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My first question concerning definition of Kaehlerian immersion. I'm motivated by the paper of V. Cortes-Special Kaehler manifolds: a survey. It is not clear from that paper how that immersion looks like, for example if we want to immerse $C^{2}$ to $C^{4}$. Is Kaehler immersion same as totally complex immersion. I can construct many examples of Lagrangian immersions but I want to construct Lagrangian immersion which is at the same time Kaehlerian then there is a problem. The next question is about construction of a Kaehlerian-Lagrangian immersion of a complex manifold into the cotangent bundle of a complex projective space in the same way as in the paper above. Is that possible, or there are some limitations?

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    $\begingroup$ Have you taken into account that Cortes's 'Kähler' manifolds are actually pseudo-Kähler? The point is that the ambient metric is not positive definite; in fact, it is of split type in his case, so there is no problem finding complex submanifolds on which the symplectic form vanishes (i.e., that are Lagrangian). Also, 'Kählerian immersion' is not quite the same as 'totally complex', because of the requirement that the metric pull back to be nondegenerate. If the ambient metric were positive definite, then, of course, one could not have submanifolds that are both complex and Lagrangian. $\endgroup$ May 6, 2013 at 0:06
  • $\begingroup$ @Robert I thought that Kaehler and totally complex immersions are equivalent because of Proposition 6 in the paper "Special complex manifolds", Alekseevsky, Cortes, Devchand. Is there a reference where we can see the examples of those immersions? $\endgroup$
    – max_lidia
    May 8, 2013 at 10:47

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