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Chern-Dold character in complex cobordisms and theta divisors

V. M. Buchstaber, A. P. Veselov

Abstract

We show that the smooth theta divisors of general principally polarised abelian varieties can be chosen as irreducible algebraic representatives of the coefficients of the Chern-Dold character in complex cobordisms and describe the action of the Landweber-Novikov operations on them. We introduce a quantisation of the complex cobordism theory with the dual Landweber-Novikov algebra as the deformation parameter space and show that the Chern-Dold character can be interpreted as the composition of quantisation and dequantisation maps. Some smooth real-analytic representatives of the cobordism classes of theta divisors are described in terms of the classical Weierstrass elliptic functions. The link with the Milnor-Hirzebruch problem about possible characteristic numbers of irreducible algebraic varieties is discussed.

Chern-Dold character in complex cobordisms and theta divisors

Abstract

We show that the smooth theta divisors of general principally polarised abelian varieties can be chosen as irreducible algebraic representatives of the coefficients of the Chern-Dold character in complex cobordisms and describe the action of the Landweber-Novikov operations on them. We introduce a quantisation of the complex cobordism theory with the dual Landweber-Novikov algebra as the deformation parameter space and show that the Chern-Dold character can be interpreted as the composition of quantisation and dequantisation maps. Some smooth real-analytic representatives of the cobordism classes of theta divisors are described in terms of the classical Weierstrass elliptic functions. The link with the Milnor-Hirzebruch problem about possible characteristic numbers of irreducible algebraic varieties is discussed.

Paper Structure

This paper contains 7 sections, 30 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Complex bordisms and cobordisms
  3. Theta divisors, Todd class and Chern-Dold character in complex cobordisms
  4. The Landweber-Novikov algebra and quantisation of complex cobordisms
  5. Real-analytic elliptic representatives
  6. Discussion: Milnor-Hirzebruch problem
  7. Acknowledgements

Key Result

Theorem 1.1

The theta divisor $\Theta^n$ of a general principally polarised abelian variety $A^{n+1}$ is a smooth irreducible projective variety, which can be taken as an algebraic representative of the coefficient $[\mathcal{B}^{2n}]$ in the Chern-Dold character.

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • ...and 34 more