Stringy Euler Criteria for Smooth Spherical Varieties

Definitions and mathematical statement

A normal irreducible variety $X$ is locally factorial if every local ring $\mathcal O_{X,x}$ is a unique factorization domain; equivalently, every Weil divisor on $X$ is Cartier. In the usual algebraically closed field convention, a spherical variety is a normal irreducible variety equipped with an action of a connected reductive algebraic group $G$ such that a Borel subgroup $B\subseteq G$ has a dense open orbit. The supplied record does not specify the ground field, its characteristic, or the acting group.

The intended meaning of a smoothness criterion is the equivalence \[ e_{\mathrm{st}}(X)=e(X) \quad\Longleftrightarrow\quad X\text{ is smooth}, \] asserted for every locally factorial spherical variety $X$ in the intended class. The problem asks to prove or disprove this universal equivalence.

The invariants require conventions and hypotheses that the record does not supply. For example, over ${\mathbb{C}}$, one possible standard convention for the ordinary Euler characteristic is \[ e(X)=\sum_{j\geq 0}(-1)^j\dim_{{\mathbb{Q}}}H_c^j(X({\mathbb{C}}),{\mathbb{Q}}). \] In the usual complex log-terminal setting, the stringy Euler characteristic is defined using a log resolution $f:Y\to X$ with simple normal crossing exceptional divisors $(E_i)_{i\in I}$. Assuming $K_X$ is ${\mathbb{Q}}$-Cartier and writing \[ K_Y=f^*K_X+\sum_{i\in I}a_iE_i, \qquad a_i>-1, \] set \[ E_J^{\circ}=\left(\bigcap_{j\in J}E_j\right) \setminus\bigcup_{i\in I\setminus J}E_i \quad(J\subseteq I), \qquad \bigcap_{j\in\varnothing}E_j=Y. \] With the empty product equal to $1$, the corresponding formula is \[ e_{\mathrm{st}}(X)=\sum_{J\subseteq I} e(E_J^{\circ})\prod_{j\in J}\frac{1}{a_j+1}. \] These conventions explain a possible precise interpretation; they are not additional hypotheses stated in the supplied source. In particular, the record does not specify whether completeness is intended or establish the applicable definition and domain of $e_{\mathrm{st}}$ for its entire quantified class. The universal statement therefore remains qualified by these unresolved formulation issues.

Short English statement

The problem asks whether a locally factorial spherical variety is smooth exactly when its stringy Euler characteristic equals its ordinary Euler characteristic. The supplied formulation leaves the ground field, Euler characteristic conventions, and hypotheses for defining the stringy invariant unspecified; these require mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002003$, source problem code catalogue entry, titled Stringy Euler Criteria for Smooth Spherical Varieties; Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2012; no individual proposer is supplied.
• Supplied report attribution: Enveloping Algebras and Geometric Representation Theory (2012), Oberwolfach Reports, pp. 733--809, DOI 10.4171/owr/2012/13. https://doi.org/10.4171/owr/2012/13. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: The Arc Space of Horospherical Varieties. https://arxiv.org/abs/1203.0671.
• Supplied literature reference: Stringy Invariants for Horospherical Varieties of Complexity One. https://arxiv.org/abs/1511.03852.
• Supplied literature assessment, checked on 2026-08-21: the equality is described as a smoothness criterion for locally factorial horospherical varieties and some further special classes, while the arbitrary locally factorial spherical case is described as remaining unproved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record labels the problem open and expressly requests independent verification of time-sensitive status and cited claims.