Equivariant embeddings of rational homology balls

Brendan Owens
University of Glasgow

Date(s) : 14/01/2019   iCal
14 h 00 min - 15 h 00 min

I will describe a family of smooth rational homology 4-balls B_{p,q}, bounded by lens spaces; these are the Milnor fibres of Wahl singularities. Smooth embeddings of B_{p,q} in other 4-manifolds arise in several contexts, for example in smoothings of singular projective varieties or in the rational blowdown surgery of Fintushel-Stern and J. Park. Khodorovskiy used Kirby calculus to give existence theorems for such embeddings, and these were generalised by Park-Park-Shin using methods from the minimal model program for algebraic 3-folds. Evans-Smith gave a beautiful result showing that the symplectic embeddings of B_{p,q} in the complex projective plane correspond to solutions of the Markov equation; this coincides precisely with a classification by Hacking-Prokhorov of projective surfaces with quotient singularities which admit a CP^2 smoothing.

I will describe a method for embedding B_{p,q} by taking double branched covers of surfaces in the 4-ball. This recovers the results of Khodorovskiy and Park-Park-Shin and also gives some interesting new embeddings. In particular we find that there are more smooth embeddings in CP^2 than those of Hacking-Prokhorov and Evans-Smith.


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