Abstract:

I will discuss two applications of Hamiltonian reductions in shifted symplectic settings.

Part I — Shifted symplectic rigidification (with H. Park).

For BGm-actions on non-positively shifted symplectic stacks, we construct symplectic rigidifications (Hamiltonian BGm-reductions) and show their functoriality under equivariant Lagrangian correspondences.

Rigidification removes scalar Gm automorphisms and yields shifted symplectic structures on the moduli schemes of stable sheaves on Calabi-Yau varieties, including rank-zero components.

As an application, we obtain reduced Donaldson—Thomas theory (DT4) on compact Calabi-Yau fourfolds via (-2)-shifted symplectic schemes and establish new reduced pairs/sheaves correspondences, including cases conjectured by Cao-Oberdieck-Toda.

Part II — Local structure theorems for 0-shifted symplectic stacks (with Y.-H. Kiem and H. Park, in progress).

We give a local structure theorem: at points with reductive stabilizers, any 0-shifted symplectic derived Artin stack is formally locally a Hamiltonian reduction of a symplectic, formally smooth formal scheme.

Étale-local variants hold assuming only non-degeneracy, without requiring closedness of the 2-forms.

These models provide a uniform framework for analyzing singularities of “singular symplectic” moduli, e.g., of sheaves/complexes on K3 surfaces.

Presentation time : 15:00-16:00.

ZOOM address : 828 570 1072.