Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls
[Antiflips, mutations, et plongements symplectiques des boules d’homologie rationelles]
Annales de l'Institut Fourier, Online first, 37 p.

La fibre de Milnor d’un lissage -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 -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.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : https://doi.org/10.5802/aif.3429
Classification : 14J29,  14J17,  53D35
Mots clés : Singularités, MMP, géométrie symplectique, variétés presque toriques
@unpublished{AIF_0__0_0_A35_0,
     author = {Evans, Jonathan D. and Urz\'ua, Giancarlo},
     title = {Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2021},
     doi = {10.5802/aif.3429},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Evans, Jonathan D.
AU  - Urzúa, Giancarlo
TI  - Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls
JO  - Annales de l'Institut Fourier
PY  - 2021
DA  - 2021///
PB  - Association des Annales de l’institut Fourier
N1  - Online first
UR  - https://doi.org/10.5802/aif.3429
DO  - 10.5802/aif.3429
LA  - en
ID  - AIF_0__0_0_A35_0
ER  - 
%0 Unpublished Work
%A Evans, Jonathan D.
%A Urzúa, Giancarlo
%T Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls
%J Annales de l'Institut Fourier
%D 2021
%I Association des Annales de l’institut Fourier
%Z Online first
%U https://doi.org/10.5802/aif.3429
%R 10.5802/aif.3429
%G en
%F AIF_0__0_0_A35_0
Evans, Jonathan D.; Urzúa, Giancarlo. Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls. Annales de l'Institut Fourier, Online first, 37 p.

[1] Bhupal, Mohan; Ono, Kaoru Symplectic fillings of links of quotient surface singularities, Nagoya Math. J., Volume 207 (2012), pp. 1-45 corrigendum in ibid. 225 (2017), p. 207-212 | Article | MR 2957141 | Zbl 1258.53088

[2] Evans, Jonathan D.; Smith, Ivan Markov numbers and Lagrangian cell complexes in the complex projective plane, Geom. Topol., Volume 22 (2018) no. 2, pp. 1143-1180 | Article | MR 3748686 | Zbl 1381.53159

[3] Evans, Jonathan D.; Smith, Ivan Bounds on Wahl singularities from symplectic topology, Algebr. Geom., Volume 7 (2020) no. 1, pp. 59-85 | Article | MR 4038404 | Zbl 1458.14054

[4] Fine, Joel; Panov, Dmitri The diversity of symplectic Calabi–Yau 6-manifolds, J. Topol., Volume 6 (2013) no. 3, pp. 644-658 | Article | MR 3100885 | Zbl 1288.14028

[5] Galkin, Sergey; Usnich, Alexandr Mutations of potentials (2010) (preprint IPMU, 10-0100)

[6] Hacking, Paul; Tevelev, Jenia; Urzúa, Giancarlo Flipping surfaces, J. Algebr. Geom., Volume 26 (2017) no. 2, pp. 279-345 | Article | MR 3606997 | Zbl 1360.14055

[7] Khodorovskiy, Tatyana Bounds on Embeddings of Rational Homology Balls in Symplectic 4-manifolds (2013) (https://arxiv.org/abs/1307.4321)

[8] Khodorovskiy, Tatyana Smooth embeddings of rational homology balls, Topology Appl., Volume 161 (2014), pp. 386-396 | Article | MR 3132378 | Zbl 1408.57030

[9] Lee, Yongnam; Park, Jongil A simply connected surface of general type with p g =0 and K 2 =2, Invent. Math., Volume 170 (2007) no. 3, pp. 483-505 | Article | MR 2357500 | Zbl 1126.14049

[10] Lekili, Yankı; Maydanskiy, Maksim The symplectic topology of some rational homology balls, Comment. Math. Helv., Volume 89 (2014) no. 3, pp. 571-596 | Article | MR 3260842 | Zbl 1315.53098

[11] Lisca, Paolo On symplectic fillings of lens spaces, Trans. Am. Math. Soc., Volume 360 (2008) no. 2, pp. 765-799 | Article | MR 2346471 | Zbl 1137.57026

[12] Looijenga, Eduard; Wahl, Jonathan Quadratic functions and smoothing surface singularities, Topology, Volume 25 (1986) no. 3, pp. 261-291 | Article | MR 842425 | Zbl 0615.32014

[13] Mori, Shigefumi On semistable extremal neighborhoods, Higher dimensional birational geometry (Kyoto, 1997) (Advanced Studies in Pure Mathematics), Volume 35, Mathematical Society of Japan, 2002, pp. 157-184 | Article | MR 1929794 | Zbl 1066.14018

[14] Nguyen Tien Zung Symplectic topology of integrable Hamiltonian systems. II. Topological classification, Compos. Math., Volume 138 (2003) no. 2, pp. 125-156 | Article | MR 2018823 | Zbl 1127.53308

[15] Owens, Brendan Equivariant embeddings of rational homology balls, Q. J. Math, Volume 69 (2018) no. 3, pp. 1101-1121 | Article | MR 3859226 | Zbl 1405.57007

[16] Park, Heesang; Park, Jongil; Shin, Dongsoo Smoothly embedded rational homology balls, J. Korean Math. Soc., Volume 53 (2016) no. 6, pp. 1293-1308 | Article | MR 3570974 | Zbl 1361.57036

[17] Park, Heesang; Shin, Dongsoo Rational homology balls in 2-handlebodies, Bull. Korean Math. Soc., Volume 54 (2017) no. 6, pp. 1927-1933 | MR 3733773 | Zbl 1420.57075

[18] Rana, Julie A boundary divisor in the moduli spaces of stable quintic surfaces, Int. J. Math., Volume 28 (2017) no. 4, 1750021, 61 pages | Article | MR 3639040 | Zbl 1386.14144

[19] Rana, Julie; Urzúa, Giancarlo Optimal bounds for T-singularities in stable surfaces, Adv. Math., Volume 345 (2019), pp. 814-844 | Article | MR 3901675 | Zbl 07021555

[20] Sepe, Daniele; Vũ Ngọc, San Integrable systems, symmetries, and quantization, Lett. Math. Phys., Volume 108 (2018) no. 3, pp. 499-571 | Article | MR 3765971 | Zbl 1390.37101

[21] Symington, Margaret Generalized symplectic rational blowdowns, Algebr. Geom. Topol., Volume 1 (2001), pp. 503-518 | Article | MR 1852770 | Zbl 0991.57027

[22] Symington, Margaret Four dimensions from two in symplectic topology, Topology and geometry of manifolds (Athens, GA, 2001) (Proceedings of Symposia in Pure Mathematics), Volume 71, American Mathematical Society, 2003, pp. 153-208 | Article | MR 2024634 | Zbl 1049.57016

[23] Urzúa, Giancarlo Identifying neighbors of stable surfaces, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 16 (2016) no. 4, pp. 1093-1122 | MR 3616327 | Zbl 1365.14054

[24] Vũ Ngọc, San On semi-global invariants for focus-focus singularities, Topology, Volume 42 (2003) no. 2, pp. 365-380 | Article | MR 1941440 | Zbl 1012.37041

[25] Wahl, Jonathan Smoothings of normal surface singularities, Topology, Volume 20 (1981) no. 3, pp. 219-246 | Article | MR 608599 | Zbl 0484.14012

Cité par Sources :