Question on Hori, Iqbal and Vafa's 'D-branes and Mirror Symmetry'

Question

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?

This post imported from StackExchange MathOverflow at 2017-04-08 22:34 (UTC), posted by SE-user Mtheorist

Comments

Because it says "normal and tangent", does it mean the fiber $T_p X \otimes_\mathbb{R} \mathbb{C}$ for $p \in L \subset X$? Just from the ambient complexified tangent bundle.

This post imported from StackExchange MathOverflow at 2017-04-08 22:34 (UTC), posted by SE-user AHusain