Symplectic rational homology ball fillings of Seifert fibered spaces

We characterize when some small Seifert fibered spaces can be the convex boundaries of symplectic rational homology balls and give strong restrictions for others to bound such manifolds. In particular, we show that the only spherical $3$-manifolds, oriented as links of the corresponding quotient singularities, which admit symplectic rational homology ball fillings, are the lens spaces $L(p^2,pq-1)$ previously identified by Lisca. In a different direction, we provide evidence for Gompf’s conjecture that Brieskorn spheres do not bound Stein domains in ${\mathbb{C}}^2$. Finally, we establish restrictions on Lagrangian disk fillings of certain Legendrian knots in small Seifert fibered spaces.