Holomorphic Symplectic Geometry — Conjecture 10

Definitions and mathematical statement

Unresolved scope. The supplied excerpt does not state the hypotheses on $X$ or the coefficient convention for $CH(X)$. In particular, it does not specify whether $X$ must be projective, irreducible holomorphic symplectic, or satisfy other conditions from the surrounding source. The formulation below is therefore a statement schema with an explicitly conditional coefficient interpretation.

Under the integral Chow interpretation, write $CH^*(X)$ for the Chow ring of algebraic cycles modulo rational equivalence with integer coefficients. For an intended smooth complex algebraic variety $X$, a line bundle $D\in\operatorname{Pic}(X)$ determines $c_1(D)\in CH^1(X)$; interpret $c_i(X)$ as the Chern class $c_i(T_X)\in CH^i(X)$ of the tangent bundle. Write \[ \operatorname{cl}_X\colon CH^*(X)\longrightarrow\bigoplus_{r\geq 0}H^{2r}(X,\mathbb Z) \] for the integral cycle-class map, and let \[ R_X=\mathbb Z\left[c_1(D),c_i(T_X)\mid D\in\operatorname{Pic}(X),\ i\geq 1\right]\subseteq CH^*(X) \] be the subring generated by divisor classes and tangent-bundle Chern classes.

Conjectural statement schema (Beauville, Conjecture 10). For every $X$ satisfying the intended, presently unspecified source hypotheses, every finite list $D_1,\ldots,D_k\in\operatorname{Pic}(X)$, and every class $z\in CH^*(X)$ that is a polynomial in $c_1(D_1),\ldots,c_1(D_k)$ and the classes $c_i(T_X)$, \[ \operatorname{cl}_X(z)=0\text{ in }H^*(X,\mathbb Z) \quad\Longrightarrow\quad z=0\text{ in }CH^*(X). \] Under the integral interpretation above, this says that $\operatorname{cl}_X|_{R_X}$ is injective. The missing hypotheses on $X$ and the intended Chow and polynomial coefficients must be confirmed before this schema becomes a fully specified conjecture; the integral cohomology target is retained exactly as supplied.

Short English statement

Beauville's supplied conjecture says that a polynomial in divisor classes and the Chern classes of $X$ vanishes modulo rational equivalence whenever its integral cohomology class vanishes. The excerpt leaves the permitted varieties $X$ and the Chow coefficient convention unspecified, so these require review.

Sources

• Local UnsolvedMath registry: permanent ID $800201$, exact source problem code catalogue entry, title Holomorphic Symplectic Geometry --- Conjecture 10. The supplied record attributes the conjecture to Beauville, with proposed year 2010.
• Supplied source: Beauville, Holomorphic Symplectic Geometry: A Problem List (2010), Conjecture 10, https://arxiv.org/abs/1002.4321. The record reports extraction method arxiv_latex and access date 2026-10-16; these are supplied provenance claims, not independently verified access or source verification.
• The local record labels the problem open and records an update on 2026-10-16. This is supplied registry status, not an independently verified mathematical or literature assessment. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.