Grothendieck's Lefschetz standard conjecture (Conjecture B) over finite fields

The strong form of Grothendieck's Lefschetz standard conjecture (Conjecture B) asserts that the inverse of the Lefschetz operator on the cohomology of a smooth projective variety is induced by an algebraic correspondence. Over finite fields the weak form (Conjecture C, algebraicity of the Kunneth components of the diagonal) is a theorem of Katz and Messing following Deligne's proof of the Riemann hypothesis, which led Milne (2020) to single out Conjecture B over finite fields as possibly the most accessible of the standard conjectures, and to pose the derived question whether Conjecture C together with the strong Lefschetz theorem already implies Conjecture B. Conjecture B is known only for special classes (abelian varieties, complex threefolds, Kummer-type hyper-Kahler varieties, and cases flowing from the Tate conjecture such as K3 surfaces), the semisimplicity conjecture for Frobenius remains open, and no proof of the C-implies-B implication is known.

Background

Grothendieck's standard conjectures, formulated in the 1960s, predict that the cohomological behavior of algebraic cycles on smooth projective varieties is governed by algebraic cycles themselves; they are prerequisites for his envisioned theory of motives and have been open in general ever since. This problem concerns the Lefschetz-type standard conjecture in the setting of varieties over finite fields highlighted by Milne (2020).

Let $k$ be an algebraically closed field and $X$ a smooth projective variety of dimension $n$ over $k$. A Weil cohomology theory assigns to $X$ graded finite-dimensional vector spaces over a coefficient field of characteristic zero, satisfying functoriality, Kunneth, Poincare duality, and cycle-class axioms. For $k=\bar{\mathbb{F}}_q$, $q=p^a$, the standard example is $\ell$-adic etale cohomology $H^*_{\mathrm{\acute{e}t}}(X,\mathbb{Q}_\ell)$ with $\ell\neq p$, the cohomology of the constant sheaf $\mathbb{Q}_\ell$ on the etale site of $X$. Every algebraic cycle (a $\mathbb{Q}$-linear combination of subvarieties) of codimension $i$ on $X$ has a cohomology class in $H^{2i}(X)(i)$, where $(i)$ denotes the $i$-fold Tate twist; the algebraic classes span a finite-dimensional $\mathbb{Q}$-subspace $C^i(X)$ of $H^{2i}(X)(i)$. An algebraic correspondence on $X$ is an algebraic cycle on $X\times X$, and it acts on cohomology by pull-push.

Fix an ample hyperplane section of $X$ and let $L$ be the Lefschetz operator, cup product with its class, raising cohomological degree by $2$. The hard (strong) Lefschetz theorem states that $L^{n-i}:H^i(X)\to H^{2n-i}(X)(n-i)$ is an isomorphism for all $i\leq n$; over finite fields this is a theorem of Deligne, proved as a consequence of his proof of the Riemann hypothesis in Weil I. Conjecture B, the strong form of the Lefschetz standard conjecture, asserts that for every $i$ with $2i\leq n$ the operator $L^{n-2i}$ induces an isomorphism $C^i(X)\xrightarrow{\sim}C^{n-i}(X)$; equivalently (Kleiman), the inverse of $L^{n-2i}:H^{2i}(X)(i)\to H^{2n-2i}(X)(n-i)$ is induced by an algebraic correspondence. Hard Lefschetz gives the injectivity of $L^{n-2i}$ on algebraic classes; a correspondence inducing the inverse restricts to an isomorphism $C^i(X)\xrightarrow{\sim}C^{n-i}(X)$, while the converse direction runs through Conjecture D (Milne 2020, Proposition 6.1 and Corollary 6.2, following Kleiman 1968, Theorem 4-1). Conjecture B is known for abelian varieties (a theorem of Lieberman recorded in Kleiman 1968), but is open in general over every algebraically closed field, including $\mathbb{C}$.

Two neighboring statements organize what is known. Conjecture C, the weak form (Kleiman's eighth version of the Lefschetz-type conjecture, which he suggested 'is, doubtless, truly weaker'), asserts that the Kunneth components of the diagonal $\Delta\subset X\times X$ — the projectors $\pi^j$ onto the degree-$j$ cohomology of $X$ furnished by the Kunneth decomposition of the class of $\Delta$ — are algebraic. Over finite fields Conjecture C is a theorem of Katz and Messing, deduced from Deligne's Riemann hypothesis. Conjecture D asserts that numerical and homological equivalence of algebraic cycles coincide, i.e. that the cup-product pairing $C^i(X)\times C^{n-i}(X)\to\mathbb{Q}$ is nondegenerate; Jannsen proved unconditionally that the category of motives defined using numerical equivalence is a semisimple tannakian category, that is, a tensor category equivalent to representations of an affine group scheme.

Milne's observation is that since Conjecture C is a theorem over finite fields while Conjecture B is open everywhere, Conjecture B over finite fields may be the most accessible of the standard conjectures. His paper proves that Conjecture B over finite fields would imply the almost-Hodge conjecture for complex abelian varieties, the full Tate conjecture (surjectivity of the cycle class map onto Frobenius-invariant classes) for abelian varieties over $\bar{\mathbb{F}}_p$, and the Hodge standard conjecture (a positivity statement) for abelian varieties in characteristic $p$; conversely, the full Tate conjecture for all varieties over $\bar{\mathbb{F}}_p$ would imply all of Grothendieck's standard conjectures over $\bar{\mathbb{F}}_p$. He also examines whether the already-proven weak form suffices: by his Proposition 6.6, if there exists a single Weil cohomology theory satisfying the strong Lefschetz theorem and Conjecture D, then every Weil cohomology theory satisfying strong Lefschetz and Conjecture C satisfies Conjecture B; the remaining gap is to choose the fibre functor on the numerical motive category so that the induced Weil cohomology satisfies strong Lefschetz.

Problem Statement

Let $p$ be a prime, $q=p^a$, and $\bar{\mathbb{F}}_q$ an algebraic closure. Prove, or refute, Grothendieck's Lefschetz standard conjecture in its strong form (Conjecture B) for smooth projective varieties over finite fields:

(Conjecture B over finite fields.) For every smooth projective variety $X$ of dimension $n$ over $\bar{\mathbb{F}}_q$, every prime $\ell\neq p$, and every $i$ with $2i\leq n$, the inverse of the Lefschetz operator

$$L^{n-2i}:H^{2i}_{\mathrm{\acute{e}t}}(X,\mathbb{Q}_\ell)(i)\to H^{2n-2i}_{\mathrm{\acute{e}t}}(X,\mathbb{Q}_\ell)(n-i),$$

where $L$ is defined by cup product with the class of an ample hyperplane section, is induced by an algebraic correspondence with rational coefficients on $X\times X$. Equivalently, $L^{n-2i}$ restricts to an isomorphism from the space of algebraic classes of codimension $i$ to that of codimension $n-i$.

In addition, resolve the derived question posed by Milne (2020) (following Kleiman's remark that the weak form is 'doubtless, truly weaker'): does Conjecture C — the algebraicity of the Kunneth components of the diagonal — together with the strong Lefschetz theorem imply Conjecture B for an arbitrary Weil cohomology theory on algebraic varieties over an algebraically closed field? Since Conjecture C and the strong Lefschetz theorem are theorems over finite fields (Katz and Messing 1974; Deligne 1974), an affirmative answer would establish Conjecture B in the above setting. The two demands are not equivalent and both are to be resolved: an affirmative answer to the implication settles Conjecture B over finite fields as well, whereas a counterexample to the implication settles only the derived question and leaves Conjecture B over finite fields open.

References

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