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 Dec 9 '18 at 8:23
complexboycomplexboy
583
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add a comment |
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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 Dec 9 '18 at 8:23
complexboycomplexboy
583
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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.
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– Ben
Dec 9 '18 at 12:47
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@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
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– complexboy
Dec 9 '18 at 13:03
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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.)
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– Ben
Dec 9 '18 at 13:12
add a comment |
$begingroup$
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 Dec 9 '18 at 8:23
complexboycomplexboy
583
$endgroup$
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 Dec 9 '18 at 8:23
complexboycomplexboy
583
asked Dec 9 '18 at 8:23
complexboycomplexboy
583
edited Dec 9 '18 at 8:43
complexboy
asked Dec 9 '18 at 8:23
complexboycomplexboy
583
asked Dec 9 '18 at 8:23
complexboycomplexboy
583
asked Dec 9 '18 at 8:23
complexboycomplexboy
583
583
$begingroup$
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.
$endgroup$
– Ben
Dec 9 '18 at 12:47
$begingroup$
@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
$endgroup$
– complexboy
Dec 9 '18 at 13:03
$begingroup$
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.)
$endgroup$
– Ben
Dec 9 '18 at 13:12
add a comment |
$begingroup$
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.
$endgroup$
– Ben
Dec 9 '18 at 12:47
$begingroup$
@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
$endgroup$
– complexboy
Dec 9 '18 at 13:03
$begingroup$
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.)
$endgroup$
– Ben
Dec 9 '18 at 13:12
$begingroup$
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.
$endgroup$
– Ben
Dec 9 '18 at 12:47
$begingroup$
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.
$endgroup$
– Ben
Dec 9 '18 at 12:47
$begingroup$
@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
$endgroup$
– complexboy
Dec 9 '18 at 13:03
$begingroup$
@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
$endgroup$
– complexboy
Dec 9 '18 at 13:03
$begingroup$
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.)
$endgroup$
– Ben
Dec 9 '18 at 13:12
$begingroup$
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.)
$endgroup$
– Ben
Dec 9 '18 at 13:12
add a comment |
1 Answer
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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 Dec 9 '18 at 10:02
BenBen
5992513
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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 Dec 9 '18 at 10:02
BenBen
5992513
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add a comment |
$begingroup$
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 Dec 9 '18 at 10:02
BenBen
5992513
$endgroup$
add a comment |
$begingroup$
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 Dec 9 '18 at 10:02
BenBen
5992513
$endgroup$
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 Dec 9 '18 at 10:02
BenBen
5992513
edited Dec 9 '18 at 11:08
answered Dec 9 '18 at 10:02
BenBen
5992513
answered Dec 9 '18 at 10:02
BenBen
5992513
answered Dec 9 '18 at 10:02
BenBen
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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.
$endgroup$
– Ben
Dec 9 '18 at 12:47
$begingroup$
@Ben that is true of course but we are interested in whether that distinction is visible at the topological level
$endgroup$
– complexboy
Dec 9 '18 at 13:03
$begingroup$
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.)
$endgroup$
– Ben
Dec 9 '18 at 13:12