ANTIFLIPS, MUTATIONS, AND UNBOUNDED SYMPLECTIC EMBEDDINGS OF RATIONAL HOMOLOGY BALLS
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2021
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Abstract
The Milnor fibre of a Q-Gorenstein smoothing of a Wahl singularity is a rational homology ball B-p,B-q. For a canonically polarised surface of general type X, it is known that there are bounds on the number p for which B-p,B-q admits a symplectic embedding into X. In this paper, we give a recipe to construct unbounded sequences of symplectically embedded B-p,B-q into surfaces of general type equipped with non-canonical symplectic forms. Ultimately, these symplectic embeddings come from Mori's theory of flips, but we give an interpretation in terms of almost tonic structures and mutations of polygons. The key point is that a flip of surfacns, as studied by Hacking, Tevelev and Urmia, can be formulated as a combination of mutations of an almost toric structure and deformation of the symplectic form.
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Singularities, MMP, symplectic geometry, almost toric manifolds