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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?

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    $\begingroup$ I don't think $R_{jjjj}$, for all $j$, determines the whole curvature tensor. What might be true (I don't remember) is that $R(X,\bar{X},X,\bar{X})$, for all $X \in T_{\mathbb{C}}$, determines the whole curvature tensor. $\endgroup$
    – Deane Yang
    Oct 28, 2011 at 13:39
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    $\begingroup$ Thanks for the correction. I'm now very confused by Siu's phrase "We now polarize the expression for $R^{(WP)}_{i \bar i i \bar i}$ to get the expression for $R^{(WP)}_{i \bar j k \bar l}$." at the beginning of paragraph 5.4 in "Curvature of the Weil-Petersson metric in the moduli space of compact Kahler-Einstein manifolds of negative first Chern class". $\endgroup$
    – Gunnar Þór Magnússon
    Oct 28, 2011 at 13:48
  • $\begingroup$ Maybe Siu is speaking figuratively, but you'll need to work out the details. Or maybe someone who actually knows the paper can comment further. $\endgroup$
    – Deane Yang
    Oct 28, 2011 at 13:53
  • $\begingroup$ @Deane - Nannicini says the same thing: "The complete expression for the Riemann tensor can now be obtained by polarization of $R_{i \bar i i \bar i}$" in "Weil-Petersson metric in the moduli space of compact polarized Kahler-Einstein manifolds of zero first Chern class". If they're not referring to the above lemma on the holomorphic sectional curvature, then what are Siu and Nannicini doing? $\endgroup$
    – Gunnar Þór Magnússon
    Oct 28, 2011 at 13:55
  • $\begingroup$ Gunnar, I know absolutely nothing about the Weil-Petersson metric, so I can't answer. There might be additional properties that matter here? Also, are they working with a fixed frame of tangent vectors and claiming that $R_{j\bar{j}j\bar{j}}$ with respect to that frame determines the whole curvature tensor? Or are they claiming that knowing $R_{j\bar{j}j\bar{j}}$ with respect to all, say, unitary frames determines the full tensor? $\endgroup$
    – Deane Yang
    Oct 28, 2011 at 14:14

1 Answer 1

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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 $Q(X)=2 R(V,\bar{V},V,\bar{V})$.

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