# Equivalent definitions of normality for complex algebraic varieties

In Kollár's article The structure of algebraic threefolds: an introduction to Mori's program he gives the following definition of a normal variety:

Definition 5.4. Let $$V \subset \mathbb{C}^n$$ be an algebraic variety and $$v \in V$$. We say $$V$$ is normal at $$v$$ if every rational function bounded in some neighborhood of $$v$$ is regular at $$v$$.

This is in contrast to the definition of normal which I have seen everywhere else: the point $$v \in V$$ is normal if the local ring $$\mathcal{O}_{V,v}$$ is integrally closed. Of course the first question that came to mind is whether these two concepts are equivalent. I intuitively see that if for the rational function $$g/h$$ we have that $$|(g/h)(w)| \to \infty$$ as $$w \to v$$, then said function cannot satisfy a monic polynomial relation with regular coefficients (although I am not 100% sure how to prove this). However the converse seems very misterious to me. Is this the way to prove they are equivalent or am I following the wrong path? Are they equivalent at all?

• Welcome new contributor. I recommend that you search for the key phrase "Zariski's Main Theorem". The discussion in Mumford's "Red Book" is particularly useful. Feb 3 at 20:44
• One thing which is relevant is that on normal varieties rational functions are regular away from their divisors of poles (see `Structure theorem' in III.8 in Mumford's red book), hence if a rational function is bounded on an open set, this set does not intersect the divisors of poles, and the function is regular. Feb 3 at 22:36
• See also Exercise 4.25 on page 141 in Eisenbud's book on commutative algebra. Feb 3 at 22:55

The fact that the integral closure is contained in the weakly holomorphic functions follows from an elementary property of roots of monic polynomials. The proof of the other inclusion uses the local parametrization theorem (which corresponds to the Noether normalization in the algebraic setting) and the Riemann extension theorem which says that that any function which is holomorphic on $$U \setminus A$$ and locally bounded near $$A$$, where $$U \subseteq \mathbb{C}^n$$ is open, and $$A$$ is an analytic subset of positive codimension.