14 problems

Non-vanishing conjecture. There exists an effective $\mathbb{R}$-divisor $E\geq 0$ such that

Let $f:X\to Y$ be an algebraic fiber space between smooth projective varieties, and suppose that the general fiber $F$ satisfies $\kappa(F)=0$. Let $H$ be an ample divisor on $Y$.…

Let $f:X\to Y$ be an algebraic fiber space, meaning that $X$ and $Y$ are smooth projective varieties and $f$ is surjective with connected fibers. Let $F$ be a generic fiber satisfy…

Let $X$ be a smooth projective variety, and let $1 \leq q \leq \dim X$. A vector bundle $\mathcal{E}$ on $X$ is pseudoeffective when the tautological line bundle…

Non-vanishing conjecture. The relative real linear system is nonempty:

Mazur's conjecture. If $e(\chi_0)=+1$ (the definite case), then

Non-vanishing conjecture for strictly nef anti-canonical divisors. One should have

Non-vanishing conjecture. One has $\kappa(L)\geq 0$.

Non-vanishing conjecture. There is an algebraic Hecke character $\chi$ of $\operatorname{GL}_1(\mathbb A_F)$, with conjugate self-dual archimedean component $\chi_\infty$, such tha…

Non-vanishing conjecture for smooth varieties. The canonical divisor $K_{X}$ has an effective $\mathbb{Q}$-linearly equivalent representative.

Non-vanishing conjecture. The stated effective representative exists.

Let $X$ be a smooth projective variety. Non-vanishing conjecture. If $K_X$ is pseudo-effective, then there exists an effective $\mathbb{Q}$-divisor $D$ such that … This is the smoo…

Non-vanishing conjecture. There exists an effective $\mathbb{K}$-divisor $D$ such that

Non-vanishing conjecture for klt pairs. Then