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Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if

  1. $P$ is Lagrangian

  2. P involutive

  3. dim$P\cap\bar P \cap TM$ is constant.

Let $P_1, P_2$ be two complex polarization, then when $\bar P_1\cap P_2$ is involutiove ?

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  • $\begingroup$ Dear @Hassan Jolany, I cannot see how the tags 'quantum-mechanics', 'mp.mathematical-physics', and 'gt.geometric-topology' are relevant for your question. Are you sure they are required? $\endgroup$ Mar 9, 2014 at 16:36
  • $\begingroup$ In fact this question come from Blattner–Kostant–Sternberg pairing in Geometric quantization. That is why , I added quantum mechanics. But I remove the tag of geometric topology $\endgroup$
    – user21574
    Mar 9, 2014 at 17:07
  • $\begingroup$ Yes, you right.My conjecture is if,$P_1,P_2$ be positive then $\bar P_1\cap P_2$ is involutive $\endgroup$
    – user21574
    Mar 10, 2014 at 11:45
  • $\begingroup$ Is Hassan Jolany no longer with us ? There are now two users $1234$, see mathoverflow.net/users/45187/1234 and mathoverflow.net/users/21574/1234. $\endgroup$ Mar 13, 2014 at 19:51

1 Answer 1

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I may be missing something, but isn't it completely obvious that $\bar P_1\cap P_2$ is always involutive?

Indeed, if $\xi$, $\eta$ are two complex vector fields taking values in $\bar P_1\cap P_2$, then $[\xi,\eta]$ takes its values in $\bar P_1$ because $\bar P_1$ is involutive, and in $P_2$ because $P_2$ is involutive, hence it takes values in $\bar P_1\cap P_2$, no?

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