Let $M$ be an almost complex manifold of complex dimension $n$. Let $E$ be a complex vector bundle over $M$ with a Hermitian metric $h$. Let $A$ be a Hermitian connection with respect to $h$. Let $d_A$ be the exterior covariant derivative of $A$, and let $F_A$ be the curvature form of $A$. Let $*$ be the Hodge star operator, where it maps $\Omega^{p,q}$ into $\Omega^{n-q,n-p}$. I have several questions about notation and computations using the above objects.
- I recall reading that one cannot use the usual local basis consisting of $\{ dz^i, d\bar{z}^j \}$ for $\Omega^{p,q}$ because of a lack of complex structure. Is this correct?
- Do we then get a (nice enough) local basis $e^1, \ldots, e^n, \bar{e}^1, \ldots, \bar{e}^n$ of $T^* M$, for some forms $\{ e^k \}$ and their complex conjugate? Does this make sense, even though we get $2n$ such forms? Or am I misunderstanding how this works?
- Say $n=2$ for simplicity. If the above is correct, what would $*(e^1 \wedge e^2)$ equal? Would it be $\pm \bar{e}^1 \wedge \bar{e}^2$? Similarly would $*e^1 =\pm e^2 \wedge \bar{e}^1 \wedge \bar{e}^2$? (Modulo some constant factors maybe.)
- As a concrete example I'd like to work out, I know $F_A$ is a $2$-form with $3$ components $F_A^{2,0}, F_A^{1,1}$, and $F_A^{0,2}$ in $\Omega^{2,0}, \Omega^{1,1}$, and $\Omega^{0,2}$ respectively. What is $* d_A * F_A ^{2,0}$ in terms of the above local basis? We can take $n=2$ for simplicity as well to give a clearer picture. (This form arises in the Yang-Mills functional if you're wondering why I'm interested.)
I've yet to find a set of notes or a textbook that covers these cases (1-3) in detail or similar examples, if you could provide a reference to something like this, that would be much appreciated as well.