The most general δgmn 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 h1,1 real parameters describing the Kähler moduli (they become complexified when the B-field two-form is added in type II string theory) and h1,2 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 δgmn 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 τ 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.
This post imported from StackExchange Physics at 2014-08-12 18:14 (UCT), posted by SE-user Luboš Motl