9 problems

Beauville–Catanese conjecture. For every integer $q\geq0$, the locus $S^q(X)$ is a finite union of torsion translates of subtori of $\operatorname{Pic}^0(X)$.

Kollár's positivity conjecture. A variety of general type and maximal Albanese dimension should satisfy

Let $X$ be a smooth projective variety of dimension $d$ with generically finite Albanese map $a:X\rightarrow A$, let $P$ be a Poincaré bundle on $X\times\widehat A$, and let…

Let $X$ be a compact Kähler or complex projective manifold that is aspherical or has a nef cotangent bundle, and let $\mathcal{M}$ be a mixed Hodge module on $X$. For an integer…

Catanese's conjecture. The dimension of the paracanonical system satisfies

Schreieder–Kotschick conjecture. The following three statements are equivalent: (1) $X$ admits a global holomorphic 1-form without zeros; (2) $X$ admits a smooth real closed 1-form…

Debarre–Pareschi–Popa conjecture. The following are equivalent: (1) $X$ is reduced, of pure dimension, and has minimal cohomology class

Pareschi--Popa's generic vanishing conjecture. The only geometrically nondegenerate GV-subschemes are the known examples of minimal class: the Brill--Noether loci $W_d(C)$ in Jacob…

Let $X$ be a smooth projective variety. A primitive variety of $\chi=0$ means, in the terminology of the source, a variety with vanishing holomorphic Euler characteristic that is p…