This post contains calculations that continue on the ideas from the previous post "Fubini--Study metric," 2017-03-05. First we suppose that γ lies on a curve C⊂P2, with the curve defined as the zero locus of a polynomial P. Taking the derivative of P on C2 gives Pz1dz1+Pz2dz2=0, which can be manipulated to give
dz2=−Pz1Pz2dz1,∂∂z2=−Pz2Pz1∂∂z1,d¯z2=−¯Pz1¯Pz2d¯z1,∂∂¯z2=−¯Pz2¯Pz1∂∂¯z1.
Using the above and an equation from the mentioned post, for e=∂∂z1+∂∂¯z1+∂∂z2+∂∂¯z2, we get
dγdt=(¯γ′1−Pz2Pz1¯γ′2)∂∂z1+(γ′1−¯Pz2¯Pz1γ′2)∂∂¯z1(2∑k,ℓ=1χkℓ(γ)dzk∧d¯zℓ)(e,e)=1+|γ2|2+¯Pz1¯Pz2¯γ1γ2+Pz1Pz2γ1¯γ2+|Pz1Pz2|2(1+|γ1|2)=1+|Pz1Pz2|2+|Pz1Pz2γ1+γ2|2,(dz1∧d¯z1)(dγdt,Idγdt)=det
Hence
g\left(\frac{d\gamma}{dt},\frac{d\gamma}{dt}\right) = \frac{\left(1 +\left|\frac{P_{z_1}}{P_{z_2}}\right|^2 + \left|\frac{P_{z_1}}{P_{z_2}}\gamma_1+\gamma_2\right|^2\right)\left|\gamma_1' - \frac{\overline{P_{z_2}}}{\overline{P_{z_1}}}\gamma_2'\right|^2}{\pi\left(1+|\gamma_1|^2+|\gamma_2|^2\right)^2}.
Now we move to \P^n, and consider X\subset \P^n a complete intersection of codimension r, or the zero set of polynomials P_1=0,\dots,P_r=0. Expressing some covectors in terms of others reduces the number of determinants we calculated above from 2n to 2(n-r). Then
\begin{align*} P_{1,z_1}dz_1 + \cdots + P_{1,z_n}dz_n & = 0, & dz_n & = c_{n,1}dz_1 + \cdots + c_{n,n-r}dz_{n-r}, \\ & \ \ \vdots & & \ \ \vdots \\ P_{r,z_1}dz_1 + \cdots + P_{r,z_n}dz_n & = 0, & dz_{n-r+1} & = c_{n-r+1,1}dz_1 + \cdots + c_{n-r+1,n-r}dz_{n-r}, \end{align*}
for the c_{i,j} some combinations of the P_{k,z_\ell}. By orthonormality of the basis vectors, and assuming that the c_{i,j} are all non-zero, we find
\frac\dy{\dy z_i} = \sum_{j=1}^{n-r} \frac1{(n-r)c_{i,j}}\frac\dy{\dy z_j},\hspace{2cm} \frac\dy{\dy \overline{z_i}} = \sum_{j=1}^{n-r} \frac1{(n-r)\overline{c_{i,j}}}\frac\dy{\dy \overline{z_j}},
for all integers n-r<i\leqslant n. This allows us to rewrite the path derivative as
\begin{align*} \frac{d\gamma}{dt} & = \sum_{i=1}^n \overline \gamma_i'\frac\dy{\dy z_i} +\gamma_i'\frac\dy{\dy \overline{z_i}} \\ & = \sum_{i=1}^{n-r} \left(\overline \gamma_i'\frac\dy{\dy z_i} +\gamma_i'\frac\dy{\dy \overline{z_i}}\right) +\sum_{i=n-r+1}^n \left(\sum_{j=1}^{n-r} \frac{\overline \gamma_i'}{(n-r)c_{i,j}}\frac\dy{\dy z_j} + \sum_{j=1}^{n-r} \frac{\gamma_i'}{(n-r)\overline{c_{i,j}}}\frac\dy{\dy \overline{z_j}}\right) \\ & = \sum_{i=1}^{n-r}\left(\overline\gamma_i' + \sum_{j=n-r+1}^n \frac{\overline\gamma_j'}{(n-r)c_{j,i}}\right)\frac\dy{\dy z_i} + \left(\gamma_i'+\sum_{j=n-r+1}^n \frac{\gamma_j'}{(n-r)\overline{c_{j,i}}}\right)\frac\dy{\dy \overline{z_i}}. \end{align*}
In the case of a curve in \P^n, when r=n-1, let c_{1,1}=1 and e = \frac\dy{\dy z_1} + \frac\dy{\dy \overline {z_1}} + \cdots + \frac\dy{\dy z_n} + \frac\dy{\dy \overline{z_n}} to get
\begin{align*} \frac{d\gamma}{dt} & = \left(\sum_{j=1}^n \frac{\overline\gamma_j'}{c_{j1}}\right)\frac\dy{\dy z_1} + \left(\sum_{j=1}^n \frac{\gamma_j'}{\overline{c_{j1}}}\right)\frac\dy{\dy \overline{z_1}},\\ \left(\sum_{k,\ell=1}^n\chi_{k\ell}(\gamma)dz_k\wedge d\overline{z_\ell}\right)(e,e) & = \sum_{k,\ell=1}^n \left(1+\sum_{i=1}^n |\gamma_i|^2\right)\delta_{k\ell} - \overline{\gamma_kc_{\ell1}}\gamma_\ell c_{k1}, \\ (dz_1\wedge d\overline{z_1})\left(\frac{d\gamma}{dt},I\frac{d\gamma}{dt}\right) & = \det \begin{bmatrix} \sum_{j=1}^n \frac{\overline\gamma_j'}{c_{j1}} & i \sum_{j=1}^n \frac{\overline\gamma_j'}{c_{j1}} \\[5pt] \sum_{j=1}^n \frac{\gamma_j'}{\overline{c_{j1}}} & -i\sum_{j=1}^n \frac{\gamma_j'}{\overline{c_{j1}}} \end{bmatrix} = -2i \left|\sum_{j=1}^n \frac{\gamma_j'}{\overline{c_{j1}}}\right|^2. \end{align*}
Hence
g\left(\frac{d\gamma}{dt},\frac{d\gamma}{dt}\right) = \frac{\left(\sum_{k,\ell=1}^n \left(1+\sum_{i=1}^n |\gamma_i|^2\right)\delta_{k\ell} - \overline{\gamma_kc_{\ell1}}\gamma_\ell c_{k1}\right)\left|\sum_{j=1}^n \frac{\gamma_j'}{\overline{c_{j1}}}\right|^2}{\pi \left(1+\sum_{i=1}^n |\gamma_i|^2\right)^2}.
The terms \overline{\gamma_kc_{\ell1}}\gamma_\ell c_{k1} may be rearranged into terms |\gamma_kc_{\ell1}-\gamma_\ell c_{k1}|^2, but it does not provide any enlightening results, similarly to the rest of this post.
dz2=−Pz1Pz2dz1,∂∂z2=−Pz2Pz1∂∂z1,d¯z2=−¯Pz1¯Pz2d¯z1,∂∂¯z2=−¯Pz2¯Pz1∂∂¯z1.
Using the above and an equation from the mentioned post, for e=∂∂z1+∂∂¯z1+∂∂z2+∂∂¯z2, we get
dγdt=(¯γ′1−Pz2Pz1¯γ′2)∂∂z1+(γ′1−¯Pz2¯Pz1γ′2)∂∂¯z1(2∑k,ℓ=1χkℓ(γ)dzk∧d¯zℓ)(e,e)=1+|γ2|2+¯Pz1¯Pz2¯γ1γ2+Pz1Pz2γ1¯γ2+|Pz1Pz2|2(1+|γ1|2)=1+|Pz1Pz2|2+|Pz1Pz2γ1+γ2|2,(dz1∧d¯z1)(dγdt,Idγdt)=det
Hence
g\left(\frac{d\gamma}{dt},\frac{d\gamma}{dt}\right) = \frac{\left(1 +\left|\frac{P_{z_1}}{P_{z_2}}\right|^2 + \left|\frac{P_{z_1}}{P_{z_2}}\gamma_1+\gamma_2\right|^2\right)\left|\gamma_1' - \frac{\overline{P_{z_2}}}{\overline{P_{z_1}}}\gamma_2'\right|^2}{\pi\left(1+|\gamma_1|^2+|\gamma_2|^2\right)^2}.
Now we move to \P^n, and consider X\subset \P^n a complete intersection of codimension r, or the zero set of polynomials P_1=0,\dots,P_r=0. Expressing some covectors in terms of others reduces the number of determinants we calculated above from 2n to 2(n-r). Then
\begin{align*} P_{1,z_1}dz_1 + \cdots + P_{1,z_n}dz_n & = 0, & dz_n & = c_{n,1}dz_1 + \cdots + c_{n,n-r}dz_{n-r}, \\ & \ \ \vdots & & \ \ \vdots \\ P_{r,z_1}dz_1 + \cdots + P_{r,z_n}dz_n & = 0, & dz_{n-r+1} & = c_{n-r+1,1}dz_1 + \cdots + c_{n-r+1,n-r}dz_{n-r}, \end{align*}
for the c_{i,j} some combinations of the P_{k,z_\ell}. By orthonormality of the basis vectors, and assuming that the c_{i,j} are all non-zero, we find
\frac\dy{\dy z_i} = \sum_{j=1}^{n-r} \frac1{(n-r)c_{i,j}}\frac\dy{\dy z_j},\hspace{2cm} \frac\dy{\dy \overline{z_i}} = \sum_{j=1}^{n-r} \frac1{(n-r)\overline{c_{i,j}}}\frac\dy{\dy \overline{z_j}},
for all integers n-r<i\leqslant n. This allows us to rewrite the path derivative as
\begin{align*} \frac{d\gamma}{dt} & = \sum_{i=1}^n \overline \gamma_i'\frac\dy{\dy z_i} +\gamma_i'\frac\dy{\dy \overline{z_i}} \\ & = \sum_{i=1}^{n-r} \left(\overline \gamma_i'\frac\dy{\dy z_i} +\gamma_i'\frac\dy{\dy \overline{z_i}}\right) +\sum_{i=n-r+1}^n \left(\sum_{j=1}^{n-r} \frac{\overline \gamma_i'}{(n-r)c_{i,j}}\frac\dy{\dy z_j} + \sum_{j=1}^{n-r} \frac{\gamma_i'}{(n-r)\overline{c_{i,j}}}\frac\dy{\dy \overline{z_j}}\right) \\ & = \sum_{i=1}^{n-r}\left(\overline\gamma_i' + \sum_{j=n-r+1}^n \frac{\overline\gamma_j'}{(n-r)c_{j,i}}\right)\frac\dy{\dy z_i} + \left(\gamma_i'+\sum_{j=n-r+1}^n \frac{\gamma_j'}{(n-r)\overline{c_{j,i}}}\right)\frac\dy{\dy \overline{z_i}}. \end{align*}
In the case of a curve in \P^n, when r=n-1, let c_{1,1}=1 and e = \frac\dy{\dy z_1} + \frac\dy{\dy \overline {z_1}} + \cdots + \frac\dy{\dy z_n} + \frac\dy{\dy \overline{z_n}} to get
\begin{align*} \frac{d\gamma}{dt} & = \left(\sum_{j=1}^n \frac{\overline\gamma_j'}{c_{j1}}\right)\frac\dy{\dy z_1} + \left(\sum_{j=1}^n \frac{\gamma_j'}{\overline{c_{j1}}}\right)\frac\dy{\dy \overline{z_1}},\\ \left(\sum_{k,\ell=1}^n\chi_{k\ell}(\gamma)dz_k\wedge d\overline{z_\ell}\right)(e,e) & = \sum_{k,\ell=1}^n \left(1+\sum_{i=1}^n |\gamma_i|^2\right)\delta_{k\ell} - \overline{\gamma_kc_{\ell1}}\gamma_\ell c_{k1}, \\ (dz_1\wedge d\overline{z_1})\left(\frac{d\gamma}{dt},I\frac{d\gamma}{dt}\right) & = \det \begin{bmatrix} \sum_{j=1}^n \frac{\overline\gamma_j'}{c_{j1}} & i \sum_{j=1}^n \frac{\overline\gamma_j'}{c_{j1}} \\[5pt] \sum_{j=1}^n \frac{\gamma_j'}{\overline{c_{j1}}} & -i\sum_{j=1}^n \frac{\gamma_j'}{\overline{c_{j1}}} \end{bmatrix} = -2i \left|\sum_{j=1}^n \frac{\gamma_j'}{\overline{c_{j1}}}\right|^2. \end{align*}
Hence
g\left(\frac{d\gamma}{dt},\frac{d\gamma}{dt}\right) = \frac{\left(\sum_{k,\ell=1}^n \left(1+\sum_{i=1}^n |\gamma_i|^2\right)\delta_{k\ell} - \overline{\gamma_kc_{\ell1}}\gamma_\ell c_{k1}\right)\left|\sum_{j=1}^n \frac{\gamma_j'}{\overline{c_{j1}}}\right|^2}{\pi \left(1+\sum_{i=1}^n |\gamma_i|^2\right)^2}.
The terms \overline{\gamma_kc_{\ell1}}\gamma_\ell c_{k1} may be rearranged into terms |\gamma_kc_{\ell1}-\gamma_\ell c_{k1}|^2, but it does not provide any enlightening results, similarly to the rest of this post.
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