16 problems

Let $A$ be an abelian group and let $X$ be a smooth quasi-projective variety of dimension $n$. Let … be the natural transform from Lawson homology to singular homology. Suslin's co…

Let ${\cal I}_{\mathbb{P}^n}(r,c)$ denote the moduli space of rank $r$ instanton sheaves of charge $c$ on $\mathbb{P}^n$. Quasi-projectivity conjecture. In general,…

Nilpotency conjecture. The fundamental group $\pi_1(X)$ is virtually nilpotent.

Virtually nilpotent fundamental-group conjecture. The fundamental group $\pi_1(X)$ is virtually nilpotent.

Let $X$ be a smooth quasi-projective variety. Define the special loci … as follows: $\mathrm{Sp}_{\mathrm{sab}}(X)$ is the Zariski closure of the union of images of non-constant ra…

Let $U$ be a smooth quasi-projective surface over a finite extension $k$ of the $p$-adic field $\mathbb{Q}_p$. Suslin's singular homology $H_0^{\operatorname{sus}}(U)$ has a degree…

Let $X$ be a quasi-projective Calabi–Yau 4-fold, and let $\curly P=\curly P_v^{(q)}(X)$ be the relevant PT$_q$ moduli space. Assume that $\curly P$ is projective, so that its virtu…

Hodge conjecture. The cycle class map

Extended Lang–Vojta conjecture. The following conditions are equivalent:

Algebraicity conjecture. The space $\operatorname{Sh}_{\varrho}(X)$ is quasi-projective and $\operatorname{sh}_{\varrho}$ is an algebraic morphism.

The Kodaira-codimension conjecture. One should have

Let $f\colon U\to V$ be a smooth projective algebraic fiber space between smooth quasi-projective varieties, and let $F$ be its general fiber. The quasi-projective additivity conje…

Persistence conjecture. If $X$ is arithmetically hyperbolic over $k$, then its base change $X_L$ is arithmetically hyperbolic over $L$. This asserts that finiteness of integral poi…

Lang–Vojta's conjecture. These three conditions are equivalent. This conjecture proposes that complex-analytic hyperbolicity, geometric log-general type, and pointed finiteness of…

Quasi-projective $K(π,1)$ conjecture. There exists a smooth, connected, quasi-projective Eilenberg–MacLane space $X$ such that

Biaynicki-Birula's conjecture. Every normal variety $X$ contains finitely many maximal quasi-projective open subvarieties.