(1) Let $M$ be a complex manifold of real dimension $2n$, and denote the line bundle of complex $(N,0)$-forms by $\Omega^{(N,0)}(M)$. When $M = CP^N$, the line bundles are indexed by the integers, and so, $\Omega^{(N,0)}(CP^N)$ must correspond to a integer. What is this integer? In the $N=1$ case, the corresponding integer is $2$. This suggests, that in general, $\Omega^{(N,0)}(CP^N)$ corresponds to $2N$. Is this true?

(2) For the anti-canonical spin$^c$ structure of $CP^N$, the spinor bundle is isomorphic to $$ S := (\Omega^{(0,0)}(CP^N) + \cdots + \Omega^{(0,N)}(CP^N)) \otimes S_N, $$ where $S_k$ is the square root of $\Omega^{(N,0)}(CP^N)$ (square root wrt tensoring as multiplication). What does the square root mean when when line bundle integer is odd? In the $N=2$ case, this is seen to reduce to $\cal{E}_{-1} \otimes \cal{E}_1$, where $\cal{E}_p$ is the line bundle corresponding to the integer $p$. Is this the $Z2$ grading on the spinor bundle? If so, what does this look like in higher dimensions?

(3) Finally, for a given spin connection $\nabla$, to define a Dirac operator we would need a Clifford representation, ie a map $$ C:(\Omega^{(1,0)} \oplus \Omega^{(0,1)}) \otimes S \to S. $$ For $N=2$, this should be given by a map $$ C: (\Omega^{(1,0)} \oplus \Omega^{(0,1)}) \otimes (\cal{E}_{-1} \oplus \cal{E}_1) \to \cal{E}_1 \oplus \cal{E}_1. $$ What is this rep? What does it look like for higher order $N$? Note: the first subindex in the second $\cal{E} \oplus \cal{E}$ above should be $-1$, I'm having tex problems when I try to write it as such though.

linecorresponding to the integer $p$"? Do you meanline bundle? In that case $\mathcal E_{-p}\otimes \mathcal E_{p}$ is the trivial line bundle... – Sándor Kovács Nov 5 '10 at 17:23