# Euler characteristic and intersection number [closed]

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.

• I wonder why this question was closed. I have observed a history of a particular user trying to close questions without any reason. Commented Oct 27, 2023 at 5:56
• Dear @CraniumClamp, my question is closed by the community because it lacks background and motivation, relevant definitions, source, possible strategies, my current progress, why the question is interesting or important, etc.
– Doug
Commented Oct 27, 2023 at 8:58
• Those are silly reasons - is what I was trying to say. Commented Oct 27, 2023 at 17:39

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.