8 problems

Let $K$ be a number field and let $Z/K$ be a smooth projective irreducible variety of dimension $d$. Define the Albanese kernel by … Here $\mathrm{CH}^d(Z)_0$ is the group of homol…

Let $X$ and $Y$ be smooth complex projective varieties. For a coherent sheaf $F$ on $X$, define its cohomological support loci by … In particular, write $V^i(X,\omega_X)$ for the l…

Let $X$ be a smooth projective variety with $\kappa(X)=0$, and let $\alpha:X\rightarrow A$ be its Albanese map, with general fibre $F$. Ueno Conjecture K. The map $\alpha$ is surje…

Let $(X,D)$ be a C-pair, meaning a pair consisting of a complex space $X$ and a boundary divisor $D$ with Campana's orbifold multiplicities, and suppose that $X$ is compact Kähler.…

Let $X$ be a smooth projective variety, and let $X'$ be a Fourier–Mukai partner of $X$, meaning that $D^b(X')$ and $D^b(X)$ are equivalent. Let $\operatorname{Alb}(X)$ and…

Let $K$ be a global field with separable closure $K^s$, and let $X^s$ be a smooth projective variety over $K^s$. Let $\operatorname{CH}_0(X^s)^0$ denote the degree-zero Chow group…

Let $X$ be a smooth, projective surface over an algebraically closed field, let $b_1$ be its first Betti number, and let $h^{1,1}_W$ be its $(1,1)$ Hodge–Witt number. Let…

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…