11 problems

Let $X$ be a smooth projective variety over the complex numbers, and let $D\subset X$ be a very general smooth ample divisor of sufficiently large degree. Write…

Cheeger–MacPherson conjecture. The hard Lefschetz theorem and all the other standard consequences of Hodge theory should hold for intersection homology theory.

Soit $X\subset \mathbb{P}^r_k$ un sous-schéma fermé du schéma projectif qui soit localement une intersection complète et dont toute composante irréductible soit de codimension…

Let $X$ be a smooth projective variety over a field $k$, let $\mathcal O_X(1)$ be an ample line bundle, and let $V\to\Gamma(X,\mathcal O_X(1))$ be a space of sections generating…

Let $X$ be a smooth projective variety and let $Y$ be a smooth ample divisor in $X$. The inclusion $Y\hookrightarrow X$ is called a cycle-theoretic $c$-equivalence when it induces…

Let $X$ be a smooth projective variety over the complex numbers, let $D\subset X$ be a very general smooth ample divisor of sufficiently large degree, and let…

Let $X$ be a smooth projective variety and let $D\subset X$ be an ample divisor. Let $\mathscr{H}\to D$ be a polarized variation of Hodge structure, and let its numerical invariant…

Hartshorne's conjecture. The stated isomorphism holds for every such $X$, $Y$, and $i$.

Let $X$ be a smooth projective variety, let $D\subset X$ be a very general smooth ample divisor of sufficiently large degree, and let $j:D\hookrightarrow X$ be the inclusion. Pushf…

Let $X$ be a smooth projective variety, let $D$ be a smooth ample divisor in $X$, and let $j:D\hookrightarrow X$ be the inclusion. Paranjape's conjecture. The restriction map … is…

Kollár's local Lefschetz conjecture. Grothendieck's local formulation remains true under weaker hypotheses than those imposed in SGA 2.