Basics on Kähler Manifolds
Riemann Metric, Hermitian Metric and Kähler Form
For a (real) manifold \(M\) of dimension \(n\), the tangent bundle is denoted by \(\mathrm{T}M\) while the tangent space at a point \(x \in M\) is denoted by \(\mathrm{T}_xM\). A Riemann metric \(g\) associates to every point \(x \in M\) a positive definite symmetric inner product \[g_x:\mathrm{T}_xM \times \mathrm{T}_xM \to \mathbb{R}\] in a smooth way. Locally around \(x\) choose set of bases \[\{\partial_{x_1}, \partial_{x_2}, \ldots, \partial_{x_n}\}\] the riemann metric \(g\) is then written as \[g=g_{ij}\mathrm{d}x^i\otimes\mathrm{d}x^j\] where \(g_{ij}=g_{ji}\) varies smoothly along any smooth chart.
For a complex manifold \(A\) of complex dimension \(n\) (therefore of real dimension \(2n\)), the complexified tangent space is the tensor product of the (real) tangent space with \(\mathbb{C}\), namely
\[ \mathrm{T}_\mathbb{C}M=\mathrm{T}M\otimes\mathbb{C}. \]
There is an associated linear operation \(J\) acting on \(TM\) over \(M\) such that \(J^2=-1\), which represents the so-called almost complex structure. The tangent space \(\mathrm{T}_xM\) is then decomposed into eigen spaces \[\mathrm{T}_xM=\mathrm{T}_xM^{1,0}\oplus\mathrm{T}_xM^{0,1}\] where \(\mathrm{T}_xM^{1,0}\) is the space of vectors \(v\) such that \(Jv=\mathrm{i}v\) and \(\mathrm{T}_xM^{0,1}\) is the space of vectors \(v\) such that \(Jv=-\mathrm{i}v\). Now the local choice of (real) bases is denoted by \[\{\partial_{z_1}, \partial_{z_2}, \ldots, \partial_{z_n}, \partial_{\bar{z_1}}, \partial_{\bar{z_2}}, \ldots, \partial_{\bar{z_n}}\}.\]
\[ \mathrm{T}_M\hookrightarrow \mathrm{T}M\otimes\mathbb{C}=\mathrm{T}^{1,0}\oplus\mathrm{T}^{0,1} \]
In this case, a Hermitian metric \(h\) turns the tangent space to each point \(x\in M\) into a Hermitian space (with respect to the almost complex structure \(J\)). The action of \(h\) is calculated as \[h()\] This amounts to say