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

Given a variety $X$ over the projective space $\mathbb{P}(V)$ and a semiorthogonal decomposition of the derived category of $X$ which is Lefschetz, i.e., compatible in a certain way with the twist by $\mathcal{O}(1)$, homological projective duality is a way to construct a triangulated category, now over the dual projective space $\mathbb{P}(V^\vee)$, also with a Lefschetz decomposition, that is in many aspects similar to the derived category of $X$ and enjoys many useful properties. This “dual” category can be constructed in a formal way, but the relations with the derived category of $X$ become much more interesting if the dual category is also described geometrically, for example in terms of some variety over the dual projective space. I will talk about basic notions of homological projective duality and I'll give a conjectural description of the dual category to $\mathrm{Gr}(2, 2n)$ in its Plücker embedding, motivated by the description Kuznetsov gave in 2005 for the case $n=3$. This is a work in progress.