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Misha Verbitsky
Misha Verbitsky
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Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic? I can deduce this statement from Remmert-Stein theorem when $\dim_R S < 2 \dim_R Z$$2\dim_R S < \dim_R Z$, where $S$ is the singular set of $Z$. Also I can deduce this statement from Skoda-El Mir theorem when $S$ is pluripolar. I suspect that it should be true in bigger generality, maybe always.

Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic? I can deduce this statement from Remmert-Stein theorem when $\dim_R S < 2 \dim_R Z$, where $S$ is the singular set of $Z$. Also I can deduce this statement from Skoda-El Mir theorem when $S$ is pluripolar. I suspect that it should be true in bigger generality, maybe always.

Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic? I can deduce this statement from Remmert-Stein theorem when $2\dim_R S < \dim_R Z$, where $S$ is the singular set of $Z$. Also I can deduce this statement from Skoda-El Mir theorem when $S$ is pluripolar. I suspect that it should be true in bigger generality, maybe always.

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Misha Verbitsky
Misha Verbitsky
  • 9.2k
  • 1
  • 28
  • 48

Real analytic subvariety in complex manifold which is complex outside of its singular set

Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic? I can deduce this statement from Remmert-Stein theorem when $\dim_R S < 2 \dim_R Z$, where $S$ is the singular set of $Z$. Also I can deduce this statement from Skoda-El Mir theorem when $S$ is pluripolar. I suspect that it should be true in bigger generality, maybe always.