Algebraic cycles and Fano threefolds of genus 7

Abstract: Let $Y$ be a very general prime Fano threefold of genus 7. We exhibit an explicit 2-cycle on $Y\times Y$ that is Abel-Jacobi trivial but non-torsion in the Chow group $A^4(Y\times Y)$. As a consequence, $Y$ does not admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. We also show that any Fano threefold has a multiplicative Chow-Künneth decomposition modulo algebraic equivalence.

• The paper constructs an explicit Abel–Jacobi trivial but nonzero cycle on the square of a very general genus 7 Fano threefold, proving that it obstructs any multiplicative Chow–Künneth decomposition.
• The paper relates the Fano threefold to a dual genus 7 curve through a Chow-motive isomorphism, transporting the nontrivial Faber–Pandharipande cycle to establish the obstruction.
• The paper proves that every Fano threefold admits an MCK decomposition modulo algebraic equivalence, showing that multiplicative structure survives there even though it fails under rational equivalence.

Overview

This paper by Robert Laterveer studies the multiplicative structure of the Chow ring of prime Fano threefolds of genus 7, within the framework of multiplicative Chow–Künneth (MCK) decompositions introduced by Shen–Vial. The central result is a negative answer to a question the author had raised previously: whether every Fano threefold of Picard number 1 admits an MCK decomposition. For a very general prime Fano threefold $Y$ of genus 7, Laterveer constructs an explicit cycle $Z_Y \in \A^4(Y\times Y)$ that is Abel–Jacobi trivial but non-zero in $\A^4(Y\times Y)$, and shows this obstructs the existence of an MCK decomposition. In contrast to the rational-equivalence failure, he proves that every Fano threefold admits an MCK decomposition modulo algebraic equivalence.

Background: MCK decompositions and the splitting property

Motivated by Beauville's splitting property conjecture — inspired by the Beauville–Voisin description of the Chow ring of K3 surfaces and Beauville's Fourier-transform splitting for abelian varieties — Shen–Vial defined an MCK decomposition as a Chow–Künneth decomposition $\{\pi_X^i\}$ of a smooth projective variety $X$ such that the small diagonal $\Delta_X^{sm}$ acts compatibly with the grading: $$\pi_X^k \circ \Delta_X^{sm} \circ (\pi_X^i \times \pi_X^j) = 0$$ whenever $i+j \neq k$. An MCK decomposition induces a bigrading on the Chow ring via $\A^i_{(j)}(X) := (\pi_X^{2i-j})_\ast \A^i(X)$.

The known landscape is mixed: hyperelliptic curves admit MCK decompositions (via the Gross–Schoen vanishing of the modified diagonal at a Weierstraß point), but very general curves of genus $\geq 3$ do not; K3 surfaces do, while certain high-degree surfaces in $Z_Y \in \A^4(Y\times Y)$0 do not. Prior work had established affirmative answers for cubic threefolds, intersections of two quadrics, quadric-cubic intersections, quartic and sextic double solids, and prime Fano threefolds of genus 8 and 10. The genus 7 case treated here is thus the first negative instance among Picard-number-one Fano threefolds, and it requires an obstruction beyond Beauville's original one (which relies on non-injectivity of $Z_Y \in \A^4(Y\times Y)$1 and vanishes when $Z_Y \in \A^4(Y\times Y)$2).

The dual curve and motivic relation

The proof rests on Mukai's model: a prime Fano threefold $Z_Y \in \A^4(Y\times Y)$3 of genus 7 ($Z_Y \in \A^4(Y\times Y)$4) is a dimensionally transverse codimension-7 linear section of the spinor tenfold $Z_Y \in \A^4(Y\times Y)$5, with Hodge numbers $Z_Y \in \A^4(Y\times Y)$6 and $Z_Y \in \A^4(Y\times Y)$7. Its projectively dual construction yields a smooth curve $Z_Y \in \A^4(Y\times Y)$8 of genus 7, the dual curve; general curves of genus 7 arise this way (precisely those with no $Z_Y \in \A^4(Y\times Y)$9).

The paper assembles the rich dictionary between $\A^4(Y\times Y)$0 and $\A^4(Y\times Y)$1: equality of the Jacobian of $\A^4(Y\times Y)$2 with the intermediate Jacobian of $\A^4(Y\times Y)$3; homological projective duality giving $\A^4(Y\times Y)$4; identifications of the surface of conics with $\A^4(Y\times Y)$5 and the Hilbert scheme of cubics with $\A^4(Y\times Y)$6; and moduli-theoretic descriptions via rank-2 stable sheaves. Crucially, the author adds a motivic statement:

$\A^4(Y\times Y)$7

in the category of Chow motives with rational equivalence. Three independent proofs are given: upgrading the homological motive isomorphism via Kimura finite-dimensionality; the Bloch–Srinivas argument from the HPD-induced isomorphism $\A^4(Y\times Y)$8; and a direct geometric argument using the Iliev–Markushevich flop diagram relating $\A^4(Y\times Y)$9 to the blow-up of a quadric along $\{\pi_X^i\}$0, combined with uniqueness of the even/odd decomposition of finite-dimensional motives.

Franchetta properties

The technical engine is a pair of Franchetta-type results for the universal family $\{\pi_X^i\}$1 of smooth linear sections of $\{\pi_X^i\}$2. First, $\{\pi_X^i\}$3 has the full Franchetta property: generically defined cycles inject into cohomology, using the fact that $\{\pi_X^i\}$4 (indeed $\{\pi_X^i\}$5). Second, generically defined cycles on $\{\pi_X^i\}$6 are exactly generated by pullbacks of $\{\pi_X^i\}$7 from the factors together with the diagonal:

$\{\pi_X^i\}$8

proved via the "stratified projective bundle" formalism of Fu–Laterveer–Vial applied to the very ample polarization of $\{\pi_X^i\}$9, plus triviality of the Chow groups of $X$0. This gives the Franchetta property for $X$1 in codimension $X$2; the main theorem shows it fails in codimension 4. A relative version of the HPD Fourier–Mukai kernel (existing over $X$3 by base-change compatibility) ensures the correspondences inducing the motivic equivalence are themselves generically defined, yielding injections $X$4 compatible with cycle class maps.

The main theorem

The explicit cycle is

$X$5

with $X$6. It is homologically trivial by direct computation of its action on cohomology. Suppose $X$7 in $X$8. Then the description of $X$9 would force the full Franchetta property for $\Delta_X^{sm}$0, hence — through the generically defined motivic injection — also for $\Delta_X^{sm}$1. But then the Faber–Pandharipande cycle

$\Delta_X^{sm}$2

would vanish, contradicting Green–Griffiths' theorem that this cycle is non-zero for the very general curve of genus $\Delta_X^{sm}$3. Moreover, since the injection sends $\Delta_X^{sm}$4 to a non-zero multiple of $\Delta_X^{sm}$5 and $\Delta_X^{sm}$6 is Albanese trivial, $\Delta_X^{sm}$7 is Abel–Jacobi trivial. This makes $\Delta_X^{sm}$8 one of the few explicitly written-down Abel–Jacobi trivial, non-torsion cycles in the literature, alongside the Faber–Pandharipande cycle itself.

The obstruction to MCK follows because Kimura finite-dimensionality forces any CK decomposition of $\Delta_X^{sm}$9 to have even pieces of the form $$\pi_X^k \circ \Delta_X^{sm} \circ (\pi_X^i \times \pi_X^j) = 0$$0; multiplicativity and self-duality would force the specific form $$\pi_X^k \circ \Delta_X^{sm} \circ (\pi_X^i \times \pi_X^j) = 0$$1, placing $$\pi_X^k \circ \Delta_X^{sm} \circ (\pi_X^i \times \pi_X^j) = 0$$2 in $$\pi_X^k \circ \Delta_X^{sm} \circ (\pi_X^i \times \pi_X^j) = 0$$3, which injects into cohomology — forcing $$\pi_X^k \circ \Delta_X^{sm} \circ (\pi_X^i \times \pi_X^j) = 0$$4, a contradiction. Hence no MCK decomposition exists for very general $$\pi_X^k \circ \Delta_X^{sm} \circ (\pi_X^i \times \pi_X^j) = 0$$5 of genus 7.

Modulo algebraic equivalence

The picture changes completely after quotienting by algebraic equivalence. For any smooth projective threefold with $$\pi_X^k \circ \Delta_X^{sm} \circ (\pi_X^i \times \pi_X^j) = 0$$6 — in particular any Fano threefold — the Bloch–Srinivas decomposition-of-the-diagonal argument provides a split motive injection into $$\pi_X^k \circ \Delta_X^{sm} \circ (\pi_X^i \times \pi_X^j) = 0$$7; tensoring three times and using coincidence of algebraic and homological equivalence for zero-cycles and divisors kills the Griffiths group $$\pi_X^k \circ \Delta_X^{sm} \circ (\pi_X^i \times \pi_X^j) = 0$$8, which suffices for multiplicativity modulo algebraic equivalence. The argument extends to odd-dimensional varieties whose Chow groups vanish below the middle dimension.

Consequently, the tautological algebra $$\pi_X^k \circ \Delta_X^{sm} \circ (\pi_X^i \times \pi_X^j) = 0$$9 injects into cohomology for all $i+j \neq k$0, equivalent (for Picard number 1) to having an MCK decomposition modulo algebraic equivalence. This stands in sharp contrast to curves: the very general genus 7 curve has no MCK decomposition even modulo algebraic equivalence, by the Ceresa cycle. The upshot is a precise failure of compatibility: the motivic injection $i+j \neq k$1 does not send $i+j \neq k$2 into $i+j \neq k$3, since the Gross–Schoen modified diagonal lies in $i+j \neq k$4, is non-zero modulo algebraic equivalence, yet its image must be cohomologically trivial while $i+j \neq k$5 injects into cohomology. The motivic equivalence between $i+j \neq k$6 and $i+j \neq k$7 therefore does not respect multiplicative structures.

Limitations and open questions

Several qualifications are in order. The negative result holds only for the very general genus 7 Fano threefold; the paper does not determine whether special members of the family (e.g., those with extra automorphisms or special moduli) admit MCK decompositions. The proof is indirect, proceeding by contradiction through the Faber–Pandharipande cycle rather than by computing $i+j \neq k$8 itself; in particular, the full structure of $i+j \neq k$9 remains unexplored, as does the Abel–Jacobi image of $\A^i_{(j)}(X) := (\pi_X^{2i-j})_\ast \A^i(X)$0. The paper leaves open whether other families of prime Fano threefolds of Picard number 1 fail MCK, and whether the genus 7 obstruction can be formulated intrinsically without reference to the dual curve. Finally, the conjectural vanishing $\A^i_{(j)}(X) := (\pi_X^{2i-j})_\ast \A^i(X)$1 for $\A^i_{(j)}(X) := (\pi_X^{2i-j})_\ast \A^i(X)$2 and $\A^i_{(j)}(X) := (\pi_X^{2i-j})_\ast \A^i(X)$3 for varieties with MCK decompositions is invoked only as background and is not addressed here.

Conclusion

The paper settles the genus 7 case of the MCK question for Picard-number-one Fano threefolds negatively, via an explicit Abel–Jacobi trivial, non-torsion cycle on $\A^i_{(j)}(X) := (\pi_X^{2i-j})_\ast \A^i(X)$4 whose non-triviality is transported from the Faber–Pandharipande cycle on the dual curve through a generically defined motivic equivalence. Combined with the positive result that all Fano threefolds admit MCK decompositions modulo algebraic equivalence, the work delineates precisely where the multiplicative structure of the Chow ring breaks down under rational equivalence while surviving under the coarser equivalence relation — and identifies the genus 7 Fano/curve duality as a setting where motivic equivalence and multiplicative compatibility diverge.