Smoothing low-dimensional cycles in algebraic cobordism

The question of smoothability of algebraic cycles was first raised by Borel and Haefliger in 1961, in the context of Betti cohomology. Although many varieties with non-smoothable cycles have been found since 1974, all known examples concern cycles whose dimension is at least half that of the ambient variety. In the complementary range, a recent breakthrough of Kollár and Voisin shows that cycles of dimension less than half that of the ambient variety are always smoothable in the Chow group, i.e., their classes can be written as linear combinations of classes of smooth subvarieties. In this talk, I will present a generalization of their result from Chow groups to algebraic cobordism, which can be viewed as the algebraic analogue of complex cobordism.