On holomorphic and anti-holomorphic sectional curvature of indefinite Kähler manifolds of real dimension \(n\geq 6\) (Q1313575)

The author studies sufficient conditions to have a Kähler manifold \(M\) of (real) \(\dim M\geq 6\) with indefinite metric and constant holomorphic sectional curvature. These conditions are expressed in terms of bounded curvatures of holomorphic planes with signature \((-,-)\) (or \((+,+)\)) in \((\text{span}\{z,Jz\})^ \perp\), for each space-like vector \(z\in T_ p M\), provided that \(n-\nu\geq 4\), where \(\nu\) is the index of \(\langle,\rangle\). If \(M\) is null holomorphically flat at \(p\in M\), then the author studies also sufficient conditions to have \(M\) of constant holomorphic curvature at \(p\). If \(M\) is null holomorphically flat at \(p\in M\) these conditions are expressed in terms of bounded from below (or above) curvatures of nondegenerate holomorphic planes in \((\text{span}\{z,Jz\})^ \perp\). The author studies these sufficient conditions in terms of bounded curvatures of anti-holomorphic planes with signature \((-,+)\) or \((+,+)\) or \((-,-)\).