Jump Loci for Motivic Complexity in Families

Definitions and mathematical statement

Let $f\colon X\to S$ be a smooth morphism of quasi-projective varieties. The source does not specify their ground field, and does not assume that $f$ is proper or projective. For a point $s\in S$, write \[ X_s=X\times_S\operatorname{Spec}\kappa(s), \qquad X_s^{\times n}=\underbrace{X_s\times_{\kappa(s)}\cdots\times_{\kappa(s)}X_s}_{n\text{ factors}} \quad(n\geq 1), \] where $\kappa(s)$ is the residue field. These are scheme-theoretic fibers; whether the intended comparison instead uses geometric fibers is unspecified. An algebraic cycle of codimension $r$ on a variety $Y$ is a finite integral linear combination of integral closed subvarieties of codimension $r$. This definition alone supplies neither an equivalence relation on cycles nor a measure of their abundance.

Source question. Describe the locus of points $s\in S$ for which the motive of $X_s$ is simpler than the motive of a very general fiber, in the stated sense that $X_s$ and its positive Cartesian powers contain more algebraic cycles.

Unresolved formulation. This is a research question whose exceptional locus is not precisely defined by the supplied record. A precise formulation requires a ground field and a choice of ordinary or geometric points; a category of motives appropriate to these possibly nonproper fibers; coefficients and an equivalence relation or realization for algebraic cycles; and a comparison that defines what it means to have more cycles. The source does not specify whether an increase in one power and one codimension suffices, or whether a stronger condition across powers is intended. It also does not specify a component of $S$, assumptions under which a very general fiber supplies a common comparison value, or the intended meaning of describing the locus. Over an appropriate uncountable algebraically closed field, a very general point usually means a point outside a countable union of proper closed subsets of a specified irreducible base; those hypotheses and the relevant exceptional subsets are not supplied here. Consequently, no numerical jump condition, algebraicity assertion, or fully quantified conjecture can be extracted without additional choices.

Short English statement

The question asks where fibers of a smooth family of quasi-projective varieties acquire additional algebraic cycles on themselves or their Cartesian powers, indicating a simpler motive than that of a very general fiber. The supplied record leaves the motivic framework, cycle comparison, and meaning of very general unspecified, so a precise exceptional locus requires further mathematical formulation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003196$, title Jump Loci for Motivic Complexity in Families, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified.
• Supplied report citation: Arithmetic Geometry (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/38, pp. 2171--2224. https://doi.org/10.4171/owr/2016/38. The record gives proposed year 2016 and the tag owr-year:2016; the supplied report title retains 2017. These bibliographic details have not been independently reconciled.
• Supplied literature source: Klingler Hodge locus conjectures. https://javier.fresan.perso.math.cnrs.fr/klingler.pdf. No further bibliographic identification is supplied.
• Supplied literature source: OWR Arithmetic Geometry report. https://doi.org/10.4171/owr/2016/38.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that algebraicity results describe Hodge loci for important pure variations, while a general motivic description of loci where fibers simplify remains unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification. It does not specify a theorem applicable to every smooth quasi-projective family in the question.
• The registry labels the statement corrected_verified, explaining that its mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This supplied editorial label does not establish that the unresolved choices above have been settled.