Deformations of hypertoric varieties and its applications

Hypertoric variety $Y(A, \alpha)$ 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 $\pi:Y(A, \alpha) \to Y(A, 0)$, and for generic $\alpha$, 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 $Y(A, \alpha)$ 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.