A concrete obstruction to resolution-independent products on intersection cohomology

Definitions and mathematical statement

T. Holm's Question $9.3$ asks whether the middle-perversity intersection cohomology of a singular Kähler quotient has a natural ring structure. "Natural" is not specified. A precise candidate requirement is independence of a chosen small resolution: for a proper small resolution $\pi:Y\to X$ from a smooth variety, use the normalized intersection-cohomology identification $T_\pi:IH^*(X;{\mathbb{Q}})\cong H^*(Y;{\mathbb{Q}})$ and transport cup product by \[ a\star_\pi b=T_\pi^{-1}(T_\pi(a)\smile T_\pi(b)). \] Small means that, for every $r>0$, the locus whose fiber dimension is at least $r$ has complex codimension greater than $2r$. The candidate asks whether these products agree for different small resolutions under their natural additive identifications, rather than merely whether the resulting rings are abstractly isomorphic.

Comment $9.4$, E. Lerman, attributes small-resolution constructions and induced products for circle reductions to E. Lerman--S. Tolman [LT00], noting their special circle-action setting. Comment $9.5$, R. Sjamaar, cites Kiem--Woolf's example with two small resolutions and concludes that a natural product appears obstructed. The supplied aid makes the sense of that discrepancy explicit.

For the Kiem--Woolf ${\mathbb{C}}^*$-action on $\mathbb P^4$ with weights $(0,1,1,-1,-1)$, take the singular quotient and its two small resolutions in the cited construction. In the aid's common rational intersection-cohomology model, \[ V=\operatorname{span}_{\mathbb{Q}}\{1,h,u,h^2,hu,h^3\},\qquad |h|=|u|=2, \] where the degree-four symbol $hu$ is a nonzero basis element, the two naturally transported products satisfy \[ u\star_+u=-h^2+2hu,\qquad u\star_-u=-h^2-2hu, \] \[ (u\star_+u)-(u\star_-u)=4hu\ne0. \] Thus cup product cannot be transported from an arbitrary small resolution in a resolution-independent way under these identifications. At the same time, the aid reports that the two abstract graded rings are isomorphic by $h\mapsto h$, $u\mapsto-u$. The source phrase "distinct cohomology rings" must therefore be understood as distinct naturally transported products here, not as abstract nonisomorphism.

The example rules out that particular resolution-independent construction. It does not prove that no product can be chosen on the vector space or that every conceivable enriched, chamber-dependent or differently natural construction is impossible. The exact quotient linearization and definitions of the common-model generators are not supplied in this record and require the cited example. The appended heading "$10$ Computations over ${\mathbb{Z}}$" belongs to the next section and supplies no further question.

Short English statement

Two small resolutions can induce different products on the same naturally identified intersection-cohomology space. The aid gives an explicit discrepancy $4hu$ in the Kiem--Woolf example, while the two rings remain abstractly isomorphic. This obstructs resolution-independent transport of cup product, rather than every possible notion of a natural product.

Sources

• Local UnsolvedMath registry, permanent ID $20001974$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[311].
• AIM workshop "Moment maps and surjectivity in various geometries," Question $9.3$, T. Holm; Comment $9.4$, E. Lerman; Comment $9.5$, R. Sjamaar: https://aimath.org/WWN/momentmaps/momentmaps.pdf. Page number $15$ and the Section $10$ heading are extraction artifacts or spillover. No PDF was opened in this session.
• E. Lerman and S. Tolman [LT00], small resolutions for $S^1$-reductions, without full title or publication data in the record.
• Kiem and Woolf, "Intersection cohomology of symplectic quotients by circle actions." The original comment says "to be published in J. London Math. Soc." and drops the initial I in extraction; the aid describes the example as published. No publication year or complete citation is supplied.
• Supplied research attempt $1$, partial_result, run status partially_solved, rigor label "proved," novelty confidence "low." Registry classification PARTIAL-PROGRESS, updated 14 August 2026. Original remarks and literature fields are empty; no proposer-field value or proposed year is supplied.