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

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 $T={\mathbb{C}}^n\mathrm{/}{\mathbb{Z}}^n$, generalized (orientable) Hantzsche–Wendt manifolds, hyperelliptic manifolds and manifolds of type $T\mathrm{/}G$, 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.