I am reading the book, Lectures on Kahler Geometry, by Andrei Moroianu. Here a link, https://books.google.com.hk/books?id=oqmroUc9E8YC&printsec=frontcover&hl=zh-CN&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false. I am confused with the proof of the Lemma 11.7 in Page 85 and I have some questions about the proof.
- Why the author claims that $(\bar{\partial}f)(X)=\frac{1}{2}\partial_{(X+iJX)}f$ ?
- In the paragraph before the lemma, the author writes that "a product $iX$ for some $X\in TM$ is identified with $JX$". But soon later he also writes that "We don't say that $iX=JX$". So my question is in what sense $iX$ is identified with $JX$.
I post below the text in Page 85 and more text concerning this question in Page 81.
Text from Page 85.
"**
11.4. Comparison of the Levi-Civita and Chern connections
** Our next aim is to express the $\bar{\partial}$ -operator on the tangent bundle of a Hermitian manifold $(M, h, J)$ in terms of the Levi-Civita connection of $h .$ In order to do so, we have to remember that $T M$ is identified with a complex vector bundle via the complex structure $J .$ In other words, a product $i X$ for some $X \in T M$ is identified with $J X$. Since this point is particularly confusing, we insist a little more on it: we don't say that $i X=J X$ on $T M$ (this actually would make no sense because $T M$ is a real bundle), we just say that the complex structure on $T M$ (which is usually denoted by $i$ on vector bundles) is, in this case, given by the tensor $J$.
LEMMA 11.7. For every section $Y$ of the complex vector bundle $(T M, J)$, the $\bar{\partial}$ -operator, as $T M$ -valued $(0,1)$ -form is given by $$ \bar{\partial}^{\nabla} Y(X):=\frac{1}{2}\left(\nabla_{X} Y+J \nabla_{J X} Y-J\left(\nabla_{Y} J\right) X\right) $$ where $\nabla$ denotes the Levi-Civita connection of any Hermitian metric $h$ on $M$
Proof. Recall first that $(\bar{\partial} f)(X)=\frac{1}{2} \partial_{(X+i J X)} f$, so $$ \begin{aligned} \bar{\partial}^{\nabla}(f Y)(X)=& f \frac{1}{2}\left(\nabla_{X} Y+J \nabla_{J X} Y-J\left(\nabla_{Y} J\right) X\right) \\ &+\frac{1}{2}\left(\left(\partial_{X} f\right) Y+\left(\partial_{J X} f\right) J Y\right)=f \bar{\partial}^{\nabla} Y(X)+\bar{\partial} f(X) Y \end{aligned} $$ which shows that the operator $\bar{\partial}^{\nabla}$ defined by (11.5) satisfies the Leibniz rule...... "
Text from Page 81.
"DEFINITION 11.1. A Hermitian metric on an almost complex manifold $(M, J)$ is a Riemannian metric $h$ such that $h(X, Y)=h(J X, J Y)$, for all $X, Y \in T M$. The fundamental 2-form of a Hermitian metric is defined by $\Omega(X, Y):=h(J X, Y)$.
The extension by C-linearity (also denoted by $h$ ) of the Hermitian metric to $T M^{\mathbb{C}}$ satisfies $$ \left\{\begin{array}{l} h(\bar{Z}, \bar{W})=\overline{h(Z, W)}, \quad \forall Z, W \in T M^{\mathbb{C}} \\ h(Z, \bar{Z})>0, \quad \text { for every non-zero complex vector } Z \\ h(Z, W)=0, \quad \forall Z, W \in T^{1,0} M \text { and } \forall Z, W \in T^{0,1} M \end{array}\right. $$
Conversely, each symmetric tensor on $T M^{\mathbb{C}}$ with these properties defines a Hermitian metric by restriction to $T M$ (exercise).
REMARK. The tangent bundle of an almost complex manifold is in particular a complex vector bundle. If $h$ is a Hermitian metric on $M$, then $H(X, Y):=h(X, Y)-i h(J X, Y)=(h-i \Omega)(X, Y)$ defines a Hermitian structure on the complex vector bundle $(T M, J)$, as defined in the previous chapter. Conversely, any Hermitian structure $H$ on $T M$ as complex vector bundle defines a Hermitian metric $h$ on $M$ by $h:=\operatorname{Re}(H)$.
REMARK. Every almost complex manifold admits Hermitian metrics. Simply choose an arbitrary Riemannian metric $g$ and define $h(X, Y):=$ $g(X, Y)+g(J X, J Y)$"