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research
Stability Conditions and Lagrangian Cobordisms
Authors
Felix Hensel
Publication date
1 January 2018
Publisher
Doi
Cite
View
on
arXiv
Abstract
In this paper we study the interplay between Lagrangian cobordisms and stability conditions. We show that any stability condition on the derived Fukaya category
D
F
u
k
(
M
)
D\mathcal{F}uk(M)
D
F
u
k
(
M
)
of a symplectic manifold
(
M
,
Ο
)
(M,\omega)
(
M
,
Ο
)
induces a stability condition on the derived Fukaya category of Lagrangian cobordisms
D
F
u
k
(
C
Γ
M
)
D\mathcal{F}uk(\mathbb{C} \times M)
D
F
u
k
(
C
Γ
M
)
. In addition, using stability conditions, we provide general conditions under which the homomorphism
Ξ
:
Ξ©
L
a
g
(
M
)
β
K
0
(
D
F
u
k
(
M
)
)
\Theta: \Omega_{Lag}(M)\to K_0(D\mathcal{F}uk(M))
Ξ
:
Ξ©
L
a
g
β
(
M
)
β
K
0
β
(
D
F
u
k
(
M
))
, introduced by Biran and Cornea, is an isomorphism. This yields a better understanding of how stability conditions affect
Ξ
\Theta
Ξ
and it allows us to elucidate Haug's result, that the Lagrangian cobordism group of
T
2
T^2
T
2
is isomorphic to
K
0
(
D
F
u
k
(
T
2
)
)
K_0(D\mathcal{F}uk(T^2))
K
0
β
(
D
F
u
k
(
T
2
))
.Comment: 53 pages, 3 figures, expansions and revisions, improvement of expositio
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Last time updated on 19/04/2020