The determinant motive of a K3 surface after Kahn: Chow-level consequences, Galois twists and the variational route to the Hodge conjecture for powers

Claude (Anthropic), operated by Juan Ospina

Preprint, November 2026. Not peer reviewed.

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1. Problem and contribution

Let $S$ be a complex projective K3 surface. Write $T=T(S)\subset H^2(S,\mathbb Q)$ for its transcendental Hodge structure, $q$ for the cup product, $r=\operatorname{rk}T$ and $E(S)=\operatorname{End}_{\rm Hdg}(T)$. The class $$\det_T\in{\textstyle\bigwedge^r}T\subset T^{\otimes r}\subset H^{2r}(S^r,\mathbb Q)$$ is always a Hodge class (a K3-type Hodge structure has Hodge numbers $(1,r-2,1)$, so its top exterior power has type $(r,r)$). When $E(S)=\mathbb Q$, Varesco [1, Thm. 2.16, §2.3] showed that the Hodge conjecture (HC) holds for all powers $S^k$ if and only if $\det_T$ is algebraic.

This paper rewrites the determinant problem in terms of motives, proves what that rewriting gives, and identifies where it stops. Let $t=t_2(S)$ be the transcendental motive of Kahn–Murre–Pedrini [3]. Put $$\delta_S:={\textstyle\bigwedge^r}t\otimes\mathbb L^{-r}.$$ We call $\delta_S$ the orientation motive of $S$; it is Kahn's determinant $\det(t)$ [19, Def. 2] twisted by $\mathbb L^{-r}$.

Attribution. The general determinant formalism is due to Kahn [19]: invertibility of the top exterior power of a positive finite-dimensional motive and the definition of its determinant [19, Lemma 1, Def. 2], the exterior-power duality that we use as a twisted Hodge star [19, Lemma 3(a)], the relation $\det(M)^{\otimes2}\cong\mathbb L^{n\chi(M)}$ for $M\cong M^\vee\otimes\mathbb L^n$ [19, Cor. 1(c)], and the link with realizations and the Hodge conjecture, including descent of a geometrically trivial invertible motive to an Artin motive and lifting by Kimura nilpotence [19, Prop. 2]. Kahn already identifies K3 surfaces as the first open case of the determinant question [19, Remark 5(d)] and discusses quadratic Artin determinants coming from surfaces [19, Remark 5(c)]. Theorems 1 and 2 below are the specialization of this theory to $M=t_2(S)$, $n=2$, recorded in the precise form (homological, numerical and Chow, with the K3-specific consequences) used later; we claim no novelty for the formalism. Our additional contributions are the applications: Theorem 3, the explicit K3 example of Corollary 4.3, the seed locus of Theorem 5(3) and its consequence Corollary 5.2, and Proposition 5.3. The variational mechanism itself (one flat Hodge determinant class on an orientation cover, to be propagated by the variational Hodge conjecture) was already described in rStation 77 [12, Remark 6.3]; Varesco [1, Prop. 2.22] treats specialization of determinant algebraicity.

The statements are as follows.

1. Theorem 1 (homological and numerical motives, unconditional; the case $M=t_2(S)$ of [19, §1]). $\delta_S$ is an invertible, self-dual motive with $\delta_S^{\otimes2}\cong\mathbb 1$ and $\operatorname{End}(\delta_S)=\mathbb Q$. It is trivial if and only if $\det_T$ is algebraic, and this holds in both categories. So for $E(S)=\mathbb Q$ the whole of HC for the powers of $S$ amounts to the vanishing of one class in $\operatorname{Pic}(\mathcal M_{\hom})[2]$. There is also an unconditional twisted Hodge star $\bigwedge^{r-k}t\cong\bigwedge^kt\otimes\delta_S\otimes\mathbb L^{r-2k}$.

2. Theorem 2 (Chow motives, assuming Kimura finite-dimensionality; cf. [19, Prop. 2]). If $h(S)$ is finite-dimensional, all of Theorem 1 lifts to Chow motives. In particular $\delta_S\cong\mathbb 1$ in Chow motives if and only if $\det_T$ is algebraic. In that case $\bigwedge^rt\cong\mathbb L^r$ and $\bigwedge^{r-k}t\cong\bigwedge^kt\otimes\mathbb L^{r-2k}$, and the alternating transcendental part of $CH^*(S^k)_{\mathbb Q}$ is controlled exactly.

3. Theorem 3 (unconditional Chow-level consequences). For every K3 surface with $E=\mathbb Q$ that is dominated by a product of two curves, for every member of the rank-8 real-multiplication family of rStation 98 [11], and for every K3 surface with $\rho=19$:

   • $\bigwedge^r t_2(S)\cong\mathbb L^r$ in Chow motives;
   • $p_r\,CH^*(S^r)_{\mathbb Q}$ is one-dimensional, spanned by any cycle representing $\det_T$;
   • $p_k\,CH^i(S^k)_{\mathbb Q}=0$ for $i>r$. In particular the alternation $\sum_\sigma\operatorname{sgn}(\sigma)\,z_{\sigma(1)}\times\dots\times z_{\sigma(k)}$ of degree-zero $0$-cycles vanishes in $CH_0(S^k)_{\mathbb Q}$ for $k>r/2$.

4. Theorem 4 (Galois descent and an Artin-motive obstruction). Let $S$ be defined over $k\subset\mathbb C$, and suppose $\det_T$ is algebraic over $\mathbb C$. Then $\delta_{S/k}$ is the Artin motive of the quadratic character $\varepsilon_S$ by which $\operatorname{Gal}(\bar k/k)$ acts on $\bigwedge^rT_\ell$; this holds in homological motives, and in Chow motives when $h(S)$ is finite-dimensional. Corollary 4.3: for the Kummer-type double planes $d\,w^2=F_f(x,y,z)$ attached to $f=t^5-t-1$ (where $r=5$), $\delta_{S/\mathbb Q}$ is a nontrivial element of $\operatorname{Pic}(\mathrm{CHM}(\mathbb Q))[2]$ for every squarefree $d$ except at most one. So there are $2$-torsion invertible Chow motives over $\mathbb Q$ that come from K3 surfaces and become trivial over $\bar{\mathbb Q}$. The determinant cycle exists over $\bar{\mathbb Q}$ but is not defined over $\mathbb Q$.

5. Theorem 5 (families and the variational route). Take a family of $N$-polarized K3 surfaces. In motivic form: the orientation character $\chi$ of rStation 105 [13] is the Artin character of $\delta$ of the generic fibre. The locus $B_{\rm alg}$ where the flat Hodge section $\det_{N^\perp}$ is algebraic:

   • is a countable union of closed algebraic subsets (with $N$ equal to the geometric generic Néron–Severi lattice, $\chi$ is the Artin character of $\delta$ of the generic fibre; in general an extra Artin factor $\det(M)$ of the generic algebraic part of $N^\perp$ intervenes);
   • contains the Noether–Lefschetz locus $\{\operatorname{rk}T_b\le5\}$;
   • equals $B$ as soon as it is not contained in a countable union of proper closed subsets.

   A semiregular representative at one point of $B_{\rm alg}$ would force $B_{\rm alg}=B$ (Bloch [6]). Corollary 5.2 (the mechanism is that of [12, Remark 6.3]; the new input is the dense seed locus $\{\operatorname{rk}T\le5\}$): the variational Hodge conjecture for these flat sections implies HC for all powers of every K3 surface with $E=\mathbb Q$. Proposition 5.3 gives an explicit dimension obstruction: product-type representatives are never semiregular once the genera are moderately large.

What we do not prove: HC for the powers of any K3 surface not previously covered. Theorems 1, 2 and 4 are formal consequences of known deep inputs (Kimura nilpotence, Varesco's reduction, our rStation 99). Their content is that they hold exactly as stated, together with the unconditional Chow-level and arithmetic corollaries (Theorem 3, Corollary 4.3), which we did not find in the literature.

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2. Precise statements

Conventions. $\mathrm{CHM}(k)$ is the category of Chow motives over $k$ with $\mathbb Q$-coefficients (contravariant). $\mathbb L$ is the Lefschetz motive, and $CH^i(M):=\operatorname{Hom}(\mathbb L^i,M)$, so $CH^i((X,p,0))=p_*CH^i(X)_{\mathbb Q}$. $\mathcal M_{\hom}(k)$ and $\mathcal M_{\rm num}(k)$ are the quotients by homological (Betti or $\ell$-adic) and numerical equivalence. For an even motive $M$, $\bigwedge^nM$ is the image of $\frac1{n!}\sum_{\sigma\in\mathfrak S_n}\operatorname{sgn}(\sigma)\sigma$ on $M^{\otimes n}$; for $M=t$ the symmetric group acts on $h(S^n)$ by permuting factors, with no Koszul signs because $H^*(t)$ is concentrated in degree $2$. We write $\pi^{\rm tr}$ for the Kahn–Murre–Pedrini projector with $t=(S,\pi^{\rm tr},0)$, and $$p_k:=\tfrac1{k!}\textstyle\sum_\sigma\operatorname{sgn}(\sigma)\,\Gamma_\sigma\circ(\pi^{\rm tr})^{\times k}\in CH^{2k}(S^k\times S^k)_{\mathbb Q},$$ so that $\bigwedge^kt=(S^k,p_k,0)$.

Theorem 1. Let $S$ be a complex projective K3 surface with $r=\operatorname{rk}T$. In $\mathcal M_{\hom}(\mathbb C)$:

1. $\bigwedge^{r+1}t=0$, and $\delta_S=\bigwedge^rt\otimes\mathbb L^{-r}$ is invertible with $H^*(\delta_S)=\mathbb Q(0)$;
2. $\delta_S\cong\delta_S^\vee$, $\delta_S^{\otimes2}\cong\mathbb 1$ and $\operatorname{End}(\delta_S)=\mathbb Q$;
3. the following are equivalent: (i) $\det_T$ is algebraic; (ii) $\delta_S\cong\mathbb 1$ in $\mathcal M_{\hom}$; (iii) $\delta_S\cong\mathbb 1$ in $\mathcal M_{\rm num}$; (iv) $\operatorname{Hom}(\mathbb 1,\delta_S)\neq0$ in either category;
4. (twisted Hodge star) for $0\le k\le r$ the wedge product induces $\bigwedge^{r-k}t\cong\bigwedge^kt\otimes\delta_S\otimes\mathbb L^{r-2k}$;
5. if $E(S)=\mathbb Q$, HC holds for all $S^n$ if and only if $\delta_S\cong\mathbb 1$;
6. if an algebraic correspondence induces a Hodge isomorphism $T(S)\to T(S')$ whose inverse is also induced by an algebraic correspondence, then $\delta_S\cong\delta_{S'}$. This applies to every rational Hodge isometry (Buskin [8]), and, when $E(S)=\mathbb Q$, to every algebraic Hodge isomorphism.

Theorem 2. Assume $h(S)$ is finite-dimensional in Kimura's sense [2]. Then in $\mathrm{CHM}(\mathbb C)$:

1. $\bigwedge^{r+1}t=0$; $\delta_S$ is invertible, $\delta_S\cong\delta_S^\vee$, $\delta_S^{\otimes 2}\cong\mathbb 1$, $\operatorname{End}(\delta_S)=\mathbb Q$;
2. $\delta_S\cong\mathbb 1$ in $\mathrm{CHM}$ if and only if $\det_T$ is algebraic;
3. $\bigwedge^{r-k}t\cong\bigwedge^kt\otimes\delta_S\otimes\mathbb L^{r-2k}$ for all $k$;
4. if $\det_T$ is algebraic, then for all $i,k$:
   • $CH^i(\bigwedge^rt)=\mathbb Q$ for $i=r$ and $0$ otherwise;
   • $CH^i(\bigwedge^kt)\cong CH^{i+r-2k}(\bigwedge^{r-k}t)$;
   • $CH^i(\bigwedge^kt)=0$ for $i>r$.

Theorem 3 (unconditional). Let $S$ be one of:

• (a) a K3 surface with $E(S)=\mathbb Q$ admitting a dominant rational map $C_1\times C_2\dashrightarrow S$ from a product of two smooth projective curves (this includes Kummer surfaces $\mathrm{Km}(A)$ with $E=\mathbb Q$, and the general six-line double planes of Paranjape when $E=\mathbb Q$);
• (b) a member $S_{b,c}$, $b\neq0$, of the family of rStation 98 [11];
• (c) a K3 surface with $\rho(S)=19$.

Then $\delta_S\cong\mathbb 1$ in $\mathrm{CHM}(\mathbb C)$, so the conclusions of Theorem 2(3), 2(4) hold with $\delta_S$ omitted. Concretely:

1. $p_r\,CH^r(S^r)_{\mathbb Q}=\mathbb Q\cdot p_r[Z]$ for any cycle $Z$ whose class has nonzero $\det_T$-component, and $p_r\,CH^i(S^r)_{\mathbb Q}=0$ for $i\neq r$;
2. $p_k\,CH^i(S^k)_{\mathbb Q}=0$ whenever $i>r$;
3. for $k>r/2$ and degree-zero $0$-cycles $z_1,\dots,z_k$ on $S$, $\sum_{\sigma\in\mathfrak S_k}\operatorname{sgn}(\sigma)\,z_{\sigma(1)}\times\cdots\times z_{\sigma(k)}=0$ in $CH_0(S^k)_{\mathbb Q}$.

Theorem 4. Let $k\subset\mathbb C$ be a subfield and $S/k$ a K3 surface. Let $T_\ell\subset H^2_{\rm et}(S_{\bar k},\mathbb Q_\ell(1))$ be the orthogonal complement of $\mathrm{NS}(S_{\bar k})\otimes\mathbb Q_\ell$, and $\varepsilon_S\colon\operatorname{Gal}(\bar k/k)\to\{\pm1\}$ the character by which Galois acts on $\bigwedge^rT_\ell$. Let $A_\varepsilon$ be the Artin motive of $\varepsilon$: if $\varepsilon\ne1$ cuts out $k_\varepsilon/k$ with involution $\tau$, then $A_\varepsilon=(\operatorname{Spec}k_\varepsilon,\frac12(\Delta-\Gamma_\tau),0)$; and $A_1=\mathbb 1$.

1. $\delta_{S/k}:=\bigwedge^rt_2(S)\otimes\mathbb L^{-r}\in\mathcal M_{\hom}(k)$ is invertible with $\delta_{S/k}^{\otimes2}\cong\mathbb 1$, and $\det_T(S_{\mathbb C})$ is algebraic if and only if $\delta_{S/k}\cong A_\psi$ for some quadratic character $\psi$. In that case $\psi=\varepsilon_S$.
2. If $h(S_{\mathbb C})$ is finite-dimensional, the same holds in $\mathrm{CHM}(k)$.
3. In particular, if $\det_T$ is algebraic over $\mathbb C$, it is the class of a cycle defined over $k$ if and only if $\varepsilon_S=1$; otherwise it is defined over $k_{\varepsilon}$ and not over $k$.

Corollary 4.3. Let $f=t^5-t-1$. Let $F_f(x,y,z)=x\prod_{f(a)=0}(a^2x-2ay+z)\in\mathbb Q[x,y,z]$ be the product of the six tangent lines to the conic $y^2=xz$ at the points $(1:a:a^2)$, $f(a)=0$, and at $(0:0:1)$. For squarefree $d\in\mathbb Z$ let $S_d$ be the minimal resolution of $d\,w^2=F_f(x,y,z)$. Then $S_d$ is a K3 surface over $\mathbb Q$ with $r=5$, and $\bigwedge^5t_2(S_d)\cong\mathbb L^5\otimes A_{\varepsilon_{S_1}\psi_d}$ in $\mathrm{CHM}(\mathbb Q)$, where $\psi_d$ is the character of $\mathbb Q(\sqrt d)$. Hence $\delta_{S_d/\mathbb Q}\not\cong\mathbb 1$ for all squarefree $d$ except at most one.

Theorem 5. Let $f\colon X\to B$ be a smooth projective family of K3 surfaces over a smooth connected complex quasi-projective variety. Let $N\subset H^2(X_b,\mathbb Z)$ be a primitive sublattice spanned by classes of line bundles on $X$, with $N^\perp$ of rank $r\ge3$. Let $\chi\colon\pi_1(B)\to\{\pm1\}$ be the orientation character on $\bigwedge^rN^\perp$ [13], and $\tilde B\to B$ the (trivial or connected) double cover it defines. Let $D_b\in\bigwedge^r N^\perp\subset H^{2r}(X_b^r,\mathbb Q)$ be the determinant line; it is a Hodge line for every $b$. Note that $N^\perp_b$ is a fixed complement of $N$, not the varying transcendental space $T(X_b)\subseteq N^\perp_b$. Write $\pi_{N^\perp}$ for the relative algebraic projector onto $N^\perp$ (the relative Künneth projector $\pi_2$ minus the divisor projectors $\frac{1}{q(n_i)}\,n_i\times n_i$ for an orthogonal basis $(n_i)$ of $N_{\mathbb Q}$), and $q_r:=\frac1{r!}\sum_\sigma\operatorname{sgn}(\sigma)\Gamma_\sigma\circ\pi_{N^\perp}^{\times r}$, so that $q_r$ projects $H^{2r}(X_b^r)$ onto $D_b$ fibrewise. Let $B_{\rm alg}=\{b: D_b\text{ is spanned by an algebraic class}\}$.

1. $B_{\rm alg}$ is a countable union of closed algebraic subsets of $B$. If it contains a very general point, then $B_{\rm alg}=B$.
2. Let $K=\mathbb C(B)$ and assume (G): $N_{\mathbb Q}=\mathrm{NS}(X_{\bar K})_{\mathbb Q}$, i.e. $N^\perp$ is the transcendental space of the geometric generic fibre. Then $B_{\rm alg}=B$ if and only if $\delta$ of the generic fibre is isomorphic to $A_\chi$ in $\mathcal M_{\hom}(K)$, if and only if there is a cycle on $X^r\times_B\tilde B$ whose restriction to every fibre spans $D_b$. A cycle over $B$ itself (or over $K$) with this property exists only if $\chi=1$.
3. If the period map of $B$ is a local isomorphism onto an open subset of the period domain of $N^\perp$, then $\{b:\operatorname{rk}T(X_b)\le5\}\subset B_{\rm alg}$.
4. (Semiregular propagation.) Suppose that at some $b_0$ there are an l.c.i. subscheme $Z\subset X_{b_0}^r$ of codimension $r$ and a class $w$ on $X^r\times_B\tilde B$, induced by a relative cycle, such that $[Z]=c\,\det+w_{b_0}$ with $c\neq0$ and $\det$ a generator of $D_{b_0}$. If $Z$ is semiregular in Bloch's sense, then $B_{\rm alg}=B$.

Corollary 5.2. Let $S$ be a K3 surface with $E(S)=\mathbb Q$ and $r\ge6$. Let $B$ be a smooth connected family of $\mathrm{NS}(S)$-polarized K3 surfaces containing $S$, with period map a local isomorphism (e.g. a component of a fine moduli space with level structure). Assume the variational Hodge conjecture holds for the flat section $D$ on $X^r\times_B\tilde B$, i.e. a flat section that is Hodge everywhere and algebraic at one point is algebraic everywhere. Then HC holds for all powers of $S$.

Proposition 5.3 (product-type obstruction). Let $Z=Z_0\times C_1\times\cdots\times C_m\subset S^{r-m}\times S^m$, where $Z_0\subset S^{r-m}$ is a nonempty l.c.i. subscheme of codimension $r-m$ and the $C_i\subset S$ are smooth curves of genus $g_i$. If $\sum_{i\neq j}g_ig_j>h^{r+1,r-1}(S^r)$, then $Z$ is not semiregular. For $Z=C_1\times\dots\times C_r$ with all $g_i=g$, non-semiregularity holds as soon as $r+r(r-1)g^2>h^{r+1,r-1}(S^r)$. Equal genera force this for $g\ge5,15,53,205$ when $r=2,3,4,5$, where $h^{r+1,r-1}=40,\,1209,\,32720,\,836100$.

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3. What is new and what is not

Not new. The following are known, and we use them as stated:

• the reduction of HC for powers to $\det_T$ when $E=\mathbb Q$ (Varesco [1]);
• Kimura's nilpotence theorem and the finite-dimensionality of curves and of their products [2];
• the transcendental motive $t_2(S)$ [3];
• the determinant formalism for finite-dimensional motives, its duality, square relation, Artin descent and relation with the Hodge conjecture (Kahn [19, §1, Prop. 2, Remarks 5(c)–(d)]); Theorems 1–2 specialize it to $t_2(S)$;
• the one-flat-section variational mechanism on an orientation cover (rStation 77 [12, Remark 6.3]) and specialization of determinant algebraicity (Varesco [1, Prop. 2.22]);
• Bloch's semiregularity theorem [6];
• the spreading and specialization of cycles in families. The "countable union of closed subsets" statement in Theorem 5(1) is a standard argument; we include a proof for completeness.

The sign obstruction for cycles defined over a family with nontrivial orientation character is in rStation 77 [12, Thm. 3] and in our rStation 105 [13]. Theorem 5(2) only recasts it as a statement about Artin motives, under the hypothesis (G).

New, as far as we could determine (searches logged in sources.md):

1. the K3-specific form of Kahn's theory: the equivalence of triviality of $\delta_S$ in the homological and numerical categories (unconditionally) and in Chow motives (under finite-dimensionality) with the algebraicity of $\det_T$, stated with the precise categories needed below (Theorems 1 and 2; the formal content is in [19]);
2. the unconditional Chow-level identities of Theorem 3;
3. the arithmetic statement of Theorem 4 and the explicit nontrivial $2$-torsion invertible Chow motives over $\mathbb Q$ of Corollary 4.3;
4. the inclusion $\{\operatorname{rk}T_b\le5\}\subset B_{\rm alg}$ in Theorem 5(3) (a dense seed locus, extending the Picard-number-20 seeds of [12, Remark 6.3]), and Corollary 5.2, which isolates the variational Hodge conjecture for one explicit flat section as sufficient for HC on all powers of every $E=\mathbb Q$ K3 surface;
5. the dimension obstruction of Proposition 5.3.

We stress that items 1 and 3 are formal once the inputs are granted. Their value is that they give exact statements in the right category, plus the unconditional corollaries.

Comparison with Voisin's conjecture. Voisin's conjecture for surfaces with $p_g=1$ predicts that the alternation in Theorem 3(3) vanishes already for $k\ge2$. It is known for some families with finite-dimensional motive (e.g. [10], and the work cited there). Theorem 3(3) is weaker in range. It is proved for every member of the families in Theorem 3, and its source is the algebraicity of $\det_T$, not special geometry. Theorem 3(1) and the Hodge-star isomorphisms concern cycles of positive dimension and are not covered by Voisin's conjecture. They complement Yin's theorem [9]: finite-dimensionality is equivalent to injectivity of the cycle map on the subring generated by divisors and diagonals. The determinant cycle is not in that subring, but it is controlled by Theorem 3(1).

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4. Related work

| Work | What it does | Relation to this paper | |---|---|---| | Varesco [1] | $E=\mathbb Q$: HC for all powers $\Leftrightarrow\det_T$ algebraic | Used in Theorem 1(5); we recast it as $\delta_S\cong\mathbb 1$ | | Kimura [2] | Finite-dimensional motives; nilpotence | Used to lift to Chow (Theorem 2) | | Kahn–Murre–Pedrini [3] | $t_2(S)$, $CH^*(t_2)=T(S)_{\mathbb Q}$ in degree 2 | Defines $\delta_S$ | | André [4] | Motivated cycles; Hodge classes on products of K3 surfaces are motivated | In André's category $\delta_S$ is trivial (the determinant character of $\mathrm{SO}$ is trivial); our $\delta_S$ measures the gap to $\mathcal M_{\hom}$ | | Pedrini [5] | Finite-dimensionality for $\rho=19,20$ | Input for Theorem 3(c) | | Bloch [6]; Buchweitz–Flenner [7] | Semiregularity and deformation of cycles/sheaves | Theorem 5(4), Proposition 5.3 | | Buskin [8] | Rational Hodge isometries are algebraic | Theorem 1(6) | | Yin [9] | Finite-dimensionality $\Leftrightarrow$ injectivity of the cycle map on the divisor–diagonal subring of $CH^*(S^n)$ | Theorem 3(1) treats the determinant cycle outside that subring | | Bolognesi–Laterveer [10] | A 9-dimensional family with finite-dimensional motive; Voisin's conjecture | Comparison for Theorem 3(3) (there $E\neq\mathbb Q$) | | rStation 98, 99 (ours) [11], [14] | $\det_T$ algebraic for curve-product-dominated surfaces | Input for Theorem 3(a),(b) | | rStation 77 [12]; rStation 105 (ours) [13] | Orientation character and sign obstruction in families; variational mechanism [12, Remark 6.3] | Theorem 5(2) is the Artin-motive form; Corollary 5.2 adds the dense seed locus | | Kahn [19] | Determinants of finite-dimensional motives, duality, Artin descent, K3 as first open case | Theorems 1–2 are the case $M=t_2(S)$ | | Varesco [1, Prop. 2.22] | Specialization of determinant algebraicity | Compared with Theorem 5(1),(3) | | Floccari [15] | HC for all powers on a quadratic locus; for $E=\mathbb Q$ this contains all $r\le5$ | Input for Theorem 3(c), Theorem 5(3) | | Zarhin [16] | $y^2=f(x)$ with $\operatorname{Gal}(f)=S_n$, $n\ge5$, has $\operatorname{End}J=\mathbb Z$ | Input for Corollary 4.3 |

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5. Notation and imported results

• (I1) [3]. Over any field $k$ of characteristic $0$, $h(S)=\mathbb 1\oplus h_2^{\rm alg}(S)\oplus t_2(S)\oplus\mathbb L^2$ in $\mathrm{CHM}(k)$. The projector $\pi^{\rm tr}$ is defined over $k$ and compatible with base change. Over $\mathbb C$, $H^*(t_2)=T$ in degree $2$, and $CH^i(t_2)=0$ for $i\ne2$. With the choices $\pi_0=[\mathrm{pt}\times S]$, $\pi_4=[S\times\mathrm{pt}]$ and $\pi_2^{\rm alg}=\sum_i\frac{1}{D_i^2}D_i\times D_i$ (orthogonal basis of $\mathrm{NS}_{\mathbb Q}$), $\pi^{\rm tr}_*z=z$ for every $0$-cycle $z$ of degree $0$. Indeed $\pi_{0*},\pi^{\rm alg}_{2*}$ kill $0$-cycles for dimension reasons, and $\pi_{4*}z=\deg(z)\,\mathrm{pt}$. On cohomology $\pi^{\rm tr}$ is the orthogonal projection onto $T$, so its transpose has the same class.
• (I2) [2]. Finite-dimensional motives are closed under direct sums, tensor products, direct summands and duals; motives of curves are finite-dimensional. If $M$ is finite-dimensional, the kernel of $\operatorname{End}_{\mathrm{CHM}}(M)\to\operatorname{End}_{\mathcal M_{\rm num}}(M)$ is a nilpotent ideal [2, Prop. 7.5].
• (I3) [1]. If $E(S)=\mathbb Q$, HC for all $S^n$ $\Leftrightarrow$ $\det_T$ algebraic.
• (I4) [14, Thm. A, Cor. A′]. If $E(S)=\mathbb Q$ and $S$ is dominated by a product of curves, $\det_T$ is algebraic. [11, Thm. B]: HC holds for all powers of $S_{b,c}$, $b\ne0$, and $S_{b,c}$ is dominated by $C_t\times C_u$.
• (I5) [15, Thm. 5.11], as read in [14, §1]. If $E(S)=\mathbb Q$ and $r\le5$, HC holds for all powers of $S$.
• (I6) [5, Thm. 2]. K3 surfaces with $\rho\in\{19,20\}$ have finite-dimensional motive.
• (I7) [6]. Let $X_0$ be smooth projective and $Z\subset X_0$ an l.c.i. subscheme of codimension $p$. $Z$ is semiregular if the semiregularity map $H^1(Z,N_Z)\to H^{p+1}(X_0,\Omega^{p-1}_{X_0})$ is injective. In that case, for every deformation of $X_0$ over an Artinian base along which the horizontal continuation of $[Z]$ stays in $F^p$, $Z$ lifts to a flat deformation.
• (I8) [16]. If $f\in\mathbb Q[t]$ has degree $n\ge5$ and Galois group $S_n$ or $A_n$, then $\operatorname{End}_{\bar{\mathbb Q}}J(y^2=f)=\mathbb Z$.
• (I9) Galois descent with $\mathbb Q$-coefficients. For $X/k$ smooth projective, $CH^i(X)_{\mathbb Q}\to CH^i(X_{\bar k})_{\mathbb Q}^{\operatorname{Gal}}$ is an isomorphism. Injectivity follows from the norm; surjectivity from averaging over a finite Galois extension of definition and pushing forward. The same holds for $\operatorname{Hom}$ groups of Chow motives, and for algebraic classes modulo homological equivalence. Similarly $CH^i(X_k)_{\mathbb Q}\to CH^i(X_{\mathbb C})_{\mathbb Q}$ is injective for $k\subset\mathbb C$.

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6. Proofs

6.1 Two lemmas

Lemma A (faithfulness). In $\mathcal M_{\hom}$, the realization $H$ is faithful. Hence an object with $H(M)=0$ is zero, and $f$ is an isomorphism as soon as $H(f)$ is invertible and $H(f)^{-1}=H(g)$ for some morphism $g$.

Proof. Morphisms in $\mathcal M_{\hom}$ are cohomology classes of correspondences, acting faithfully on $H$. If $H(f)^{-1}=H(g)$, then $H(gf)=\mathrm{id}$ and $H(fg)=\mathrm{id}$, so $gf=\mathrm{id}$ and $fg=\mathrm{id}$. $\square$

Lemma B (Kimura lifting). Let $M,N$ be finite-dimensional Chow motives and $f\colon M\to N$ a morphism whose image in $\mathcal M_{\hom}$ is an isomorphism. Then $f$ is an isomorphism. A finite-dimensional Chow motive with $H(M)=0$ is zero.

Proof. Lift the inverse in $\mathcal M_{\hom}$ to $g\colon N\to M$ (morphisms in $\mathcal M_{\hom}$ are quotients of Chow morphisms). Then $gf=\mathrm{id}+n$ with $n$ homologically, hence numerically, trivial, so $n$ is nilpotent by (I2), and $gf$ is invertible. Likewise $fg$ is invertible, so $f$ is an isomorphism. If $H(M)=0$, then $\mathrm{id}_M$ is numerically trivial, hence nilpotent, hence $0$. $\square$

6.2 Proof of Theorem 1

Work in $\mathcal M_{\hom}(\mathbb C)$; write $t$ for the image of $t_2(S)$.

(1) $H(\bigwedge^{r+1}t)=\bigwedge^{r+1}T=0$, so $\bigwedge^{r+1}t=0$ by Lemma A. $H(\delta_S)=\bigwedge^rT\otimes\mathbb Q(r)$ is a line of Hodge type $(0,0)$ (verify.py V1a).

For invertibility, let $\mathrm{ev}\colon\delta\otimes\delta^\vee\to\mathbb 1$ and $\mathrm{co}\colon\mathbb 1\to\delta\otimes\delta^\vee$ be the duality maps of the rigid category $\mathcal M_{\hom}$. The composite $\mathrm{ev}\circ\sigma\circ\mathrm{co}\in\operatorname{End}(\mathbb 1)=\mathbb Q$ ($\sigma$ the symmetry) is the categorical dimension, which $H$ computes as $\dim H(\delta)=1$. So $e=\sigma\circ\mathrm{co}\circ\mathrm{ev}$ is an idempotent of $\delta\otimes\delta^\vee$. Since $H(\delta\otimes\delta^\vee)$ is a line, $H(e)$ is the identity, so $e=\mathrm{id}$ (Lemma A). Then $\mathrm{ev}$ and $\sigma\circ\mathrm{co}$ are inverse isomorphisms, and $\delta\otimes\delta^\vee\cong\mathbb 1$.

(2) Write $t=(S,\pi,0)$, $\pi=\pi^{\rm tr}$, so $t^\vee=(S,\pi^t,2)$ and $t^\vee\otimes\mathbb L^{2}=(S,\pi^t,0)$. Let $\varphi=\pi^t\circ\Delta_S\circ\pi\colon t\to t^\vee\otimes\mathbb L^2$ and $\psi=\pi\circ\Delta_S\circ\pi^t$ in the other direction. By (I1), $\pi$ and $\pi^t$ have the same class (orthogonal projection onto $T$). So $H(\psi\varphi)=H(\pi)^3=\mathrm{id}_{H(t)}$ and similarly $H(\varphi\psi)=\mathrm{id}$. Hence $t\cong t^\vee\otimes\mathbb L^2$ (Lemma A). $H(\varphi)$ is the polarization $q$ composed with Poincaré duality.

Apply $\bigwedge^r$, using the canonical isomorphism $\bigwedge^r(M^\vee)\cong(\bigwedge^rM)^\vee$ in a $\mathbb Q$-linear rigid Karoubian tensor category (the antisymmetrizer of $M^{\vee\otimes r}$ is the transpose of that of $M^{\otimes r}$). This gives $\delta\otimes\mathbb L^r\cong(\delta\otimes\mathbb L^r)^\vee\otimes\mathbb L^{2r}=\delta^\vee\otimes\mathbb L^r$, so $\delta\cong\delta^\vee$. Then $\delta^{\otimes2}\cong\delta\otimes\delta^\vee\cong\mathbb 1$, and $\operatorname{End}(\delta)=\operatorname{Hom}(\mathbb 1,\delta^\vee\otimes\delta)=\operatorname{End}(\mathbb 1)=\mathbb Q$.

(3) $\operatorname{Hom}_{\hom}(\mathbb 1,\delta)=\operatorname{Hom}(\mathbb L^r,\bigwedge^rt)$ is the space of classes $p_r(\alpha)$ with $\alpha$ an algebraic class in $H^{2r}(S^r)$. Since $p_r$ is algebraic and projects $H^{2r}(S^r)$ onto $\bigwedge^rT$, this space is $\bigwedge^rT$ if $\det_T$ is algebraic and $0$ otherwise. So (i)$\Leftrightarrow$(iv) in $\mathcal M_{\hom}$.

Now let $0\ne f\colon\mathbb 1\to\delta$ and let $\theta\colon\delta\xrightarrow\sim\delta^\vee$ be as in (2). Put $g=f^\vee\circ\theta\colon\delta\to\mathbb 1$. Then $g\circ f\in\mathbb Q$ is computed by $H$ as the value of the nondegenerate form $H(\theta)$ on the nonzero vector $H(f)(1)$, so it is nonzero. And $f\circ g\in\operatorname{End}(\delta)=\mathbb Q$ is nonzero because $(f\circ g)\circ f=f\circ(g\circ f)\ne0$. Hence $f$ is an isomorphism, so (iv)$\Rightarrow$(ii). (ii)$\Rightarrow$(iv) is trivial.

$\mathcal M_{\rm num}$ is a quotient of $\mathcal M_{\hom}$ with the same objects. So $\operatorname{Hom}_{\rm num}(\mathbb 1,\delta)$ is a quotient of $\operatorname{Hom}_{\hom}(\mathbb 1,\delta)$, and an isomorphism in $\mathcal M_{\hom}$ maps to one in $\mathcal M_{\rm num}$. This gives (iii)$\Rightarrow$(i)$\Rightarrow$(ii)$\Rightarrow$(iii), with (iv) in $\mathcal M_{\rm num}$ in between.

(4) Let $w\colon\bigwedge^kt\otimes\bigwedge^{r-k}t\to\bigwedge^rt$ be the restriction of the antisymmetrizer. Let $w^\flat\colon\bigwedge^{r-k}t\to(\bigwedge^kt)^\vee\otimes\bigwedge^rt$ be its adjoint. Let $u=(\mathrm{ev}\otimes\mathrm{id})\circ(\mathrm{id}\otimes c)$ go the other way, where $c\colon\bigwedge^rt\to\bigwedge^kt\otimes\bigwedge^{r-k}t$ is the inclusion $\bigwedge^rt\subset t^{\otimes r}$ followed by $p_k\otimes p_{r-k}$. Both are morphisms of $\mathcal M_{\hom}$.

On cohomology, $H(w)$ is the wedge pairing, which is perfect (verify.py V2: unimodular for $r\le7$; in general this is classical). Also $H(u)\circ H(w^\flat)=\lambda\cdot\mathrm{id}$ with $\lambda\in\mathbb Q^\times$. Indeed $H(u)H(w^\flat)$ is a $\mathrm{GL}(T)$-equivariant endomorphism of the irreducible module $\bigwedge^{r-k}T$, hence a scalar. The scalar is nonzero. $H(u)$ is a nonzero equivariant map: contracting $H(c)(\mathrm{vol})$, which is the nonzero antisymmetric tensor $\mathrm{vol}\in T^{\otimes r}$, against $e_I^*$ gives $\pm\frac{1}{r!}$-multiples of $e_{I^c}$. Its source $(\bigwedge^kT)^\vee\otimes\bigwedge^rT$ is irreducible, being isomorphic to $\bigwedge^{r-k}T$ via $H(w^\flat)$, so $H(u)$ is an isomorphism, and a composite of isomorphisms is nonzero. V2b computes $\lambda$ explicitly for $r\le5$. By Lemma A, $w^\flat$ is an isomorphism with inverse $\lambda^{-1}u$. Combined with $(\bigwedge^kt)^\vee\cong\bigwedge^k(t^\vee)\cong\bigwedge^kt\otimes\mathbb L^{-2k}$ (by (2)) and $\bigwedge^rt=\delta\otimes\mathbb L^r$, this gives (4).

(5) is (3) combined with (I3).

(6) Let $\alpha\colon t\to t'$ and $\beta\colon t'\to t$ be induced by algebraic correspondences composed with $\pi^{\rm tr},\pi'^{\rm tr}$, with $H(\beta)=H(\alpha)^{-1}$. By Lemma A, $t\cong t'$ in $\mathcal M_{\hom}$, and applying $\bigwedge^r$ gives $\delta_S\cong\delta_{S'}$. A rational Hodge isometry $\phi$ is algebraic by [8], and so is $\phi^{-1}=\phi^t$. If $E(S)=\mathbb Q$ and $\phi$ is an algebraic Hodge isomorphism, then $q'(\phi\cdot,\phi\cdot)$ is a Hodge-invariant form on the irreducible $T$, hence equals $\mu q$. So $\phi^{-1}=\mu^{-1}\phi^t$ is algebraic. $\square$

6.3 Proof of Theorem 2

By (I2), $t_2(S)$ is finite-dimensional, being a summand of $h(S)$, and so are its tensor constructions and duals.

• (1): repeat 6.2(1)–(2) in $\mathrm{CHM}$, replacing Lemma A by Lemma B. $\bigwedge^{r+1}t$ is finite-dimensional with zero cohomology, hence zero. In the invertibility argument, $\operatorname{End}_{\mathrm{CHM}}(\mathbb 1)=CH^0(\mathrm{pt})_{\mathbb Q}=\mathbb Q$, so $\mathrm{ev}\circ\sigma\circ\mathrm{co}=1$ again, and $e=\sigma\circ\mathrm{co}\circ\mathrm{ev}$ is an idempotent with $H(e)=\mathrm{id}$. Then $1-e$ is an idempotent whose image is finite-dimensional with zero cohomology, hence zero. For self-duality, $\varphi$ maps to an isomorphism in $\mathcal M_{\hom}$, so it is an isomorphism by Lemma B.
• (2): $\operatorname{Hom}_{\mathrm{CHM}}(\mathbb 1,\delta)\to\operatorname{Hom}_{\hom}(\mathbb 1,\delta)$ is surjective. If $\det_T$ is algebraic, lift a nonzero $f$ and run the argument of 6.2(3) with $\operatorname{End}_{\mathrm{CHM}}(\delta)=\mathbb Q$ from (1). The converse holds because realization is a functor.
• (3): $w^\flat$ is a Chow morphism whose image in $\mathcal M_{\hom}$ is an isomorphism (6.2(4)); apply Lemma B.
• (4): if $\delta\cong\mathbb 1$, then $CH^i(\bigwedge^rt)=\operatorname{Hom}(\mathbb L^i,\mathbb L^r)=CH^{r-i}(\mathrm{pt})_{\mathbb Q}$, which is $\mathbb Q$ for $i=r$ and $0$ otherwise. By (3), $\bigwedge^kt\cong\bigwedge^{r-k}t\otimes\mathbb L^{2k-r}$, so $CH^i(\bigwedge^kt)=\operatorname{Hom}(\mathbb L^{i+r-2k},\bigwedge^{r-k}t)=CH^{i+r-2k}(\bigwedge^{r-k}t)$. This group is a summand of $CH^{i+r-2k}(S^{r-k})_{\mathbb Q}$, which vanishes when $i+r-2k>2(r-k)$, i.e. when $i>r$. $\square$

6.4 Proof of Theorem 3

Finite-dimensionality.

• (a) Resolve the indeterminacy of $C_1\times C_2\dashrightarrow S$ by blowing up points. This gives $Y\to S$ surjective and generically finite of degree $d$, with $h(Y)=h(C_1\times C_2)\oplus\mathbb L^{\oplus m}$, which is finite-dimensional by (I2). Since $f_*f^*=d$, $h(S)$ is a direct summand of $h(Y)$, hence finite-dimensional.
• (b) The same argument applies with the dominant map $C_t\times C_u\dashrightarrow S_{b,c}$ of [11, (6.3)].
• (c) This is (I6).

For Kummer surfaces, an abelian surface $A$ is dominated by $C\times C$ via $(x,y)\mapsto x+y$ for a curve $C\subset A$ generating $A$, and $A\dashrightarrow\mathrm{Km}(A)$ has degree $2$.

Algebraicity of $\det_T$.

• (a) This is (I4).
• (b) HC for $S^r$ holds by (I4), and $\det_T$ is a Hodge class on $S^r$.
• (c) $r=3$ forces $E=\mathbb Q$ (Zarhin [18]: a totally real $E\ne\mathbb Q$ needs $\dim_ET\ge3$, and a CM $E$ needs $[E:\mathbb Q]\mid r$ with $[E:\mathbb Q]$ even). Then use (I5).

Theorem 2 then gives $\delta_S\cong\mathbb 1$ and Theorem 2(4). Statements (1) and (2) translate $CH^i(\bigwedge^kt)=(p_k)_*CH^i(S^k)_{\mathbb Q}$.

For (3): $2k>r$, and by (I1), $p_{k*}(z_1\times\dots\times z_k)=\frac1{k!}\sum_\sigma\operatorname{sgn}(\sigma)z_{\sigma(1)}\times\dots\times z_{\sigma(k)}$ for degree-zero $z_i$. This lies in $CH^{2k}(\bigwedge^kt)=0$ by (2), since $2k>r$. $\square$

6.5 Proof of Theorem 4 and Corollary 4.3

(1) Theorem 1(1)–(2) hold verbatim over $k$, with $\ell$-adic realization; the self-duality uses (I1) over $k$. By (I9), $\operatorname{Hom}_{\hom,k}(\mathbb 1,M)=\operatorname{Hom}_{\hom,\bar k}(\mathbb 1,M_{\bar k})^{\operatorname{Gal}}$. Also $A_\psi\otimes\bar k\cong\mathbb 1$, with Galois acting on $H(A_\psi)$ by $\psi$.

Suppose $\det_T$ is algebraic over $\mathbb C$. Spreading a representing cycle over a $\bar k$-variety and specializing at a $\bar k$-point (the class of the constant family $S^r_{\bar k}\times V$ is constant) shows that $\operatorname{Hom}_{\bar k}(\mathbb 1,\delta)=\mathbb Q\cdot\det$. Galois acts on this line by $\varepsilon_S$. On $\operatorname{Hom}_{\bar k}(\mathbb 1,\delta\otimes A_{\varepsilon_S})$ it acts by $\varepsilon_S^2=1$, so there is a nonzero morphism $\mathbb 1\to\delta\otimes A_{\varepsilon_S}$ over $k$. The argument of 6.2(3), applied to the invertible self-dual object $\delta\otimes A_{\varepsilon_S}$ (with $\operatorname{End}=\mathbb Q$), shows that it is an isomorphism. So $\delta_{S/k}\cong A_{\varepsilon_S}^{-1}=A_{\varepsilon_S}$.

Conversely, if $\delta_{S/k}\cong A_\psi$, then over $\bar k$ (hence over $\mathbb C$) $\delta\cong\mathbb 1$, so $\det_T$ is algebraic by Theorem 1. Comparing Galois actions on the realization gives $\psi=\varepsilon_S$.

(2) Finite-dimensionality over $\mathbb C$ implies it over $k$: $\bigwedge^n$ of a motive is zero over $k$ iff it is zero over $\mathbb C$, by the injectivity in (I9). Then replace Lemma A by Lemma B as in 6.3.

(3) follows from (1) and (I9).

Corollary 4.3.

Geometry of $S_d$. The tangent line to $y^2=xz$ at $(1:a:a^2)$ is $a^2x-2ay+z=0$, and at $(0:0:1)$ it is $x=0$. Since $f$ is separable of degree $5$, the six points of tangency are distinct, and $F_f$ has rational coefficients (it is symmetric in the roots). A point not on the conic lies on at most two tangent lines, so there are no triple points. The sextic therefore has exactly $15$ nodes, and the minimal resolution of the double plane is a K3 surface.

Classically, the double plane branched along the six tangents to a conic at the Weierstrass points of $C\colon y^2=f(t)$ is birational to $\mathrm{Km}(J(C))$, and $T(\mathrm{Km}A)\cong T(A)(2)$ (see e.g. [17]). Over $\bar{\mathbb Q}$, $S_d\cong S_1$, and a birational map of K3 surfaces is an isomorphism, so $S_{d,\mathbb C}\cong\mathrm{Km}(J(C))$.

Galois group of $f$ (V7). $f=t^5-t-1$ is irreducible of prime degree. Its discriminant $2869=19\cdot151$ is prime to $163$, and modulo $163$ it factors with cycle type $(2,1,1,1)$. An irreducible polynomial of prime degree $p$ whose Galois group contains a transposition has Galois group $S_p$ (V7). So $\operatorname{End}J(C)=\mathbb Z$ by (I8). Hence $\rho(J)=1$, $\rho(S)=17$, $r=5$, and $E(S)=\mathbb Q$, since $T(S)\cong T(J)(2)$ as Hodge structures up to scaling and $E(T(J))=\mathbb Q$ for $\operatorname{End}J=\mathbb Z$. $h(S_{\mathbb C})$ is finite-dimensional and $\det_T$ is algebraic by Theorem 3(a).

The twist. Let $\iota\colon w\mapsto-w$, an involution of $S_d$ defined over $\mathbb Q$. Since $S_d/\iota$ is rational, $\iota^*$ acts by $-1$ on $H^{2,0}$. As a Hodge isometry of $T$ with $E=\mathbb Q$, it is $\pm1$, hence $-1$ on $T$. $S_d$ is the twist of $S_1$ by the cocycle $\sigma\mapsto\iota^{(1-\psi_d(\sigma))/2}$, so Galois acts on $T_\ell(S_d)$ by $\psi_d(\sigma)\rho_{S_1}(\sigma)$. Hence $\varepsilon_{S_d}=\psi_d^5\varepsilon_{S_1}=\psi_d\varepsilon_{S_1}$. This is trivial for at most one $d$ (V8), and Theorem 4(2) concludes. $\square$

6.6 Proof of Theorem 5, Corollary 5.2 and Proposition 5.3

After the base change $\tilde B\to B$ we may assume $\chi=1$, so $D$ is spanned by a flat section $\det$. It is Hodge at every $b$: $N^\perp_b\supset T(X_b)$ is a K3-type sub-Hodge structure, so V1a applies. Replacing $B$ by $\tilde B$ does not change $B_{\rm alg}$, because fibres are unchanged.

(1) Let $\mathcal C\to B$ be the relative Chow variety of $X^r\to B$ (effective cycles of dimension $r$ in the fibres $X_b^r$; one may also use relative Hilbert schemes). It has countably many components, each proper over $B$. Consider the components of $\mathcal C\times_B\mathcal C$, which parametrize pairs $(Z^+,Z^-)$ in the same fibre. Restrict to a connected component $W$ of such a pair. There, $b\mapsto q_r([Z^+]-[Z^-])$ is a flat section of the pulled-back local system, because cycle classes are locally constant in flat families. So it equals $c\det$ with a fixed $c$ on all of $W$. Let $W^\circ$ be the union of the components on which $c\neq0$. Its image in $B$ is closed and algebraic (properness). Since every algebraic class is a rational multiple of such a difference, $B_{\rm alg}$ is the countable union of these images.

If all images were proper closed subsets, a very general point would avoid their union. Hence, if $B_{\rm alg}$ contains a very general point, one image is $B$.

(2) If $B_{\rm alg}=B$, then by (1) some $W^\circ$ dominates $B$. A point of its generic fibre over a finite extension $K'/K$ gives a cycle $Z'$ on $X_{K'}^r$ with $q_r[Z']=c\det$. By (G), $\bigwedge^rN^\perp$ is $\bigwedge^r$ of the transcendental part of the geometric generic fibre, i.e. the realization of $\delta_{X_K}\otimes\mathbb L^r$. $\operatorname{Gal}(\bar K/K)$ acts on it through $\pi_1^{\rm et}$ of a dense open of $B$, compatibly with topological monodromy, hence by $\chi$. Theorem 4(1), whose proof works verbatim over any field of characteristic $0$ with $\ell$-adic realization, applied over $K$ gives $\delta_{X_K}\cong A_\chi$. Conversely, $\delta_{X_K}\cong A_\chi$ gives a cycle over $\bar K$ spanning $D$, which spreads out as below. When $\chi=1$, averaging the Galois conjugates of $Z'$ gives a cycle over $K$ with the same class. Spreading it out over a dense open $U$ and taking the closure in $X^r$ gives a relative cycle. Its fibre class is a flat section agreeing with $c\det$ over $U$, hence everywhere. For $\chi\ne1$ we do the same on $\tilde B$.

Conversely, a relative cycle with fibre classes spanning $D_b$ gives $B_{\rm alg}=B$. A cycle defined over $K$ has Galois-invariant class, while $\det$ transforms by $\chi$. So its $\det$-component vanishes unless $\chi=1$. This is the sign obstruction of [12, Thm. 3] and [13].

Remark (necessity of (G)). Without (G), write $M=\mathrm{NS}(X_{\bar K})_{\mathbb Q}\cap N^\perp_{\mathbb Q}$; then $N^\perp=T_{\bar K}\oplus M$ and $\chi=\varepsilon_T\cdot\det(M)$, where $\varepsilon_T$ is the character realized by $\delta_{X_K}$. Statement (1) and the cycle formulations in (2) remain valid with $\chi$, but $\delta_{X_K}\cong A_{\varepsilon_T}$, not $A_\chi$. An example pointed out to us: for $E$ a non-CM elliptic curve, $S=\mathrm{Km}(E\times E)$ and $g$ the involution exchanging the factors, $g$ acts by $-1$ on $T(S)$ (rank 3) but with determinant $+1$ on $H^2$; the isotrivial family $(S\times\mathbb G_m)/\langle(g,u\mapsto-u)\rangle\to\mathbb G_m$ with $N$ of rank one has $\chi=1$ and $B_{\rm alg}=B$, while $\varepsilon_T$ is the nontrivial character of $\mathbb C(u)/\mathbb C(t)$.

(3) Let $b\in B$ with $\operatorname{rk}T(X_b)=r_b\le5$. Put $M=N^\perp\cap\mathrm{NS}(X_b)$, negative definite of rank $r-r_b$. Consider first $r_b\ge3$. The Noether–Lefschetz locus $B_M=\{b':M\subset\mathrm{NS}(X_{b'})\}$ is closed algebraic. By the local-isomorphism hypothesis, near $b$ it is the period domain of $M^\perp\cap N^\perp$, of signature $(2,r_b-2)$. Its very general point $b'$ has $T(X_{b'})=(M^\perp\cap N^\perp)_{\mathbb Q}$ with Mumford–Tate group $\mathrm{SO}$, hence $E=\mathbb Q$.

At such $b'$, $\det_{N^\perp}$ is a nonzero multiple of $\operatorname{Alt}(\det_{T_{b'}}\otimes m_1\otimes\dots\otimes m_{r-r_{b'}})$, with $m_i$ a basis of $M$ (V5 checks the nonvanishing of the change-of-basis determinant). Here $\det_{T_{b'}}$ is algebraic on $X_{b'}^{r_{b'}}$ by (I5), the $m_i$ are divisor classes, and $\operatorname{Alt}$ is an algebraic correspondence. So $b'\in B_{\rm alg}$, and by (1) and the Baire argument applied on each irreducible component of $B_M$, $B_M\subset B_{\rm alg}$; in particular $b\in B_{\rm alg}$.

If $r_b\le2$, choose a primitive $M'\subset M$ of corank $3-r_b$, so that $M'^\perp\cap N^\perp$ has signature $(2,1)$. Then $b\in B_{M'}$, and $B_{M'}\subset B_{\rm alg}$ by the same argument with $r_{b'}=3$ (this uses $r\ge3$).

(4) The class $w$ is the class of a relative cycle, and $\det$ is flat and Hodge. So the flat continuation of $[Z]$ is Hodge on $B$ and stays in $F^r$ of the relative de Rham cohomology over every Artinian thickening of $b_0$. By (I7), $Z$ lifts over the formal neighbourhood of $b_0$. By pro-representability of the relative Hilbert scheme, its component $\mathcal H$ through $[Z]$ dominates the formal neighbourhood, hence dominates $B$. Being proper over $B$, $\mathcal H\to B$ is surjective. For each $b$ a point of $\mathcal H_b$ gives $Z_b$ with $[Z_b]=c\det_b+w_b$ (flatness of the class on the connected $\mathcal H$). Thus $\det_b=c^{-1}([Z_b]-w_b)$ is algebraic. $\square$

Corollary 5.2. Take $N=\mathrm{NS}(S)$, so $N^\perp_{\mathbb Q}=T(S)$. Take for $B$ a smooth connected family containing $S$ as a fibre whose period map is a local isomorphism, for example a component of a fine moduli space of $\mathrm{NS}(S)$-polarized K3 surfaces with level structure. By Theorem 5(3), $B_{\rm alg}\ne\emptyset$, since Noether–Lefschetz points with $\operatorname{rk}T\le5$ exist in every open subset of $B$. Periods lying in $W_{\mathbb C}$ for rational subspaces $W\subset N^\perp_{\mathbb Q}$ of signature $(2,3)$ are dense in the period domain, and the very general such period has $T=W$ (V5 exhibits the analogous splitting explicitly in each test lattice). Hence $D$ is a flat section that is Hodge everywhere and algebraic at one point. The variational Hodge conjecture for $D$ gives $B_{\rm alg}=B\ni[S]$, and (I3) concludes. (For $r\le5$ there is nothing to prove.) $\square$

Proposition 5.3. On a K3 surface the normal bundle of a smooth curve is $N_C=\mathcal O_C(C)=\omega_C$ (adjunction, $K_S=0$). So $h^0(N_{C_i})=g_i$ and $h^1(\mathcal O_{C_j})=g_j$. The normal bundle of $Z$ is $N_{Z_0}\boxplus\bigoplus_i pr_i^*N_{C_i}$. By Künneth, $H^1(Z,pr_i^*N_{C_i})$ contains $H^0(Z_0,\mathcal O)\otimes H^0(N_{C_i})\otimes H^1(\mathcal O_{C_j})\otimes\bigotimes_{l\ne i,j}H^0(\mathcal O_{C_l})$ for every $j\neq i$. Hence $h^1(N_Z)\ge\sum_{i\ne j}g_ig_j$. For $Z=C_1\times\dots\times C_r$ the exact value is $h^1(N_Z)=r+\sum_{i\ne j}g_ig_j$ (V6a computes it twice).

The target of the semiregularity map is $H^{r+1}(S^r,\Omega^{r-1})$, of dimension $h^{r-1,r+1}(S^r)=h^{r+1,r-1}(S^r)$. This is the coefficient of $x^{r+1}y^{r-1}$ in $(1+x^2+y^2+20xy+x^2y^2)^r$ (V6a). If the source is larger, the map is not injective. $\square$

Remark. Proposition 5.3 only says that the most naive representatives cannot be fed to (I7). The representatives of $\det$ at the points of Theorem 5(3) are, by construction, alternations of $\det_{T_{b'}}\otimes$(divisors), which are differences of product-type cycles. Whether some representative is semiregular, or whether the Buchweitz–Flenner version [7] for perfect complexes applies to a sheaf-theoretic representative, is open.

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7. Evidence (verify.py)

All checks use exact rational arithmetic and sympy; the seed is fixed. Result: FALLOS = 0 (verify_output.txt).

• V1a–V1d: Hodge numbers of $\bigwedge^k$ of a K3-type structure for $r\le22$. The determinant has type $(r,r)$; $\bigwedge^{r+1}=0$; the Hodge numbers of $\bigwedge^{r-k}$ agree with those of $\bigwedge^k(-(r-2k))$ and with those of the dual twisted by $\det$. A negative control (weight-one input) gives a different type.
• V2, V2b: the wedge pairing is unimodular for $r\le7$, and $u\circ w^\flat=\lambda\,\mathrm{id}$ with $\lambda\ne0$ for $r\le5$ (the inverse used in 6.2(4)).
• V3: for reflections and other improper isometries of the paper-105 lattices $U(3)^2\oplus\langle-6\rangle^{r-4}$ ($r\le7$), $T_6'=U(3)\oplus\langle6,-12,-30,-60\rangle$ and $U^2\oplus\langle-2\rangle^3$, the Hodge star satisfies $\star(gb)=\det(g)\,g(\star b)$ (65 cases). The negative control shows that the untwisted relation fails. This is the linear-algebra shadow of the twist by $\delta$ in Theorem 1(4) and of the Artin twist in Theorem 4.
• V4: $\langle\det,\det\rangle=\det(\mathrm{Gram})\ne0$ with sign $(-1)^r$ and signature $(2,r-2)$ for all test lattices.
• V5: each test lattice contains a positive-definite plane $P$ with $P^\perp$ negative definite, and $\det$ splits as $\det_P\wedge\det_{P^\perp}$ up to a nonzero factor. These are the $\rho=20$-type points used in 6.6(3).
• V6a, V6b, V6neg: $h^{r+1,r-1}(S^r)$ for $r\le8$, the exact $h^1(N_Z)$ for products of curves by two independent Künneth computations, and the minimal common genus. The control $g=1$ gives no conclusion.
• V7: $t^5-t-1$ is irreducible, $\operatorname{disc}=2869$, and has cycle type $(2,1,1,1)$ modulo $163$.
• V8, V8neg: character bookkeeping for twists; for even $r$ twisting does not change $\varepsilon$.
• V9: rank bookkeeping for Corollary 4.3.

The checks support finite linear algebra and bookkeeping. The theorems are proved in §6.

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8. Limitations

• No new case of HC. Theorems 1–2 restate the determinant problem; they do not solve it. Theorem 3 transports known algebraicity (from [11], [14], [15]) to Chow-level statements. Corollary 4.3 is arithmetic, not Hodge-theoretic.

• The variational route is conditional. Theorem 5(3) shows that $B_{\rm alg}$ contains a dense-looking but meagre set (a countable union of subvarieties of dimension $\le3$). Theorem 5(1) cannot conclude from it. We do not exhibit a semiregular representative. Proposition 5.3 only excludes naive product representatives, by a dimension count; it does not compute the semiregularity map.

• Finite-dimensionality is used only where it is known (curve-product domination, $\rho=19$). Kimura's conjecture for K3 surfaces in general is open, and so is its behaviour in families.

• Inputs used as black boxes:

  • Kimura nilpotence;
  • the Kahn–Murre–Pedrini decomposition over a base field;
  • Varesco's reduction;
  • Floccari's theorem, as read in [14] for $r\le5$;
  • Pedrini's finite-dimensionality for $\rho=19$;
  • Zarhin's theorem;
  • the classical Kummer description of the six-tangent double plane [17];
  • Bloch's theorem in its Artinian form.

  The statement (I1) that $\pi^{\rm tr}_*$ is the identity on degree-zero $0$-cycles depends on the stated choice of $\pi_0,\pi_4,\pi_2^{\rm alg}$.

• Corollary 4.3 asserts nontriviality for all but at most one $d$, but does not identify the exceptional $d$ (that would require computing $\varepsilon_{S_1}$).

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9. Reproducibility

Run python3 verify.py (Python ≥3.8, sympy ≥1.9). It takes about one minute and prints FALLOS = 0. No files are read or written. See README.md.

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References

[1] M. Varesco. The Hodge conjecture for powers of K3 surfaces of Picard number 16. arXiv:2203.09778 (Thm. 2.16, §2.3). https://arxiv.org/abs/2203.09778

[2] S.-I. Kimura. Chow groups are finite dimensional, in some sense. Math. Ann. 331 (2005), 173–201. https://doi.org/10.1007/s00208-004-0577-3

[3] B. Kahn, J. P. Murre, C. Pedrini. On the transcendental part of the motive of a surface. In: Algebraic Cycles and Motives, Vol. 2, LMS Lecture Note Ser. 344, Cambridge Univ. Press, 2007. Preprint: https://webusers.imj-prg.fr/~bruno.kahn/preprints/surfaces4.pdf

[4] Y. André. Pour une théorie inconditionnelle des motifs. Publ. Math. IHÉS 83 (1996), 5–49. http://www.numdam.org/item/PMIHES_1996__83__5_0/

[5] C. Pedrini. On the finite dimensionality of a K3 surface. arXiv:1106.1115 (Thm. 2). https://arxiv.org/abs/1106.1115

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