| Does a given compact
3-manifold bound a rational homology disk (or QHD), i.e., a 4-manifold
with no non-trivial rational homology? This problem has only recently
been completely solved for lens spaces. The algebro-geometric analogue
is to determine normal surface singularities with a smoothing whose
Milnor fibre is a QHD, i.e., with Milnor number 0. We showed in the
early ’80’s that for cyclic quotient singularities (whose links are
lens spaces), these smoothings occur exactly for type p^2/pq−1 (0 <
q < p, (p, q) = 1). We found other examples as well; all were
weighted homogeneous rational singularities, and included 3
triply-infinite families, each based on one of the spherical triples
(3, 3, 3), (2, 4, 4), and (2, 3, 6). The search for “symplectic fillings” of 3-manifolds attracted interest of symplectic topologists in the last decade, motivated by Seiberg-Witten theory and “rational blow-down” of Fintushel-Stern. A 2008 paper of Stipsicz, Szabó, and Wahl (SSW) gave strong restrictions on the possible resolution dual graphs of surface singularities which could admit such a smoothing. A recent paper of Bhupal-Stipsicz used (SSW) to prove that for star-shaped resolution graphs (i.e., weighted homogeneous singularities), our old list of examples with QHD smoothings is complete. This list of surface singularities is mysterious, and not known to arise in any other context. Our recent theorem yields the unexpected result that for each example, one may choose the total space of the QHD smoothing (a 3-dimensional singularity) to be log-terminal. This is a very restrictive class of singularities which arises naturally in Mori theory, for study of questions about higher-dimensional non-singular projective varieties. The result implies that the smoothings themselves are “Q-Gorenstein”, an important property of smoothings first studied by Kollár and Shepherd-Barron in 1988. The method involves introducing a “graded discrepancy” for isolated weighted-homogeneous singularities. |