The tangent bundle of an almost complex manifold (Q2739252)

The question the authors address here is whether one can endow in a natural way the tangent bundle \(TM\) of an almost complex manifold \((M,J)\) with an almost complex structure. The almost complex structure they construct on \(TM\), is characterized by simple properties: (a) The projection \(\pi :TM\rightarrow M\) is holomorphic; (b) The embedding \(\varepsilon :M\rightarrow TM\) of \(M\) as the zero section is holomorphic; (c) If \((N,I)\) is another almost complex manifold and \(\Phi :N\rightarrow M\) is a holomorphic map then \(d\Phi :TN\rightarrow TM\) is also holomorphic, etc. They treat also the question whether \(TM\rightarrow M\) is an almost holomorphic vector bundle.