41 problems

Let $\chi$ be a primitive quadratic Dirichlet character, and let $L(s,\chi)$ denote its Dirichlet $L$-function. Chowla's conjecture. … The conjecture asserts non-vanishing at the c…

Graded-core conjecture. One has

Let $X$ be a smooth projective variety. Denote its Kodaira dimension by $\kappa(X)$ and its numerical dimension by $\nu(X)$. Non-vanishing conjecture. If … then … This conjecture a…

Let $L(s,\chi)$ be a primitive Dirichlet $L$-function, with central point $s=\frac12$. Dirichlet central-point non-vanishing conjecture. Every primitive Dirichlet $L$-function sati…

Non-vanishing conjecture. Then

Let $(X,B,\mathbf{M})$ be a projective log canonical generalized pair over the complex numbers, with generalized log canonical divisor $K_X+B+\mathbf{M}_X$. A divisor is numericall…

Work over the complex numbers. Let $(X,B)$ be a projective log canonical pair, where $X$ is a normal projective variety and $B$ is a boundary divisor, and let $K_X+B$ be its log ca…

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 $D$ an effective divisor on $X$. The Campana–Peternell conjecture asserts that if $mK_X-D$ is pseudo-effective for some $m\geq 1$, then…

Let $X$ be a smooth complex projective variety over $\mathbb{C}$, and let $K_X$ denote its canonical divisor. A divisor is pseudo-effective if its numerical class lies in the closu…

Let $\chi$ be a primitive Dirichlet character, and let $L(s,\chi)$ denote its Dirichlet $L$-function. The central point is $s=\frac{1}{2}$. Non-vanishing conjecture. The function…

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…

A primitive Dirichlet $L$-function is an $L$-function $L(s,\chi)$ associated to a primitive Dirichlet character $\chi$. Its central point is $s=\frac{1}{2}$. Central non-vanishing…

Let $(X,\Delta)$ be a klt pair with nef log-canonical divisor $K_X+\Delta$. The Non-Vanishing conjecture. There exists an integer $m\geq 1$ such that $m(K_X+\Delta)$ is Cartier and…

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

Non-vanishing conjecture. If $K_X+B$ is pseudo-effective over $Z$, then

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

Non-vanishing conjecture. The counting function satisfies

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

Let $(X,B)$ be a klt pair such that $K_X+B$ is pseudoeffective, and let $L$ be a nef divisor on $X$ with $K_X+B+L$ nef and abundant. Non-vanishing conjecture. The divisor $K_X+B+L$…

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

In the setting of log pairs, let $(X,\Delta)$ be a log canonical pair whose log canonical divisor $K_X+\Delta$ is pseudo-effective. Non-vanishing conjecture for log canonical pairs…

Erdős's conjecture. Whenever the series

Erdős's conjecture. One should have

Weak non-vanishing and weak abundance for g-pairs. 1. There exists an effective $R$-divisor $N$ such that