Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$-spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on ${\mathbb{C}}^n$ and for high powers of ample holomorphic line bundles over compact complex manifolds.

We establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain Zariski-dense subset of the biprojective hypersurface $$x_1{y}_1^2+\cdots +x_4{y}_4^2=0$$ in ${\mathbb{P}}^3\times {\mathbb{P}}^3$. This confirms the modified Manin conjecture for this variety, in which the removal of a “thin” set of rational points is allowed.

We determine internal characterizations for the case when a tensor category is (super) Tannakian over fields of positive characteristic. This generalizes the corresponding characterizations in characteristic $0$ by Deligne. We also explore notions of Frobenius twists in tensor categories in positive characteristic.