Euler characteristic of the complex torus

Question

In his book An Introduction to Invariants and Moduli, Mukai claims (Chapter 12.2) that $$\chi(X,L) = \frac{(c_1(L))^n}{n!}$$ for any line bundle $L$ on the complex torus $X$. Now, it is clear that $T_X$ is trivial, so by HRR: $$\chi(X,L) = \int_{X} e^{c_1(L)} = \int_X \sum_{m\geq 0} \frac{(c_1(L))^m}{m!}.$$ But why does it follows that this is equal to $\frac{(c_1(L))^n}{n!}$? In particular, what is $n$? Is it the degree of the line bundle?

Answer 1

That $\int_X$ means pairing with the fundamental class of $X$, which is a homology class in $H_{2n}(X)$ where $n$ is the complex dimension of $X$. This pairing only sees the part of the integrand in $H^{2n}(X)$ and the rest is irrelevant; because $T_X$ is trivial this is $\frac{c_1(L)^n}{n!}$ only. (Strictly speaking there's a slight abuse of notation going on here where Mukai is identifying a Chern class with a Chern number.)