t-SNE - how variance is set and how it affects dense vs sparse clusters in HD space

When learning about t-SNE, I found a resource saying width of the normal curve (a gaussian centered at $x_i$) depends on the density of data near the point of interest. Which is why we do the normalization with $\sum_{k\neq i} e^{(-||x_i - x_k||^2/2\sigma^2)}$ in $p_{j|i}= \frac { e^{(-||x_i - x_j||^2/2\sigma^2)}} {\sum_{k\neq i} e^{(-||x_i - x_k||^2/2\sigma^2)}}$.

I know that the gaussian's width depends on the variance, ${\sigma}^2$. However there was no mention of calculating the variance and I read that variance is set as input to the algorithm (using 'perplexity'). So I feel a bit confused.

Are we calculating the variance for each point/gaussian (still depending on perplexity somehow) and getting that for denser clusters it is smaller and for sparser clusters greater? If that is so, I get why the normalization argument makes sense..