Examples of non-positively curved Kähler manifolds (Q1922779)

The author constructs three new sequences of compact complex surfaces which admit Kähler metrics with quasi-negative sectional curvature (i.e., nonpositive everywhere and negative somewhere). In order to do this, one starts with a compact complex ball quotient which contains a totally geodesic divisor \(D\). If \(Y\) is a finite branched covering of \(X\) with ramifications \(R\) over \(D\), then one can add a suitably chosen metric, which supports in a neighborhood of the divisor \(R\), to the pullback of the canonical metric of \(X\). The sum metric is then quasi-negatively curved.