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In the case that (M, OM ) is a complex manifold, F is the sheaf of holomorphic functions on M . §27 Locally free sheaf. A sheaf E is locally free and of rank r if there is an open covering . One can show that (isomorphism classes of) locally free {Uj } such that E|Uj ∼ = OU⊕r j sheaves of rank r over a manifold M are in one-to-one correspondence with (isomorphism classes of) vector bundles of rank r over M . The sheaf E corresponding to a certain vector bundle E is given by the sheaf dual to the sheaf of sections of E.

These deformations are called relative deformations. The Kodaira-Spencer theory of gravity, on the other hand, is a theory which describes the deformation of the total complex structure of a Calabi-Yau manifold as a result of closed string interactions. 1 Deformation of compact complex manifolds §1 Deformation of complex structures. Consider a complex manifold M covered by patches Ua on which there are coordinates za = (zai ) together with transition functions fab on nonempty intersections Ua ∩ Ub = ∅ satisfying the compatibility condition (cf.

30 Complex Geometry §27 K¨ ahler potential. Given a K¨ ahler manifold (M, g) with K¨ ahler form J, it follows i ¯ ¯ from dJ = (∂ + ∂)igi¯ dz ∧ d¯ z = 0 that ∂gl¯ ∂gi¯ = l ∂z ∂z i ∂gl¯ ∂gi¯ = . 8) l ∂ z¯ ∂ z¯i ¯ ¯ . This Thus, we can define a local real function K such that g = ∂ ∂K and J = i∂ ∂K function is called the K¨ ahler potential of g. 8). §28 Examples. A simple example is the K¨ ahler metric on m obtained from the K¨ ahler 1 i ¯ ı z z¯ , which is the complex analog of (Ê2m , δ). Also easily seen is the potential K = 2 ahler: since J is a fact that any orientable complex manifold M with dim M = 1 is K¨ real two-form, dJ has to vanish on M .