In order to describe physics of 4 dimensional space-time starting from 10 dimensional space, we consider that the extra 6 dimensional space has very tiny size (about the Planck length). This is called the comopactification in string theory. Then the symmetry of theory requires this internal space to be <PERSON> flat: \begin{equation} R_{IJ}=0 \end{equation} where $I$ and $J$ run from 0 to 6.
Since C-Y manifolds have the structure of the complex manifolds, it is appropriate to use the indices of the complex manifolds, i.e. $i,j=1,2,3$. The metric on C-Y manifolds is of type (1,1) $G_{i,\bar j}$ and we obtain the closed (1,1)-form from the metric \begin{equation} \omega =\sqrt{-1}G_{i\bar j} dz^i\wedge d\bar z^{\bar j} . \end{equation} And there exists nowhere vanishing holomorphic 3-form $\Omega_{klm}$ since the <PERSON> tensor vanishes.
In what follows we consider the moduli of C-Y manifolds. If the geometric object can be continuously deformed with its geometric properties preserved, we call the parameter of this deformation moduli. Suppose that the metric $G_{IJ}(y)$ is given on a C-Y manifold $M$ where $y$ is the local coordinate of $M$. And assume that the metric change $G_{IJ}+g_{IJ}$ and this new metric also gives RIcci flat. Then taking the first order of the deformation of metric, we have \begin{equation} \Delta_6 g_{IJ}(y)=0 \end{equation} where $\Delta_6$ is a 6 dimensional Laplacian. Thus, the deformation of the metric which preserves the definition of C-Y manifolds is given by the eigenfuntion of the Laplacian with its eigenvalue zero. 578
In general, the eigenvalues of $-\Delta_6$ take zero and positive values: \begin{equation} -\Delta_6 f_{IJ}^\alpha(y)=m_\alpha^2 f_{IJ}^\alpha(y), \,\,\,\alpha=1,2,\cdots. \end{equation} Let us consider the case that the deformation of metric $g_{IJ}$ is of type (1,1), $\delta g_{i\bar j}$, and of type (2,0) , $\delta g_{i j}$, respectively. Generally, the metric on a K\"ahler manifold is of type (1,1) and $g_{i\bar j}$ gives the deformation preserving this type. This is called the deformation of K\"ahler structure. The deformation of K\"ahler structure is described by a solution of the equation \begin{equation} \Delta_6 \omega_{i j}(y)=0, \end{equation} i.e. it is given by a harmonic (1,1)-form. The number of a harmonic (1,1)-form is provided by the Hodge number $h_{1,1}$ of the manifold $M$. On the other hand, the type (2,0) deformation of metric implies the deformation of complex structure on a complex manifold. Using the complex conjugate $\bar \Omega$ of $\Omega$ , we obtain \begin{equation} \chi_{i\bar j \bar k} \equiv \delta g_{ij} G^{j\bar k} \bar \Omega_{\bar k\bar l\bar m}. \end{equation} Therefore the deformation of complex structure is described by harmonic (1,2)-form and the degree of freedom of the deformation becomes the Hodge number $h_{1,2}=h_{2,1}$. As we see, C-Y manifolds have two kinds of the deformation parameters, the K\"ahler parameters and the parameters of complex structure, which are called moduli, and the K\"ahler parameters correspond to the degree of the change of the size and the parameters of complex structure correspond to that of the deformation of the shape. 526