The main result is a classification of the crepant partial resolutions of the nilpotent cone: the paper says that every one is represented by a Hamiltonian reduction, and that the correspondence with the relative interiors of cones in dominant-weight space preserves order. The same work says the associated base-changed universal Poisson deformations can be built as GIT quotients at the matching stability parameter.
This is a proof-based investigation of algebraic-geometric objects associated with a simply connected semisimple group over the complex numbers, together with a chosen maximal torus. Its subject is mathematical schemes, varieties, morphisms and quotient constructions rather than an empirical population.
One common space behind the constructions
The starting point is an identification of the Hamiltonian Cox space of the nilpotent cone with the affine closure of T*(G/U), the cotangent bundle of base affine space. That identification is used to compare quotient models as the stability parameter changes.
The technical engine is variation of geometric invariant theory, or GIT. In this argument, inclusions between the semistable loci—the portions admitted by a chosen stability rule—induce projective morphisms between quotient models. This is how changes in a parameter produce maps between the resulting varieties.
A classification by regions
The reductions are not presented merely as a selection of examples. The claimed correspondence is exhaustive: every crepant partial resolution of the nilpotent cone appears among them, with an order-preserving bijection to the relative interiors of cones in dominant-weight space. The geometry is therefore organized by regions of the parameter space, while the ordering among those regions tracks the ordering among the resolution models.
The same stability label is carried into deformation theory. For each crepant projective partial resolution, the base change of its universal Poisson deformation is realized as a GIT quotient of the affine cotangent-bundle closure at that same parameter. The result links the original resolution models and their deformation spaces within one quotient framework.
Several structural results fill out the picture. The zero fiber of the torus moment map is an integral Cohen-Macaulay scheme, and regular dominant weights form a GIT chamber for the torus action on that fiber.
A geometric map of the models
At the level of birational geometry, the real linearization map identifies the wall-and-chamber decomposition of the dominant-weight cone with the Mori fan of the Springer resolution. The paper also describes the Springer resolution as a relative Mori dream space over the nilpotent cone.
The partial-resolution varieties tied to parabolic subgroups have a similarly strong internal structure: they are normal local complete intersections, and therefore Gorenstein and Cohen-Macaulay, while also being Q-factorial.
What the preprint establishes
The supplied document is labeled arXiv:2608.25994v1 [math.AG] and dated 26 Aug 2026. Its scope is the stated simply connected semisimple setting over the complex numbers with the chosen torus; the result is a theorem-and-proof construction within that framework.
The acknowledgments disclose that a draft was checked by generative artificial intelligence and that minor errors were corrected. They also say the first author is supported by EPSRC grants EP-W013053-1 and EP-R034826-1, while the second author is supported by an AMS-Simons Travel Grant.
Paper data and sources
Original title: Crepant partial resolutions of the nilpotent cone via Hamiltonian reduction
Authors: Gwyn Bellamy, Tom Gannon
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text