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Dec 22, 2021 at 3:10 comment added Jason Starr When $Y$ is the projective line and $X$ and $D$ are both Fano manifolds, the Cohen-Jones-Segal conjecture predicts a Lefschetz property: the usual Lefschetz property for $D$ inside $X$ implies a Lefschetz property for the double loop spaces, and then you apply Cohen-Jones-Segal. Please feel free to e-mail me for more details (I lectured about this a couple of times recently).
Dec 22, 2021 at 2:44 history asked user127776 CC BY-SA 4.0