Degeneration of spectral sequences and complex Lagrangian submanifolds

There is a local-to-global $\mathrm{Ext}$ spectral sequence $\mathrm{E}_2^{p,q} = \mathrm{H}^p(\mathrm{L}, \Omega^q_\mathrm{L}) \Rightarrow \mathrm{Ext}^{p+q}(i_*\mathscr{O}_\mathrm{L}, i_*\mathscr{O}_{\mathrm{L}})$ for a smooth Lagrangian subvariety in a hyperk\"ahler variety. We prove its degeneration on $\mathrm{E}_{2}$, and various generalisations thereof.