On the geography of surfaces - simply connected minimal surfaces with positive index
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.
simply connected surface of general type
scientific article; zbMATH DE number 3957298