Extended Hyperbolicity

TL;DR

This paper develops a new framework for constructing hyperbolic models of complex spaces using model category theory, defining holotopy groups, and analyzing their properties to understand hyperbolicity in a homotopical context.

Contribution

It introduces a definition of hyperbolic simplicial sheaves extending classical concepts, proves the existence of hyperbolic models, and relates holotopy groups to hyperbolicity.

Findings

Hyperbolic models exist for any simplicial sheaf.

Vanishing of holotopy groups in positive degrees is necessary for hyperbolicity.

Hyperbolic models of complex projective spaces are not hyperbolic themselves.

Abstract

Given a complex space $X$, we cosidered the problem of finding a {\it hyperbolic model} of $X$. This is an object $\ip(X)$ with a morphism $i:X\to \ip(X)$ in such a way that $\ip(X)$ is ``hyperbolic'' in a suitable sense and $i$ is as close as possible to be an isomorphism. Using the theory of model categories, we found a definition of hyperbolic simplicial sheaf (for the strong topology) that extends the classical one of Brody for complex spaces. We prove the existence of hyperbolic models for any simplicial sheaf. Furthermore, the morphism $i$ can be taken to be a cofibration and an affine weak equivalence (in an algebraic setting, Morel and Voevodsky called it an $\aff$ weak equivalence). Imitating one possible definition of homotopy groups for a topological space, we defined the {\it holotopy} groups for a simplicial sheaf and showed that their vanishing in ``positive'' degrees is a necessary condition for a sheaf to be hyperbolic. We deduce that if $X$ is a complex space with a non zero holotopy group in positive degree, then its hyperbolic model (that in general will only be a simplicial sheaf) cannot be weakly equivalent to a hyperbolic complex space (in particular is not itself hyperbolic). We finish the manuscript by applying these results and a {\it topological realization functor}, constructed in the previous section, to prove that the hyperbolic models of the complex projective spaces cannot be weakly equivalent to hyperbolic complex spaces.