The following is Huybrechts' Complex Geometry, Exercise 4.4.2:
Show that for a base-point free line bundle $L$ on a compact complex manifold $X$ the integral $\int_X c_1(L)^n$ is non-negative.
I've been considering the vector bundle $E = L^{\oplus n}$, where $n$ is the dimension of $M$. Suppose $L$ has $m$ global sections which form a base-point free linear system. For $m \le n$, $E$ has a nowhere vanishing global section, whence we see that it contains the trivial bundle as a sub-bundle. Consequently by the Whitney product formula, $c_n(E) = 0$. Moreover, $c(E) = (1 + c_1(L))^n$, whence $c_1(L)^n = 0$ and thus the required integral is $0$.
However, I'm not sure what to do $m>n$. To that end, I would be grateful if someone could provide a hint.
Thanks.
Best Answer
I think it is better to think of the situation geometrically rather than trying to apply the Whitney sum formula.
The base point free line bundle $ L $ induces a holomorphic map $ f : X \rightarrow P = \mathbb{P} (H^0(X,L)^{\vee}) $ such that $ L = f^* O(1) $, where $ O(1) $ is the dual of the tautological line bundle on $ P $. So $ c_1(L) = f^* c_1(O(1)) $ by functoriality of the Chern class. But $ c_1(O(1)) $ is given by the cohomology class of the Fubini-Study form $ \omega_{FS} $ on $ P $, a positive definite form. So the question reduces to show that $ \int_X f^* (\omega_{FS}^n) \ge 0 $ which is clear enough: locally this form on $ X $ is either zero (in the case when $ df $ is not injective, i.e. in the case of a critical point) or takes positive values (in the case $ df $ is injective so that $ f $ is a local diffeomorphism onto its image in the neighborhood of a point).
Edit: I made a little mistake. $ c_1(O(1)) $ is $ i /2 \pi $ times the curvature form, so is $ \omega_{FS} $. The factor is subsumed. The proof goes through without any change though.