How is the Inner Product Expansion in Holomorphic Bundles Derived?

[math6]

let x a point on complex manifold X, z_j a coordinate system at x , E a holomorphic bundle and let h_α be a holomorphic frame of E. After replacing h_α by suitable linear combinations with constant coefficients we may assume that h_ α is an orthonormal basis of E_{x}. Then an inner product <h_α, h_β > have an expansion :

<h_α, h_β > = δ_αβ + Ʃ ( a_jαβ z_j + a'_jαβ z_j) + O( |z|^2 ) (1)
for some complex coefficients a_jαβ and a'_jαβ .

I would like to understandd how did he get the expression (1) ?

Thnx for your answers..

 

[jvicens]

Thank you for your question. The expression (1) is obtained using the Gram-Schmidt process, which is a method for orthonormalizing a set of vectors in a vector space. In this case, the vectors are the holomorphic frames h_α of the holomorphic bundle E.

To explain the process, let's start with an arbitrary basis h_1, h_2, ..., h_n of the vector space E_x at the point x. We can then apply the Gram-Schmidt process to obtain an orthonormal basis h'_1, h'_2, ..., h'_n. This process involves finding the orthogonal projection of each vector onto the subspace spanned by the previously orthonormalized vectors.

In our case, since the basis h_α is already holomorphic, we can simplify the process by using the fact that the inner product <h_α, h_β> is holomorphic as well. This means that we can write it as a power series in the coordinates z_j, with complex coefficients a_jαβ and a'_jαβ, as shown in (1). The constant term δ_αβ represents the Kronecker delta, which is 1 if α = β and 0 otherwise.

I hope this helps to clarify how the expression (1) is obtained. Please let me know if you have any further questions or need more explanation.