On the topology of some quasi-projective surfaces

TL;DR

This paper explores the topological properties of quasi-projective surfaces with isolated singularities in complex projective space, focusing on fundamental groups, Galois coverings, homotopy groups, and Hodge structures.

Contribution

It provides new insights into the topology of quasi-projective surfaces, analyzing their fundamental groups, homotopy groups, and Hodge structures.

Findings

Analysis of fundamental groups and Galois coverings

Description of second homotopy groups

Investigation of mixed Hodge structures on cohomology

Abstract

Let $X$ be surface with isolated singularities in the complex projective space $P^3$ and let denote $Y$ the smooth part of $X$. In this note we discuss some aspects of the topology of such quasi-projective surfaces $Y$: the fundamental groups and the associated Galois coverings, the second homotopy groups and the mixed Hodge structure on the first cohomology group.