| Abstract: | The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle (T_{mathbb {C}}E=T'Eoplus T''E) and its link, via the tangent anchor map, with the complexified tangent bundle (T_{mathbb {C}}(T'M)=T'(T'M)oplus T''(T'M)). A holomorphic Lie algebroid structure is emphasized on (T'E). A special study is made for integral curves of a spray on (T'E). Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on (T'E). In the second part of section two, we study the holomorphic prolongation (mathcal {T}'E) of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on (T'M) induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from (T'M). |
