Say $Z$ is an anlatyic set in a complex manifold $M$. Is it true that irreducible components of $Z$ are just the closures in $Z$ of connected components of $Z\backslash sing(Z)$?
I need this for a paper I am reading. In general, can someone reccomend a good book about complex analytic geometry? I don't want to waste a lot of time figuring these things out without at least a good source for all these notions.
Thank you! Benny
[1] The answer to your question is yes.There is an important theorem on decomposition analytic subset into irreducible components:(Cf E.M.Chirka's Complex Analytic Sets chapter 5.4.Theorem1)
Let $A$ be an analytic subset of a complex manifold $\Omega$.Then:
(1)every irreducible component $A$ has the form $\overline S$,where $S$ is a connected component of $regA=A\backslash singA$;
(2)if $reg A=\bigcup_{j\in J}S_j$ is the decomposition into connected connected components ($J$ is finite or countable, $S_j\bigcap S_k=\varnothing$ for $j\not= k$),then $A=\bigcup_{j\in J}\overline {S_j}$,and this is the decomposition of $A$ into ieeducible components;
(3) the decomposition of $A$ into irreducible components is locally finite,i.e.for every compact $K\subset\Omega$ there is only a finite (or empty) set of indices $j\in J$ such that $K\bigcap\overline {S_j}$ is nonempty.
[2] Demailly's Complex Analytic and Differential Geometry is wonderful.I also recommend E.M.Chirka's Complex Analytic Sets,it's worth reading and full of a lot of classical theorems.