Positive projectively flat manifolds are locally conformally flat-Kähler Hopf manifolds (Q2031902)

This paper focuses on projectively flat metrics on a complex manifold \(M\) of complex dimension \(n\), which is defined as a Hermitian metric \(h\) on the holomorphic tangent bundle \(T_M\) of \(M\) which satisfies the equation \(F_h = \mathrm{tr}_h F_h \cdot \mathrm{Id}_{T_M}/n\). If \(h\) is projectively flat then so is any metric that is conformally equivalent to it. Assuming that a projectively flat metric exists on \(M\), the author defines the following notions for the conformal class \(\{ h \}\) of projectively flat metrics on \(M\): \(\{ h \}\) is defined to be negative (resp.~zero, positive) projectively flat if and only if \(\{ h \}\) contains a representative whose Chern scalar curvature is negative (resp.~zero, positive). The author proves that a compact Hermitian manifold is positive projectively flat if and only if it is a locally conformally flat-Kähler Hopf manifold endowed with a metric globally conformal to the Boothby metric, and that there are no negative projectively flat metrics on compact complex manifold of complex dimension \(n \ge 2\). It is also proved that a metric is zero projectively flat if and only if it is projectively flat and balanced, or it is a globally conformally Chern flat metric, when \(n \ge 2\). These results are applied to prove the following: the only similarity Hopf manifolds which admit projectively flat metrics are locally conformally flat-Kähler Hopf manifolds endowed with the Boothby metric; a Hermitian metric on a compact Hermitian manifold with \(n \ge 3\) that is astheno-Kähler and projectively flat is Kähler and zero projectively flat.