On the geography of surfaces of general type with fixed fundamental group
| dc.contributor.advisor | Urzúa Elia, Giancarlo A. | |
| dc.contributor.author | Troncoso Igua, Sergio | |
| dc.contributor.other | Pontificia Universidad Católica de Chile. Facultad de Matemáticas | |
| dc.date.accessioned | 2020-06-26T12:27:22Z | |
| dc.date.available | 2020-06-26T12:27:22Z | |
| dc.date.issued | 2020 | |
| dc.description | Tesis (Doctor en Matemática)--Pontificia Universidad Católica de Chile, 2020 | |
| dc.description.abstract | In this thesis, we study the geography of complex surfaces of general type with respect to the topological fundamental group. The understanding of this general problem can be coarsely divided into geography of simply-connected surfaces and geography of non-simply-connected surfaces. The geography of simply-connected surfaces was intensively studied in the eighties and nineties by Persson, Chen, and Xiao among others. Due to their works, we know that the set of Chern slopes c2 1/c2 of simply-connected surfaces of general type is dense in the interval [1/5, 2]. The last result which closes the density problem for this type of surfaces happened in 2015. Roulleau and Urzúa showed the density of the Chern slopes in the interval [1, 3]. This completes the study since accumulation points of c2 1/c2 belong to the interval [1/5, 3] by the Noether’s inequality and the Bogomolov-Miyaoka-Yau inequality for complex surfaces. The geography of non-simply-connected surfaces is well understood only for small Chern slopes. Indeed, because of works of Mendes, Pardini, Reid, and Xiao, we know that for c2 1/c2 ∈ [1/5, 1/3] the fundamental group is either finite with at most nine elements, or the fundamental (algebraic) group is commensurable with the fundamental (algebraic) group of a curve. Furthermore, a well-known conjecture of Reid states that for minimal surfaces of general type with c2 1/c2 < 1/2 the topological fundamental group is either finite or it is commensurable with the fundamental group of a curve. Due to Severi-Pardini’s inequality and a theorem of Xiao, Reid’s conjecture is true, at least in the algebraic sense for irregular surfaces or surfaces having an irregular étale cover. Keum showed with an example in his doctoral thesis that Reid’s conjecture cannot be extended over 1/2. For higher slopes essentially there are no general results. In this thesis, we prove that for any topological fundamental group G of a given non-singular complex projective surface, the Chern slopes c2 1(S)/c2(S) of minimal non-singular projective surfaces of general type S with π1(S) ' G are dense in the interval [1, 3]. It remains open the question for non-simplyconnected surfaces in the interval [1/2, 1]. | |
| dc.format.extent | 82 páginas | |
| dc.identifier.doi | 10.7764/tesisUC/MAT/29424 | |
| dc.identifier.uri | https://doi.org/10.7764/tesisUC/MAT/29424 | |
| dc.identifier.uri | https://repositorio.uc.cl/handle/11534/29424 | |
| dc.language.iso | en | |
| dc.nota.acceso | Contenido completo | |
| dc.rights | acceso abierto | |
| dc.subject.ddc | 514.3 | |
| dc.subject.dewey | Matemática física y química | es_ES |
| dc.subject.other | Superficies algebraicas | es_ES |
| dc.subject.other | Grupos topológicos | es_ES |
| dc.title | On the geography of surfaces of general type with fixed fundamental group | es_ES |
| dc.type | tesis doctoral | |
| sipa.codpersvinculados | 13222 | |
| sipa.codpersvinculados | 237891 |