5 problems

Ambro–Kawamata conjecture. One has

Let $X$ be a smooth projective variety of dimension $n$ over the complex numbers, and let $L$ be a nef and big line bundle on $X$. Thus $(X,L)$ is a quasi-polarized manifold. Iones…

Weighted-complete-intersection semigroup conjecture. For every positive integer $k$ such that $h\mid k$ and $k>\delta(d;a)$, there exist natural numbers $x_0,\ldots,x_n$ satisfying

Ambro--Kawamata effective non-vanishing conjecture. Then

Let $(d;a)=(d_1,\ldots,d_c;a_0,\ldots,a_n) \in \mathbb N^c \times \mathbb N^{n+1}$ be a regular pair such that $a_i \ne 1$ and $d_j \ne a_i$ for any $i,j$. Assume $c \le n$ and…