Multiplicativity of Tautological Chow Projections

Definitions and mathematical statement

Let $\mathcal A_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$, as described in the supplied record. Write $\mathrm{CH}^{*}(\mathcal A_g)$ for its graded Chow ring, with intersection product. The base, coefficient ring, stack versus coarse-space convention, and intended range of $g$ are not specified in the supplied excerpt.

Let $T_g\subseteq\mathrm{CH}^{*}(\mathcal A_g)$ denote the subspace called tautological in the source, and let \[ \pi_g:\mathrm{CH}^{*}(\mathcal A_g)\longrightarrow T_g \] denote the tautological projection referred to as equation $(1)$ there. The supplied record does not include that equation, a definition of $T_g$, or the construction characterizing $\pi_g$. Thus these symbols refer to the intended source construction; they do not specify a uniquely determined projection from the supplied evidence alone. A projection onto a subspace is understood to restrict to the identity on that subspace.

Question, conditional on the missing source definitions. With the inherited intersection product on $T_g$, is $\pi_g$ a ring homomorphism? Its multiplicativity condition is \[ \pi_g(\alpha\cdot\beta)=\pi_g(\alpha)\cdot\pi_g(\beta) \qquad\text{for all }\alpha,\beta\in\mathrm{CH}^{*}(\mathcal A_g). \] Calling this a ring homomorphism presupposes that $T_g$ is closed under the inherited product and that the projection has the appropriate additive structure. If ring homomorphisms are required to preserve the identity, the condition also includes $\pi_g(1)=1$; this follows from the projection property if $1\in T_g$. These structural conventions require confirmation from the omitted construction. The record also does not specify whether the question concerns a fixed $g$ or every $g$ in an intended range.

The surrounding excerpt describes joint work with R. Pandharipande and H.-H. Tseng on maps $C\to E$, where $C$ has genus $g$, $E$ is an elliptic curve, and both curves vary in moduli. It mentions three approaches connected to three open problems: intersection theory on $\mathcal A_g$, Gromov--Witten theory of the universal elliptic curve $\mathcal E\to\mathcal M_{1,1}$, and genus-one Gromov--Witten theory of the Hilbert scheme of points in ${\mathbb{C}}^2$. This context supplies no further definition of the projection in the present record.

Short English statement

The question asks whether projecting Chow classes on $\mathcal A_g$ onto the source's tautological subspace preserves multiplication. The defining projection and several domain conventions are omitted, so this is a conditional statement requiring the original construction. The supplied dated curation labels the problem partially solved; that assessment is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006276$, title Multiplicativity of Tautological Chow Projections, exact source problem code catalogue entry. The record gives proposed year 2025 and identifies no proposer.
• Recent Trends in Algebraic Geometry (2025), Oberwolfach Reports, pp. 1451--1516, DOI 10.4171/owr/2025/28. Supplied report reference: https://doi.org/10.4171/owr/2025/28. The excerpt calls this Question 1, refers to a projection in equation $(1)$, and says that the question was introduced in reference [3]. Neither equation $(1)$ nor the bibliographic identity of [3] is supplied.
• Supplied literature source: Iribar López, Noether--Lefschetz cycles, https://doi.org/10.1017/fmp.2025.10018.
• Supplied literature source labelled "original OWR report": https://doi.org/10.4171/owr/2025/28.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that the general multiplicativity question remains unresolved, while Iribar López proves the homomorphism property in several cases and for a natural subring generated by Noether--Lefschetz data. This is supplied dated curation, not an independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims. The cases and subring are not specified in the supplied evidence.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked, the question isolated, and its notation repaired. These are supplied source-verification claims, not verification performed for this draft. The surrounding excerpt attributes the enumerative work to joint work with R. Pandharipande and H.-H. Tseng without identifying the speaker in the supplied text.