Chern-Weil Theory for $p_1$

Question

I'm studying Riemannian manifolds that admit an almost-complex structure, thus $$3\tau+2\chi=c_1^2,$$ where $\tau$ is the signature, $\chi$ is the euler characteristic and $c_1$ is the first Chern class.

I know from Hizerbruch theorem that $$3\tau=p_1,$$ where $p_1$ is the first Pontryagin class. Notice that $\chi=c_2$, thus $$p_1=c_1^2-2c_2.$$ But I know from Chern-Weil Theory that $$c_1=\frac{iTr(F_A)}{2\pi},$$

$$c_2=\frac{Tr(F_A^2)-Tr(F_A)^2}{8\pi^2}$$ and $$p_1=-\frac{Tr(F_A^2)}{8\pi^2}$$. Here, $F_A^2=F_A\wedge F_A$ and $c_1^2=c_1\wedge c_1$. Computing, we have: $$c_1^2-2c_2=-\frac{Tr(F_A)^2}{4\pi^2}+\frac{Tr(F_A)^2-Tr(F_A^2)}{4\pi^2}=-\frac{Tr(F_A^2)}{4\pi^2}=2p_1.$$ What did I miss?

Answer 1

It follows from the definition $p_k(E) := (-1)^kc_{2k}(E \otimes \mathbb{C})$ that the formula $$p_1 = c_1^2 - 2c_2$$ is valid for all complex vector bundles (rather than just the tangent bundle of an almost complex surface).

The problem in your Chern-Weil calculation is essentailly that you are confusing the trace of a complex and a real matrix. A complex $n \times n$ matrix $M$ can be turned into a real $2n \times 2n$ matrix $M_\mathbb{R}$, and the traces are related by $$ {\rm Tr} \, M_\mathbb{R} = 2 \, {\rm Re} \, {\rm Tr} \, M . $$ If you take a complex connection $A$ on a complex vector bundle $E$, and let $A_\mathbb{R}$ be the induced connection on the underlying real vector bundle $E_\mathbb{R}$, then the correct interpretation of the Chern-Weil formula is that $p_1(E_\mathbb{R})$ is represented by $$ -\frac{{\rm Tr}(F_{A_\mathbb{R}}^2)}{8\pi^2} = -{\rm Re}\frac{{\rm Tr}(F_A^2)}{4\pi^2} . $$ Since the cohomology class ${\rm Tr}(F_A^2)$ is real that corrects the final step in your calculation.