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