This is standard material but since we are considering the issue of kurtosis, I thought to include this computation. For lognormal one has . So the moments
And kurtosis is
In other words here kurtosis increases with variance of $\log(\sigma_t)$. The kurtosis of returns is produced by any stochastic volatility model. Our concern is at the moment kurtosis of volatility jumps themselves.
In particular, this argument can be used to show that if then because that
In particular the variance of volatility measures kurtosis of returns.