I’m studying Kähler geometry. I have several quiet basic questions:
- The Kähler metric is usually denoted by $g_{\mathbb{C}}=\sum _{i,j}g_{i\bar{j}}dz^i\otimes d\bar{z}^j$, Kähler form is defined to be $\omega(\cdot,\cdot)=g(J\cdot,\cdot)$. However, many materials, like in Yau’s proof to Calabi conjecture and Gang Tian’s book Canonical metrics in Kähler geometry, $\omega=\frac{\sqrt{-1}}{2}g_{i\bar{j}}dz^i\wedge d\bar{z}^j$.
Here is what confusing me: $g_\mathbb{C}=g_{i\bar{j}}dz^i\otimes d\bar{z}^j + g_{j\bar{i}}d\bar{z}^i\otimes dz^j$ then, $\omega= \sqrt{-1}g_{i\bar{j}}dz^i\otimes d\bar{z}^j -\sqrt{-1}g_{j\bar{i}}d\bar{z}^i\otimes dz^j=\sqrt{-1}g_{i\bar{j}}(dz^i\otimes d\bar{z}^j -d\bar{z}^j\otimes dz^i)= \sqrt{-1}g_{i\bar{j}}dz^i\wedge d\bar{z}^j $, always missing $\frac{1}{2}$.
I have the following attempts:
(1) John Lee’s Smooth manifold page 358 says there are two different conventions of wedge. If $dz^i\wedge d\bar{z}^j=\frac{1}{2}(dz^i\otimes d\bar{z}^j -d\bar{z}^j\otimes dz^i)$, the coefficient $\frac{1}{2}$ does exist;
(2) $dz^i\wedge d\bar{z}^j$ still defined to be $dz^i\otimes d\bar{z}^j -d\bar{z}^j\otimes dz^i$, however the convention $g_{\mathbb{C}}$ stands for $\frac{1}{2}g_{i\bar{j}}dz^i\otimes d\bar{z}^j + \frac{1}{2}g_{j\bar{i}}d\bar{z}^i\otimes dz^j$.
$\textbf{My frist question:}$ does (1) and (2) makes sense? or it really doesn’t matter.
- I’m following Tian’s book to compute the curvature tensor from $\mathbb{C}$-extending the real case: $R(u,v)w=\nabla_u\nabla_vw-\nabla_v\nabla_uw-\nabla_{[u,v]}w$, and $R(u,v,w,z)=g_{\mathbb{C}}(R(u,v)z,w)$, where $u,v, w,z\in T_\mathbb{C}M$: $R_{i\bar{j}k\bar{l}}:=R(\partial_i,\partial_\bar{j}, \partial_k,\partial_\bar{l})=g_{\mathbb{C}}(\nabla_i\nabla_\bar{j}\partial_\bar{l},\partial_k)=g_\mathbb{C}(\nabla_i(g^{q\bar{p}}g_{q\bar{j},\bar{l}}\partial_\bar{p}),\partial_k)=g_{\mathbb{C}}(g^{q\bar{p}}g_{q\bar{j},i\bar{l}}\partial_\bar{p}-g^{q\bar{r}}g_{s\bar{r},i}g^{s\bar{p}}g_{q\bar{j},\bar{l}}\partial_\bar{p},\partial_k)=g_{k\bar{j},i\bar{l}}-g^{q\bar{r}}g_{k\bar{r},i}g_{q\bar{j},\bar{l}}$, in my computation, the sign of second order term now is positive, it’s wrong.
Here is what I thought:
(1) most books define the postion to raise or lower to be the second or the fourth, in this way, I can get the right sign. However I don’t see why this corresponds to the real case.
$\textbf{My second question:}$ is my computation wrong? or the Tian’s book is wrong? or there is some definition difference I didn’t notice.
$\textbf{My third question:}$ is it true $R_{i\bar{j}k\bar{l}}=-R_{\bar{j}ik\bar{l}}$ & $R_{i\bar{j}k\bar{l}}=-R_{\bar{j}ik\bar{l}}$? I don’t see any difference from the real case, however I can’t remember from where I read it lost antisymmetry in the first and second, third and fourth position. I want to make sure of it.
Thanks!