# Complex structure on the set of prequantization line bundles

+ 2 like - 0 dislike
687 views

For geometric quantization, the set of equivalence classes of prequantization line bundles of a quantizable symplectic manifold $(M, ω)$ is parametrized by $H^1(M, S^1)$ which represents the equivalence classes of ﬂat Hermitian line bundles.

When $M$ is a compact Riemann surface, we can construct a complex structure on $H^1(M, \mathbb{R})$ by Hodge $*$ operator and this structure induces a complex structure on $H^1(M, S^1)=H^1(M, \mathbb{R})/H^1(M, \mathbb{Z})$.

My question is as follows:

Does $H^1(M, S^1)$ admit a complex structure when $M$ is a compact Kähler manifold?

If not, which condition $M$ should satisfy to construct a complex structure on $H^1(M, S^1)$?

This post imported from StackExchange MathOverflow at 2014-10-14 10:50 (UTC), posted by SE-user Bo Liu
retagged Oct 14, 2014
Yes. The complex structure on $H^1(M,\mathbb{R})$ can be defined via the Hodge decomposition $H^1(M,\mathbb{R})\otimes \mathbb{C}=H^{1,0}\oplus H^{0,1}$, and this holds on any compact Kähler manifold.

This post imported from StackExchange MathOverflow at 2014-10-14 10:50 (UTC), posted by SE-user abx
Thank you! For this case, could the complex structure on $H^1(M,\mathbb{R})$ give a complex structure on $H^1(M,\mathbb(S^1))$? And are there some references about this?

This post imported from StackExchange MathOverflow at 2014-10-14 10:50 (UTC), posted by SE-user Bo Liu
Yes to the first question. Reference: any introduction to complex geometry, e.g. Griffiths-Harris or Voisin.

This post imported from StackExchange MathOverflow at 2014-10-14 10:50 (UTC), posted by SE-user abx
I see, topologically, $H^1(M,S^1)$ here is just the Jacobian $\mathrm{Pic}^0(M)$, the kernel of $C_1:\mathrm{Pic}(M)\rightarrow H^2(M,\mathbb{Z})$, a complex torus of dimension b_1(M). Is it right?

This post imported from StackExchange MathOverflow at 2014-10-14 10:50 (UTC), posted by SE-user Bo Liu
Perfectly right.

This post imported from StackExchange MathOverflow at 2014-10-14 10:50 (UTC), posted by SE-user abx

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar\varnothing$sicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.