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For a compact Kähler manifold of dimension $2n$, is there a classification of the homological $n$-cycles which are supported in a compact Lagrangian submanifold?

The main example for this question is a projective manifold with the symplectic structure coming from Fubini-study metric. Using symplectic Picard-Lefschetz theory one can describe many Lagrangians which are vanishing cycles, and explicit computations of the monodromy group helps a lot.

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    $\begingroup$ I just heard recently a lecture by Jake Salomon in which it is stated that the question whether the classes of Lagransian submanifolds generate the whole middle cohomology for CY manifolds is open, and is sort of Mirror Symmetric to the Hodge conjecture on algebraic cycles, so it should't be easy. I'm not sure if this is precisely what you mean or if I understood correctly what he said, though. $\endgroup$
    – S. carmeli
    Commented Apr 24, 2018 at 16:25
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    $\begingroup$ For n=2 this question was studied by Schoen-Wolfson. The answer is yes if certain conical singularities are allowed. $\endgroup$
    – Chris Woodward
    Commented Apr 25, 2018 at 4:17
  • $\begingroup$ Thanks a lot. Actually, the question arose after I took a course in Hodge theory by H. Movasati (see w3.impa.br/%7Ehossein/myarticles/hodgetheory.pdf). So the idea come what would be the equivalent of Hodge conjecture in symplectic geometry. Even there is no conjectural description of such cycles. $\endgroup$
    – D. L Garcia
    Commented Apr 26, 2018 at 18:39

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