected fibres between two quasi-projective manifolds. Assume that $K_{X_y}$ is pseudo-effective for general $y\in Y$, and tha…

Let $f:Y\to T$ be a smooth morphism from a smooth affine threefold $Y$ onto a smooth affine curve $T$, and suppose that every closed fiber $Y_t$ has an $\mathbb{A}^1$-fibration of…