71 problems

Lefschetz standard conjecture. The inverse of this hard Lefschetz isomorphism is induced by an algebraic correspondence; assuming this conjecture, the Hodge conjecture for…

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

Grothendieck standard conjecture A. The restriction of $L^{m-2j}$ gives an isomorphism $C^j(X)\to C^{m-j}(X)$; equivalently, $\Lambda$ maps $C^{m-j}(X)$ into $C^j(X)$. This is one…

Grothendieck's standard conjecture $B(X)$. The homological correspondence $*$ is algebraic.

Let $S$ be a regular scheme of dimension $1$, let $f:X\to S$ be projective and flat with $X$ regular of dimension $n+1$, and let $\widehat{\operatorname{Ch}}^i(X)$ be the modified…

In a perfect field $k$, let $X$ be a smooth proper $k$-scheme. Write $\pi_X^i$ for the $i$th Künneth projector of the crystalline cohomology $H^*_{\operatorname{crys}}(X)$, let…

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 a smooth projective variety, let $L$ be the Lefschetz operator on $H^*(X;\Q)$, and let $\Lambda$ be its adjoint with respect to the Hodge inner product. A cycle $\beta$…

Standard Conjectures. The Lefschetz Standard Conjecture claims that $L$ induces an isomorphism on $A^*(X)$, while the Hodge Standard Conjecture claims that the cup product on primi…

Let $Y$ be an equidimensional smooth projective variety over a field $k$, let $L$ be an ample divisor class, put $n=\dim Y$, and let $A^{j}(Y)$ be the image of the cycle class map…

The standard conjecture $A(Y,L)$. For $0\leq i<n/2$, multiplication by $L^{n-2i}$ induces an isomorphism

Let $Y$ be an equidimensional smooth projective variety over a field $k$, let $L$ be an ample divisor class, and write $n=\dim Y$. Hard Lefschetz and the Lefschetz decomposition de…

For a smooth projective $\overline{k}$-variety $Y$ with an ample line bundle $L$, Poincaré duality, the Hard Lefschetz theorem, and the Lefschetz decomposition define a modified pa…

Let $X$ be a regular, projective, flat scheme over $\operatorname{Spec}\mathbb Z$ of absolute dimension $d+1$, and let $\overline{M}$ be an arithmetically ample hermitian line bund…

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$, and let $A^p(X)$ be the real vector space of algebraic cycles of codimension $p$ mod…

Arithmetic analogues of Grothendieck's standard conjectures. For a suitable choice of $k$, if $2p\leq d+1$, then (a) the homomorphism

The index conjecture. The pairing

Standard conjecture A. This map is an isomorphism.

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 $d$ over a field $k$, let $\mathcal L$ be an ample divisor on $X$, and for $0\leq n\leq d/2$ let … be the primitive part of the…

Let $X$ be a smooth projective variety, choose an inner product on $H^*(X)$ for which the different generalized eigenspaces of Frobenius are orthogonal, and let $B$ be the resultin…

Let $X$ be a smooth projective variety and let $Z$ be an algebraic cycle on $X$ of codimension $k$. Grothendieck–Bombieri's standard conjecture D. If $Z$ is numerically trivial, th…

The standard conjecture of Hodge type. For all $0\leqslant i\leqslant n/2$, the pairing

Lefschetz standard conjecture. The inverse of this map is induced by an algebraic correspondence. This is one of Grothendieck's standard conjectures; over the complex numbers, it i…

The Hodge standard conjecture. The pairing