Grothendieck's standard conjectures

Let $k$ be a field and let $H^*$ be a Weil cohomology theory for smooth projective $k$-varieties with coefficients in a field $F$ of characteristic $0$. Examples include $\ell$-adic cohomology of $X_{\bar k}$ for a prime $\ell\neq\operatorname{char}k$, and Betti or algebraic de Rham cohomology when $\operatorname{char}k=0$. Let $X$ be a smooth projective variety of dimension $n$ over $k$.

• $\mathcal Z^i(X)$ is the $\mathbb Q$-vector space of algebraic cycles of codimension $i$ on $X$.
• $\gamma_X\colon\mathcal Z^i(X)\to H^{2i}(X)$ is the cycle class map, with Tate twists suppressed.
• $A^i(X)\subseteq H^{2i}(X)$ is the $\mathbb Q$-span of $\gamma_X(\mathcal Z^i(X))$.

A linear map $\phi\colon H^*(X)\to H^*(X)$ is algebraic if some cycle $\Gamma\in\mathcal Z^*(X\times X)$ induces it:

$$\phi(x)=p_{2*}\bigl(\gamma_{X\times X}(\Gamma)\cup p_1^*x\bigr).$$

Fix a smooth hyperplane section $W$ and let $L(x)=\gamma_X(W)\cup x$. By hard Lefschetz, $L^{n-m}\colon H^m(X)\to H^{2n-m}(X)$ is an isomorphism for $m\le n$. This gives the Lefschetz decomposition

$$H^m(X)=\bigoplus_{j\ge 0}L^j P^{m-2j}(X),\qquad P^m(X)=\ker\bigl(L^{n-m+1}\colon H^m(X)\to H^{2n-m+2}(X)\bigr)\quad(m\le n).$$

The Lefschetz operator $\Lambda$ is defined by $\Lambda(L^ja)=L^{j-1}a$ for $a\in P^m(X)$ and $j\ge1$, and $\Lambda(a)=0$ for primitive $a$.

The standard conjectures assert that the following hold for every $k$, $H^*$, and $X$ as above:

• $B(X)$, Lefschetz type. $\Lambda$ is algebraic.
• $C(X)$, Künneth type. For each $0\le m\le 2n$, the projector $\pi^m\colon H^*(X)\to H^m(X)\subseteq H^*(X)$ is algebraic. Equivalently, each Künneth component of the diagonal class $\gamma_{X\times X}(\Delta_X)$ is algebraic.
• $D(X)$, homological equals numerical. For every $i$ and every $z\in\mathcal Z^i(X)$, $\gamma_X(z)=0$ if and only if $\deg(z\cdot z')=0$ for every $z'\in\mathcal Z^{n-i}(X)$.
• $\mathrm{Hdg}(X)$, Hodge type. For every $i$ with $2i\le n$, the $\mathbb Q$-valued form

$$\langle x,y\rangle=(-1)^i\,\operatorname{tr}_X\bigl(L^{n-2i}(x\cup y)\bigr)$$

is positive definite on $A^i_{\mathrm{prim}}(X)=A^i(X)\cap P^{2i}(X)$.

In characteristic $0$, Hodge theory proves $\mathrm{Hdg}(X)$. The open problem is the full set of assertions for all varieties over all fields. $B$, $C$ and $D$ are open in characteristic $0$, and all four are open in positive characteristic. [1, 2, 3]

Background

Grothendieck stated these conjectures in the mid-1960s and published them in 1969 [1]. They had two aims. The first was the Weil conjectures: $B$ and $\mathrm{Hdg}$ together imply the Riemann hypothesis for varieties over finite fields, by an argument modeled on Serre's Kähler analogue. The second was a theory of pure motives. Under $D$, motives for homological equivalence coincide with numerical motives. With $C$, the category becomes a semisimple, graded Tannakian category whose motivic Galois group is a reductive group. Kleiman's 1968 exposition [2] fixed the now-standard formulation, and his 1994 survey [3] updates it.

The conjectures are tightly linked. $B(X)$ implies $C(X)$, and $B$ together with $\mathrm{Hdg}$ implies $D$. Over $\mathbb C$, $\Lambda$ is induced by a Hodge class on $X\times X$, so the Hodge conjecture implies $B$. Deligne proved the Weil conjectures in 1974 by other means, so the standard conjectures are now motivated chiefly by the theory of motives and algebraic cycles.