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Buchstaber, Ochanine, Krichever, and Witten genera

Mikhail Kornev

Abstract

We introduce a new class of formal group laws whose modulus square construction yields Buchstaber's family of polynomials. This class is related to, but does not coincide with, the family of formal group laws associated with the Krichever genus. We compute the values of the corresponding Hirzebruch genus on theta divisors and complex projective spaces, describe its relation to the Ochanine, Krichever, and Witten genera, and show how this construction gives examples not arising from Hirzebruch's elliptic genera of level $n$.

Buchstaber, Ochanine, Krichever, and Witten genera

Abstract

We introduce a new class of formal group laws whose modulus square construction yields Buchstaber's family of polynomials. This class is related to, but does not coincide with, the family of formal group laws associated with the Krichever genus. We compute the values of the corresponding Hirzebruch genus on theta divisors and complex projective spaces, describe its relation to the Ochanine, Krichever, and Witten genera, and show how this construction gives examples not arising from Hirzebruch's elliptic genera of level .
Paper Structure (3 sections, 9 theorems, 37 equations)

This paper contains 3 sections, 9 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. The Modulus Square Construction
  3. The Buchstaber Genus

Key Result

Proposition 1

Let $G$ be an abelian group and $\sigma: g\mapsto -g$ an involution. Denote by $X$ the orbit space $G/\sigma$ with points $[g, -g]$. Then $X$ carries an involutive commutative two-valued coset group structure with operation with neutral element $[e, e]$, and $e$ is a unit.

Theorems & Definitions (11)

  • Proposition 1
  • Definition 1
  • Proposition 2
  • Definition 2
  • Proposition 3
  • Theorem 1: Buchstaber75
  • Theorem 2
  • Corollary 2.1
  • Proposition 4
  • Proposition 5
  • ...and 1 more