Towards homological projective duality for $\mathrm{Gr}(2, 2n)$

Abstract: Consider a Grassmannian $\mathrm{Gr}(2, V)$ for an even-dimensional vector space $V$. Its derived category of coherent sheaves has a Lefschetz exceptional collection with respect to the Pl\"ucker embedding. We consider a variety $X_1$ of pairs consisting of a degenerate $2$-form on $V$ and a line in its kernel. Note that $X_1$ is generically a $\mathbb{P}^1$-fibration over the Pfaffian variety of degenerate $2$-forms on $V$. We construct an exceptional collection of coherent sheaves on $X_1$ such that the subcategory of $D^b_{\mathrm{coh}}(X_1)$ generated by that collection is conjecturally equivalent to the homologically projectively dual category of the Grassmannian.