18 problems

Let $X$ be a $d$-dimensional normal irreducible projective variety with at worst log-terminal singularities, and let $\phi$ be a family of bounded motivic invariants of type…

Let $X$ be an arithmetic variety, meaning a regular scheme projective and flat over $\mathbb Z$ whose generic fibre $X_{\mathbb Q}$ is smooth over $\mathbb Q$. Let…

Grothendieck's standard conjecture of Hodge type. For any $i \le d/2$, the symmetric bilinear form

The Lefschetz standard conjecture. For each $k\leq n$, there exists an algebraic self-correspondence

Let $\mathcal X$ be a smooth projective scheme of relative dimension $d$ over a field or over the ring of integers of a $p$-adic field, satisfying the Künneth standard conjecture,…

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$, and let $\ell$ be a prime distinct from the characteristic of $k$. Numerical equival…

Let $X$ be a complex projective variety of dimension $3$, and let … be the Friedlander–Mazur cycle class map, with rational coefficients. Surjectivity conjecture. The map…

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbf{K}$, and write the Künneth decomposition of the diagonal class as … with $\pi_i\in H^i(X)\otimes H^{2n-i}(X)$.…

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbf{K}$, and for each $r\in\mathbf{Q}_{>0}$ let $\gamma_r$ be the homological correspondence acting on $H^i(X)$ by…

Perfectness conjecture. The intersection pairing on algebraic cohomology classes is perfect. This is presented as one of the paper's analogues of Grothendieck's standard conjecture…

Let $X\to S$ be the projective family in the paper, let $X_s$ be its special fiber, let $X_\eta$ be its generic fiber, and let $H^*_{\psi}(X_s)$ and $H^*_{\phi}(X_s)$ denote the ne…

Standard conjecture of Künneth type. There exists a unique decomposition

Grothendieck's standard conjectures. The following assertions hold: (1) there exist correspondences $\Lambda_j\in CH^j(Y\times Y)$ such that $\Lambda_{j*}:H^{2d_Y-j}(Y)\to H^j(Y)$…

Chow–Künneth conjecture. There exist correspondences $\pi_{j,X}\in CH^d(X\times X)$ satisfying

Let $X/k$ be a projective smooth variety, set $d=\dim X$, and let … be the $\mathbb{Q}$-subspace spanned by cohomology classes of codimension-$d$ cycles on $X\times_k X$ that are f…

Let $k$ be a field, let $\mathrm{H}^*$ be a Weil cohomology theory with coefficients in a characteristic-zero field $F$, and let $M_{hom}(k)_F$ be the category of homological motiv…

Supporting-cycle conjecture. There exists a codimension-$k$ cycle $Z'$ on $X$ with rational coefficients, supported on $Y$, such that $[Z']=[Z]$ in $H^{2k}(X,\mathbb{Q})$. The pape…

Let $X$ be a smooth projective variety of dimension $r$, let $H^*(X)$ be a Weil cohomology theory, and let $L$ be the Lefschetz operator. Let $\Lambda$ be the linear operator deter…