The Tate conjecture for divisors on irreducible holomorphic symplectic varieties over finite fields without restrictions on the characteristic

François Charles (Inventiones mathematicae, 2013) proved the Tate conjecture for K3 surfaces over finite fields of characteristic $p \ge 5$ and, more generally, for divisors on the reduction of a complex projective irreducible holomorphic symplectic variety of dimension $2n$ with second Betti number greater than 3, under the restrictions $p \ge 5$ , $p > 2n$ and $p$ prime to the polarization degree $d$ , together with good reduction over an unramified extension of $\mathbb{Q}_p$ . The surface case has since been settled in every characteristic (Madapusi Pera for odd characteristic; Kim and Madapusi Pera in characteristic 2). The open problem, explicitly anticipated by Charles, is to remove the restrictions on the characteristic in the higher-dimensional case: prove the Tate conjecture for divisors on reductions of irreducible holomorphic symplectic varieties of dimension at least 4 — at minimum of $K3^{[n]}$ -type — at arbitrary primes, including $p \le 2n$ , $p$ dividing $d$ , and $p \in \{2,3\}$ , and ultimately for such varieties over finitely generated fields without the good-reduction hypothesis.

Background

The Tate conjecture for divisors asserts that for a smooth projective variety $X$ over a finite field $k$ and a prime $\ell \ne \mathrm{char}(k)$ , the cycle class map from divisor classes with $\mathbb{Q}_\ell$ -coefficients onto the classes fixed by the Galois group $\mathrm{Gal}(\bar{k}/k)$ in the second $\ell$ -adic étale cohomology of $X$ is surjective:

\mathrm{CH}^1(X) \otimes \mathbb{Q}_\ell \to H^2_{\mathrm{\acute{e}t}}(X_{\bar{k}}, \mathbb{Q}_\ell(1))^{\mathrm{Gal}(\bar{k}/k)}.

A K3 surface is a smooth projective surface with trivial canonical bundle and no nonconstant global 1-forms; its second cohomology is a rank-22 lattice. Historically the conjecture was proved for ordinary K3 surfaces (Nygaard), that is, K3 surfaces whose formal Brauer group has height one, for K3 surfaces of finite height (Nygaard and Ogus), and for supersingular K3 surfaces admitting an elliptic fibration (Artin); a K3 surface is called supersingular when its formal Brauer group (the smooth formal group naturally associated with the coherent cohomology group $H^2(X, \mathcal{O}_X)$ ) has infinite height, a condition conjectured by Artin to force the Picard number to be 22. Maulik proved Artin's conjecture under a large-prime assumption (for a polarization of degree $2d$ , that $p > 2d + 4$ ), using semistable reduction for K3 surfaces in equal positive characteristic and automorphic forms on orthogonal groups. Charles removed the restriction on the size of the prime for surfaces: the paper proves Artin's conjecture, and hence the Tate conjecture, for all K3 surfaces over finite fields of characteristic $p \ge 5$ . The engine behind the surface theorem is a statement for varieties of arbitrary dimension. An irreducible holomorphic symplectic variety (also called a compact hyperkähler manifold) is a simply connected smooth complex projective variety whose space of holomorphic 2-forms is spanned by a single everywhere-non-degenerate form; K3 surfaces are the 2-dimensional examples, and the Hilbert scheme of $n$ points on a K3 surface represents the deformation type called $K3^{[n]}$ -type. The second cohomology of such a variety carries the Beauville–Bogomolov quadratic form $q$ , an integral structure generalizing the intersection pairing on a K3 surface; for $K3^{[n]}$ -type varieties the second Betti number is 22 when $n = 1$ and 23 when $n \ge 2$ . Charles's Theorem 4 states: if $Y$ is a complex projective irreducible holomorphic symplectic variety of dimension $2n$ with second Betti number $b_2 > 3$ , $h$ the class of an ample line bundle with $d = h^{2n}$ , and $p \ge 5$ is a prime with $p \nmid d$ and $p > 2n$ , such that $Y$ is defined over a finite unramified extension of $\mathbb{Q}_p$ with good reduction at $p$ and $q$ is non-degenerate modulo $p$ on the primitive part of $H^2(Y, \mathbb{Z})$ , then the reduction $X$ of $Y$ at $p$ satisfies the Tate conjecture for divisors. For $K3^{[n]}$ -type varieties the lattice condition follows from the others (Corollary 5 of the paper). The proof goes through the Kuga–Satake construction, which attaches to the weight-2 Hodge structure of K3 type on $H^2(Y)$ an abelian variety whose cohomology contains a copy of $H^2(Y)$ ; this correspondence is not known to be algebraic in general (a special case of the Hodge conjecture), but its existence suffices to build mixed-characteristic analogues of Noether–Lefschetz loci (the loci in a parameter space along which extra algebraic cohomology classes appear) and to lift Galois-invariant classes to characteristic zero. The three restrictions have distinct technical origins. The condition $p > 2n$ guarantees that the Hodge-to-de Rham spectral sequence of the reduction degenerates at $E_1$ , via the classical lifting argument that requires the characteristic to exceed the dimension. The conditions $p \nmid d$ and non-degeneracy of $q$ modulo $p$ keep the integral lattice theory of the Beauville–Bogomolov form usable. The unramified good-reduction setup allows the Kuga–Satake map to be compared with integral canonical models of orthogonal Shimura varieties (Shimura varieties attached to orthogonal groups, parametrizing polarized Hodge structures of K3 type) at hyperspecial primes, that is, primes at which the underlying integral lattice has good reduction and such integral models are well understood. Charles avoids semistable reduction — which is not available for higher-dimensional irreducible holomorphic symplectic varieties in equal positive characteristic — but the inductive lifting argument still depends on the listed restrictions; the paper states that the author strongly expects an extension of the method to relax the hypotheses on $p$ , and leaves this open. After 2013 the surface case was completed in every characteristic: Madapusi Pera proved the Tate conjecture for K3 surfaces over finitely generated fields of odd characteristic with no good-reduction or lattice hypothesis; Kim and Madapusi Pera settled characteristic 2 using 2-adic integral canonical models built from Dieudonné displays (linear-algebraic objects that encode $p$ -divisible groups over bases of characteristic $p$ and substitute for crystalline data at bad primes); and Charles gave a geometric proof that also covers K3 surfaces of Picard number at least 2 over arbitrary finite fields. In contrast, for dimension at least 4 the restrictions of Theorem 4 still stand: neighboring later results, such as the Tate conjecture for even-dimensional Gushel–Mukai varieties (varieties cut out by quadrics from a cone over a Veronese variety, whose Hodge theory closely parallels that of K3 surfaces) in characteristic $p \ge 5$ (Fu and Moonen) and the description of supersingular O'Grady sixfolds (the six-dimensional exceptional deformation class of irreducible holomorphic symplectic varieties, discovered by O'Grady) in characteristic $> 2$ (Fu, Li and Zou), retain comparable characteristic restrictions. Removing them for $K3^{[n]}$ -type varieties of dimension at least 4 is the open problem stated below.

Problem Statement

Remove the restrictions on the characteristic from Charles's Theorem 4 — the extension explicitly anticipated in the source paper ('We strongly expect that an extension of our method might relax the hypotheses on $p$ even in the higher-dimensional case'). Prove that the reduction $X$ at a prime $p$ of a complex projective irreducible holomorphic symplectic variety $Y$ of dimension $2n \ge 4$ with second Betti number $b_2 > 3$ , where $Y$ is defined over a finite unramified extension of $\mathbb{Q}_p$ and has good reduction at $p$ , satisfies the Tate conjecture for divisors — without the hypotheses $p \ge 5$ , $p > 2n$ and $p \nmid d$ (where $d = h^{2n}$ for the class $h$ of an ample line bundle), and without the auxiliary assumption that the Beauville–Bogomolov form is non-degenerate modulo $p$ on the primitive lattice. In particular, the statement must cover varieties of $K3^{[n]}$ -type of dimension at least 4 at primes $p \le 2n$ , at primes dividing the polarization degree $d$ , and at $p \in \{2,3\}$ ; hypotheses of a genuinely lattice-theoretic nature are admissible only insofar as they do not exclude these primes for the class of $K3^{[n]}$ -type varieties. The stronger natural target, achieved for surfaces by Madapusi Pera's odd-characteristic theorem, is: prove the Tate conjecture for divisors on varieties of $K3^{[n]}$ -type over finitely generated fields of arbitrary characteristic, with no hypothesis of good or unramified reduction. The common objective of all admissible forms is a single theorem making the Tate conjecture for divisors on these higher-dimensional hyperkähler varieties independent of restrictions on the characteristic; the completion boundary is that no prime or polarization-degree condition remains for the $K3^{[n]}$ -type class. The dimension-2 case is already settled in all characteristics and is not part of what remains.

References

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