The Chern–Dold character and the Milnor–Hirzebruch problem

The talk is devoted to problems and results related to the Chern numbers of complex, algebraic and toric manifolds. Our goal is to present a recent result of Buchstaber and Veselov: The exponential generating series of complex cobordism classes of theta divisors of principally polarized Abelian varieties realizes the exponential of the formal group of geometric cobordisms. This result is based on Buchstaber's construction (1970) of the Chern–Dold character in the theory of complex cobordism. We will discuss applications of this result to well-known problems in algebraic topology and algebraic geometry, including the hitherto open Milnor–Hirzebruch problem (1958) on Chern numbers of irreducible smooth algebraic manifolds.