Full proof of the Xu–Yau tree-sum conjectures in Kähler tensor calculus: closed-form characterization of the curvature and pointed tree sets for arbitrary index orderings

On a Kähler manifold, high-order local invariants can be written in two normal forms: as partial derivatives of the metric (or of a function) in Kähler normal coordinates, or as polynomials in covariant derivatives of the curvature tensor. The conversion between the two forms underlies computations of Bergman kernel, Berezin transform, star product, and heat kernel coefficients, but the naive recursive conversion explodes in complexity with the order. Hao Xu and Shing-Tung Yau ('Trees and tensors on Kähler manifolds', 2013) introduced a tree formalism that organizes the conversion as a signed sum over decorated trees and conjectured (Conjectures 2.3 and 3.2) closed-form tree-sum formulas for the forward conversion $D$ for arbitrary sequences of barred and unbarred indices; the conjectures include an explicit characterization of the admissible tree sets $\mathscr{T}_R$ and $\mathscr{T}_f$ and the assertion that each tree arises with multiplicity one. The formulas were verified by exhaustive enumeration only for $k\le 11$ (metric case) and for small $k$ in the function case (cross-checked at $k=6$ via the Ricci formula), and proofs exist only for special index orderings (alternating and grouped). The open problem is to prove the two conjectures in full generality: give a closed-form membership criterion for the tree sets attached to an arbitrary index ordering, prove multiplicity one, and establish the tree-sum identities for all orders. A further open direction posed in the same paper is the extension of the tree formalism to general Riemannian manifolds via undirected trees.

Background

Setting. A Kähler manifold $(M,g)$ is a complex manifold with a Hermitian metric $g_{i\bar{j}}$ whose associated real 2-form is closed; equivalently, around every point there are holomorphic Kähler normal coordinates in which $g_{i\bar{j}}=\delta_{ij}$ and all 'pure' derivatives $g_{i\bar{j}k_1\cdots k_r}$ and $g_{i\bar{j}\bar{l}_1\cdots\bar{l}_r}$ (differentiation in only unbarred, or only barred, directions) vanish at the center. The mixed derivatives encode curvature: $R_{i\bar{j}k\bar{l}}=-g_{i\bar{j}k\bar{l}}+g^{p\bar{q}}g_{p\bar{j}\bar{l}}g_{i\bar{q}k}$, and iterating this identity expresses every partial derivative of the metric as a polynomial in the curvature tensor, its covariant derivatives, and the inverse metric.

Two normal forms and the conversion problem. Local curvature invariants on Kähler manifolds arise naturally in two shapes: in metric-jet form (partial derivatives $g_{i\bar{j}\beta_1\cdots\beta_k}$ of the metric, or $f_{\beta_1\cdots\beta_k}$ of a smooth function, where each $\beta_i$ is either an unbarred holomorphic index $a_i$ or a barred antiholomorphic index $\bar{b}_i$) and in curvature form (covariant derivatives $R_{a\bar{b}a\bar{b}/\gamma_1\cdots\gamma_r}$ and $f_{/\beta_1\cdots\beta_k}$, where $/$ denotes covariant differentiation). The conversion $D$ sends a partial-derivative expression to its canonical curvature-form expression, and $P$ is the inverse conversion. These conversions are the technical bottleneck in a circle of problems: the Tian–Yau–Zelditch/Catlin asymptotic expansion of the Bergman kernel — the reproducing kernel of the space of holomorphic sections of the $k$-th power of an ample line bundle, with $B_k(z)\sim k^n\sum_m b_m(z)k^{-m}$ and coefficients $b_m$ that are curvature polynomials computed order by order — the Berezin transform (quantize a function to a Toeplitz operator and take its covariant symbol back) and Berezin–Toeplitz star products (the deformation-quantization product on functions induced by Toeplitz operators), higher Laplace–Beltrami operators $L_m$ (iterated complex traces of high-order covariant derivatives), and heat kernel coefficients (the coefficients in the small-time expansion of the heat trace). Different methods deliver coefficients in different normal forms, and hand conversion via the recurrence becomes prohibitively laborious and error-prone beyond low order.

The tree formalism. Xu and Yau organized the conversion combinatorially. A decorated tree is a finite directed tree whose vertices carry ordered tuples of external half-edges (legs), each leg carrying a barred or unbarred index; a tree is stable if every ordinary vertex has at least two outward and at least two inward half-edges. Admissible trees are the stable decorated trees obtained from a one-vertex tree by the following four local operations: (i) split a vertex into two; (ii) add an outward or an inward half-edge to a vertex; (iii) let a half-edge act on another half-edge of the same direction, that is, combine the two half-edges on a new vertex; (iv) let a half-edge act on an edge, that is, insert the half-edge in the middle of that edge. A weight function $\omega$ on the half-edges fixes the index order at each vertex. For a vertex $v$ of $T$, write $\mathcal{A}(v)$ and $\mathcal{B}(v)$ for the sets of its ascendants and descendants, so that $u\in\mathcal{A}(v)$ if there is a directed path from $u$ to $v$, and $u\in\mathcal{B}(v)$ if there is a directed path from $v$ to $u$. In the metric case the weight of a half-edge is

$$\omega(\gamma)=\begin{cases} j, & \gamma=\beta_j \text{ an external leg},\\ \min\{j\mid \bar{b}_j\in\mathcal{A}(u)\}, & \gamma=\bar{b}_e \text{ the inward half-edge carried by the edge } e=uv,\\ \min\{j\mid a_j\in\mathcal{B}(w)\}, & \gamma=a_e \text{ the outward half-edge carried by the edge } e=vw; \end{cases}$$

in the pointed (function) case the same expression is used except that an edge half-edge whose side contains the distinguished vertex $\bullet$ has weight $0$:

$$\omega(\gamma)=\begin{cases} j, & \gamma=\beta_j,\\ 0, & \gamma=\bar{b}_e\ (e=uv) \text{ and } \bullet\in\mathcal{A}(u),\\ \min\{j\mid \bar{b}_j\in\mathcal{A}(u)\}, & \gamma=\bar{b}_e\ (e=uv) \text{ and } \bullet\notin\mathcal{A}(u),\\ 0, & \gamma=a_e\ (e=vw) \text{ and } \bullet\in\mathcal{B}(w),\\ \min\{j\mid a_j\in\mathcal{B}(w)\}, & \gamma=a_e\ (e=vw) \text{ and } \bullet\notin\mathcal{B}(w). \end{cases}$$

Thus an external leg $\beta_j$ has weight $j$, while a half-edge carried by an edge inherits the smallest index position occurring on its side of that edge (in the pointed case, a half-edge whose side contains the distinguished vertex has weight $0$). At an ordinary vertex $v$, the two outward half-edges of smallest weight become the first two unbarred indices of the curvature tensor $R_{a_{m_1}\bar{b}_{n_1}a_{m_2}\bar{b}_{n_2}/\gamma_1\gamma_2\cdots}$ decorating $v$, the two inward half-edges of smallest weight become its first two barred indices, and the remaining half-edges, sorted by increasing weight, become the covariant-derivative indices; at a distinguished vertex, the half-edges sorted by weight give the covariant derivative $f_{/\gamma_1\gamma_2\cdots}$. Acting with the indices $\beta_5,\dots,\beta_k$ on the seed tree carrying $a_1,\bar{b}_2,a_3,\bar{b}_4$ generates the tree set $\mathscr{T}_R(a_1\bar{b}_2 a_3\bar{b}_4\beta_5\cdots\beta_k)$; for functions, acting with $\beta_1,\dots,\beta_k$ on a single distinguished vertex generates the set $\mathscr{T}_f(\beta_1\cdots\beta_k)$ of pointed admissible trees (stable decorated trees with a distinguished root vertex, at which the covariant derivative of $f$ sits).

The conjectures. The paper conjectures (Conjecture 2.3, metric case, $k\ge 4$; Conjecture 3.2, function case) that the conversion is the tree sum

$$D\big(g_{a_1\bar{b}_2 a_3\bar{b}_4 \beta_5\cdots\beta_k}\big)=\sum_{T\in\mathscr{T}_R(\cdots)}(-1)^{|V(T)|}\prod_{e\in E(T)}g^{a_e\bar{b}_e}\prod_{v\in V(T)}R_v,\qquad D(f_{\beta_1\cdots\beta_k})=\sum_{T\in\mathscr{T}_f(\cdots)}(-1)^{|V(T)|-1}\prod_{e\in E(T)}g^{a_e\bar{b}_e}\prod_{v\in V(T)}R_v,$$

where $|V(T)|$ is the number of vertices, $E(T)$ the edge set, $g^{a_e\bar{b}_e}$ the inverse metric, and $R_v$ the covariant curvature tensor decorating $v$ (for $\mathscr{T}_f$: at the distinguished vertex, the covariant derivative of $f$), with indices ordered by $\omega$. Two features make the conjectures nontrivial beyond the identities themselves: the tree sets are defined by a recursion whose output has no known closed-form description for a general index ordering, and the conjectures assert that each tree arises with multiplicity one — different generation paths could a priori reach the same tree, which would require bookkeeping or cancellation; moreover, the paper notes that some admissible trees cancel in pairs because of sign differences arising from derivatives of the inverse metric, so $\mathscr{T}_R$ is in general a proper subset of the admissible trees, and no general criterion for which trees cancel is known. The paper verified Conjecture 2.3 for all $k\le 11$ by exhaustive enumeration, characterized all two- and three-vertex trees with multiplicity one, and cross-checked the $k=8$ alternating metric case against the independently derived weight-three Bergman coefficient and the $k=6$ function case against a direct Ricci-formula computation of $L_3f-L_1^3f$. The reverse conversions $P$ are proved as theorems in the same paper (Theorems 4.1–4.3, with brief proofs), and the special alternating and grouped orderings of the forward conjectures are proved in the companion paper — but the general curvature-form conjectures remain open.

Problem Statement

Prove Conjectures 2.3 and 3.2 of the source paper in full generality, for a Kähler manifold $(M,g)$ in Kähler normal coordinates at a point (so $g_{i\bar{j}}=\delta_{ij}$ and $g_{i\bar{j}k_1\cdots k_r}=g_{i\bar{j}\bar{l}_1\cdots\bar{l}_r}=0$ at the center):

(a) Metric case (Conjecture 2.3). For every $k\ge 4$ and every sequence $\beta_5,\ldots,\beta_k$, each $\beta_i$ either unbarred ($a_i$) or barred ($\bar{b}_i$), give an explicit closed-form characterization of the set $\mathscr{T}_R(a_1\bar{b}_2 a_3\bar{b}_4\beta_5\cdots\beta_k)$ of admissible trees (stable decorated trees generated from the one-vertex tree carrying $a_1,\bar{b}_2,a_3,\bar{b}_4$ by successively applying the four operations — split a vertex, add a half-edge, let a half-edge act on a half-edge of the same direction, let a half-edge act on an edge — with the indices $\beta_5,\ldots,\beta_k$ in that order, under the weight function $\omega$ of the background ordering indices), prove that the generation procedure produces each tree with multiplicity one, and deduce

$$D\big(g_{a_1\bar{b}_2 a_3\bar{b}_4 \beta_5\cdots\beta_k}\big)=\sum_{T\in\mathscr{T}_R(a_1\bar{b}_2 a_3\bar{b}_4\beta_5\cdots\beta_k)}(-1)^{|V(T)|}\prod_{e\in E(T)}g^{a_e\bar{b}_e}\prod_{v\in V(T)}R_v,$$

where $D$ is the canonical conversion of the partial metric derivative into covariant curvature tensors and $R_v$ is the curvature tensor decorating $v$ with indices ordered by $\omega$. Note that $\mathscr{T}_R$ is not the set of all admissible trees with the given external legs: the paper observes that some admissible trees cancel in pairs because of sign differences arising from derivatives of the inverse metric, and such trees do not appear in $\mathscr{T}_R$; a general cancellation criterion is part of what is missing. The object to be characterized in (a) is this tree set $\mathscr{T}_R$ — the index set of the tree sum above — and not the larger collection of all admissible trees with those external legs.

(b) Function case (Conjecture 3.2). The analogous statement for $D(f_{\beta_1\cdots\beta_k})$ for every $k\ge 0$ and every ordering $\beta_1,\ldots,\beta_k$: a closed-form characterization of the pointed admissible tree set $\mathscr{T}_f(\beta_1\cdots\beta_k)$ (trees generated from the single distinguished vertex by the same operations — the paper notes that operation (iv), a half-edge acting on an edge, does not occur in this case — under the weight function modified so that an edge half-edge whose side contains the distinguished vertex has weight $0$), multiplicity one, and the tree-sum identity with sign $(-1)^{|V(T)|-1}$, where the distinguished vertex carries the covariant derivative of $f$ with its half-edges sorted by weight.

For a general index ordering the tree sets are still described only by the recursion: what is missing is a membership criterion, readable off without running the recursion, from which the sets $\mathscr{T}_R$ and $\mathscr{T}_f$ follow, a general proof of multiplicity one, and the specialization of that criterion to the known cases: the alternating ordering in (a) (stable decorated rooted trees whose root carries $\bar{b}_2,\bar{b}_4$, with the weight-monotonicity condition of the paper's Remark 2.6), the grouped ordering in (b) (pointed rooted trees with the minimum condition along directed edges), the two- and three-vertex characterizations, and the enumerated cases with $k\le 11$. Because the conjectures are universal over all $k$ and all index orderings, (a) and (b) are independent claims.

A further open direction posed in the same paper, not required for resolution of (a)–(b): extend the tree formalism to general (not necessarily Kähler) Riemannian manifolds, where the barred/unbarred structure disappears and one expects undirected decorated trees.

References

1. Hao Xu and Shing-Tung Yau, "Trees and tensors on Kähler manifolds", Annals of Global Analysis and Geometry 44, no. 2 (2013) 151–168, DOI 10.1007/s10455-012-9361-x, https://doi.org/10.1007/s10455-012-9361-x. [source paper]
2. Hao Xu, "A closed formula for the asymptotic expansion of the Bergman kernel", Communications in Mathematical Physics 314, no. 3 (2012) 555–585, DOI 10.1007/s00220-012-1531-y, arXiv:1103.3060, https://arxiv.org/abs/1103.3060. [adjacent: admissible-tree machinery and partial tree-sum formula, Lemma 2.5]
3. Hao Xu, "Bergman kernel and Kähler tensor calculus", Pure and Applied Mathematics Quarterly 9 (2013), no. 3, 507–546, DOI 10.4310/pamq.2013.v9.n3.a7, https://doi.org/10.4310/pamq.2013.v9.n3.a7. [companion paper: proves the special-ordering cases]
4. Hao Xu, "Weyl invariant polynomial and deformation quantization on Kähler manifolds", Journal of Geometry and Physics 76 (2014) 124–135, DOI 10.1016/j.geomphys.2013.10.022, arXiv:1208.2463, https://arxiv.org/abs/1208.2463. [adjacent: all-order tree formula in the metric-jet normal form, Theorem 2.5]
5. Kefeng Liu and Hao Xu, "Heat kernel coefficients on Kähler manifolds", Reviews in Mathematical Physics 37, no. 6 (published online November 2024; print issue July 2025), article 2450047, DOI 10.1142/S0129055X24500478, arXiv:1311.5510, https://arxiv.org/abs/1311.5510. [citing work: Lemma 2.11 proves the all-order pointed-tree formula for the reverse conversion]
6. Miroslav Engliš and Hao Xu, "Higher Laplace–Beltrami operators on bounded symmetric domains", Acta Mathematica Sinica, English Series 34, no. 8 (2018) 1297–1312, DOI 10.1007/s10114-018-8162-y, https://doi.org/10.1007/s10114-018-8162-y. [citing work: graph manipulation of Kähler curvature on bounded symmetric domains]
7. Miroslav Engliš and Hao Xu, "Forelli–Rudin construction and asymptotic expansion of Szegö kernel on Reinhardt domains", Osaka Journal of Mathematics 52, no. 4 (2015) 905–927, DOI 10.18910/57651, https://doi.org/10.18910/57651. [citing work of the companion paper: graph-theoretic closed formula for Szegö-kernel invariant coefficients on Reinhardt domains]