Topological and arithmetic characteristics about products of projective lines with complex tori

TL;DR

This paper investigates the topological and arithmetic properties of surfaces formed by the product of a projective line and a complex torus, revealing their fundamental group structures, irregularities, and classification via Chern numbers.

Contribution

It introduces a detailed analysis of the fundamental groups, irregularities, and classification of surfaces constructed from products of projective lines and complex tori, expanding understanding of their geometric properties.

Findings

Fundamental groups of Galois covers have abelian subgroups of rank m(2n-1).

Irregularity of these surfaces is at least 2mn-1.

Chern numbers are used to compute the surfaces' index and classify them.

Abstract

In this paper, we study non-planar degeneracies with cylindrical configurations. They could be constructed by the product $\mathbb{CP}^1 \times T$ of the projective plane and a complex torus with embedding $(m,n)$. We prove that their fundamental groups of Galois covers have an abelian subgroup of rank $m(2n-1)$ respectively, and the irregularity of these surfaces are at least $2mn-1$. Furthermore, we also use Chern numbers to compute the index of such surfaces and classify them.