# Issues with the $\overline{ \partial}$-operator and the almost complex structure of a hermitian manifold

I'm working through "Lecture on Kahler Geometry" by Andrei Moroianu, and am stuck on Lemma 11.7 (p. 85).

The lemma says:

For every section $$Y$$ of the complex vector bundle $$(TM, J)$$ the $$\overline{\partial}$$-operator, as a $$TM$$-valued $$(0,1)$$-form is given by

$$\overline{\partial}^\nabla Y (X) := \frac{1}{2} (\nabla_X Y + J \nabla_{JX}Y + J(\nabla_YJ)X)$$ where $$\nabla$$ denotes the Levi-Civita connection of any Hermitian metric $$h$$ on $$M$$.

The proof starts by proving the Leibniz rule, recalling that $$(\overline{\partial} f) (X) = \frac{1}{2} \partial_{(X+iJX)} f$$, then

$$(\overline{\partial}^\nabla f Y)(X) = f \frac{1}{2} (\nabla_X Y + J \nabla_{JX}Y + J(\nabla_YJ)X) + \frac{1}{2} ((\partial_Xf)Y + (\partial_{JX}f)JY) = f \overline{\partial}^\nabla Y(X) + \overline{\partial}f (X)Y$$

My question is how $$\overline{\partial}f (X)Y = (\frac{1}{2} ((\partial_Xf)Y + (\partial_{JX}f)JY))$$ when we're given $$(\overline{\partial} f) (X) = \frac{1}{2} \partial_{(X+iJX)} f$$?

Any help would be greatly apperiated.

• Are you missing some equal signs? Commented Apr 1, 2021 at 0:55
• Yes, I just fixed it. Sorry about that. Commented Apr 1, 2021 at 1:15

The Dolbeault operator $$\overline{\partial}$$ can be thought of as the $$(0,1)$$--part of the flat connection $$d$$, i.e., the exterior derivative. An import property of a connection $$\nabla$$ is that $$\nabla_{X+Y} Z = \nabla_X Z + \nabla_Y Z,$$ for tangent vectors $$X,Y,Z$$, and for smooth functions $$f$$, $$\nabla_{f X} Y = f \nabla_X Y.$$
Now, if $$(\overline{\partial}f)(X) : = \frac{1}{2}\partial_{X + \sqrt{-1} J X} (f)$$, then $$(\overline{\partial} f)(X)Y = \frac{1}{2}\partial_{X+\sqrt{-1} JX}(f)Y = \frac{1}{2}\partial_X (f)Y + \frac{\sqrt{-1}}{2}\partial_{JX}(f)Y = \frac{1}{2}\partial_X(f) + \frac{1}{2}\partial_{JX}(f)JY.$$