Crystalline Variational Tate Conjecture beyond divisors: higher-codimension cycles and non-projective smooth proper families

Morrow (JEMS, 2019) formulated the Crystalline Variational Tate Conjecture: for a smooth proper family of varieties in characteristic $p>0$ , a rational algebraic cycle class on a fibre should extend to a cycle on the total family exactly when its rational crystalline cohomology class lifts, equivalently lifts to the Frobenius eigenspace, equivalently is flat for the natural connection on the higher direct images. Morrow proved the conjecture for divisors on projective families and an infinitesimal variant in all codimensions, and used the divisor case to reduce the Tate conjecture for divisors over finite fields to surfaces. The conjecture as stated remains open in its two remaining parts: cycles of codimension $i\ge2$ on global families, and smooth proper families that are not projective (where even the equivalence of the three cohomological conditions is unproved). No proof and no counterexample is known; adjacent questions — the $\ell$ -adic ( $\ell\neq p$ ) variational Tate conjecture, a reduction of the full Tate conjecture to surfaces, and the optimal power of $p$ killing torsion obstructions — also remain open, while the $\mathbb{Q}_p$ -coefficient analogue is known to be false by a counterexample of Lazda–Pál.

Background

An algebraic cycle of codimension $i$ on a smooth variety is a formal integer combination of irreducible subvarieties of that codimension; cycles modulo rational equivalence form the Chow group $\mathrm{CH}^i$ , and every Weil cohomology theory (a cohomology theory of varieties over a field behaving like singular cohomology of complex manifolds) receives a cycle class map from it. The Tate conjecture over a finite field $k$ — a central open problem in arithmetic geometry — predicts that the cycle class map $\mathrm{CH}^i(X)_{\mathbb{Q}}\to H^{2i}_{\mathrm{\acute{e}t}}(X_{\bar k},\mathbb{Q}_\ell(i))^{\mathrm{Gal}(\bar k/k)}$ is surjective for $\ell\neq p$ , i.e. that classes fixed by Galois are spanned by algebraic cycles.

Crystalline cohomology is the Weil cohomology for characteristic- $p$ varieties built from the de Rham complex of infinitesimal thickenings: for a $k$ -variety $X$ over a perfect field $k$ of characteristic $p>0$ it produces modules $H^n_{\mathrm{crys}}(X/W)$ over the Witt ring $W=W(k)$ (the ring of infinite $p$ -typical Witt vectors of $k$ , a complete discrete valuation ring of characteristic $0$ with residue field $k$ ), and after inverting $p$ the rational groups $H^n_{\mathrm{crys}}(X)=H^n_{\mathrm{crys}}(X/W)\otimes_W K$ , $K=W[1/p]$ , carry a Frobenius operator $\phi$ induced by the absolute Frobenius $x\mapsto x^p$ . The crystalline cycle class map $\mathrm{cl}^i:\mathrm{CH}^i(X)_{\mathbb{Q}}\to H^{2i}_{\mathrm{crys}}(X)$ lands in the $p^i$ -eigenspace $H^{2i}_{\mathrm{crys}}(X)^{\phi=p^i}$ .

Grothendieck, in his 1968 exposé on crystals and de Rham cohomology, posed a variational (relative) form of the Hodge and Tate conjectures: given a smooth proper family $f:X\to S$ and a cycle $z$ on a fibre $X_s$ , cohomological conditions on the class of $z$ should characterize when $z$ extends cohomologically to the whole family. In characteristic $0$ the de Rham version for divisors follows easily from the exponential sequence; in characteristic $p$ the $\ell$ -adic ( $\ell\neq p$ ) version remains poorly understood, and this led Morrow to study a $p$ -adic crystalline version, Conjecture 0.1 of the source paper.

Conjecture 0.1 (the Crystalline Variational Tate Conjecture) is the following statement. Let $f:X\to S$ be a smooth, proper morphism of smooth $k$ -varieties, let $s\in S$ be a closed point, and let $z\in\mathrm{CH}^i(X_s)_{\mathbb{Q}}$ be a rational cycle with crystalline cycle class $c=\mathrm{cl}^i(z)\in H^{2i}_{\mathrm{crys}}(X_s)$ . The following four conditions are conjectured to be equivalent: (deform) there exists $\tilde z\in\mathrm{CH}^i(X)_{\mathbb{Q}}$ with $\mathrm{cl}^i(\tilde z)|_{X_s}=c$ ; (crys) $c$ lifts to $H^{2i}_{\mathrm{crys}}(X)$ ; (crys- $\phi$ ) $c$ lifts to the eigenspace $H^{2i}_{\mathrm{crys}}(X)^{\phi=p^i}$ ; and (flat) $c$ lifts to $H^0_{\mathrm{crys}}(S,R^{2i}f_*\mathcal{O}_{X/K})$ . Here $R^{2i}f_*\mathcal{O}_{X/K}$ is the convergent F-isocrystal of higher direct images (an F-isocrystal is a $K$ -vector-space crystal equipped with a Frobenius structure), and (flat) asks that $c$ be horizontal, i.e. parallel for the Gauss–Manin-type connection on this F-isocrystal, the analogue for crystalline cohomology of the natural connection that the cohomology of any smooth family carries; the condition sits between (crys) and the others via the crystalline Leray spectral sequence, the spectral sequence computing the crystalline cohomology of the total space from that of base and fibres.

What the paper proves: (i) the conjecture for divisors ( $i=1$ ) when $f$ is projective (Theorem 1.4) — a crystalline analogue of the classical fixed-part theorem (Theorem 2.6; in characteristic zero, the fixed-part theorem identifies the cohomology classes on a fibre that extend to the whole family with those flat for the Gauss–Manin connection) shows (crys), (crys- $\phi$ ), (flat) equivalent for projective $f$ , and the hard implication (crys- $\phi$ ) $\Rightarrow$ (deform) is reduced to a local statement over power-series rings $k[[t_1,\dots,t_m]]$ (Theorem 0.5), proved using topological cyclic homology (a stable-homotopy refinement of Hochschild homology, an algebraic homology theory closely related to differential forms) and logarithmic de Rham–Witt sheaves (sheaves combining de Rham differentials with Witt-vector arithmetic, adapted to varieties carrying a normal-crossings divisor); (ii) an infinitesimal variant in all codimensions (Theorem 0.6): over $A=k[[t_1,\dots,t_m]]$ a class in $K_0$ (the Grothendieck group of vector bundles) on the special fibre lifts to all infinitesimal thickenings if and only if its crystalline Chern character (the ring homomorphism from $K_0$ to rational crystalline cohomology) lifts to the Frobenius eigenspace; (iii) consequences, including the reduction of the Tate conjecture for divisors over a finite field to surfaces (Corollary 0.3) and a recovery of Artin's classical theorem on the Picard groups (groups of line bundles) of supersingular K3 families (Corollary 0.4; a supersingular K3 surface is a K3 surface in characteristic $p$ whose Picard number attains the maximum possible value $22$ ). Lazda–Pál subsequently proved a semistable analogue of the divisor statement (semistable meaning, roughly, that the singular fibres have at worst normal-crossings singularities) and showed that the analogous statement with $\mathbb{Q}_p$ -coefficients instead of $\mathbb{Q}$ is false, obstructing $\mathbb{Q}_p$ -linear approaches.

What remains open is precisely the conjecture as stated beyond the divisor/projective case: codimension $i\ge2$ in the global setting, and smooth proper families that are not projective, where even the equivalence of the three cohomological conditions is unproved because the fixed-part degeneration of the crystalline Leray spectral sequence currently requires projectivity.

Problem Statement

Prove, or disprove, the Crystalline Variational Tate Conjecture (Conjecture 0.1 of Morrow, JEMS 2019) in the cases not settled by the source paper. In the source formulation: let $k$ be a perfect field of characteristic $p>0$ , let $f:X\to S$ be a smooth, proper morphism of smooth $k$ -varieties, let $s\in S$ be a closed point, and let $z\in\mathrm{CH}^i(X_s)_{\mathbb{Q}}$ be a rational cycle of codimension $i$ with crystalline cycle class $c=\mathrm{cl}^i(z)\in H^{2i}_{\mathrm{crys}}(X_s)$ . Determine whether the following four conditions are equivalent for all $i\ge1$ and all such $f$ : (deform) there exists $\tilde z\in\mathrm{CH}^i(X)_{\mathbb{Q}}$ with $\mathrm{cl}^i(\tilde z)|_{X_s}=c$ ; (crys) $c$ lifts to $H^{2i}_{\mathrm{crys}}(X)$ ; (crys- $\phi$ ) $c$ lifts to the eigenspace $H^{2i}_{\mathrm{crys}}(X)^{\phi=p^i}$ ; (flat) $c$ lifts to $H^0_{\mathrm{crys}}(S,R^{2i}f_*\mathcal{O}_{X/K})$ , the horizontal sections for the natural connection on the higher direct images. The open cases are: (a) codimension $i\ge2$ on global families, where only an infinitesimal variant over $k[[t_1,\dots,t_m]]$ (for $K_0$ -classes) is known; and (b) codimension $i=1$ for smooth proper families that are not projective, where even the equivalence of (crys), (crys- $\phi$ ), and (flat) is unproved. The conjecture as stated covers all codimensions and all smooth proper families, with $\mathbb{Q}$ -coefficients: a single family, closed point, and rational cycle class satisfying some of the four conditions while violating another would refute it. The adjacent questions — the $\ell$ -adic ( $\ell\neq p$ ) variational Tate conjecture, a reduction of the full all-codimension Tate conjecture over finite fields to surfaces, and the optimal power of $p$ killing the bounded torsion in the infinitesimal lifting theorems — are separate open problems and lie outside the scope of this task.

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