We prove that if a contact manifold (M,ξ) is supported by a planar open
book, then Euler characteristic and signature of any Stein filling of (M,ξ)
is bounded. We also prove a similar finiteness result for contact manifolds
supported by spinal open books with planar pages. Moving beyond the geography
of Stein fillings, we classify fillings of some lens spaces.Comment: Minor typos fixe