I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson metric $h$, show that it is Kahler, and obtain results on the holomorphic sectional curvature by heroic calculations.
From what I can piece together, the curvature estimates go as follows: 1) show that the holomorphic sectional curvature, given by $R_{jjjj}$, is negative, 2) use a polarization trick to calculate the general tensor $R_{jklm}$, 3) then a miracle occurs, 4) so the holomorphic sectional curvature $R_{jjkk}$ is negative.
I'm trying to fill in the gaps in my understanding between the first and fourth steps, and I'm stuck on the second one. For Kahler manifolds the holomorphic sectional curvature determines the entire curvature tensor (see for example Lemma 7.19 of Zheng's "Complex differential geometry"), so I'm perfectly willing to believe that knowing $R_{jjjj}$ lets us calculate $R_{jklm}$. The problem is that I don't know how to do it.
This is a purely algebraic calculation, so I imagine it's written down somewhere, but the only results I've found are of the same type as in Zheng's book, i.e. they show that these calculations are theoretically possible but don't say how to do them.
Question: Is there a reference where this calculation is done explicitly?
[Edit] As Deane pointed out in the comments, one needs to know $R(X,\bar X,X,\bar X)$ for all holomorphic tangent fields $X$ to know the curvature tensor, just knowing $R_{j\bar jj\bar j}$ doesn't cut it. This makes two phrases from Siu's and Nannicini's papers a bit mysterious:
Siu: "We now polarize the expression for $R_{i\bar i i\bar i}$ to get the expression for $R_{i \bar j k \bar l}$." in "Curvature of the Weil-Petersson metric in the moduli space of compact Kahler-Einstein manifolds of negative first Chern class" (beginning of paragraph 5.4).
Nannicini: "The complete expression for the Riemann tensor can now be obtained by polarization of $R_{i\bar ii\bar i}$" in "Weil-Petersson metric in the moduli space of compact polarized Kahler-Einstein manifolds of zero first Chern class" (the page before Theorem 1).
Revised question: What exactly are Siu and Nannicini doing, if not applying the lemma on the holomorphic sectional curvature?
Best Answer
The explicit polarization formula is the following, taken from this paper of Bishop and Goldberg.
Working with real tangent vectors (instead of $(1,0)$ vectors, but it's easy to switch from one point of view to the other) the holomorphic sectional curvature of a unit vector $X$ is $Q(X)=R(X,JX,JX,X)$, and a general sectional curvature $R(X,Y,Y,X)$ can be expressed as
$$R(X,Y,Y,X)=\frac{1}{32}(3Q(X+JY)+3Q(X-JY)-Q(X+Y)-Q(X-Y)-4Q(X)-4Q(Y) ).$$
If you want to switch back to vectors of type $(1,0)$, you can call $V=\frac{1}{\sqrt{2}}(X-iJX)$ and then you have that $Q(X)=2 R(V,\bar{V},V,\bar{V})$.