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Andrej Bauer
Andrej Bauer
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Also like affine varieties, we have:

Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.

For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables, Theorem 5.3.9. For the converse, use Cartan B mentioned above. (Added: I don't mean to be too pedantic, but Cartan says that $X$ is Stein iff sheaf cohomology vanishes for coherent analytic coefficients; it is certainly not true otherwise.) Second edit: I originally suggested a homological argument for the converse, but it's much easier than that: an argument is contained on pp 109-110 of Hörmander, immediately after the definition of Stein manifold.

Also like affine varieties, we have:

Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.

For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables, Theorem 5.3.9. For the converse, use Cartan B mentioned above. (Added: I don't mean to be too pedantic, but Cartan says that $X$ is Stein iff sheaf cohomology vanishes for coherent analytic coefficients; it is certainly not true otherwise.) Second edit: I originally suggested a homological argument for the converse, but it's much easier than that: an argument is contained on pp 109-110 of Hörmander, immediately after the definition of Stein manifold.

Also like affine varieties, we have:

Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.

For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables, Theorem 5.3.9. For the converse, an argument is contained on pp 109-110 of Hörmander, immediately after the definition of Stein manifold.

added 238 characters in body
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Donu Arapura
Donu Arapura
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Also like affine varieties, we have:

Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.

For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables, Theorem 5.3.9. For the converse, use Cartan B mentioned above. ( use Cartan B mentioned above. (Added: I don't mean to be too pedantic, but Cartan says that $X$ is Stein iff sheaf cohomology vanishes for coherent analytic coefficients; it is certainly not true otherwise.) AddedSecond edit:: I don't mean to be too pedantic I originally suggested a homological argument for the converse, but Cartan saysit's much easier than that $X$: an argument is contained on pp 109-110 of Hörmander, immediately after the definition of Stein iff sheaf cohomology vanishes for coherent analytic coefficients; it is certainly not true otherwisemanifold.)

Also like affine varieties, we have:

Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.

For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables, Theorem 5.3.9. For the converse, use Cartan B mentioned above. (Added: I don't mean to be too pedantic, but Cartan says that $X$ is Stein iff sheaf cohomology vanishes for coherent analytic coefficients; it is certainly not true otherwise.)

Also like affine varieties, we have:

Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.

For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables, Theorem 5.3.9. For the converse, use Cartan B mentioned above. (Added: I don't mean to be too pedantic, but Cartan says that $X$ is Stein iff sheaf cohomology vanishes for coherent analytic coefficients; it is certainly not true otherwise.) Second edit: I originally suggested a homological argument for the converse, but it's much easier than that: an argument is contained on pp 109-110 of Hörmander, immediately after the definition of Stein manifold.

added 184 characters in body
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Donu Arapura
Donu Arapura
  • 35.2k
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Also like affine varieties, we have:

Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.

For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables, Theorem 5.3.9. For the converse, use Cartan B mentioned above. (Added: I don't mean to be too pedantic, but Cartan says that $X$ is Stein iff sheaf cohomology vanishes for coherent analytic coefficients; it is certainly not true otherwise.)

Also like affine varieties, we have:

Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.

For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables, Theorem 5.3.9. For the converse, use Cartan B mentioned above.

Also like affine varieties, we have:

Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.

For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables, Theorem 5.3.9. For the converse, use Cartan B mentioned above. (Added: I don't mean to be too pedantic, but Cartan says that $X$ is Stein iff sheaf cohomology vanishes for coherent analytic coefficients; it is certainly not true otherwise.)

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Donu Arapura
Donu Arapura
  • 35.2k
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