The Milnor-Hirzebruch problem, complex cobordisms, and theta divisors

Victor Buchstaber, Steklov Institute

Online Talk

Let $A^{n+1} = \mathbb{C}^{n+1}/\Gamma$ be a principally polarised abelian variety. The space of holomorphic sections of its canonical line bundle $L$ is one-dimensional and generated by the classical Riemann $\theta$-function. According to the Andreotti-Mayer theorem (1967), for a generic principally polarised abelian variety, the theta divisor $\Theta^n \subset A^{n+1}$ given by $\theta(z,\tau)=0$ is a smooth irreducible algebraic variety of general type. The talk is focused on the following result of Buchstaber-Veselov (2020), which is based on the construction of the Chern-Dold character in the theory of complex cobordism (Buchstaber, 1970) The exponential generating series of the complex cobordism classes of the theta divisors $[\Theta^n],\, n= 0,1,2,\ldots,$ realizes the exponential of the formal group law of geometric cobordisms}. 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 varieties.