Trees and tensors on Kähler manifolds

Abstract

We present an organized method to convert between partial derivatives of metrics (functions) and covariant derivatives of curvature tensors (functions) on Kähler manifolds. Basically, it reduces the highly recursive computation in tensor calculus to the enumeration of certain trees with external legs.

Appendices

Appendix 1: Computations of $D(g_{a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6 a_7\bar{b}_8})$

In this appendix, we use (10) to compute $g^{a_1\bar{b}_2}g^{a_3\bar{b}_4}g^{a_5\bar{b}_6}g^{a_7\bar{b}_8}D(g_{a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6 a_7\bar{b}_8}),$ which is the key term in the weight three coefficient of the asymptotic expansion of the Bergman kernel (cf. [13]). We will express it in terms of the following basis of $15$ tensors used by Engliš [1].

$$\begin{aligned} \sigma _1&=\rho ^3,&\sigma _2&=\rho R_{i\bar{j}}R_{j\bar{i}},&\sigma _3&=\rho R_{i\bar{j} k\bar{l}}R_{j\bar{i} l\bar{k}}, \\ \sigma _4&=R_{i\bar{j}}R_{k\bar{l}} R_{j\bar{i} l\bar{k}},&\sigma _5&=R_{i\bar{j}}R_{k\bar{i} l\bar{m}}R_{j\bar{k} m\bar{l}},&\sigma _6&=R_{i\bar{j}}R_{j\bar{k}}R_{k\bar{i}},\\ \sigma _7&=R_{i\bar{j} k\bar{l}}R_{j\bar{i} m\bar{n}}R_{l\bar{k} n\bar{m}},&\sigma _8&=\rho \Delta \rho ,&\sigma _9&=R_{i\bar{j}}R_{j\bar{i}/k\bar{k}},\\ \sigma _{10}&=R_{i\bar{j} k\bar{l}}R_{j\bar{i} l\bar{k}/m\bar{m}},&\sigma _{11}&=\rho _{/i}\rho _{/\bar{i}},&\sigma _{12}&=R_{i\bar{j}/ k}R_{j\bar{i}/\bar{k}},\\ \sigma _{13}&=R_{i\bar{j} k\bar{l}/m}R_{j\bar{i} l\bar{k}/\bar{m}},&\sigma _{14}&=\Delta ^2 \rho ,&\sigma _{15}&=R_{i\bar{j} k\bar{l}}R_{j\bar{m} l \bar{n}}R_{m \bar{i} n\bar{k}}. \end{aligned}$$

It is understood that $(i,\bar{i}),\,(j, \bar{j}),$ etc. are paired indices to be contracted. We need three more tensors:

$$\begin{aligned} \widetilde{\sigma }_9=R_{i\bar{j}}\rho _{j\bar{i}},\qquad \widetilde{\sigma }_{10}=R_{i\bar{j} k\bar{l}}R_{j\bar{i}/l\bar{k}}, \qquad \widehat{\sigma }_{10}=R_{i\bar{j} k\bar{l}}R_{j\bar{i}/\bar{k} l}. \end{aligned}$$

By the Ricci formula (7),

$$\begin{aligned} \widetilde{\sigma }_9=\sigma _9+\sigma _4-\sigma _6,\qquad \widetilde{\sigma }_{10}=\widehat{\sigma }_{10}=\sigma _{10}+2\sigma _7-\sigma _5-\sigma _{15}. \end{aligned}$$

(17)

By (1) and (10), we have

$$\begin{aligned} D(g_{a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6 a_7\bar{b}_8})&= D(-\partial _{a_5\bar{b}_6 a_7\bar{b}_8}R_{a_1\bar{b}_2 a_3\bar{b}_4}) +D\big (\partial _{a_5\bar{b}_6 a_7\bar{b}_8}(g^{m\bar{n}}g_{m\bar{b}_2\bar{b}_4}g_{a_1\bar{n} a_3})\big )\\&= \sum _{T\in {\fancyscript{T}_R^{\prime }}}R_T+\sum _{T\in {\fancyscript{T}_R^{\prime \prime }}}R_T, \end{aligned}$$

where ${\fancyscript{T}_R^{\prime }}$ and ${\fancyscript{T}_R^{\prime \prime }}$ are subsets of ${\fancyscript{T}}_R(a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6 a_7\bar{b}_8)$ as defined in Remark 2.7. The unique one-vertex tree of ${\fancyscript{T}}_R(a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6 a_7\bar{b}_8)$ belongs to ${\fancyscript{T}_R^{\prime }}.$ The two-vertex and three-vertex trees of ${\fancyscript{T}}_R$ are listed in Tables 8 and 9, respectively, where we also list values after contractions by $g^{a_1\bar{b}_2}g^{a_3\bar{b}_4}g^{a_5\bar{b}_6}g^{a_7\bar{b}_8}$ and label a tree $T$ by a dagger symbol $\dag$ if and only if it belongs to ${\fancyscript{T}_R^{\prime \prime }},$ i.e., $\mathcal{B }(a_1,a_3)\cap \mathcal{A }(\bar{b}_2,\bar{b}_4)=\emptyset .$

Summing up the trees without $\dag$ in Tables 8 and 9, and using (17), we get

$$\begin{aligned} \sum _{T\in {\fancyscript{T}_R^{\prime }}}R_T&=-\sigma _{14}+6\sigma _9+\widetilde{\sigma }_9+2\widehat{\sigma }_{10}+2\widetilde{\sigma }_{10}+8\sigma _{12} -(4\sigma _4+2\sigma _5+2\sigma _6+2\sigma _7) \\&=-3\sigma _4-6\sigma _5-3\sigma _6+6\sigma _7+7\sigma _9+4\sigma _{10}+8\sigma _{12} -\sigma _{14}-4\sigma _{15}. \end{aligned}$$

Summing up the trees with $\dag$ in Tables 8 and 9, and using (17), we get

$$\begin{aligned} \sum _{T\in {\fancyscript{T}_R^{\prime \prime }}}R_T&=2\sigma _{10}+\widehat{\sigma }_{10}+\widetilde{\sigma }_{10}+2\sigma _{12}+3\sigma _{13}-(4\sigma _5+4\sigma _7) \\&=-6\sigma _{5}+4\sigma _{10}+2\sigma _{12}+3\sigma _{13}-2\sigma _{15}. \end{aligned}$$

Note that each tree $T$ is multiplied by a factor $(-1)^{|V(T)|}.$ The results agree with those of [13, Appendix A] and the computations are even simpler than [15, §5] in that we do not need to commute indices in the one-vertex tree.

Table 8 Two-vertex trees of ${\fancyscript{T}}_R(a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6 a_7\bar{b}_8)$

Table 9 Three-vertex trees of ${\fancyscript{T}}_R(a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6 a_7\bar{b}_8)$

Appendix 2: Computations of $L_3f-L_1^3f$

In this appendix, we obtain a formula of $L_3f-L_1^3f$ by computing

$$\begin{aligned} D(f_{a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6})=f_{/a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6}+\cdots \quad \text{ and}\quad D(f_{a_1 a_2 a_3\bar{b}_4 \bar{b}_5\bar{b}_6})=f_{/a_1 a_2 a_3\bar{b}_4 \bar{b}_5\bar{b}_6}+\cdots \end{aligned}$$

using (13). These are also key terms in the computation of the weight three coefficient of the asymptotic expansion of the Berezin transform (cf. [14]).

We will express final result as a linear combination of the following basis, the first nine of which were used by Engliš [1].

$$\begin{aligned} \sigma _1&=\Delta ^3 f,&\sigma _2&=R_{i\bar{j}}(\Delta f)_{/j\bar{i}},&\sigma _3&=R_{i\bar{j} k\bar{l}}f_{/j\bar{i} l\bar{k}},&\sigma _4&=R_{i\bar{j}/\bar{k}}f_{/j\bar{i} k},\\ \sigma _5&=R_{i\bar{j}/ k}f_{/j\bar{i} \bar{k}},&\sigma _6&=R_{i\bar{j} k\bar{l}}R_{j\bar{i} m\bar{k}}f_{/l\bar{m}},&\sigma _7&=R_{i\bar{j} k\bar{l}}R_{j\bar{i}}f_{/l\bar{k}},&\sigma _8&=\rho _{/i\bar{j}}f_{/j\bar{i}},\\ \sigma _9&\!=\!R_{i\bar{j}}R_{k\bar{i}}f_{/j\bar{k}},&\sigma _{10}&\!=\!R_{i\bar{j} k\bar{l}}R_{j\bar{i}/\bar{k}}f_{/l},&\sigma _{11}&\!=\!R_{i\bar{j} k\bar{l}/\bar{m}}R_{j\bar{i} l\bar{k}}f_{/m},&\sigma _{12}&\!=\!R_{i\bar{j}/\bar{k}}R_{j\bar{i}} f_{/k},\\ \sigma _{13}&=R_{i\bar{j}}\rho _{/\bar{i}}f_{/j},&\sigma _{14}&=\rho _{/\bar{i}}f_{/ij\bar{j}},&\sigma _{15}&=R_{i\bar{j}/k\bar{i}\bar{k}}f_{/j},&\sigma _{16}&=\rho _{/\bar{i}\bar{j}}f_{/ij}. \end{aligned}$$

We introduce six more tensors that will appear in our computation.

$$\begin{aligned} \widetilde{\sigma }_2&=R_{i\bar{j}}f_{/j\bar{i}k\bar{k}},&\widetilde{\sigma }_{3}&=R_{i\bar{j} k\bar{l}}f_{/j\bar{i}\bar{k} l},&\widetilde{\sigma }_{8}&=R_{i\bar{j}/ k\bar{k}}f_{/j\bar{i}},\\ \widehat{\sigma }_2&=R_{i\bar{j}}f_{/jk\bar{i}\bar{k}},&\widehat{\sigma }_{3}&=R_{i\bar{j} k\bar{l}}f_{/jl\bar{i}\bar{k}},&\widehat{\sigma }_{4}&=R_{i\bar{j}/\bar{k}}f_{/jk\bar{i}}. \end{aligned}$$

Lemma 4.1

Under the above notations, we have

$$\begin{aligned} \widetilde{\sigma }_2&=\sigma _2-\sigma _7+\sigma _9,&\widetilde{\sigma }_{3}&=\sigma _3,&\widetilde{\sigma }_{8}&=\sigma _{8}-\sigma _7+\sigma _9,\end{aligned}$$

(18)

$$\begin{aligned} \widehat{\sigma }_2&=\sigma _2+\sigma _9+\sigma _{12},&\widehat{\sigma }_{3}&=\sigma _3+\sigma _6+\sigma _{11},&\widehat{\sigma }_{4}&=\sigma _4+\sigma _{10}. \end{aligned}$$

(19)

Proof

All these equations were proved by the Ricci formula (7).

$$\begin{aligned} \widetilde{\sigma }_2=R_{i\bar{j}}f_{/k\bar{i} j\bar{k}}&=R_{i\bar{j}}(\Delta f)_{/\bar{i} j}+R_{i\bar{j}}R_{k\bar{l} j\bar{k}}f_{/l\bar{i}} -R_{i\bar{j}}R_{l\bar{i} j\bar{k}}f_{/k\bar{l}}\\&=\sigma _2+\sigma _9-\sigma _7,\\ \widetilde{\sigma }_3&=R_{i\bar{j} k\bar{l}}f_{/j\bar{i} l\bar{k}}-R_{i\bar{j} k\bar{l}}R_{j\bar{m} l\bar{k}}f_{/m\bar{i}}+R_{i\bar{j} k\bar{l}}R_{m\bar{i} l\bar{k}}f_{/j\bar{m}}\\&=\sigma _3-\sigma _6+\sigma _6=\sigma _3, \end{aligned}$$

One may use [15, Theorem 4.7] to derive the formula of $\widetilde{\sigma }_8.$ Proofs of other equations are similar, and we omit the details. $\square$

The $14$ trees of ${\fancyscript{T}}_f(a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6)$ are listed in Table 10, as well as their values after contractions by $g^{a_1\bar{b}_2}g^{a_3\bar{b}_4}g^{a_5\bar{b}_6}.$ By (13) and Lemma 4.1, we have

$$\begin{aligned}&g^{a_1\bar{b}_2}g^{a_3\bar{b}_4}g^{a_5\bar{b}_6}D(f_{a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6})=g^{a_1\bar{b}_2}g^{a_3\bar{b}_4}g^{a_5\bar{b}_6}\sum _{T\in {\fancyscript{T}}_f(a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6)} R_T\nonumber \\&=L_1^3f-(\sigma _2+2\widetilde{\sigma }_2+\sigma _3+\widetilde{\sigma }_3+2 \sigma _4+2\sigma _5+\widetilde{\sigma }_8)+\sigma _6+\sigma _7+\sigma _9\nonumber \\&=L_1^3f-3\sigma _2-2\sigma _3-2\sigma _4-2\sigma _5+\sigma _6+4\sigma _7-\sigma _8-2 \sigma _9. \end{aligned}$$

(20)

Note that in the second equation, each pointed tree $T$ is multiplied by a factor $(-1)^{|V(T)|-1}.$

Table 10 Pointed trees of ${\fancyscript{T}}_f(a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6)$

Remark 4.5

Here, we correct a typo error in [14, p. 256]. In the formulae of $\tau _1,\dots ,\tau _9$ in terms of $\sigma _1,\dots ,\sigma _9,$ all of the coefficients of $\sigma _2,\sigma _3,\sigma _4,\sigma _5,, \sigma _8$ should be changed to minus signs.

On the other hand, the computation of $g^{a_1\bar{b}_4}g^{a_2\bar{b}_5}g^{a_3\bar{b}_6}D(f_{a_1a_2 a_3\bar{b}_4 \bar{b}_5\bar{b}_6})$ is more laborious, as there are many more trees to be considered. The $28$ two-vertex trees and $18$ three-vertex trees in ${\fancyscript{T}}_f(a_1a_2a_3\bar{b}_4\bar{b}_5\bar{b}_6),$ together with their values after contractions by $g^{a_1\bar{b}_4}g^{a_2\bar{b}_5}g^{a_3\bar{b}_6},$ are listed in Tables 11 and 12, respectively. By (13) and Lemma 4.1, we have

$$\begin{aligned}&g^{a_1\bar{b}_4}g^{a_2\bar{b}_5}g^{a_3\bar{b}_6}D(f_{a_1a_2 a_3\bar{b}_4 \bar{b}_5\bar{b}_6})=g^{a_1\bar{b}_4}g^{a_2\bar{b}_5}g^{a_3\bar{b}_6}\sum _{T\in {\fancyscript{T}}_f(a_1a_2 a_3\bar{b}_4 \bar{b}_5\bar{b}_6)} R_T\nonumber \\&=L_3f-(6\widehat{\sigma }_2+3\widehat{\sigma }_3+6\widehat{\sigma }_4+3 \sigma _5+3\widetilde{\sigma }_8\nonumber \\&\quad +3\sigma _{14}+\sigma _{15}+3\sigma _{16}) +(3\sigma _6+3\sigma _7+3\sigma _9+2\sigma _{10}+2\sigma _{11}+4 \sigma _{12}+\sigma _{13})\nonumber \\&=L_3f-6\sigma _2-3\sigma _3-6\sigma _4-3\sigma _5+6\sigma _7-3 \sigma _8-6\sigma _9-4\sigma _{10}\nonumber \\&\quad -\sigma _{11}-2\sigma _{12}+ \sigma _{13}-3\sigma _{14}-\sigma _{15}-3\sigma _{16}. \end{aligned}$$

(21)

Since $g^{a_1\bar{b}_2}g^{a_3\bar{b}_4}g^{a_5\bar{b}_6}D(f_{a_1\bar{b}_2 a_3\bar{b}_4 a_5\bar{b}_6})=g^{a_1\bar{b}_4}g^{a_2\bar{b}_5}g^{a_3\bar{b}_6}D(f_{a_1a_2 a_3\bar{b}_4 \bar{b}_5\bar{b}_6}),$ we may subtract (21) from (20) to get

$$\begin{aligned} L_3f-L_1^3f&= 3\sigma _2+\sigma _3+4\sigma _4+\sigma _5+\sigma _6-2\sigma _7+2\sigma _8+4\sigma _9+4\sigma _{10}\nonumber \\&\qquad +&\sigma _{11}+2\sigma _{12} -\sigma _{13}+3\sigma _{14}+\sigma _{15}+3\sigma _{16}. \end{aligned}$$

(22)

We also computed $L_3f-L_1^3f$ by using the Ricci formula (7) directly, which gave the same result as (22).

Table 11 Two-vertex trees of ${\fancyscript{T}}_f(a_1a_2a_3\bar{b}_4\bar{b}_5\bar{b}_6)$

Table 12 Three-vertex trees of ${\fancyscript{T}}_f(a_1a_2a_3\bar{b}_4\bar{b}_5\bar{b}_6)$