1. A Little Complex Analysis
We want to introduce the notion of a ‘Fubini-Study’ metric which is important in Complex Manifold Theory and Differential Geometry (and the associated theories such as Mathematical Physics). But first we need to introduce a little Complex Analysis. The source is of course Griffiths and Harris. Let M be a complex manifold, any point, and a holomorophic co-ordinate system around p. There are three different notions of a tangent space to M at p,which we now describe:
- is the usual real tangent space to M at p,when we consider M a real manifold of dimension 2n. can be realized as the space of linear derivations on the ring of real-valued -functions in a neighbourhood of p; if we write , .
- is called the complexified tangent space to M at p. It can be realized as the space of linear derivations in the ring of complex valued -functions on M around p. We can write
- is called the holomorphic tangent space to M at p. It can be realized as the subspace of consisting of derivations that vanish on antiholomorphic functions (i.e. F such that T is holomorphic), and so is independent of the holomorphic co-ordinate system chosen. The subspace is called the antiholomorphic tangent space to M at p; clearly
Now we consider some Calculus on Complex Manifolds. Let M be a complex manifold of dimension n. A hermitian metric on M is given by a positive definite hermitian inner product on the holomorphic tangent space at z for each ,
depending smootly on z – that is, such that for local co-ordinates z on M the function
Writing in terms of the basis for , the hermitian metric is given by So let us describe the Fubini-Study Metric Let be co-ordinates on and denote by the standard projection map. Let be an open set and a lifting of U, i.e. a holomorphic map with ; consider the differential form
If is another lifting, then with f a nonzero holomorphic function, so that
= Therefore is independent of the lifting chosen; since liftings always exist locally, is a globally defined differential form in . (By the sheaf properties of differential forms) Clearly is of type (1,1). To see that is positive, first note that the unitary group acts transitively on and leaves the form positive everywhere if it is positive at one point. Now let be co-oridnates on the open set in and use the lifting on ; we have (after some substitutions
At the point , \\ Thus defines a particular hermitian metric on the projective complex space called the Fubini-Study metric. That was the aim of the article!