Maximal directional operators along algebraic varieties

We establish the sharp growth order, up to epsilon losses, of the L2-norm of the maximal directional averaging operator along a finite subset V of a polynomial variety of arbitrary dimension m, in terms of cardinality. This is an extension of the works by Córdoba, for one-dimensional mani-folds, Katz for the circle in two dimensions, and Demeter for the 2-sphere. For the case of directions on the two-dimensional sphere we improve by a factor of√ log N on the best known bound, due to Demeter, and we obtain a sharp estimate for our model operator. Our results imply new L2-estimates for Kakeya type maximal functions with tubes pointing along polynomial directions. Our proof tech-nique is novel and in particular incorporates an iterated scheme of polynomial partitioning on varieties adapted to directional operators, in the vein of Guth, Guth-Katz, and Zahl.