# Does Kähler structure on X imply Kähler structure on the loop space of X?

Does Kähler structure on $X$ imply Kähler structure on the loop space ($LX$) of $X$? Since the loop space of $X$ is the space of maps from the circle $S^1$ to $X$, I suspect one may use the pullback via the evaluation map $e:LX\rightarrow X$ of the closed Kähler form $\omega$ on $X$ to obtain a closed two-form $e^*\omega$ on $LX$. Am I correct?