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## Generating the Fukaya categories of Hamiltonian G-manifolds

Research output: Contribution to journalJournal article

Published

### Standard

In: Journal of the American Mathematical Society, Vol. 32, No. 1, 01.01.2019, p. 119-162.

Research output: Contribution to journalJournal article

### Harvard

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

### APA

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

### Vancouver

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

### Author

Evans, Jonathan David ; Lekili, YankI. / Generating the Fukaya categories of Hamiltonian G-manifolds. In: Journal of the American Mathematical Society. 2019 ; Vol. 32, No. 1. pp. 119-162.

### Bibtex

@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",

}

### RIS

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 -