The universal Poisson deformation space of hypertoric varieties and its applications

2019/01/25 Fri 10:30 - 12:00
長岡 高広

Recently, (holomorphic) symplectic varieties are extensively studied in algebraic geometry and geometric representation theory. In this talk, we will focus on hypertoric varieties among symplectic varieties. A hypertoric variety is an analogue of toric variety, and we can study its geometry from the associated combinatorial object, hyperplane arrangements (like polytopes in toric geometry). Classically, Brieskorn—Grothendiek described the simultaneous resolution of semiuniversal deformation of ADE type surface singularity. Recently, Namikawa generalized this to the existence of universal (Poisson) deformation (UPD) for all (conical) symplectic varieties. Since this is the abstract existence theorem, the concrete construction of UPD for each symplectic varieties is important. In this talk, we will describe UPD for hypertoric varieties concretely. In first application, we will classify affine hypertoric varieties by the associated combinatorial objects. On the other hand, in general, it is known that UPD (precisely, its associated Weyl group) encodes some information on all projective crepant resolutions of symplectic varieties. So, in second application, we will count the number of different crepant resolutions of some hypertoric varieties.