# Calabi-Yau condition, moduli and Lichnerowicz equation

I have a conceptual confusion about the metric moduli of Calabi-Yau manifolds, when I was reading Calabi-Yau compactification.

As I understand, the metric moduli is parametrized by infinitesimal deformation of the metric that preserve Calabi-Yau condition (Ricci-flatness or equivalently, admit one covariantly constant spinor). So we would have the Lichnerowicz equation of the deformation metric: $$\nabla^l \nabla_l\ \delta g_{mn} - [\nabla^l,\nabla_m]\ \delta g_{ln} - [\nabla^l, \nabla_n]\ \delta g_{lm} = 0$$

And moreover, the deformation is a deformation of complex structure: the compensating coordinate transformation is not holomorphic. Or more specifically, the metric can be no longer written in a Hermitian form. But since the deformation preserves Calabi-Yau condition, it should still be Calabi-Yau, so it is still Kahler.

Thus, is the deformed metric still Calabi-Yau? If it is, how to understand the emergence of non-Hermitian component in the metric?

The most general $\delta g_{mn}$ that preserve the Ricci-flatness on the original Calabi-Yau backgrounds is the sum of several components: the pure infinitesimal diffeomorphisms (I won't discuss those because they're physically vacuous and trivial); changes of the Kähler moduli; and changes of the complex structure.
A Calabi-Yau three-fold has $h_{1,1}$ real parameters describing the Kähler moduli (they become complexified when the $B$-field two-form is added in type II string theory) and $h_{1,2}$ complex parameters describing the complex structure moduli. These two integers are interchanged for the manifold related by the mirror symmetry.
The Kähler moduli describe different ways to choose the Ricci-flat metric on the manifold that are compatible with a fixed, given complex structure. The different solutions may locally be derived from the Kähler potential $K$ which can have many forms. These moduli effectively describe the proper areas of 2-cycles.
The complex structure deformations change the complex structure – and the corresponding spinor – but the new, deformed manifold still has a complex structure, just a different one! The metric after an infinitesimal variation would have a non-Hermitian component with respect to the old complex structure but with respect to the new, deformed complex structure, it is still perfectly Hermitian! Calabi-Yaus are always Kähler, $SU(3)$ (for 6 real dimensions) complex manifolds with a purely Hermitian metric in some appropriate complex coordinates, and a deformation still keeps the manifold in the set of Calabi-Yaus.
The simplest example to test your $\delta g_{mn}$ intuition is 1-complex-dimensional or 2-real-dimensional Calabi-Yaus, two-tori. The complex structure is changing the ratios of the two periods and the angle in between, namely the $\tau$ complex structure parameter. The Kähler modulus is the overall area of the two torus – the overall scaling of the whole 2-dimensional metric. You may easily see that all these transformation keep the manifold flat, and therefore Ricci-flat.