Categorical Plücker Formula and Homological Projective Duality

Abstract

Kuznetsov’s homological projective duality (HPD) theory [K4] is one of the most active and powerful recent developments in the homological study of algebraic geometry. The fundamental theorem of HPD systematically compares derived categories of dual linear sections of a pair of HP-dual varieties $(X,X^{♮})$.In this paper we generalize the fundamental theorem of HPD beyond linear sections. More precisely, we show that for any two pairs of HP-duals $(X,X^{♮})$ and $(T,T^{♮})$ which intersect properly, there exist semiorthogonal decompositions of the derived categories $D(X\cap T)$ and $D(X^{♮}\cap T^{♮})$ into primitive and ambient parts, and that there is an equivalence of primitive parts ${}^{\mathrm{prim}}D(X\cap T)\simeq D(X^{♮}\cap T^{♮}{)}^{\mathrm{p}rim}$.