Euler characteristic and intersection number

Question

Let $X$ be a complex projective manifold of pure (complex) dimension $n$. Denote its canonical line bundle by $K_X$. Is there a relation between $\int_Xc_1(K_X)^n$ and the topological Euler characteristic of $X$? Because both are integer invariants of the manifold, I would guess that they agree.

Answer 1

Note that $c_1(K_X) = -c_1(X)$ so $\int_Xc_1(K_X)^n = (-1)^n\int_Xc_1(X)^n$.

When $n = 1$, we have $\int_Xc_1(K_X) = -\int_Xc_1(X) = -\int_Xe(X) = -\chi(X)$ where $e(X)$ denotes the Euler class of $X$.

When $n = 2$, we have $\int_Xc_1(K_X)^2 = \int_Xc_1(X)^2 = 2\chi(X) + 3\sigma(X)$ where $\sigma(X)$ denotes the signature of $X$.

When $n \geq 3$, the quantity $\int_Xc_1(K_X) = (-1)^n\int_Xc_1(X)^n$ is not topological (i.e., for such $n$, it depends on the complex structure, not just the underlying manifold $X$). This follows from Theorem 2 of Topologically invariant Chern numbers of projective varieties by Kotschick. So any relationship between $\int_Xc_1(K_X)^n$ and $\chi(X)$ would have to involve other terms which depend on the complex structure.