normal complex analytic variety

Let $V$ be a local complex analytic variety (or a complex analytic space). A point $p\\in V$ is if and only if every weakly holomorphic function through $V$ extends to beholomorphic in $V$ near $p .$

In particular, if $V\\subset{\\mathbb{C}}^{n}$ is a complex analytic subvariety, it is normal at $p$ if and only if every weakly holomorphic function through $V$ extends to be holomorphic in a neighbourhood of $p$ in ${\\mathbb{C}}^{n}$ .

To see that this definition is equivalent to the usual one, that is, that $V$ is normalat $p$ if and only if ${\\mathcal{O}}_{p}$ (the ring of germs of holomorphic functions at $p$ )is integrally closed, we need the following theorem. Let ${\\mathcal{M}}_{p}$ be the total quotient ring of ${\\mathcal{O}}_{p}$ , that is, the ring of germs of meromorphic functions.

Theorem.

Let $V$ be a local complex analytic variety. Then ${\\mathcal{O}}_{p}^{w}(V)$ is the integralclosure of ${\\mathcal{O}}_{p}(V)$ in ${\\mathcal{M}}_{p}.$

References

1 Hassler Whitney..Addison-Wesley, Philippines, 1972.

单词	NormalComplexAnalyticVariety
释义	normal complex analytic variety Let $V$ be a local complex analytic variety (or a complex analytic space). A point $p\\in V$ is if and only if every weakly holomorphic function through $V$ extends to beholomorphic in $V$ near $p .$ In particular, if $V\\subset{\\mathbb{C}}^{n}$ is a complex analytic subvariety, it is normal at $p$ if and only if every weakly holomorphic function through $V$ extends to be holomorphic in a neighbourhood of $p$ in ${\\mathbb{C}}^{n}$ . To see that this definition is equivalent to the usual one, that is, that $V$ is normalat $p$ if and only if ${\\mathcal{O}}_{p}$ (the ring of germs of holomorphic functions at $p$ )is integrally closed, we need the following theorem. Let ${\\mathcal{M}}_{p}$ be the total quotient ring of ${\\mathcal{O}}_{p}$ , that is, the ring of germs of meromorphic functions. Theorem. Let $V$ be a local complex analytic variety. Then ${\\mathcal{O}}_{p}^{w}(V)$ is the integralclosure of ${\\mathcal{O}}_{p}(V)$ in ${\\mathcal{M}}_{p}.$ References 1 Hassler Whitney..Addison-Wesley, Philippines, 1972.
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