Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 429independent. Introducing local complex coordinates in the charts U i onM, the φ i can be expressed as maps from U i to an open set in C n 2 , withφ j ◦ φ −1i being a holomorphic map from C n 2 to C n 2 . Clearly, n must beeven for this to make sense. In local complex coordinates, we recall thata function h : C n 2 → C n 2 is holomorphic if h(z 1 , ¯z 1 , ..., z n 2 , ¯z n 2 ) is actuallyindependent of all the ¯z j .In a given patch on any even–dimensional manifold, we can always introducelocal complex coordinates by, for instance, forming the combinationsz j = x j + ix n 2 +j , where the x j are local real coordinates on M. The realtest is whether the transition functions from one patch to another — whenexpressed in terms of the local complex coordinates — are holomorphicmaps. If they are, we say that M is a complex manifold of complex dimensiond = n/2. The local complex coordinates with holomorphic transitionfunctions give M with a complex structure (see [Greene (1996)]).Fig. 3.12 The charts for a complex manifold M have complex coordinates (see text forexplanation).Given a smooth manifold with even real dimension n, it can be a difficultquestion to determine whether or not a complex structure exists. On theother hand, if some smooth manifold M does admit a complex structure,we are not able to decide whether it is unique, i.e., there may be numerousinequivalent ways of defining complex coordinates on M.Now, in the same way as a homeomorphism defines an equivalence betweentopological manifolds, and a diffeomorphism defines an equivalencebetween smooth manifolds, a biholomorphism defines an equivalence betweencomplex manifolds. If M and N are complex manifolds, we considerthem to be equivalent if there is a map φ : M → N which in additionto being a diffeomorphism, is also a holomorphic map. That is, when expressedin terms of the complex structures on M and N respectively, φ is