Let $(X, \omega)$ be a compact Kähler manifold. The cohomology class represented by $$\text{Ric}(\omega) = \frac{1}{2\pi} \text{Ric}_{i \overline{j}} dz^i \wedge d\overline{z}^j$$ is called the first Chern class of $X$, denoted $c_1(X) : = c_1(-K_X)$, where $-K_X$ is the anti-canonical bundle of $X$. We say that $(X, \omega)$ is Fano if $c_1(X)$ is positive, i.e., the representative: $\text{Ric}_{i\overline{j}}$ is a positive-definite matrix.
The existence of Kähler--Einstein metrics on Fano manifolds is known to be equivalent to an algebro-geometric notion of stability by the work of Tian and Chen--Donaldson--Sun. Moreover, there are obstructions given by Matsushima and Futaki relating to the Lie algebra of holomorphic vector fields.
Question: Are there topological obstructions to the existence of Kähler--Einstein metrics on Fano manifolds?