Linear Algebra Question in a Complex Geometry Proof

[cogito²]

My brain has ceased functioning while trying to understand a small aspect of the proof of Wirtinger's inequality on page 175 in Demailly's book: (http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf).

On the second line of that proof he says to "choose an oriented R-basis such that blah blah blah..." Basically he chooses an R-basis (e_1, e_2, ...) which is orthonormal wrt the real part of the hermitian metric, such that the imaginary part of the metric is non-zero except for in pairs. What I mean is that I am h(e_1, e_2), I am h(e_3,e_4), ... can be non-zero, but all other pairs (for example I am h(e_1,e_3)) are zero.

I'm having trouble seeing how this is done. I mean if I knew that the real submanifold were complex analytic ahead of time I could do it (which is done a little later in the proof), but I'm confused otherwise...this is basically just a linear algebra question so I just figured if someone here just sees this clearly they might save me a lot of head ache.

My explanation of the problem is pretty bad, but I figure since it's just the first couple lines in the proof, it might be easier to let the pdf to the talking...though if any of you guys disagree, I can try to clarify the question...

 

[pat_connell]

but that would take a while.The way Demailly chooses the R-basis is by first considering an orthonormal basis (e_1, e_2, ...) with respect to the real part of the hermitian metric. Then he shifts the basis vectors e_i so that they are orthonormal with respect to the full hermitian metric. Since the hermitian metric is a symmetric matrix, this can be done by applying a unitary transformation. Specifically, if h(e_i,e_j) is the hermitian metric, then the new basis vectors v_i can be written as: v_i = e_i + ∑_{j≠i} (h(e_i,e_j)/h(e_j,e_j)) e_j This way, for all i≠j, we have h(v_i,v_j) = 0. However, it is possible that some of the diagonal entries h(v_i,v_i) are non-zero. To fix this, we just need to scale the basis vectors so that h(v_i,v_i) = 1 for all i. This is the final R-basis that Demailly uses in his proof.