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Explicit formulas for the Hattori-Stong theorem and applications

Ping Li, Wangyang Lin

TL;DR

This work delivers explicit, closed-form coefficients for the Chern-character expansions that underlie the Hattori–Stong integrality relations, enabling direct computation of when Chern-number data realize stably almost-complex structures. It also derives an explicit evenness criterion for the signature of $4k$-dimensional stably almost-complex manifolds in terms of Chern numbers, using symmetric-function and genus-theoretic methods. By combining these results, the authors show that the signature is even under broad Chern-number patterns and in particular exclude certain stably almost-complex structures on rational projective planes. The methods provide a concrete combinatorial toolkit—rooted in partitions, Stirling numbers, and Bernoulli numbers—to translate abstract index-theoretic constraints into tangible number-theoretic conditions with geometric consequences.

Abstract

We employ combinatorial techniques to present an explicit formula for the coefficients in front of Chern classes involving in the Hattori-Stong integrability conditions. We also give an evenness condition for the signature of stably almost-complex manifolds in terms of Chern numbers. As an application, it can be showed that the signature of a $2n$-dimensional stably almost-complex manifold whose possibly nonzero Chern numbers being $c_n$ and $c_ic_{n-i}$ is even, which particularly rules out the existence of such structure on rational projective planes. Some other related results and remarks are also discussed in this article.

Explicit formulas for the Hattori-Stong theorem and applications

TL;DR

This work delivers explicit, closed-form coefficients for the Chern-character expansions that underlie the Hattori–Stong integrality relations, enabling direct computation of when Chern-number data realize stably almost-complex structures. It also derives an explicit evenness criterion for the signature of -dimensional stably almost-complex manifolds in terms of Chern numbers, using symmetric-function and genus-theoretic methods. By combining these results, the authors show that the signature is even under broad Chern-number patterns and in particular exclude certain stably almost-complex structures on rational projective planes. The methods provide a concrete combinatorial toolkit—rooted in partitions, Stirling numbers, and Bernoulli numbers—to translate abstract index-theoretic constraints into tangible number-theoretic conditions with geometric consequences.

Abstract

We employ combinatorial techniques to present an explicit formula for the coefficients in front of Chern classes involving in the Hattori-Stong integrability conditions. We also give an evenness condition for the signature of stably almost-complex manifolds in terms of Chern numbers. As an application, it can be showed that the signature of a -dimensional stably almost-complex manifold whose possibly nonzero Chern numbers being and is even, which particularly rules out the existence of such structure on rational projective planes. Some other related results and remarks are also discussed in this article.
Paper Structure (11 sections, 15 theorems, 58 equations)

This paper contains 11 sections, 15 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background notation
  4. Main results
  5. Preliminaries
  6. Symmetric functions
  7. Stirling numbers of the second kind
  8. Proof of Theorem \ref{['first main result theorem']}
  9. Proof of Theorem \ref{['second main result theorem']}
  10. Proof of Theorem \ref{['third main result theorem']}
  11. Remarks on integral/rational projective planes

Key Result

Theorem 1.1

Given a positive integer $n$ and a set of integers $\{c_{\lambda}~|~\text{$\lambda:$ partitions of weight $n$}\}$, they can be realized as Chern numbers of some $2n$-dimensional stably almost-complex manifold if and only if they satisfy the integral conditions (integrality).

Theorems & Definitions (30)

  • Theorem 1.1: Atiyah-Hirzebruch, Hattori-Stong
  • Theorem 2.1
  • Remark 2.2
  • Example 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Definition 3.1
  • Theorem 3.2: Doubilet
  • ...and 20 more