In the paper mentioned above, on page 19, the physics of A-type supersymmetry is related to a Lagrangian submanifold $\gamma$ of a Kaehler manifold $X$. In particular, the phrase "...holomorphic components of normal and tangent of $\gamma$..." is used.

What does one mean by the holomorphic component of a tangent vector of a Lagrangian submanifold? A Lagrangian submanifold does not necessarily have complex structure, and does not even need to be even-dimensional, so how can a tangent vector of the Lagrangian submanifold have a holomorphic component?