Infinitesimal Hodge conjecture

(Joint with H. Esnault and M. Kerz) It is difficult to talk about infinitesimal deformations of algebraic cycles, but the work of Thomason on the K-theory of singular schemes makes it possible, essentially by defining cycle classes via the chern character applied to vector bundles. Using this approach, we show that the cohomology class of an algebraic cycle remains algebraic under infinitesimal deformation if and only if it remains Hodge. A similar result is proven in mixed characteristic using crystalline cycle classes.