On the geometry of K\"ahler--Frobenius manifolds and their classification

TL;DR

This paper demonstrates that flat compact K"ahler manifolds have a Frobenius manifold structure, leading to their classification and revealing connections to Calabi--Yau manifolds, complex tori, and theta functions, with implications for Chern's conjecture.

Contribution

It introduces the concept of K"ahler--Frobenius manifolds, classifies all such manifolds, and explores their geometric and number-theoretic properties.

Findings

K"ahler--Frobenius manifolds include Calabi--Yau and complex tori

Classification of all flat compact K"ahler manifolds with Frobenius structure

Connection between K"ahler--Frobenius manifolds and theta functions

Abstract

The purpose of this article is to show that flat compact K\"ahler 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 K\"ahler--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 K\"ahler pre-Frobenius manifolds holds. Lastly, we establish that certain classes of K\"ahler-Frobenius manifolds share a direct relationship with theta functions which are important objects in number theory as well as complex analysis.