On the geography of surfaces - simply connected minimal surfaces with positive index
type: Item
instance of: scholarly article
title: On the geography of surfaces - simply connected minimal surfaces with positive index
zbMATH Open document ID: 0595.14027
DOI: 10.1007/BF01457284
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164241
review text: To each minimal surface X of general type, we can assign two fundamental numerical invariants - the Chern invariants \(c^ 2_ 1(X)\) and \(c_ 2(X)\). As \(c^ 2_ 1(X)+c_ 2(X)=12\chi ({\mathcal O}_ X)\), the invariants \((c^ 2_ 1(X),\chi ({\mathcal O}_ X))\) are equivalent to \((c^ 2_ 1,c_ 2)\). Surfaces with \(8\chi <c^ 2_ 1\leq 9\chi\) are called surfaces with positive index. It had been conjectured that any simply connected surface of general type has negative index. This conjecture was disproven until 1984. In this paper, using \textit{Xiao's} example (''An example of hyperelliptic surfaces with positive index'', to appear in Northeast. Math. J.), two main theorems are proven: Theorem 1. Let x, y be integers satisfying \((352/89)x+140\cdot 2x^{2/3}<y<(18644/2129)x-365\cdot 7x^{2/3}\), \(x>C\), where C is a large constant. There exists a simply connected minimal surface X of general type with \(c^ 2_ 1(X)=y\), \(\chi\) (\({\mathcal O}_ X)=x\). Furthermore, X can be represented by a surface with a hyperelliptic fibration. Theorem 2. Le x, y be integers satisfying \(8x<y\leq (17700/2011)x-33000\), \(x\leq 41169\), y is even. Then there exists a minimal surface X of general type with \(c^ 2_ 1(X)=y\), \(\chi\) (\({\mathcal O}_ X)=x\). Furthermore, X can be represented by a surface with a hyperelliptic fibration.
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 3957298
zbMATH Keywords: Chern invariants
zbMATH Keywords: surfaces with positive index
zbMATH Keywords: simply connected surface of general type
author: Zhi-Jie Chen
MaRDI profile type: Publication
cites work: Q5749404
cites work: Global moduli for surfaces of general type
cites work: A Class of Minimal Surfaces in the Unknown Region of Surface Geography
cites work: Q3936840
cites work: Surfaces fibrées en courbes de genre deux
OpenAlex ID: W2911750485
description: scientific article; zbMATH DE number 3957298