  In this paper, we study deformations of nonsingular Poisson varieties,
deformations of Poisson invertible sheaves and simultaneous deformations of
nonsingular Poisson varieties and Poisson invertible sheaves, which extend flat
deformation theory of nonsingular varieties and invertible sheaves. In an
appendix, we study deformations of Poisson vector bundles. We identify
first-order deformations and obstructions.
