Moishezon manifold vs proper complex variety
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Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is bimeromorophic to the analytification of a smooth proper complex variety, so for example fundamental groups have to be the same)?
Does there exist a smooth proper complex variety whose analytification is not homotopy equivalent to the analytification of a smooth projective complex variety?
dg.differential-geometry at.algebraic-topology complex-geometry
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up vote
8
down vote
favorite
Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is bimeromorophic to the analytification of a smooth proper complex variety, so for example fundamental groups have to be the same)?
Does there exist a smooth proper complex variety whose analytification is not homotopy equivalent to the analytification of a smooth projective complex variety?
dg.differential-geometry at.algebraic-topology complex-geometry
New contributor
By the way, not every Moishezon manifold comes from a proper complex variety; it might be an algebraic space instead of a scheme. There is an example due to Kleiman which is a quotient of Hironaka's example.
– Ben
15 hours ago
@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
– complexboy
15 hours ago
Sorry, I'm a bit off today. I misunderstood something you wrote.. (I thought you were suggesting hat every Moishezon manifold is the analytification of a proper variety.)
– Ben
14 hours ago
add a comment |
up vote
8
down vote
favorite
up vote
8
down vote
favorite
Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is bimeromorophic to the analytification of a smooth proper complex variety, so for example fundamental groups have to be the same)?
Does there exist a smooth proper complex variety whose analytification is not homotopy equivalent to the analytification of a smooth projective complex variety?
dg.differential-geometry at.algebraic-topology complex-geometry
New contributor
Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is bimeromorophic to the analytification of a smooth proper complex variety, so for example fundamental groups have to be the same)?
Does there exist a smooth proper complex variety whose analytification is not homotopy equivalent to the analytification of a smooth projective complex variety?
dg.differential-geometry at.algebraic-topology complex-geometry
dg.differential-geometry at.algebraic-topology complex-geometry
New contributor
New contributor
edited 19 hours ago
New contributor
asked 19 hours ago
complexboy
533
533
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New contributor
By the way, not every Moishezon manifold comes from a proper complex variety; it might be an algebraic space instead of a scheme. There is an example due to Kleiman which is a quotient of Hironaka's example.
– Ben
15 hours ago
@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
– complexboy
15 hours ago
Sorry, I'm a bit off today. I misunderstood something you wrote.. (I thought you were suggesting hat every Moishezon manifold is the analytification of a proper variety.)
– Ben
14 hours ago
add a comment |
By the way, not every Moishezon manifold comes from a proper complex variety; it might be an algebraic space instead of a scheme. There is an example due to Kleiman which is a quotient of Hironaka's example.
– Ben
15 hours ago
@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
– complexboy
15 hours ago
Sorry, I'm a bit off today. I misunderstood something you wrote.. (I thought you were suggesting hat every Moishezon manifold is the analytification of a proper variety.)
– Ben
14 hours ago
By the way, not every Moishezon manifold comes from a proper complex variety; it might be an algebraic space instead of a scheme. There is an example due to Kleiman which is a quotient of Hironaka's example.
– Ben
15 hours ago
By the way, not every Moishezon manifold comes from a proper complex variety; it might be an algebraic space instead of a scheme. There is an example due to Kleiman which is a quotient of Hironaka's example.
– Ben
15 hours ago
@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
– complexboy
15 hours ago
@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
– complexboy
15 hours ago
Sorry, I'm a bit off today. I misunderstood something you wrote.. (I thought you were suggesting hat every Moishezon manifold is the analytification of a proper variety.)
– Ben
14 hours ago
Sorry, I'm a bit off today. I misunderstood something you wrote.. (I thought you were suggesting hat every Moishezon manifold is the analytification of a proper variety.)
– Ben
14 hours ago
add a comment |
1 Answer
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accepted
Edit: I just realised that the OP asked for proper, not projective varieties. As it stands, it is still possible that Oguiso's Moishezon Calabi-Yau threefold is homotopy equivalent to a proper varitey, so the question remains open.
The only example I am aware of is due to Oguiso, Two remarks on Calabi-Yau Moishezon threefolds, J. reine angew. Math. 452 (1994). There, you find a construction of a Moishezon threefold $X$ with $H^2(X) = mathbb{Z}c$ where $c^3 = 0$. In particular, $X$ is not even homotopy equivalent to a Kähler manifold.
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
Edit: I just realised that the OP asked for proper, not projective varieties. As it stands, it is still possible that Oguiso's Moishezon Calabi-Yau threefold is homotopy equivalent to a proper varitey, so the question remains open.
The only example I am aware of is due to Oguiso, Two remarks on Calabi-Yau Moishezon threefolds, J. reine angew. Math. 452 (1994). There, you find a construction of a Moishezon threefold $X$ with $H^2(X) = mathbb{Z}c$ where $c^3 = 0$. In particular, $X$ is not even homotopy equivalent to a Kähler manifold.
add a comment |
up vote
4
down vote
accepted
Edit: I just realised that the OP asked for proper, not projective varieties. As it stands, it is still possible that Oguiso's Moishezon Calabi-Yau threefold is homotopy equivalent to a proper varitey, so the question remains open.
The only example I am aware of is due to Oguiso, Two remarks on Calabi-Yau Moishezon threefolds, J. reine angew. Math. 452 (1994). There, you find a construction of a Moishezon threefold $X$ with $H^2(X) = mathbb{Z}c$ where $c^3 = 0$. In particular, $X$ is not even homotopy equivalent to a Kähler manifold.
add a comment |
up vote
4
down vote
accepted
up vote
4
down vote
accepted
Edit: I just realised that the OP asked for proper, not projective varieties. As it stands, it is still possible that Oguiso's Moishezon Calabi-Yau threefold is homotopy equivalent to a proper varitey, so the question remains open.
The only example I am aware of is due to Oguiso, Two remarks on Calabi-Yau Moishezon threefolds, J. reine angew. Math. 452 (1994). There, you find a construction of a Moishezon threefold $X$ with $H^2(X) = mathbb{Z}c$ where $c^3 = 0$. In particular, $X$ is not even homotopy equivalent to a Kähler manifold.
Edit: I just realised that the OP asked for proper, not projective varieties. As it stands, it is still possible that Oguiso's Moishezon Calabi-Yau threefold is homotopy equivalent to a proper varitey, so the question remains open.
The only example I am aware of is due to Oguiso, Two remarks on Calabi-Yau Moishezon threefolds, J. reine angew. Math. 452 (1994). There, you find a construction of a Moishezon threefold $X$ with $H^2(X) = mathbb{Z}c$ where $c^3 = 0$. In particular, $X$ is not even homotopy equivalent to a Kähler manifold.
edited 17 hours ago
answered 18 hours ago
Ben
5992513
5992513
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By the way, not every Moishezon manifold comes from a proper complex variety; it might be an algebraic space instead of a scheme. There is an example due to Kleiman which is a quotient of Hironaka's example.
– Ben
15 hours ago
@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
– complexboy
15 hours ago
Sorry, I'm a bit off today. I misunderstood something you wrote.. (I thought you were suggesting hat every Moishezon manifold is the analytification of a proper variety.)
– Ben
14 hours ago