2
$\begingroup$

Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My question is: do the spaces of G-invariants of those two spaces have equal dimension? Of course one should assume something about the nature of the action of G on X, in particular so that both dimensions are finite. If G acts transitively, an invariant secion is automatically smooth, and this is a correct and easy statement . What if G acts with finitely many orbits which are geometrically reasonable? What if we are in an algebraic variety-algebraic group setting?

$\endgroup$
7
  • $\begingroup$ I am not sure what you mean in your last sentence, but in the setting of algebraic geometry this is definitely false : a very ample line bundle has a lot of sections while its dual has none. $\endgroup$
    – abx
    Jan 26, 2015 at 7:01
  • $\begingroup$ Could you please give an example over the reals? $\endgroup$
    – Dima
    Jan 26, 2015 at 9:04
  • $\begingroup$ Take the projective space and the line bundle $\mathcal{O}(1)$. $\endgroup$
    – abx
    Jan 26, 2015 at 9:36
  • $\begingroup$ Sorry, I don't understand. What is the group $G$ in this example? $\endgroup$
    – Dima
    Jan 26, 2015 at 14:57
  • $\begingroup$ The trivial group. $\endgroup$
    – abx
    Jan 26, 2015 at 15:16

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.