Euler-Characteristic Growth of Stable Quiver Moduli

Definitions and mathematical statement

Fix a quiver $Q$ and generic stability $\Theta$, and write $M_d^{\mathrm{st}}(Q)$ for the moduli space of $\Theta$-stable representations of dimension vector $d$. In a standard complex slope convention, for nonzero $e$ put $\mu_\Theta(e)=\Theta(e)/\sum_i e_i$; stability requires smaller slope for every proper nonzero subrepresentation, and $d$ is $\Theta$-coprime if no proper nonzero coordinate subvector has the same slope as $d$. The source's coefficient field, sign and genericity conventions are not supplied, so this explains one usual setting rather than selecting all omitted hypotheses.

Let $\chi$ denote the intended Euler characteristic (in the complex setting, the topological alternating Betti-number sum). The exact proposed asymptotic is \[ \log\chi\left(M_d^{\mathrm{st}}(Q)\right) \sim C_Q\left(\dim M_d^{\mathrm{st}}(Q)-1\right) \] for large $\Theta$-coprime dimension vectors, with $C_Q\in{\mathbb{R}}$. The factor on the right is dimension minus one; no square root, different normalization or alternative growth law is substituted.

For the usual relation-free finite-quiver stable quotient, when the stable locus is nonempty with scalar stabilizers, its dimension is \[ \dim M_d^{\mathrm{st}}(Q)=1-\langle d,d\rangle_Q, \qquad \langle d,d\rangle_Q=\sum_i d_i^2-\sum_{a:i\to j}d_id_j. \] The asymptotic still requires a specified family of dimension vectors: norm tending to infinity, slopes approaching a ray, and uniformity over directions are different statements. Exact positive multiples of one vector need not stay $\Theta$-coprime. The source specifies none of these limiting conventions.

Further, the real logarithm requires positive Euler characteristic; nonempty stable moduli, positivity, and eventual nonzero right side are not explicitly assumed. Standard $f\sim g$ means $f/g\to1$ and is not defined with an identically zero comparison function, so allowing $C_Q=0$ requires clarification rather than silently changing the claim to a quotient limit. Whether the constant may depend on $\Theta$ or the direction of $d$, despite its notation $C_Q$, is also unspecified.

The supplied literature triage checked 21 August 2026 reports substantial localization/degeneration special cases and bounds but says the stated general asymptotic has not been verified. Those results are not used here to certify this exact formula or fill its missing positivity and limit hypotheses.

Short English statement

Does the logarithm of the Euler characteristic of stable quiver moduli grow like a constant times their dimension minus one? The source leaves the large-vector regime, positivity and constant dependence unclear, and permits a zero constant incompatible with ordinary ratio asymptotics. Its supplied special-case results do not verify the general stated formula.

Sources

• Local UnsolvedMath registry, permanent ID 30000485, exact source code catalogue entry; Euler-Characteristic Growth of Stable Quiver Moduli. Proposed year 2006, status partially_solved; corrected_verified record retains the exact logarithm and dimension-minus-one asymptotic.
• Supplied report citation: Interactions between Algebraic Geometry and Noncommutative Algebra (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/23, pp. 1319--1384, https://doi.org/10.4171/owr/2006/23. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21: "original OWR report 23/2006", https://doi.org/10.4171/owr/2006/23; Weist, Localization in quiver moduli spaces, https://arxiv.org/abs/0903.5442; Weist, Euler characteristic of Kronecker moduli spaces, https://arxiv.org/abs/1203.2740. This dated OWR-derived curation requires independent verification of time-sensitive status and cited claims.