9 problems

Let $X\subseteq\mathbb{P}^{4}$ be a threefold hypersurface of degree $d\geq 6$, and let $C\subseteq X$ be any curve. Griffiths–Harris conjecture. One expects … The strongest form o…

Let $X$ be a smooth projective separably rationally connected variety of dimension $d$ over an algebraically closed field. The groups … and, when $X$ is defined over the complex nu…

Let $X$ be a smooth projective variety over $\mathbb C$. The integral Hodge conjecture concerns whether every integral Hodge class … is algebraic. Integral Hodge conjecture. Every…

The integral Hodge conjecture. $Z^{2i}(X)=0$.

Let $X$ be a smooth complex projective rationally connected variety of dimension at least four. The integral Hodge conjecture concerns whether integral Hodge classes are represente…

Let $X$ be a smooth complex projective variety and let $k$ be a nonnegative integer. The cycle class map is … An integral Hodge class is an integral cohomology class of type…

Let $X$ be a smooth quartic threefold over ${\mathbf R}$. A rational curve on $X$ has trivial Borel–Haefliger class when its Borel–Haefliger class in…

Let $\delta\geq 6$, and let $X\subset {\mathbf P}^4_{\mathbf C}$ be a very general hypersurface of degree $\delta$. A Griffiths--Harris conjecture. The degree of every curve on $X$…

Let $X$ be a smooth projective complex variety. The cycle map is … where $H^{*,*}(X)\subset H^{2*}(X;\mathbb C)$ is the submodule generated by $(*,*)$-forms. Integral Hodge conject…