Both the Fano variety $F$ of lines in a general cubic fourfold and the "Debarre–Voisin" fourfolds $Y_\sigma$ introduce in [DV] are smooth, four-dimensional, hyper-Kähler subvarieties of Grassmannians (G(2,6) and G(6,10), respectively).
Strikingly, both satisfy a similar Chern class identity: \begin{align*}c_2(\mathcal{T}_F) &= 5 c_1^2(\mathscr{E}_2) - 8 c_2(\mathscr{E}_2) \\ c_2(\mathcal{T}_{Y_\sigma}) &= 5 c_1^2(\mathscr{E}_6) - 8 c_2(\mathscr{E}_6) \quad \text{(see [DV, Lem. 4.5])}\end{align*} Where $\mathscr{E}_k$ is (the appropriate restriction of) the dual of the tautological subbundle on the Grassmannian $G(k,n)$.
Does this have an easy explanation? Thank you.