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I'm told that $-\mathbb{C}P^2$$\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $-\mathbb{C}$$\overline{\mathbb{C}}$ is still a complex manifold and biholomorphic to $\mathbb{C}$.

This makes me wonder, if $X$ is complex manifold is there a general criterion for when $-X$$\overline{X}$ also has a complex structure? For example, it seems that if $X$ is an affine variety than simply replacing $i$ with $-i$ gives $-X$$\overline{X}$ a complex structure and $X, -X$$X, \overline{X}$ are biholomorphic.

EDIT: the last claim is wrong; see BCnrd's comments below and Dmitri's example. Also, as explained by Dmitri and BCnrd, $X$ should be taken to have even complex dimension.

Another question: if $X$ and $-X$$\overline{X}$ both have complex structures, are they necessarily biholomorphic? Edit: No per Dmitri's answer below.

I'm told that $-\mathbb{C}P^2$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $-\mathbb{C}$ is still a complex manifold and biholomorphic to $\mathbb{C}$.

This makes me wonder, if $X$ is complex manifold is there a general criterion for when $-X$ also has a complex structure? For example, it seems that if $X$ is an affine variety than simply replacing $i$ with $-i$ gives $-X$ a complex structure and $X, -X$ are biholomorphic.

EDIT: the last claim is wrong; see BCnrd's comments below and Dmitri's example. Also, as explained by Dmitri and BCnrd, $X$ should be taken to have even complex dimension.

Another question: if $X$ and $-X$ both have complex structures, are they necessarily biholomorphic? Edit: No per Dmitri's answer below.

I'm told that $\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $\overline{\mathbb{C}}$ is still a complex manifold and biholomorphic to $\mathbb{C}$.

This makes me wonder, if $X$ is complex manifold is there a general criterion for when $\overline{X}$ also has a complex structure? For example, it seems that if $X$ is an affine variety than simply replacing $i$ with $-i$ gives $\overline{X}$ a complex structure and $X, \overline{X}$ are biholomorphic.

EDIT: the last claim is wrong; see BCnrd's comments below and Dmitri's example. Also, as explained by Dmitri and BCnrd, $X$ should be taken to have even complex dimension.

Another question: if $X$ and $\overline{X}$ both have complex structures, are they necessarily biholomorphic? Edit: No per Dmitri's answer below.

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solbap
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I'm told that $\overline{\mathbb{C}P^2}$$-\mathbb{C}P^2$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $\overline{\mathbb{C}}$$-\mathbb{C}$ is still a complex manifold and biholomorphic to $\mathbb{C}$.

This makes me wonder, if $X$ is complex manifold is there a general criterion for when $\overline{X}$$-X$ also has a complex structure? For example, it seems that if $X$ is an affine variety than simply replacing $i$ with $-i$ gives $\overline{X}$$-X$ a complex structure and $X, \overline{X}$$X, -X$ are biholomorphic.

EDIT: the last claim is wrong; see BCnrd's comments below and Dmitri's example. Also, as explained by Dmitri and BCnrd, $X$ should be taken to have even complex dimension.

Another question: if $X$ and $\overline{X}$$-X$ both have complex structures, are they necessarily biholomorphic? Edit: No per Dmitri's answer below.

I'm told that $\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $\overline{\mathbb{C}}$ is still a complex manifold and biholomorphic to $\mathbb{C}$.

This makes me wonder, if $X$ is complex manifold is there a general criterion for when $\overline{X}$ also has a complex structure? For example, it seems that if $X$ is an affine variety than simply replacing $i$ with $-i$ gives $\overline{X}$ a complex structure and $X, \overline{X}$ are biholomorphic.

EDIT: the last claim is wrong; see BCnrd's comments below and Dmitri's example. Also, as explained by Dmitri and BCnrd, $X$ should be taken to have even complex dimension.

Another question: if $X$ and $\overline{X}$ both have complex structures, are they necessarily biholomorphic?

I'm told that $-\mathbb{C}P^2$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $-\mathbb{C}$ is still a complex manifold and biholomorphic to $\mathbb{C}$.

This makes me wonder, if $X$ is complex manifold is there a general criterion for when $-X$ also has a complex structure? For example, it seems that if $X$ is an affine variety than simply replacing $i$ with $-i$ gives $-X$ a complex structure and $X, -X$ are biholomorphic.

EDIT: the last claim is wrong; see BCnrd's comments below and Dmitri's example. Also, as explained by Dmitri and BCnrd, $X$ should be taken to have even complex dimension.

Another question: if $X$ and $-X$ both have complex structures, are they necessarily biholomorphic? Edit: No per Dmitri's answer below.

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solbap
solbap
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I'm told that $\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $\overline{\mathbb{C}}$ is still a complex manifold and biholomorphic to $\mathbb{C}$.

This makes me wonder, if $X$ is complex manifold is there a general criterion for when $\overline{X}$ also has a complex structure? For example, it seems that if $X$ is an affine variety than simply replacing $i$ with $-i$ gives $\overline{X}$ a complex structure and $X, \overline{X}$ are biholomorphic. Do all Stein manifolds

EDIT: the last claim is wrong; see BCnrd's comments below and Dmitri's example. Also, as explained by Dmitri and BCnrd, $X$ should be taken to have this property?even complex dimension.

Another question: if $X$ and $\overline{X}$ both have complex structures, are they necessarily biholomorphic?

I'm told that $\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $\overline{\mathbb{C}}$ is still a complex manifold and biholomorphic to $\mathbb{C}$.

This makes me wonder, if $X$ is complex manifold is there a general criterion for when $\overline{X}$ also has a complex structure? For example, it seems that if $X$ is an affine variety than simply replacing $i$ with $-i$ gives $\overline{X}$ a complex structure and $X, \overline{X}$ are biholomorphic. Do all Stein manifolds have this property?

Another question: if $X$ and $\overline{X}$ both have complex structures, are they necessarily biholomorphic?

I'm told that $\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $\overline{\mathbb{C}}$ is still a complex manifold and biholomorphic to $\mathbb{C}$.

This makes me wonder, if $X$ is complex manifold is there a general criterion for when $\overline{X}$ also has a complex structure? For example, it seems that if $X$ is an affine variety than simply replacing $i$ with $-i$ gives $\overline{X}$ a complex structure and $X, \overline{X}$ are biholomorphic.

EDIT: the last claim is wrong; see BCnrd's comments below and Dmitri's example. Also, as explained by Dmitri and BCnrd, $X$ should be taken to have even complex dimension.

Another question: if $X$ and $\overline{X}$ both have complex structures, are they necessarily biholomorphic?

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solbap
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