Chern-Class Generators for Wonderful Compactifications

Definitions and mathematical statement

Let $G/H$ be an adjoint symmetric space and let $X$ be its wonderful compactification. In the usual complex-algebraic setting, $G$ is a connected semisimple group of adjoint type, the symmetric subgroup is attached to an involution of $G$, and $X$ is a smooth projective $G$-variety with open orbit $G/H$. Its boundary is a simple-normal-crossings union of smooth $G$-stable divisors, and the $G$-orbit closures are their appropriate intersections. The exact choice of fixed-point subgroup versus its normalizer in the term "adjoint symmetric space" is not specified in this record and must be fixed by the source convention.

The question asks for explicit Chern-class generators of the ordinary rational cohomology ring \[ H^*(X({\mathbb{C}}),{\mathbb{Q}}). \] Concretely, one seeks geometrically specified algebraic vector bundles $E_1,\ldots,E_s$ on $X$ and specified Chern classes $c_j(E_a)\in H^{2j}(X({\mathbb{C}}),{\mathbb{Q}})$ generating this graded ${\mathbb{Q}}$-algebra. Equivalently, for the chosen bundles and indices, the homomorphism \[ {\mathbb{Q}}[t_{a,j}]\longrightarrow H^*(X({\mathbb{C}}),{\mathbb{Q}}), \qquad t_{a,j}\longmapsto c_j(E_a),\qquad \deg t_{a,j}=2j, \] should be surjective. "Explicit" requires identifying the bundles and classes from the geometry or symmetric-space data; mere existence of some finite vector-space basis is not the requested description. Relations among the generators may help, but a full presentation is not explicitly demanded.

The source does not restrict the question to minimal rank or to a particular symmetric pair. Ordinary rational cohomology, equivariant cohomology, and equivariant Chow rings are distinct targets. The supplied literature assessment checked 21 August 2026 reports substantial explicit formulas in minimal-rank and other wonderful-compactification cases, but describes the general explicit generating-set problem as unresolved. Those special and equivariant results are not silently promoted to the unrestricted ordinary-cohomology statement.

Short English statement

Identify explicit vector bundles whose Chern classes generate the rational cohomology of a wonderful compactification of an adjoint symmetric space. The supplied assessment records substantial special-case and equivariant formulas, while retaining the general generating-set question.

Sources

• Local UnsolvedMath registry, permanent ID 30000120, exact source code catalogue entry; Chern-Class Generators for Wonderful Compactifications. Proposed year 2004, status partially_solved; corrected_verified extraction removes only a redundant problem label.
• Supplied report citation: Komplexe Analysis (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/42, pp. 2171--2216, https://doi.org/10.4171/owr/2004/42. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21: "Brion--Joshua", https://arxiv.org/abs/0705.1035; "Strickland", https://doi.org/10.1016/j.jalgebra.2006.01.021; "OWR report", https://doi.org/10.4171/owr/2004/42. This dated OWR-derived curation requires independent verification of time-sensitive status and cited claims.