Geography of spin surfaces (Q1922749)

The Chern invariants \(c_1^2\), \(\chi\) of a minimal compact complex surface \(X\) of general type satisfy some well-known inequalities stemming from Noether and Bogomolov, Miyaoka and Yau. It is believed that those and the positivity of the invariants are the only restrictions for \(X\), i.e. all possible invariants do occur. The authors sharpen this problem considering simply connected spin surfaces \(S\). These can only occur for points \((c_1^2,\chi)\) satisfying extra conditions coming from the behaviour of the intersection form and from geometrical considerations. The authors describe the geography of simply connected spin surfaces considering three sectors of the admissible region: the first is bounded below by the line \(y=2x-6\) (from Noether's inequality), the third is bounded above by a curve asymptotic to the line \(y=8x\).