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I am reading "Complex Geometry" by Huybrechts. He defines an analytic hypersurface of a complex manifold $X$ as an analytic subvariety $Y\subset X$ of codimension one i.e. dim$(Y)=$ dim$(X)-1.$ Then he defines a divisor $D$ on $X$ is a formal linear combination $D=\sum a_i[Y_i]$ with $Y_i\subset X$ irreducible hypersurfaces and $a_i\in\mathbb{Z}$ such that the sum is locally finite. I want to know the definition of irreducible analytic hypersurfaces and why it's important to consider the sum of irreducible hypersurfaces for the definition.

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1.An analytic subvariety $A$ is called reducible if $A=A_1\cup A_2$ where $A_1,A_2$ are also analytic subvarieties in $X$,distinct from $A$,if A cannot be represented in this form,it is called irreducible.

2.The sum is defined formally.It's important because we probably can define and research more,such as the degree of the divisor...Otherwise,we just discuss one single irreducible hypersurface sounds a little boring...

By the way,maybe we could imagine the sumgeometrically as the union of the irreducible divisors occurring with non-zero coefficients and "multiplicities" given by the coefficients $a_i$.(Although it maybe hard to imagine...).

When we consider effective(all coefficents of $[Y_i]$ are non-negative) Cartier divisors on schemes,we can indeed add them by taking the subscheme defined locally by the product of defining equations for these two divisors. This indeed give precise geometric meaning to the formal sum!

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  • $\begingroup$ For $X$ a compact complex manifold then the irreducible hypersurface is represented locally as the vanishing set of an analytic function, but globally it is only one of the zeros of a meromorphic function (it is represented globally by an ideal : (the closure of) the common zero of finitely many meromorphic functions) $\endgroup$
    – reuns
    Jan 10, 2020 at 2:28

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