- 1507.05842
Accepted author manuscript, 464 KB, PDF document

- http://www.ams.org/journals/jams/2019-32-01/S0894-0347-2018-00909-9/
Final published version

Research output: Contribution to journal › Journal article

Published

**Generating the Fukaya categories of Hamiltonian G-manifolds.** / Evans, Jonathan David; Lekili, YankI.

Research output: Contribution to journal › Journal article

Evans, JD & Lekili, Y 2019, 'Generating the Fukaya categories of Hamiltonian G-manifolds', *Journal of the American Mathematical Society*, vol. 32, no. 1, pp. 119-162. https://doi.org/10.1090/jams/909

Evans, J. D., & Lekili, Y. (2019). Generating the Fukaya categories of Hamiltonian G-manifolds. *Journal of the American Mathematical Society*, *32*(1), 119-162. https://doi.org/10.1090/jams/909

Evans JD, Lekili Y. Generating the Fukaya categories of Hamiltonian G-manifolds. Journal of the American Mathematical Society. 2019 Jan 1;32(1):119-162. https://doi.org/10.1090/jams/909

@article{c8db1f2f2e154db7990b54fe009dd178,

title = "Generating the Fukaya categories of Hamiltonian G-manifolds",

abstract = "Abstract: Let $ G$ be a compact Lie group, and let $ k$ be a field of characteristic $ p \geq 0$ such that $ H^*(G)$ has no $ p$-torsion if $ p>0$. We show that a free Lagrangian orbit of a Hamiltonian $ G$-action on a compact, monotone, symplectic manifold $ X$ split-generates an idempotent summand of the monotone Fukaya category $ \mathcal {F}(X; k)$ if and only if it represents a nonzero object of that summand (slightly more general results are also provided). Our result is based on an explicit understanding of the wrapped Fukaya category $ \mathcal {W}(T^*G; k)$ through Koszul twisted complexes involving the zero-section and a cotangent fibre and on a functor $ D^b \mathcal {W}(T^*G; k) \to D^b\mathcal {F}(X^{-} \times X; k)$ canonically associated to the Hamiltonian $ G$-action on $ X$. We explore several examples which can be studied in a uniform manner, including toric Fano varieties and certain Grassmannians.",

keywords = "Fukaya categories, symplectic geometry, Hamiltonian group actions",

author = "Evans, {Jonathan David} and YankI Lekili",

year = "2019",

month = "1",

day = "1",

doi = "10.1090/jams/909",

language = "English",

volume = "32",

pages = "119--162",

journal = "Journal of the American Mathematical Society",

issn = "1088-6834",

publisher = "American Mathematical Society",

number = "1",

}

TY - JOUR

T1 - Generating the Fukaya categories of Hamiltonian G-manifolds

AU - Evans, Jonathan David

AU - Lekili, YankI

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Abstract: Let $ G$ be a compact Lie group, and let $ k$ be a field of characteristic $ p \geq 0$ such that $ H^*(G)$ has no $ p$-torsion if $ p>0$. We show that a free Lagrangian orbit of a Hamiltonian $ G$-action on a compact, monotone, symplectic manifold $ X$ split-generates an idempotent summand of the monotone Fukaya category $ \mathcal {F}(X; k)$ if and only if it represents a nonzero object of that summand (slightly more general results are also provided). Our result is based on an explicit understanding of the wrapped Fukaya category $ \mathcal {W}(T^*G; k)$ through Koszul twisted complexes involving the zero-section and a cotangent fibre and on a functor $ D^b \mathcal {W}(T^*G; k) \to D^b\mathcal {F}(X^{-} \times X; k)$ canonically associated to the Hamiltonian $ G$-action on $ X$. We explore several examples which can be studied in a uniform manner, including toric Fano varieties and certain Grassmannians.

AB - Abstract: Let $ G$ be a compact Lie group, and let $ k$ be a field of characteristic $ p \geq 0$ such that $ H^*(G)$ has no $ p$-torsion if $ p>0$. We show that a free Lagrangian orbit of a Hamiltonian $ G$-action on a compact, monotone, symplectic manifold $ X$ split-generates an idempotent summand of the monotone Fukaya category $ \mathcal {F}(X; k)$ if and only if it represents a nonzero object of that summand (slightly more general results are also provided). Our result is based on an explicit understanding of the wrapped Fukaya category $ \mathcal {W}(T^*G; k)$ through Koszul twisted complexes involving the zero-section and a cotangent fibre and on a functor $ D^b \mathcal {W}(T^*G; k) \to D^b\mathcal {F}(X^{-} \times X; k)$ canonically associated to the Hamiltonian $ G$-action on $ X$. We explore several examples which can be studied in a uniform manner, including toric Fano varieties and certain Grassmannians.

KW - Fukaya categories

KW - symplectic geometry

KW - Hamiltonian group actions

U2 - 10.1090/jams/909

DO - 10.1090/jams/909

M3 - Journal article

VL - 32

SP - 119

EP - 162

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

SN - 1088-6834

IS - 1

ER -