9 problems

Let $X$ be a smooth projective variety that is generically finite over an abelian variety, and let $P_2(X)$ denote its second plurigenus. Kollár's characterization conjecture. If ……

Let $\pi:X\rightarrow\Delta$ be a holomorphic family of compact Kähler manifolds with fiber $X_t$, where $\Delta=\{t\in\mathbf{C}\mid |t|<1\}$. For a compact complex manifold $Y$,…

Local constancy conjecture. The pluri-genera $P_k(\mathscr{X}_s)$ are locally constant as functions of $s\in S$.

Siu's conjecture. The $m$-genus $P_m(X_t)$ is independent of $t\in\Delta$. The problem is widely open for smooth Kähler families, although deformation invariance is known in severa…

Let $s_0=2$ and $s_m=s_{m-1}(s_{m-1}-1)+1$ for $m\geq 1$ be Sylvester's sequence. For a positive integer $n$, let $X$ be the $n$-fold from the quasi-smooth Calabi–Yau hypersurface…

Let $s_0=2$ and $s_m=s_{m-1}(s_{m-1}-1)+1$ for $m\geq 1$ be Sylvester's sequence. For an integer $n\geq 2$, set … Let $X$ be a general hypersurface of degree $d$ in … It is a quasi…

Deformation invariance of plurigenera. The plurigenera $h^0(X_s,\omega_{X_s}^{[r]})$ are independent of $s\in S$ for every $r$, in some neighborhood of $0\in S$.

Let $\kappa$ be a field, let $s$ be the log point of $\kappa$, and let $X/s$ be a proper SNCL scheme of pure dimension $d$. For $n,r\in\mathbb Z_{\geq 1}$, define the log geometric…

K-poly-stability conjecture for invariance of Fano plurigenera. The quantity $P_m(X_t)$ is independent of $t\in B$ for every $m$. This is presented as an analogue of Siu's invarian…