Non-compact complex foliation on a compact manifold
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Let $M$ be a compact smooth manifold and let $mathscr F$ be a foliation on $M$ such that each leaf $ Fin mathscr F$ is a non-compact complex manifold.
Is it true that a function $f:Ftomathbb C$ is holomorphic iff $f$ is a constant?
differential-geometry manifolds complex-geometry
asked Nov 15 at 23:31
Amrat A
31317
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Let $M$ be a compact smooth manifold and let $mathscr F$ be a foliation on $M$ such that each leaf $ Fin mathscr F$ is a non-compact complex manifold.
Is it true that a function $f:Ftomathbb C$ is holomorphic iff $f$ is a constant?
differential-geometry manifolds complex-geometry
asked Nov 15 at 23:31
Amrat A
31317
As it stands, you're asking if a function on a single leaf (which can be endowed with some complex structure) must be constant if it's holomorphic. Take a foliation where the leaves are diffeomorphic to $Bbb R^2congBbb C$, and choose a complex structure for one of the leaves. You certainly can take nonconstant holomorphic functions on that particular copy of $Bbb C$.
– Ted Shifrin
Nov 17 at 17:49
Thanks Ted! So if we assume that $M$ is homogeneous under a Lie group action and the leaves are orbits and all of them are biholomorphic. Does that change anything?
– Amrat A
Nov 17 at 19:53
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $M$ be a compact smooth manifold and let $mathscr F$ be a foliation on $M$ such that each leaf $ Fin mathscr F$ is a non-compact complex manifold.
Is it true that a function $f:Ftomathbb C$ is holomorphic iff $f$ is a constant?
differential-geometry manifolds complex-geometry
asked Nov 15 at 23:31
Amrat A
31317
Let $M$ be a compact smooth manifold and let $mathscr F$ be a foliation on $M$ such that each leaf $ Fin mathscr F$ is a non-compact complex manifold.
Is it true that a function $f:Ftomathbb C$ is holomorphic iff $f$ is a constant?
differential-geometry manifolds complex-geometry
differential-geometry manifolds complex-geometry
asked Nov 15 at 23:31
Amrat A
31317
asked Nov 15 at 23:31
Amrat A
31317
asked Nov 15 at 23:31
Amrat A
31317
asked Nov 15 at 23:31
Amrat A
31317
asked Nov 15 at 23:31
Amrat A
31317
31317
As it stands, you're asking if a function on a single leaf (which can be endowed with some complex structure) must be constant if it's holomorphic. Take a foliation where the leaves are diffeomorphic to $Bbb R^2congBbb C$, and choose a complex structure for one of the leaves. You certainly can take nonconstant holomorphic functions on that particular copy of $Bbb C$.
– Ted Shifrin
Nov 17 at 17:49
Thanks Ted! So if we assume that $M$ is homogeneous under a Lie group action and the leaves are orbits and all of them are biholomorphic. Does that change anything?
– Amrat A
Nov 17 at 19:53
add a comment |
As it stands, you're asking if a function on a single leaf (which can be endowed with some complex structure) must be constant if it's holomorphic. Take a foliation where the leaves are diffeomorphic to $Bbb R^2congBbb C$, and choose a complex structure for one of the leaves. You certainly can take nonconstant holomorphic functions on that particular copy of $Bbb C$.
– Ted Shifrin
Nov 17 at 17:49
Thanks Ted! So if we assume that $M$ is homogeneous under a Lie group action and the leaves are orbits and all of them are biholomorphic. Does that change anything?
– Amrat A
Nov 17 at 19:53
As it stands, you're asking if a function on a single leaf (which can be endowed with some complex structure) must be constant if it's holomorphic. Take a foliation where the leaves are diffeomorphic to $Bbb R^2congBbb C$, and choose a complex structure for one of the leaves. You certainly can take nonconstant holomorphic functions on that particular copy of $Bbb C$.
– Ted Shifrin
Nov 17 at 17:49
As it stands, you're asking if a function on a single leaf (which can be endowed with some complex structure) must be constant if it's holomorphic. Take a foliation where the leaves are diffeomorphic to $Bbb R^2congBbb C$, and choose a complex structure for one of the leaves. You certainly can take nonconstant holomorphic functions on that particular copy of $Bbb C$.
– Ted Shifrin
Nov 17 at 17:49
Thanks Ted! So if we assume that $M$ is homogeneous under a Lie group action and the leaves are orbits and all of them are biholomorphic. Does that change anything?
– Amrat A
Nov 17 at 19:53
Thanks Ted! So if we assume that $M$ is homogeneous under a Lie group action and the leaves are orbits and all of them are biholomorphic. Does that change anything?
– Amrat A
Nov 17 at 19:53
add a comment |
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As it stands, you're asking if a function on a single leaf (which can be endowed with some complex structure) must be constant if it's holomorphic. Take a foliation where the leaves are diffeomorphic to $Bbb R^2congBbb C$, and choose a complex structure for one of the leaves. You certainly can take nonconstant holomorphic functions on that particular copy of $Bbb C$.
– Ted Shifrin
Nov 17 at 17:49
Thanks Ted! So if we assume that $M$ is homogeneous under a Lie group action and the leaves are orbits and all of them are biholomorphic. Does that change anything?
– Amrat A
Nov 17 at 19:53