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Affinization map of cotangent bundle is proper/projective?

Given a cotangent bundle of a projective variety $Y=T^*X,$ do we know that its affinization map, $$Y \rightarrow \mathrm{Spec}(H^0(Y,\mathscr{O}_Y))$$ is proper or projective?
3 votes
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99 views

Displacing a conormal Lagrangian from the zero section

I was told that the conormal bundle $\nu^*K$ of a knot $K\subset S^3$ can be displaced from the zero section $0_{S^3}$ in $T^*S^3.$ Having no intuition about whether/how often this happens in general, …