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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls
AU - Evans, Jonny
AU - Urzua, Giancarlo
PY - 2022/12/31
Y1 - 2022/12/31
N2 - The Milnor fibre of a Q-Gorenstein smoothing of a Wahl singularity is a rational homology ball B_{p,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,q} admits a symplectic embedding into X. In this paper, we give a recipe to construct unbounded sequences of symplectically embedded B_{p,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 toric structures and mutations of polygons. The key point is that a flip of surfaces, as studied by Hacking, Tevelev and Urzúa, can be formulated as a combination of mutations of an almost toric structure and deformation of the symplectic form.
AB - The Milnor fibre of a Q-Gorenstein smoothing of a Wahl singularity is a rational homology ball B_{p,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,q} admits a symplectic embedding into X. In this paper, we give a recipe to construct unbounded sequences of symplectically embedded B_{p,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 toric structures and mutations of polygons. The key point is that a flip of surfaces, as studied by Hacking, Tevelev and Urzúa, can be formulated as a combination of mutations of an almost toric structure and deformation of the symplectic form.
KW - Singularities
KW - MMP
KW - symplectic geometry
KW - almost toric manifolds
U2 - 10.5802/aif.3429
DO - 10.5802/aif.3429
M3 - Journal article
VL - 71
SP - 1807
EP - 1843
JO - Annales de L'Institut Fourier
JF - Annales de L'Institut Fourier
SN - 0373-0956
IS - 5
ER -