279 problems

Let $X$ be a normal complex algebraic variety. Suppose that $\pi_1(X)$ is virtually nilpotent. Campana's conjecture. Then $\pi_1(X)$ is virtually at most $2$-step nilpotent. This c…

Let $X$ be a smooth complex projective variety. Its fundamental group is large if, for every closed irreducible positive-dimensional subvariety $Z\subset X$, the image … is infinit…

Let $M$ be an open manifold with nonnegative Ricci curvature, and let $\widetilde M$ be its universal cover. The manifold $\widetilde M$ has Euclidean volume growth if there is a c…

Kollár's conjecture. If

Let $k$ be a finite field of characteristic $\ell\ne p$, let $X$ be a geometrically connected smooth curve over $k$, and let … be the exact sequence for the arithmetic fundamental…

Abhyankar's conjecture. The finite quotient groups of the étale fundamental group of $\mathbb{G}_m$ are precisely the finite groups $G$ such that

Let $X$ be a compact Kähler manifold with infinite fundamental group, and let $\widetilde{X}$ be its universal cover. Carlson–Toledo conjecture. There exists a finite étale cover…

Let $M^n$ be a manifold with nonnegative sectional curvature. A group is virtually abelian if it has an abelian subgroup of finite index. Fukaya–Yamaguchi conjecture. The fundament…

Let $X$ be an algebraic surface of general type, and let its index be positive, meaning that $c_1^2(X)-2c_2(X)>0$. Bogomolov's conjecture. The fundamental group $\pi_1(X)$ should b…

Let $X$ be a smooth projective variety. Its fundamental group is big if the image of the fundamental group of every positive-dimensional subvariety in $\pi_1(X)$ is infinite. Kollá…

Let $S$ be an orientable surface, let $M$ be a 3-manifold, and let $f\colon S\to M$ be a map. The induced homomorphism is … Simple Loop Conjecture. If $f_*$ is not injective, then…

Let $(X_n)_{n\geq 1}$ be the tower of normal projective surfaces obtained by successive $\mathbb{Z}/3$-Galois covers, and let $\widetilde{X}_n\to X_n$ be the smooth minimal resolut…

Let $X$ be an irreducible smooth projective variety defined over an algebraically closed field $k$ of characteristic $p>0$, and let $x\in X$. The étale fundamental group is denoted…

Abelianity conjecture. The fundamental group $\pi_1(X)$ is virtually abelian whenever $X$ is special.

Let $M$ be a compact, connected, $P^2$-irreducible $3$-manifold. Say that $M$ has almost-compact coverings if, for every finitely generated subgroup $H$ of $\pi_1(M)$, the cover of…

Let $X$ be a smooth projective variety of Kodaira dimension zero. Kollár's conjecture. The fundamental group $pi_1(X)$ is virtually abelian. This predicts strong restrictions on th…

Let $S$ be a connected $2$-manifold without boundary, and let a discrete group $G$ act on $S$ effectively, properly discontinuously, and with isolated fixed points. Assume that…

Let $C\subset\mathbf{P}^2$ be an irreducible sextic plane curve with no toric decomposition of type $(2,3)$. Let $\Delta_C(t)$ denote its Alexander polynomial, and let $\ell$ be a…

Almost abelianity conjecture. The fundamental group

Let $n\ge 4$ and let $(M^n,g)$ be a complete Riemannian manifold with $\operatorname{Ric}>0$. A free abelian subgroup of $\pi_1(M^n)$ is an abelian subgroup isomorphic to…

Let $f:\mathscr{X}\to\mathbb{D}$ be a smooth projective family over the unit disk, and write $X_t=f^{-1}(t)$. A fiber has big fundamental group when the image of the fundamental gr…

Let $(M^n,g)$ be a compact space with nonnegative Ricci curvature, and let $\pi_1(M)$ denote its fundamental group. A group is uniformly almost abelian if it has an abelian subgrou…

Reid's conjecture. The fundamental group $\pi_1(X)$ is either finite or commensurable with the fundamental group of a curve.

Rudyak's conjecture. There should be a Grossmann–Whitehead type inequality

Let $C$ be a proper smooth curve of genus $g$ over a field $k$ finitely generated over ${\mathbb Q}$. Choose a geometric point $\mathfrak b x$ of $C$, set…