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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.12607 |
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| _version_ | 1866929508770119680 |
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| author | Li, Tian-Jun Ning, Shengzhen |
| author_facet | Li, Tian-Jun Ning, Shengzhen |
| contents | In a series of work [Wor22], [Wor21] and [CW20], algebraic capacities were introduced in an algebraic manner for polarized algebraic surfaces and applied to the symplectic embedding problems. In this paper, we give a reformulation of algebraic capacities in terms of only a tamed pair of symplectic form and almost complex structure. We show that they actually only depend on the cohomology class of the symplectic form for a rational manifold. Since it is not known that any symplectic form on a rational manifold is Kähler, this novel formulation potentially is more general on a rational manifold. Additionally, for manifolds with $b^+=1$, we derive asymptotic results that are parallel to the context of ECH(Embedded Contact Homology) and algebraic settings. When assuming $c_1\cdot [ω]>0$ on rational manifolds, we further introduce a sequence of tropical polynomials which will succinctly describe those capacities viewed as functions over the domain parametrizing such symplectic forms. As an application, we give a purely symplectic proof of the correspondence between algebraic capacities and ECH capacities for smooth toric surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_12607 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Algebraic capacities as tropical polynomials over the reduced $c_1$-positive symplectic cone Li, Tian-Jun Ning, Shengzhen Symplectic Geometry In a series of work [Wor22], [Wor21] and [CW20], algebraic capacities were introduced in an algebraic manner for polarized algebraic surfaces and applied to the symplectic embedding problems. In this paper, we give a reformulation of algebraic capacities in terms of only a tamed pair of symplectic form and almost complex structure. We show that they actually only depend on the cohomology class of the symplectic form for a rational manifold. Since it is not known that any symplectic form on a rational manifold is Kähler, this novel formulation potentially is more general on a rational manifold. Additionally, for manifolds with $b^+=1$, we derive asymptotic results that are parallel to the context of ECH(Embedded Contact Homology) and algebraic settings. When assuming $c_1\cdot [ω]>0$ on rational manifolds, we further introduce a sequence of tropical polynomials which will succinctly describe those capacities viewed as functions over the domain parametrizing such symplectic forms. As an application, we give a purely symplectic proof of the correspondence between algebraic capacities and ECH capacities for smooth toric surfaces. |
| title | Algebraic capacities as tropical polynomials over the reduced $c_1$-positive symplectic cone |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/2303.12607 |