2022/03/15 by Jacovskis, Augustinas, Lin, Xun, Liu, Zhiyu +1
#14D20 #14D23 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 14F05 #secondary 14J45
paper · doi:10.48550/arxiv.2203.08187
Let X be a smooth Fano variety and Ku(X) the Kuznetsov component. Torelli theorems for Ku(X) says that it is uniquely determined by a polarized abelian variety attached to it. An infinitesimal Torelli theorem for X says that the differential of the period map is injective. A categorical variant of infinitesimal Torelli theorem for X says that the morphism H1(X,TX)\xrightarrowη HH2(Ku(X)) is injective. In the present article, we use the machinery of Hochschild (co)homology to relate the three Torelli-type theorems for smooth Fano varieties via a commutative diagram. As an application, we first prove infinitesimal categorical Torelli theorem for a class of prime Fano threefolds. Then we prove a restatement of the Debarre-Iliev-Manivel conjecture infinitesimally.