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
asked 19 hours ago
complexboy
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up vote
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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
asked 19 hours ago
complexboy
533
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complexboy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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
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
asked 19 hours ago
complexboy
533
New contributor
complexboy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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
dg.differential-geometry at.algebraic-topology complex-geometry
asked 19 hours ago
complexboy
533
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complexboy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 19 hours ago
complexboy
533
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complexboy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited 19 hours ago
asked 19 hours ago
complexboy
533
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complexboy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 19 hours ago
complexboy
533
asked 19 hours ago
complexboy
533
533
New contributor
complexboy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
complexboy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
complexboy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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
1
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up vote
4
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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.
answered 18 hours ago
Ben
5992513
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.
answered 18 hours ago
Ben
5992513
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.
answered 18 hours ago
Ben
5992513
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.
answered 18 hours ago
Ben
5992513
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.
answered 18 hours ago
Ben
5992513
edited 17 hours ago
answered 18 hours ago
Ben
5992513
answered 18 hours ago
Ben
5992513
answered 18 hours ago
Ben
5992513
5992513
add a comment |
add a comment |
complexboy is a new contributor. Be nice, and check out our Code of Conduct.
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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