Complex sprays and complex curves (Q5949006)

One considers a complex manifold \(M\) and its holomorphic tangent bundle \({\mathcal O}M\). First, the notion of \textit{complex spray} is defined following the usual notion of real spray as a homogeneous semispray. If \((z,v)=(z^{i},v^{i})\) are the local coordinates on \({\mathcal O}M\) then a complex spray \(X\) is expressed as \(X=v^{i}\frac{\partial }{\partial z^{i}}-b^{i}(z,v)\frac{\partial }{\partial v^{i}}\), the functions \(b^{i}(z,v)\) being (2,0)-homogeneous. Second, the notion of \textit{spray complex curve} associated to \(X\) is introduced and several equivalent formulations are obtained. Necessary and sufficient conditions for \(M\) to admit spray complex curves for \(X\) through each point and each direction are given. As a remarkable particular case, the horizontal radial vector field \(X\) associated to a complex Finsler metric is discussed.