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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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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 -