Multivariate normal distribution for integral points on varieties

Given a variety over $\mathbb{Q}$, we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erd\H{o}s-Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.