Speaker: Takahiro Nagaoka Title: The universal covers of hypertoric varieties and Bogomolov's decomposition Abstract: Projective toric varieties have been extensively studied by using the associated polytopes. Similarly, hypertoric varieties can be studied by looking at the associated hyperplane arrangements. In this talk, we consider the (singular) universal covers of affine hypertoric varieties, and we show that taking the universal cover corresponds to taking the gsimplificationh of the associated hyperplane arrangement. As an application, we give a necessary and sufficient condition for the uniqueness of (holomorphic) symplectic structures on hypertoric varieties, and we establish the analogue of Bogomolovfs decomposition. Moreover, as a byproduct of this application, we can show that if two (affine or) smooth hypertoric varieties are $\mathbb{C}$-equivariant isomorphic then they are also (maximal hamiltonian torus) $T$-action equivariant. This implies that the combinatorial classification actually gives the $\mathbb{C}^*$-equivariant classification as just algebraic varieties.