Holonomic poisson geometry of Hilbert schemes
File(s)
Published version
Author(s)
Matviichuk, Mykola
Pym, Brent
Schedler, Travis
Type
Journal Article
Abstract
We undertake a detailed study of the geometry of Bottacin’s Poisson structures on Hilbert schemes of points in Poisson surfaces, ie smooth complex surfaces equipped with an effective anticanonical divisor. We focus on three themes that, while logically independent, are linked by the interplay between (characteristic) symplectic leaves and deformation theory. Firstly, we construct the symplectic groupoids of the Hilbert schemes and develop the classification of their symplectic leaves, using the methods of derived symplectic geometry. Secondly, we establish local normal forms for the Poisson brackets, and combine them with a toric degeneration argument to verify that Hilbert schemes satisfy our recent conjecture characterizing holonomic Poisson manifolds in terms of the geometry of the modular vector field. Finally, using constructible sheaf methods, we compute the space of first-order Poisson deformations when the anticanonical divisor is reduced and has only quasihomogeneous singularities. (The latter is automatic if the surface is projective.) Along the way, we find a tight connection between the Poisson geometry of the Hilbert schemes and the finite-dimensional Lie algebras of affine transformations, which is mediated by syzygies. In particular, we find that the Hilbert scheme has a natural subvariety that serves as a global counterpart of the nilpotent cone, and we prove that the Lie algebras of affine transformations have holonomic dual spaces — the first such series of Lie algebras to be discovered.
Date Issued
2025-06-27
Date Acceptance
2024-06-28
Citation
Geometry and Topology, 2025, 29 (4), pp.2047-2103
ISSN
1465-3060
Publisher
Mathematical Sciences Publishers (MSP)
Start Page
2047
End Page
2103
Journal / Book Title
Geometry and Topology
Volume
29
Issue
4
Copyright Statement
© 2025 The Authors, under license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open.
License URL
Subjects
COHOMOLOGY
DEFORMATIONS
Mathematics
MODULES
Physical Sciences
POINTS
Science & Technology
SHEAVES
Publication Status
Published
Date Publish Online
2025-06-27