On fillings of contact links of quotient singularities. (English) Zbl 07824483
The broad goal of this paper is to study symplectic fillings of contact links of isolated quotient singularities in any dimension, with the primary tool of this investigation being symplectic cohomology in multiple flavors. The filling question for links of isolated singularities is interesting because these contact manifolds are always strongly fillable, but links of singularities which are not smoothable need not admit Weinstein fillings.
The author obtains a great deal of information on the fundamental groups of symplectic fillings of any isolated quotient singularity, with a particularly striking application being the affirmative resolution of a conjecture of Y. Eliashberg [“Questions and open problems”, in Contact topology in higher dimensions, American Institute of Mathematics Workshop (2012)]. Namely, the simplest quotient singularity \(\mathbb{C}^n/(\mathbb{Z}/2)\) has link \((\mathbb{RP}^{2n-1},\xi_{\mathrm{std}})\), allowing the author to prove the following theorem.
Theorem B. For \(n\geq 3\), \((\mathbb{RP}^{2n-1},\xi_{\mathrm{std}})\) is not exactly fillable.
This conjecture was previously resolved by the author [Geom. Topol. 25, No. 6, 3013–3052 (2021; Zbl 1486.53090)] whenever \(n\) is not a power of 2, using topological obstructions involving the total Chern class of \(\xi_{\mathrm{std}}\). (And the case where \(n\) is odd was resolved by P. Ghiggini and K. Niederkrüger-Eid [J. Fixed Point Theory Appl. 24, No. 2, Paper No. 37, 18 p. (2022; Zbl 1503.53155)].) But these obstructions vanish for \(n=2^k\), requiring new techniques. To obtain the more general result, the author uses symplectic cohomology for exact orbifolds to deduce information about the intersection form of a putative exact filling of \((\mathbb{RP}^{2n-1},\xi_{\mathrm{std}})\). For \(n\neq 4\) this information yields a non-integer Chern number which allows one to conclude that no such filling exists, while the case \(n=4\) requires still more work.
Because links of isolated quotient singularities always admit strong symplectic fillings, Theorem B yields an example of a strongly-but-not-exactly fillable contact manifold in every odd dimension. The author conjectures that such contact manifolds exist in abundance, and that many can be constructed as links of quotient singularities.
Conjecture E. If \(\mathbb{C}^n/G\) is an isolated terminal singularity, then the contact link \((S^{2n-1},\xi_{\mathrm{std}})\) is not exactly fillable.
The author establishes this conjecture in several cases, including all where \(n=3\).
Finally, because the primary tool of this paper is the symplectic cohomology of exact orbifolds, some of its conclusions apply to exact orbifold fillings, in addition to the usual smooth manifold fillings. For instance, the author obtains the following diffeomorphism classification for exact orbifold fillings of some real projective spaces:
Theorem I. Let \(n\geq 3\) be an odd number. Then any exact orbifold filling of \((\mathbb{RP}^{2n-1},\xi_{\mathrm{std}})\) is diffeomorphic to \(\mathbb{D}^{2n}/(\mathbb{Z}/2)\) as an orbifold with boundary.
This result is a simple case of the author’s more general conjecture that the exact orbifold fillings of \((S^{2n-1}/G,\xi_{\mathrm{std}})\), where \(\mathbb{C}^n/G\) is an isolated terminal singularity, share a common diffeomorphism type.
The author obtains a great deal of information on the fundamental groups of symplectic fillings of any isolated quotient singularity, with a particularly striking application being the affirmative resolution of a conjecture of Y. Eliashberg [“Questions and open problems”, in Contact topology in higher dimensions, American Institute of Mathematics Workshop (2012)]. Namely, the simplest quotient singularity \(\mathbb{C}^n/(\mathbb{Z}/2)\) has link \((\mathbb{RP}^{2n-1},\xi_{\mathrm{std}})\), allowing the author to prove the following theorem.
Theorem B. For \(n\geq 3\), \((\mathbb{RP}^{2n-1},\xi_{\mathrm{std}})\) is not exactly fillable.
This conjecture was previously resolved by the author [Geom. Topol. 25, No. 6, 3013–3052 (2021; Zbl 1486.53090)] whenever \(n\) is not a power of 2, using topological obstructions involving the total Chern class of \(\xi_{\mathrm{std}}\). (And the case where \(n\) is odd was resolved by P. Ghiggini and K. Niederkrüger-Eid [J. Fixed Point Theory Appl. 24, No. 2, Paper No. 37, 18 p. (2022; Zbl 1503.53155)].) But these obstructions vanish for \(n=2^k\), requiring new techniques. To obtain the more general result, the author uses symplectic cohomology for exact orbifolds to deduce information about the intersection form of a putative exact filling of \((\mathbb{RP}^{2n-1},\xi_{\mathrm{std}})\). For \(n\neq 4\) this information yields a non-integer Chern number which allows one to conclude that no such filling exists, while the case \(n=4\) requires still more work.
Because links of isolated quotient singularities always admit strong symplectic fillings, Theorem B yields an example of a strongly-but-not-exactly fillable contact manifold in every odd dimension. The author conjectures that such contact manifolds exist in abundance, and that many can be constructed as links of quotient singularities.
Conjecture E. If \(\mathbb{C}^n/G\) is an isolated terminal singularity, then the contact link \((S^{2n-1},\xi_{\mathrm{std}})\) is not exactly fillable.
The author establishes this conjecture in several cases, including all where \(n=3\).
Finally, because the primary tool of this paper is the symplectic cohomology of exact orbifolds, some of its conclusions apply to exact orbifold fillings, in addition to the usual smooth manifold fillings. For instance, the author obtains the following diffeomorphism classification for exact orbifold fillings of some real projective spaces:
Theorem I. Let \(n\geq 3\) be an odd number. Then any exact orbifold filling of \((\mathbb{RP}^{2n-1},\xi_{\mathrm{std}})\) is diffeomorphic to \(\mathbb{D}^{2n}/(\mathbb{Z}/2)\) as an orbifold with boundary.
This result is a simple case of the author’s more general conjecture that the exact orbifold fillings of \((S^{2n-1}/G,\xi_{\mathrm{std}})\), where \(\mathbb{C}^n/G\) is an isolated terminal singularity, share a common diffeomorphism type.
Reviewer: Austin Christian (San Luis Obispo)
MSC:
| 57R17 | Symplectic and contact topology in high or arbitrary dimension |
| 57R18 | Topology and geometry of orbifolds |
| 55N32 | Orbifold cohomology |
| 53D42 | Symplectic field theory; contact homology |
| 14B05 | Singularities in algebraic geometry |
Keywords:
Weinstein fillings; orbifold fillings; symplectic cohomology; Chern number; quotient singularitiesReferences:
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