Eisenstein–Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation

We show that for an arbitrary totally complex number field $L$ the (regularized) critical $L$-values of algebraic Hecke characters of $L$ divided by certain periods are algebraic integers. This relies on a new construction of an equivariant coherent cohomology class with values in the completion of the Poincaré bundle on an abelian scheme $\mathcal{A}$. From this we obtain a cohomology class for the automorphism group of a CM abelian scheme $\mathcal{A}$ with values in some canonical bundles, which can be explicitly calculated in terms of Eisenstein--Kronecker series. As a further consequence, using an infinitesimal trivialization of the Poincaré bundle, we construct a $p$-adic measure interpolating the critical $L$-values in the ordinary case. This generalizes previous results for CM fields by Damerell, Shimura and Katz and settles the algebraicity and $p$-adic interpolation in the remaining open cases of critical values of Hecke $L$-functions.