### On symplectic fillings of normal complex surface singularities

Date
06/17/2015 - 06/27/2015
Location
CAMP
Research Background and Goal In recent years, one of most active research areas in 4-manifold theory is to classify symplectic fillings of certain 3-manifolds equipped with a natural contact structure. Among them, people have long studied symplectic fillings of the link of a normal complex surface singularity. But the complete answer is far from reach. For example, P. Lisca classified symplectic fillings of cyclic quotient singularities whose corresponding link is lens space (Trans. of AMS, vol. 360 (2008)), and A. Nemethi and P.Popescu-Pampu identified the correspondence between the symplectic fillings in Lisca's classification and the Milnor fibers for cyclic quotient singularities (Proc. of LMS, vol. 101 (2010)). Furthermore, M. Bhupal and K. Ono tried to extend these results, so that they classified all possible symplectic fillings of quotient surface singularities (Nagoya Math. J., vol. 207 (2012)). In this project, we'd like to investigate the correspondence between the symplectic fillings in Bhupal-Ono's classification and the Milnor fibers of quotient surface singularities. Main Information Let (X, 0) be a germ of a normal surface singularity which is assumed to be embedded in (CN, 0). If B is a small ball centered at origin, then a small neighborhood X∩B of the singularity is contractible and is homeomorphic to the cone over its boundary L:= X∩∂B. The smooth compact 3-manifold L is called the link of the singularity. The main goal in the research of symplectic fillings of a normal surface singularity is to classify all possible symplectic 4-manifolds with the boundary L satisfying certain compatibility conditions with the natural contact structure on L. Note that the link L of a normal surface singularity carries a canonical contact structure which is also known as the Milnor fillable contact structure. On the other hand, the Milnor fiber of a smoothing of the singularity, i.e., a nearby fiber of a smoothing of the singularities, has a natural Stein (and hence symplectic) structure, and so it provides a natural example of such a symplectic filling. In this project we approach the symplectic fillings of a normal surface singularity in two different ways – A topological viewpoint and an algebro-geometric viewpoint. We first investigate this problem in topological aspect by using a blowing-up and a rational blow-down surgeries from rational surfaces. We also study this problem in algebro-geometric aspect by investigating minimal resolutions and P-resolutions of the singularity. And then, by comparing the results obtained from both sides, we identify the correspondence between the symplectic fillings in Bhupal-Ono's classification and the Milnor fibers of quotient surface singularities. Research Expected Effect The output of this project would solve a longstanding open problem on the symplectic fillings of quotient surface singularities. Furthermore, the technique we obtained by combining techniques from topology and algebraic geometry would be very powerful to study symplectic fillings, so that it will play a crucial role in the forthcoming study of 4-manifolds. Research Expected Result The result would be published in a high-ranked SCI journal.