3
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

See Part I of Sergeev's book... (he talks more about Lie groups) - see also Henrich Falk's diplomarbeit (2009, Berlin) page 26

$\endgroup$
  • $\begingroup$ Could you kindly point out the section I should be looking for, I can't seem to find it. $\endgroup$ – Meer Ashwinkumar Nov 28 '15 at 17:01
  • $\begingroup$ @MeerAshwinkumar An even more explict description is in the second reference... $\endgroup$ – Igor Rivin Nov 28 '15 at 18:39
  • 2
    $\begingroup$ Igor, your references do not really resolve the question which OP is asking. The issue of integrability of the almost complex structure is a delicate one: For the loop space of a 3-dimensional Riemannian manifold (the case I thought about) the answer is that the almost complex structure is formally integrable but, in general, not integrable, see Lempert's paper. What happens when the target is Kahler I am not sure. $\endgroup$ – Misha Nov 28 '15 at 23:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.