arXiv subjects

Rodolfo Aguilar Aguilar

Publications and source records attributed to Rodolfo Aguilar Aguilar.

4 recordsLinked to original sources

arXiv2024

On homology planes and contractible $4$-manifolds

We call a non-trivial homology sphere a Kirby-Ramanujam sphere if it bounds both a homology plane and a Mazur or Poénaru manifold. In 1980, Kirby found the first example by proving that the boundary of the Ramanujam surface bounds a Mazur manifold and it has remained a single example since then. By tracing their initial step, we provide the first additional examples and we present three infinite families of Kirby-Ramanujam spheres. Also, we show that one of our families of Kirby-Ramanujam spheres is diffeomorphic to the splice of two certain families of Brieskorn spheres. Since this family of Kirby-Ramanujam spheres bound contractible $4$-manifolds, they lie in the class of the trivial element in the homology cobordism group; however, both splice components are separately linearly independent in that group.

math.GT↗

arXiv2024

Around Homology Planes: Old and new

We survey some old and new results about acyclic (affine) complex surfaces, also called homology planes. We ask several questions and leave open directions for future research.

math.AG↗

arXiv2023

The nilpotent quotients of normal quasi-projective varieties with proper quasi-Albanese map

We show that if $X$ is a normal complex quasi-projective variety, the quasi-Albanese map of which is proper, then the torsionfree nilpotent quotients of $π_1(X)$ are, up to a controlled finite index, the same ones as those of the normalisation of its quasi-Albanese image. When $X$ is quasi-Kähler smooth, we get the same conclusion, but only for the smooth models of the quasi-Albanese image. In this second case, the proof is elementary, as the one given in [Cam95] for $X$ compact. In the normal quasi-projective case, the étale Galois cover of $X$ associated to the nilpotent completion of $π_1(X)$ is thus holomorphically convex. This is proved in the smooth case by $3$ other methods in [GGK22], which motivated the present text. When $X$ is `special' in the sense of [Cam11], we deduce that the torsion free nilpotent quotients of $π_1(X)$ are abelian. Examples show that this property fails (as first observed in [CDY22]) when the quasi-Albanese map is not proper. This leads to replace our previous `Abelianity conjecture' in the compact case by an `Nilpotency conjecture' in the non-compact quasi-Kähler context.

math.AG↗

arXiv2021

Fundamental groups of partial compactifications of arrangements and homology planes

Following our previous work, we develop an algorithm to compute a presentation of the fundamental group of certain partial compactifications of the complement of a complex arrangement of lines in the projective plane. It applies, in particular, to homology planes arising from arrangements of lines. This permits to determine the infinitude of the fundamental group of homology planes of log-general type that may be the first examples of this type.

math.AG↗