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This is true. Alex Isaev pointed out to me that it is Theorem 4.6 from `A geometric criterion for algebraic varieties' by Rudin. It also maybe follows from proposition A local version is proposition 4 of Serre's GAGA paper combined with Cartan's theorem A.

This is true. Alex Isaev pointed out to me that it is Theorem 4.6 from `A geometric criterion for algebraic varieties' by Rudin. It also maybe follows from proposition 4 of Serre's GAGA paper combined with Cartan's theorem A.

This is true. Alex Isaev pointed out to me that it is Theorem 4.6 from `A geometric criterion for algebraic varieties' by Rudin. A local version is proposition 4 of Serre's GAGA paper.

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This is true. Alex Isaev pointed out to me that it is Theorem 4.6 from `A`A geometric criterion for algebraic varieties' by Rudin. It also maybe follows from proposition 4 of Serre's GAGA paper combined with Cartan's theorem A.

This is true. Alex Isaev pointed out to me that it is Theorem 4.6 from `A geometric criterion for algebraic varieties' by Rudin. It also maybe follows from proposition 4 of Serre's GAGA paper.

This is true. Alex Isaev pointed out to me that it is Theorem 4.6 from `A geometric criterion for algebraic varieties' by Rudin. It also maybe follows from proposition 4 of Serre's GAGA paper combined with Cartan's theorem A.

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This is true. Alex Isaev pointed out to me that it is Theorem 4.6 from `A geometric criterion for algebraic varieties' by Rudin. It also maybe follows from proposition 4 of Serre's GAGA paper.