On Symplectic Fillings of Lens Spaces

Let C be the contact structure naturally induced on the lens space L(p,q) by the standard contact structure on the three–sphere. We obtain a complete classification of the symplectic fillings of (L(p,q),C) up to orientation-preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorphism types of the smoothings of complex two-dimensional cyclic quotient singularities.