Abstract for the talk on 26.09.2018 (09:00 h)Seminar on Nonlinear Algebra
Takahiro Nagaoka (Kyoto University)
Deformations of hypertoric varieties and its applications
Hypertoric variety Y (A,α) is a (holomorphic) symplectic variety, which is deﬁned as a Hamiltonian reduction of complex vector space by torus action. This is an analogue of toric variety. Actually, its geometric properties can be studied through the associated hyperplane arrangements (instead of polytopes). By deﬁnition, there exists a projective morphism π : Y (A,α) → Y (A,0), and for generic α, this gives a crepant resolution of aﬃne hypertoric variety Y (A,0). In general, for a (conical) symplectic variety and its crepant resolution, Namikawa showed the existence of the universal Poisson deformation space of them. We construct the universal Poisson deformation space of hypertoric varieties Y (A,α) and Y (A,0). We will explain this construction. In application, we can classify aﬃne hypertoric varieties by the associated matroids. If time permits, we will also talk about applications to counting crepant resolutions of aﬃne hypertoric varieties. This talk is based on my master thesis.