Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Complex structure on the set of prequantization line bundles

+ 2 like - 0 dislike
1027 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
asked Jul 26, 2014 in Theoretical Physics by Bo Liu (10 points) [ no revision ]
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

Your answer

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):
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$\hbar$ysicsOve$\varnothing$flow
Then 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).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...