Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls

Evans, Jonathan D.; Urzúa, Giancarlo

Annales de l'Institut Fourier, Tome 71 (2021) no. 5, pp. 1807-1843.

La fibre de Milnor d’un lissage $\mathbb{Q}$-Gorenstein d’une singularité de Wahl est une boule d’homologie rationelle $B_{p,q}$. Si $X$ est une surface de type général polarisée canoniquement, l’ensemble des entiers $p$ pour lesquels il existe un plongement symplectique de $B_{p,q}$ dans $X$ est borné. Dans cet article, nous montrons comment construire une suite non-bornée de boules d’homologie rationnelles plongées symplectiquement dans des surfaces de type général munies de formes symplectiques non-canoniques. Ces plongements proviennent de la théorie de Mori sur les flips, mais nous les interprétons en termes de structures presque toriques et de mutations de polygones. Un flip de surfaces tel que ceux étudiés par Hacking, Tevelev et Urzúa peut être décomposé en une succession de mutations de structure presque torique et de déformations de la forme symplectique.

The Milnor fibre of a $\mathbb{Q}$-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.

Classification : 14J29, 14J17, 53D35
Keywords: Singularities, MMP, symplectic geometry, almost toric manifolds
Mot clés : Singularités, MMP, géométrie symplectique, variétés presque toriques

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