Milnor fibers and symplectic fillings of quotient surface singularities

Abstract

We determine a one-to-one correspondence between Milnor fibers and minimal symplectic fillings of a quotient surface singularity (up to diffeomorphism type) by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3-folds and Pinkham's negative weight smoothing. As by-products, we show that: -- Milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions. For this, we classify minimal symplectic fillings of a quotient surface singularity up to diffeomorphism. -- Any symplectic filling of a quotient surface singularity is obtained by a sequence of rational blow-downs from a special resolution (so-called the maximal resolution) of the singularity, which is an analogue of the one-to-one correspondence between the irreducible components of the reduced versal deformation space and the so-called $P$-resolutions of a quotient surface singularity.

arXiv comment58 pages, 28 figures; Added two subsections (4.3, 4.4) in Section 4; Modified slightly statements of Proposition 5.4, Theorem 5.5, Theorem 10.5, Corollary 11.5; Modified slightly the other parts to improve the exposition. To appear in Advances in Mathematics