The topology of nonsingular projective variety and its corresponding manifold.
0
A nonsingular projective variety is a manifold, then if the topology of the variety can be identified with the topology on the corresponding manifold? in other words, if any open set $U$ of the nonsingular projective variety can be seen as an open set of the corresponding manifold and vice verse?
algebraic-geometry complex-geometry
asked Dec 4 '18 at 21:20
66666666
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add a comment |
0
A nonsingular projective variety is a manifold, then if the topology of the variety can be identified with the topology on the corresponding manifold? in other words, if any open set $U$ of the nonsingular projective variety can be seen as an open set of the corresponding manifold and vice verse?
algebraic-geometry complex-geometry
asked Dec 4 '18 at 21:20
66666666
1,270620
2
Over $mathbb{C}$ (or more generally a topological field) there is a correspondence. Any Zariski open set is open in the euclidean sense, as they are complements of closed sets (which are closed in both topologies).However the converse is not true, there are (many) euclidean open sets which are not open in the Zariski topology. However over $mathbb{C}$ you have the so-called complex topology, which does indeed give a smooth projective variety the structure of a complex compact manifold.
– DKS
Dec 4 '18 at 22:11
@DKS can you tell me the reference of the correspondence?
– 6666
Dec 5 '18 at 0:25
One such reference is given in "The Red Book of Varieties and Schemes" by Mumford. A quick google also revealed these notes.
– DKS
Dec 5 '18 at 2:15
@DKS but I think that's for affine case, not for projective case
– 6666
Dec 5 '18 at 3:10
Look at Mumford's book. Its section 10 of chapter 1.
– DKS
Dec 5 '18 at 13:05
add a comment |
0
0
0
A nonsingular projective variety is a manifold, then if the topology of the variety can be identified with the topology on the corresponding manifold? in other words, if any open set $U$ of the nonsingular projective variety can be seen as an open set of the corresponding manifold and vice verse?
algebraic-geometry complex-geometry
asked Dec 4 '18 at 21:20
66666666
1,270620
A nonsingular projective variety is a manifold, then if the topology of the variety can be identified with the topology on the corresponding manifold? in other words, if any open set $U$ of the nonsingular projective variety can be seen as an open set of the corresponding manifold and vice verse?
algebraic-geometry complex-geometry
algebraic-geometry complex-geometry
asked Dec 4 '18 at 21:20
66666666
1,270620
asked Dec 4 '18 at 21:20
66666666
1,270620
asked Dec 4 '18 at 21:20
66666666
1,270620
asked Dec 4 '18 at 21:20
66666666
1,270620
asked Dec 4 '18 at 21:20
66666666
1,270620
1,270620
2
Over $mathbb{C}$ (or more generally a topological field) there is a correspondence. Any Zariski open set is open in the euclidean sense, as they are complements of closed sets (which are closed in both topologies).However the converse is not true, there are (many) euclidean open sets which are not open in the Zariski topology. However over $mathbb{C}$ you have the so-called complex topology, which does indeed give a smooth projective variety the structure of a complex compact manifold.
– DKS
Dec 4 '18 at 22:11
@DKS can you tell me the reference of the correspondence?
– 6666
Dec 5 '18 at 0:25
One such reference is given in "The Red Book of Varieties and Schemes" by Mumford. A quick google also revealed these notes.
– DKS
Dec 5 '18 at 2:15
@DKS but I think that's for affine case, not for projective case
– 6666
Dec 5 '18 at 3:10
Look at Mumford's book. Its section 10 of chapter 1.
– DKS
Dec 5 '18 at 13:05
add a comment |
2
Over $mathbb{C}$ (or more generally a topological field) there is a correspondence. Any Zariski open set is open in the euclidean sense, as they are complements of closed sets (which are closed in both topologies).However the converse is not true, there are (many) euclidean open sets which are not open in the Zariski topology. However over $mathbb{C}$ you have the so-called complex topology, which does indeed give a smooth projective variety the structure of a complex compact manifold.
– DKS
Dec 4 '18 at 22:11
@DKS can you tell me the reference of the correspondence?
– 6666
Dec 5 '18 at 0:25
One such reference is given in "The Red Book of Varieties and Schemes" by Mumford. A quick google also revealed these notes.
– DKS
Dec 5 '18 at 2:15
@DKS but I think that's for affine case, not for projective case
– 6666
Dec 5 '18 at 3:10
Look at Mumford's book. Its section 10 of chapter 1.
– DKS
Dec 5 '18 at 13:05
2
2
Over $mathbb{C}$ (or more generally a topological field) there is a correspondence. Any Zariski open set is open in the euclidean sense, as they are complements of closed sets (which are closed in both topologies).However the converse is not true, there are (many) euclidean open sets which are not open in the Zariski topology. However over $mathbb{C}$ you have the so-called complex topology, which does indeed give a smooth projective variety the structure of a complex compact manifold.
– DKS
Dec 4 '18 at 22:11
Over $mathbb{C}$ (or more generally a topological field) there is a correspondence. Any Zariski open set is open in the euclidean sense, as they are complements of closed sets (which are closed in both topologies).However the converse is not true, there are (many) euclidean open sets which are not open in the Zariski topology. However over $mathbb{C}$ you have the so-called complex topology, which does indeed give a smooth projective variety the structure of a complex compact manifold.
– DKS
Dec 4 '18 at 22:11
@DKS can you tell me the reference of the correspondence?
– 6666
Dec 5 '18 at 0:25
@DKS can you tell me the reference of the correspondence?
– 6666
Dec 5 '18 at 0:25
One such reference is given in "The Red Book of Varieties and Schemes" by Mumford. A quick google also revealed these notes.
– DKS
Dec 5 '18 at 2:15
One such reference is given in "The Red Book of Varieties and Schemes" by Mumford. A quick google also revealed these notes.
– DKS
Dec 5 '18 at 2:15
@DKS but I think that's for affine case, not for projective case
– 6666
Dec 5 '18 at 3:10
@DKS but I think that's for affine case, not for projective case
– 6666
Dec 5 '18 at 3:10
Look at Mumford's book. Its section 10 of chapter 1.
– DKS
Dec 5 '18 at 13:05
Look at Mumford's book. Its section 10 of chapter 1.
– DKS
Dec 5 '18 at 13:05
add a comment |
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2
Over $mathbb{C}$ (or more generally a topological field) there is a correspondence. Any Zariski open set is open in the euclidean sense, as they are complements of closed sets (which are closed in both topologies).However the converse is not true, there are (many) euclidean open sets which are not open in the Zariski topology. However over $mathbb{C}$ you have the so-called complex topology, which does indeed give a smooth projective variety the structure of a complex compact manifold.
– DKS
Dec 4 '18 at 22:11
@DKS can you tell me the reference of the correspondence?
– 6666
Dec 5 '18 at 0:25
One such reference is given in "The Red Book of Varieties and Schemes" by Mumford. A quick google also revealed these notes.
– DKS
Dec 5 '18 at 2:15
@DKS but I think that's for affine case, not for projective case
– 6666
Dec 5 '18 at 3:10
Look at Mumford's book. Its section 10 of chapter 1.
– DKS
Dec 5 '18 at 13:05