Symplectic rational homology balls and antiflips

Jonny Evans (University of Lancaster)

Monday 14th October 16:00-17:00 Maths 311B

Abstract

(Joint work with Giancarlo Urzua). The Milnor fibre of the cyclic quotient singularity 1/p^2(1,pq-1) is a symplectic rational homology ball B_{p,q}. You can show that if B_{p,q} embeds symplectically into a canonically polarised suface of general type then p is bounded above by an expression involving K_X^2. By contrast, if you allow other (noncanonical) symplectic forms then you can break this bound and find sequences of embeddings with p unbounded. Similar phenomena have been observed in the smooth category by many authors (including Khodorovskiy, Owens, Park, Park and Shin). There is a "magic" argument for this, which invokes Mori theory, but there is also a very concrete way to construct these embeddings using almost toric symplectic geometry. I will review the basics of almost toric geometry and explain how to use this to construct the symplectic embeddings.

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