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On the geometry of Kähler--Frobenius manifolds and their classification

Noémie. C. Combe

Abstract

The purpose of this article is to show that flat compact Kähler manifolds exhibit the structure of a Frobenius manifold, a structure originating in 2D Topological Quantum Field Theory and closely related to Joyce structure. As a result, we classify all such manifolds. It can be deduced that Kähler--Frobenius manifolds include certain Calabi--Yau manifolds, complex tori $T=\mathbb{C}^n/\mathbb{Z}^n$, generalized (orientable) Hantzsche--Wendt manifolds, hyperelliptic manifolds and manifolds of type $T/G$, where $G$ is a finite group acting on $T$ freely and containing no translations. An explicit study is provided for the two-dimensional case. Additionally, we can prove that Chern's conjecture for Kähler pre-Frobenius manifolds holds. Lastly, we establish that certain classes of Kähler-Frobenius manifolds share a direct relationship with theta functions which are important objects in number theory as well as complex analysis.

On the geometry of Kähler--Frobenius manifolds and their classification

Abstract

The purpose of this article is to show that flat compact Kähler manifolds exhibit the structure of a Frobenius manifold, a structure originating in 2D Topological Quantum Field Theory and closely related to Joyce structure. As a result, we classify all such manifolds. It can be deduced that Kähler--Frobenius manifolds include certain Calabi--Yau manifolds, complex tori , generalized (orientable) Hantzsche--Wendt manifolds, hyperelliptic manifolds and manifolds of type , where is a finite group acting on freely and containing no translations. An explicit study is provided for the two-dimensional case. Additionally, we can prove that Chern's conjecture for Kähler pre-Frobenius manifolds holds. Lastly, we establish that certain classes of Kähler-Frobenius manifolds share a direct relationship with theta functions which are important objects in number theory as well as complex analysis.

Paper Structure

This paper contains 47 sections, 26 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Plan with results
  4. Frobenius bundles
  5. Frobenius bundles
  6. Recollections
  7. Hermitian WDVV equation
  8. Frobenius manifolds: prerequisites, recollections, notations
  9. Affine connections
  10. Affine structures
  11. Affine manifold
  12. Crystallographic groups
  13. Metrics
  14. Frobenius Algebra bundles
  15. Algebras
  16. ...and 32 more sections

Key Result

Theorem 2.1.0.1

Let $(M,g)$ be a Kähler manifold. Then, the following statements are equivalent:

Figures (2)

  • Figure 1: Relations between spaces
  • Figure 2: pre-Frobenius versus Frobenius algebras

Theorems & Definitions (57)

  • Theorem 2.1.0.1
  • Definition 2.3.0.1
  • Definition 2.5.5.1
  • Remark 2.5.5.1
  • Lemma 3.1.0.1
  • proof
  • Lemma 3.2.2.1
  • proof
  • Definition 3.2.4.1
  • Lemma 3.2.5.1
  • ...and 47 more