On the real homotopy type of compact complex surfaces

TL;DR

This paper investigates the algebraic and topological properties of compact complex surfaces, showing vanishing Massey products beyond length three and providing explicit presentations of their fundamental groups, especially in non-Kähler cases.

Contribution

It establishes the vanishing of higher Massey products and explicitly describes the Malcev completion for these surfaces, revealing new algebraic structures.

Findings

Massey products of degree one classes vanish beyond length three

The Malcev completion is generated by quadratic and cubic relations

Explicit presentations are provided for non-Kähler surfaces

Abstract

We show that on a compact complex surface all Massey products of cohomology classes in degree one vanish beyond length three. Dually, the real Malcev completion of the fundamental group is homogeneously presented by quadratic and cubic relations. In the non-K\"ahler case, we give an explicit presentation, which is determined by the first Betti number of the surface.