Gaussian likelihood geometry of projective varieties

We explore the maximum likelihood degree of a homogeneous polynomial $F$ on a projective variety $X$, $\mathrm{MLD}_X(F)$, which generalizes the concept of Gaussian maximum likelihood degree. We show that $\mathrm{MLD}_X(F)$ is equal to the count of critical points of a rational function on $X$, and give different geometric characterizations of it via topological Euler characteristic, dual varieties, and Chern classes.