Question

Let $X$ be an algebraic variety, $S \subset X$ its singular locus, and $x \in S$. Say that $x$ is good if for any neighborhood $U$ of $x$, any top differential form $\omega$ on $U \setminus S$ and some resolution of singularities $U' \rightarrow U$, the pullback of $\omega$ to $U'$ is regular.

An obvious necessary condition is that $S$ is of co-dimension at least 2. An example of a good point is $(0,0,0)$ in the variety defined by $x^2+y^2+z^2=0$. An example of a singular point which is not good is $(0,0,0)$ in the variety defined by $x^4+y^4+z^4=0$.

Here is the question:

Is the notion of good point related to other properties of singular points?

We are mainly interested in the case when $X$ is complete intersection.

Answer 1

The condition you want is very close to having rational singularities.

Definition: A variety $X$ of characteristic zero has rational singularities if it is:

1. Cohen-Macaulay and,
2. for some resolution of singularities $\pi : Y \to X$ we have $\pi_* \omega_Y = \omega_X$.

Now, let me explain why condition 2. is essentially the one you described. If $S$ is codimension 2, then $\omega_X$ is actually defined to be the pushforward of $\Omega^{\dim X}_{X\setminus S}$. In other words, $\omega_X$ is exactly those top forms coming from the smooth locus. Since $Y$ is a resolution and hence smooth, $\Omega^{\dim Y}_Y = \omega_Y$.

The statement that $\pi_* \omega_Y = \omega_X$ simply says that any form from the smooth locus of $X$, also comes from a form on $Y$. Which is exactly what you want.

It is worth noting that this definition is independent of the choice of resolution. In particular, if it holds for one resolution, it holds for all.

For further reading I would suggest Singularities of Pairs by János Kollár.