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In the excellent book From Holomorphic Functions to Complex Manifolds by Klaus Fritzsche and Hans Grauert, if I follow the definition and properties of analytic subsets and the definition of a complex submanifold $A$ of a complex manifold $X$, then $A$ is necessary closed in $X$ :

"From the definition (of analytic subset) it follows that $A$ is a closed subset of $X$."

"An analytic set $A\subset X$ is called regular of codimension $d$ at a point $p\in A$ if..."

"If $A$ is regular at every point, $A$ is called a complex submanifold of $X$."

However, a few pages further, there are a few exercises with mentions of "closed submanifolds". For example, one of them starts like this : "Let $f:X\to Y$ be a holomorphic map and $Z\subset Y$ a closed submanifold..."

Am I missing something?

Thanks in advance for any information.

Philippe Malot
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  • You didn't give us the definition in the book, but merely what apparently they say follows from the definition. My guess is that somehow you've taken that line out of context. Surely any definition should allow that if $U\subset X$ is an analytic open subset, then $U$ is a a complex submanifold. – Matt Apr 25 '13 at 21:41
  • It is conceivable that ‘submanifold’ means closed submanifold. I attended a lecture course on complex manifolds using that definition. – Zhen Lin Apr 25 '13 at 21:46
  • Check p160-161 here : http://books.google.fr/books?id=jSeRz36zXIMC&printsec=frontcover&dq=from%20holomorphic%20functions%20to%20complex%20manifolds&hl=en&sa=X&ei=y6R5UfrdBYiX7QaPpICgAg&ved=0CC0Q6AEwAA#v=onepage&q=analytic%20subsets&f=false – Philippe Malot Apr 25 '13 at 21:51
  • I see. Analytic implies closed because you are (locally) the zero set of analytic (and hence continuous) functions. It is possible that $U$ is open and analytic, but this also implies that $U$ is a whole connected component of $X$. I assume the emphasis on "closed" is just because some authors (maybe even these ones in other parts of the book?) talk separately about "submanifolds," "complex submanifolds," and "analytic subsets." I couldn't find this exercise, but presumably you can use this weaker notion of closed submanifold. – Matt Apr 25 '13 at 22:41

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