The monotone wrapped Fukaya category and the open-closed string map
The monotone wrapped Fukaya category and the open-closed string map
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单调包裹的 Fukaya 范畴和开闭弦图
DOI：
10.1007/s00029-016-0255-9
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
I. Smith
中科院分区：
文献类型：
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作者：
Alexander F. Ritter;I. Smith
We build the wrapped Fukaya category $${\mathcal {W}}(E)$$W(E) for any monotone symplectic manifold E, convex at infinity. We define the open-closed and closed-open string maps, $${\mathrm {OC}}:{\mathrm {HH}}_*({\mathcal {W}}(E))\rightarrow { SH }^*(E)$$OC:HH∗(W(E))→SH∗(E) and $${\mathrm {CO}}: { SH }^*(E)\rightarrow {\mathrm {HH}}^*({\mathcal {W}}(E))$$CO:SH∗(E)→HH∗(W(E)). We study their algebraic properties and prove that the string maps are compatible with the $$c_1({ TE })$$c1(TE)-eigenvalue splitting of $${\mathcal {W}}(E)$$W(E). We extend Abouzaid’s generation criterion from the exact to the monotone setting. We construct an acceleration functor $${\mathcal {AF}}: {\mathcal {F}}(E)\rightarrow {\mathcal {W}}(E)$$AF:F(E)→W(E) from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map $$c^*:{ QH }^*(E)\rightarrow { SH }^*(E)$$c∗:QH∗(E)→SH∗(E). We define the $${ SH }^*(E)$$SH∗(E)-module structure on the Hochschild (co)homology of $${\mathcal {W}}(E)$$W(E) which is compatible with the string maps (this was proved independently for exact convex symplectic manifolds by Ganatra). The module and unital algebra structures, and the generation criterion, also hold for the compact Fukaya category $${\mathcal {F}}(E)$$F(E), and also hold for closed monotone symplectic manifolds. As an application, we show that the wrapped category of $${\mathcal {O}}(-k) \rightarrow \mathbb {C}\mathbb {P}^m$$O(-k)→CPm is proper (cohomologically finite) for $$1\le k \le m$$1≤k≤m. For any monotone negative line bundle E over a closed monotone toric manifold B, we show that $${ SH }^*(E)\ne 0$$SH∗(E)≠0, $${\mathcal {W}}(E)$$W(E) is non-trivial and E contains a non-displaceable monotone Lagrangian torus $${\mathcal {L}}$$L on which $${\mathrm {OC}}$$OC is non-zero.
We build the wrapped Fukaya category $${\mathcal {W}}(E)$$W(E) for any monotone symplectic manifold E, convex at infinity. We define the open-closed and closed-open string maps, $${\mathrm {OC}}:{\mathrm {HH}}_*({\mathcal {W}}(E))\rightarrow { SH }^*(E)$$OC:HH∗(W(E))→SH∗(E) and $${\mathrm {CO}}: { SH }^*(E)\rightarrow {\mathrm {HH}}^*({\mathcal {W}}(E))$$CO:SH∗(E)→HH∗(W(E)). We study their algebraic properties and prove that the string maps are compatible with the $$c_1({ TE })$$c1(TE)-eigenvalue splitting of $${\mathcal {W}}(E)$$W(E). We extend Abouzaid’s generation criterion from the exact to the monotone setting. We construct an acceleration functor $${\mathcal {AF}}: {\mathcal {F}}(E)\rightarrow {\mathcal {W}}(E)$$AF:F(E)→W(E) from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map $$c^*:{ QH }^*(E)\rightarrow { SH }^*(E)$$c∗:QH∗(E)→SH∗(E). We define the $${ SH }^*(E)$$SH∗(E)-module structure on the Hochschild (co)homology of $${\mathcal {W}}(E)$$W(E) which is compatible with the string maps (this was proved independently for exact convex symplectic manifolds by Ganatra). The module and unital algebra structures, and the generation criterion, also hold for the compact Fukaya category $${\mathcal {F}}(E)$$F(E), and also hold for closed monotone symplectic manifolds. As an application, we show that the wrapped category of $${\mathcal {O}}(-k) \rightarrow \mathbb {C}\mathbb {P}^m$$O(-k)→CPm is proper (cohomologically finite) for $$1\le k \le m$$1≤k≤m. For any monotone negative line bundle E over a closed monotone toric manifold B, we show that $${ SH }^*(E)\ne 0$$SH∗(E)≠0, $${\mathcal {W}}(E)$$W(E) is non-trivial and E contains a non-displaceable monotone Lagrangian torus $${\mathcal {L}}$$L on which $${\mathrm {OC}}$$OC is non-zero.
拉格朗日Floer上同调的计算及应用
DOI：
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发表时间：
2010
期刊：
影响因子：
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作者：
渡部敏明;長倉大輔;K. Fuakya
通讯作者：
K. Fuakya
拉格朗日弗洛尔理论中的体积变形
DOI：
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发表时间：
2011
期刊：
影响因子：
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作者：
Burnham;D. & Sekivama. K;K. Ono
通讯作者：
K. Ono
DOI：
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发表时间：
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期刊：
Homological Mirror symmetry (ed. A. Kapustin, M. Kreuzer, K. -G. Schlesinger) Birkhauser.論文集 (掲載決定)
影响因子：
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作者：
Kenji Fukaya;P. Seidel I. Smith
通讯作者：
P. Seidel I. Smith