# On a variational theorem of Gauduchon and torsion-critical manifolds

Dongmei Zhang  Dongmei Zhang. School of Mathematical Sciences, Chongqing
Normal University, Chongqing 401331, China<EMAIL_ADDRESS>and  Fangyang
Zheng  Fangyang Zheng. School of Mathematical Sciences, Chongqing Normal
University, Chongqing 401331, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In 1984, Gauduchon considered the functional of $L^{2}$-norm of his torsion
$1$-form on a compact Hermitian manifold. He obtained in [14] the Euler-
Lagrange equation for this functional, and showed that in dimension $2$ the
critical metrics must be balanced (namely with vanishing torsion $1$-form). In
this note we extend his result to higher dimensions, and show that critical
metrics are balanced in all dimensions. We also consider the $L^{2}$-norm of
the full Chern torsion, and show by examples that there are critical points of
this functional that are not Kähler.

###### Key words and phrases:

Hermitian manifold; Chern connection; variation; Gauduchon’s torison $1$-form;
torsion-critical manifolds

###### 2010 Mathematics Subject Classification:

53C55 (primary), 53C05 (secondary)

Zheng is partially supported by National Natural Science Foundations of China
with the grant No.12071050 and 12141101, Chongqing grant cstc2021ycjh-
bgzxm0139, and is supported by the 111 Project D21024.

###### Contents

  1. 1 Introduction and statement of results
  2. 2 Proof of Proposition 1
  3. 3 Torsion-critical manifolds
  4. 4 Existence and non-existence of torsion-critical metrics

Dongmei Zhang and Fangyang Zheng

##  1\. Introduction and statement of results

Let $(M^{n},g)$ be a compact Hermitian manifold of complex dimension $n$.
Denote by $\omega$ its Kähler form. Gauduchon’s torsion $1$-form, which is the
trace of the torsion tensor $T$ of the Chern connection $\nabla$, is the
global $(1,0)$-form $\eta$ on $M^{n}$ defined by
$\partial(\omega^{n-1})=-\eta\wedge\omega^{n-1}$. In the celebrated work [14],
Gauduchon considered the functional

${\mathcal{G}}(g)=V^{\frac{1-n}{n}}\int_{M}|\eta|^{2}dv,\ \ \ \ \ \
g\in{\mathcal{H}}_{M},$

where ${\mathcal{H}}_{M}$ is the set of all Hermitian metrics on the complex
manifold $M^{n}$, $dv=\frac{\omega^{n}}{n!}$ the volume form, and
$V=\int_{M}dv$ the volume. Note that the factor in front of the integral is
simply to make the functional independent of the scaling of metric (by
constant multiples). Equivalently, one can drop this $V$ factor but restrict
to the subset of all Hermitian metrics with unit volume. He obtained the
Euler-Lagrange equation of ${\mathcal{G}}$:

(1)
$\sqrt{-1}(\partial\overline{\eta}-\overline{\partial}\eta-\eta\wedge\overline{\eta})=a\,\omega$

where $a\geq 0$ is a constant. This is formula (48bis) of [14], where it was
written in terms of the Lee form $\theta$, which is just
$-(\eta+\overline{\eta})$. Here we rewrote it in the equivalent but more
convenient complex form. He also proved [14, Theorem III.4] that, when $n=2$,
any critical point of ${\mathcal{G}}$ must be balanced, namely, with $\eta=0$.

In this note, we extend his theorem to general dimensions:

######  Proposition 1.

If a compact Hermitian manifold $(M^{n},g)$ satisfies equation (1), then
$\eta=0$. That is, the only critical points of the Gauduchon functional
$\,{\mathcal{G}}$ are balanced metrics.

Next, we mimic Gauduchon’s functional and consider the $L^{2}$-norm of the
Chern torsion tensor $T$:

(2) ${\mathcal{F}}(g)=V^{\frac{1-n}{n}}\int_{M}|T|^{2}dv,\ \ \ \ \ \
g\in{\mathcal{H}}_{M}.$

Here $T$ is defined by $\,T(x,y)=\nabla_{x}y-\nabla_{y}x-[x,y]\,$ for any
vector fields $x$, $y$ on $M^{n}$. For convenience, let us introduce the
following terminology

######  Definition.

A compact Hermitian manifold $(M^{n},g)$ is said to be torsion-critical, if
$g$ is a critical point of ${\mathcal{F}}$.

Since $T=0$ if and only if $g$ is Kähler, we see that all Kähler metrics are
torsion-critical, and they are the absolute minimum of the functional
${\mathcal{F}}$. So the point is to understand torsion-critical metrics that
are not Kähler. It turns out that unlike the case of Gauduchon functional,
there do exist non-Kähler metrics that are torsion-critical for each $n\geq
3$.

In order the describe the equations obeyed by torsion-critical metrics, we
need to fix some notations first. Given a Hermitian manifold $(M^{n},g)$, let
$\\{e_{1},\ldots,e_{n}\\}$ be a local unitary frame of type $(1,0)$ tangent
vectors, and let $\\{\varphi_{1},\ldots,\varphi_{n}\\}$ be the unitary coframe
of $(1,0)$-forms dual to $e$. Under the frame $e$, the components of $T$ are:

$T(e_{i},\overline{e}_{k})=0,\ \ \ \ T(e_{i},e_{k})=\sum_{r}T^{r}_{ik}e_{r},\
\ \ \ \ \ \ \ \ \ 1\leq i,k\leq n.$

We have $\eta=\sum_{i}\eta_{i}\varphi_{i}$, $\ \eta_{i}=\sum_{r}T^{r}_{ri}$.
So $\eta$ is the trace of $T$. Under any unitary local frame $e$ and for any
$1\leq i,j\leq n$, let us denote by

(3) $A_{i\bar{j}}=\sum_{r,s}T^{r}_{is}\overline{T^{r}_{js}},\ \ \
B_{i\bar{j}}=\sum_{r,s}T^{j}_{rs}\overline{T^{i}_{rs}},\ \ \
\phi_{i}^{j}=\sum_{r}T^{j}_{ir}\overline{\eta}_{r},\ \ \ \
\xi_{i}^{j}=\sum_{r}T^{j}_{ir,\,\bar{r}}$

where the index after comma stands for covariant derivative with respect to
Chern connection. Define

$\displaystyle\sigma_{1}=\sqrt{-1}\sum_{i,j}A_{i\bar{j}}\varphi_{i}\wedge\overline{\varphi}_{j},\
\ \ \ \ \
\sigma_{2}=\sqrt{-1}\sum_{i,j}B_{i\bar{j}}\varphi_{i}\wedge\overline{\varphi}_{j},$
$\displaystyle\phi=\sqrt{-1}\sum_{i,j}\phi_{i}^{j}\varphi_{i}\wedge\overline{\varphi}_{j},\
\ \ \ \ \ \ \
\xi=\sqrt{-1}\sum_{i,j}\xi_{i}^{j}\varphi_{i}\wedge\overline{\varphi}_{j}.$

Clearly, $\sigma_{1}$, $\sigma_{2}$, $\phi$, $\xi$ are independent of the
choice of local unitary frames, hence are globally defined $(1,1)$-forms on
$M^{n}$, with $\sigma_{1}\geq 0$, $\sigma_{2}\geq 0$. Let us denote by

$|T|^{2}:=\sum_{i,j,k}|T^{j}_{ik}|^{2}$

and call it the square norm of the torsion tensor $T$. Then the trace of
$\sigma_{1}$ or $\sigma_{2}$ with respect to the Kähler form
$\omega=\sqrt{-1}\sum_{i}\varphi_{i}\wedge\overline{\varphi}_{i}$ is equal to
$|T|^{2}$, so $\sigma_{1}=0$ (or $\sigma_{2}=0$) if and only if $g$ is Kähler.
We have the following

######  Proposition 2.

A compact Hermitian manifold $(M^{n},g)$ is torsion-critical if and only if

(4)
$2\sigma_{1}-\sigma_{2}+2(\phi+\bar{\phi})-2(\xi+\bar{\xi})=(|T|^{2}-\frac{n-1}{n}b)\,\omega,\
\ \ \ \ \ b=V^{-1}\int_{M}|T|^{2}dv.$

That is, $g$ is a critical point of the functional ${\mathcal{F}}$ if and only
if it satisfies equation $(\ref{eq:tor-crit})$.

Note that when $n=2$, $\eta$ and $T$ carry the same amount of information, to
be more precise one has $|T|^{2}=2|\eta|^{2}$ when $n=2$, so ${\mathcal{F}}$
and ${\mathcal{G}}$ are essentially the same. So by the aforementioned theorem
of Gauduchon we know that

######  Corollary 1 ([14]).

For $n=2$, any torsion-critical metric must be balanced, hence Kähler.

In general dimensions, the trace of $\phi$ with respect to $\omega$ is
$|\eta|^{2}=\sum_{r}|\eta_{r}|^{2}$, while the trace of $\xi$ is the global
function $\chi=\sum_{r}\eta_{r,\,\bar{r}}$. Since

$\sqrt{-1}\partial\overline{\partial}\omega^{n-1}=(|\eta|^{2}-\chi)\omega^{n},$

we know that $\chi$ is a real-valued function on $M^{n}$, and
$|\eta|^{2}-\chi=0$ identically if and only if the metric is Gauduchon. By
taking trace with respect to $\omega$ on both sides of (4), we get the
following result which is due to Angella, Istrati, Otiman, and Tardini, [4,
Proposition 17]:

######  Corollary 2 ([4]).

Denote by $[g]$ the conformal class of $g$, namely, the set of all Hermitian
metrics on $M^{n}$ conformal to $g$. Then $g$ is a critical point of
${\mathcal{F}}|_{[g]}$ if and only if

(5) $4\big{(}|\eta|^{2}-\chi\big{)}=(n-1)\big{(}|T|^{2}-b\big{)},$

where $b$ is the average value of $|T|^{2}$. In particular, the metric will be
Gauduchon if and only if $|T|^{2}$ is a constant.

Note that the Euler-Lagrange equation of ${\mathcal{F}}|_{[g]}$ is just the
trace of that of ${\mathcal{F}}$. In particular, all torsion-critical metrics
satisfy (5). To see that equation (5) is the same as (18) of [4], we note that
under any unitary coframe $\varphi$, it holds that
$\partial\omega=\sqrt{-1}\sum_{i,j,k}T^{j}_{ik}\varphi_{i}\wedge\varphi_{k}\wedge\overline{\varphi}_{j}$.
Hence $|d\omega|^{2}=2|\partial\omega|^{2}=|T|^{2}$, and our functional
${\mathcal{F}}$ here is the same as the functional ${\mathcal{A}}$ studied in
[4]. For the Lee form $\theta=-(\eta+\overline{\eta})$, one has
$d^{\ast}\theta=2(\chi-|\eta|^{2})$.

In the paper [4], the authors studied the variational problem for a number of
interesting geometric functionals, with a special emphasis on their
restriction on conformal classes. The papers [2] and [3] also give excellent
discussions on related problems.

We are particularly interested in the set ${\mathcal{H}}^{tc}_{n}$ of
$n$-dimensional non-Kähler, torsion-critical manifolds, and especially its
subset of all balanced ones, for $n\geq 3$. At present, we have very limited
understanding about this class, except knowing that it is non-empty on one
hand and highly restrictive on the other hand, in the sense that many familiar
types of special Hermitian metrics are not in it (unless Kähler).

Denote by $\nabla^{s}$ the Strominger connection [20] of $(M^{n},g)$. It is
also known as Bismut connection [7] in many literature. A Hermitian manifold
is called Strominger torsion parallel, or STP in short, if $\nabla^{s}T=0$. We
have the following

######  Proposition 3.

Suppose $(M^{n},g)$ is a compact STP manifold. Then it will be torsion-
critical if and only it is balanced and with $\sigma_{2}=c\,\omega$ for some
constant $c$.

######  Corollary 3.

For any $k\geq 3$, the complex Lie group $SO(k,{\mathbb{C}})$, equipped with a
compatible metric, is balanced, non-Kähler, SPT, and with $\sigma_{2}$ equal
to a constant multiple of $\omega$. So any compact quotient of it is a non-
Kähler, balanced, torsion-critical manifold.

Since the product of torsion-critical manifolds are torsional-critical, so by
taking product, we know that the set ${\mathcal{H}}^{tc}_{n}$ (and actually
its subset of balanced ones) is non-empty for all $n\geq 3$. This settles the
non-emptiness of the class. To illustrate its restrictiveness, we make the
following observation:

######  Proposition 4.

Let $(M^{n},g)$ be a compact Hermitian manifold belonging to one of the
following

  1. (i)

locally conformally Kähler, or

  2. (ii)

Strominger Kähler-like, or

  3. (iii)

complex nilmanifold with nilpotent $J$ in the sense of [9], or

  4. (iv)

all the complex nilmanifolds or Calabi-Yau type complex solvmanifolds in
dimension $3$, listed by [5, Table 1 and 2]

Then $g$ cannot be torsion-critical unless it is already Kähler.

Since the class seems to be so restrictive, it would be very attempting to try
to classify it, at least in low dimensions or for some special types of
Hermitian metrics. For instance, one could try to

Classify all balanced, non-Kähler, torsion-critical manifolds in dimension
$3$.

Classify all torsion-critical manifolds that are Chern flat [8].

Classify all torsion-critical manifolds that are pluriclosed, namely,
$\partial\overline{\partial}\omega=0$.

Similar to balanced manifolds, pluriclosed manifolds (also known as strong
Kähler with torsion, or SKT for short in many literature) are an important
class of special Hermitian metrics that are widely studied. We refer the
readers to the excellent paper [11] for more discussion.

The variational method is a classical approach in geometry and analysis, with
a long and rich history. Various types of special non-Kähler metrics are also
widely studied in Hermitian geometry. As a limited sampler, we refer the
readers to the following papers and the references therein for more
discussions: [1], [6], [10], [12], [13], [15], [16], [17], [19], [21], [22],
[23], [24], [26], [27], [28], and [32].

This short article is organized as follows. In §2, we will give a proof to
Proposition 1. In §3, we will deduce the Euler-Lagrange equation, namely,
Proposition 2. In §3, we will prove Propositions 3 and 4.

##  2\. Proof of Proposition 1

Let $(M^{n},g)$ be a compact Hermitian manifold of complex dimension $n\geq
2$. Let us first verify that equation (1) is indeed the Euler-Lagrange
equation of ${\mathcal{G}}$, namely, it is the same as formula (48bis) of
[14].

Recall that the Lee form $\theta$ is the $1$-form on $M^{n}$ defined by
$d\omega^{n-1}=\theta\wedge\omega^{n-1}$, where $\omega$ is the Kähler form.
So clearly, one has

$\theta=-(\eta+\overline{\eta}),\ \ \ \ \mbox{hence}\ \ \ \
J\theta=\sqrt{-1}(\overline{\eta}-\eta).$

Therefore,

$(dJ\theta)^{1,1}=\sqrt{-1}(\partial\overline{\eta}-\overline{\partial}\eta),\
\ \ \ \ \ \theta\wedge J\theta=-2\sqrt{-1}\eta\wedge\overline{\eta}.$

So by Gauduchon’s formula (48bis) in [14], we get equation (1).

Now assume that $g$ is a critical point of the Gauduchon functional
${\mathcal{G}}$. So we have equation (1), which is in complex form thus being
more convenient. Gauduchon’s proof of his Theorem III.4 in [14] for the $n=2$
case is an integration-by-part argument, using Stokes’ theorem repeatedly.
Below we give a brief account of his proof for readers convenience.

###### Gauduchon’s proof of Proposition 1 in the $n=2$ case.

Let $(M^{2},g)$ be a compact Hermitian surface with $g$ satisfying equation
(1), where the constant $a$ is equal to $\frac{1}{2V}\int_{M}|\eta|^{2}dv$.
Rewrite the equation as

$L:=\sqrt{-1}(\partial\overline{\eta}-\overline{\partial}\eta)=a\,\omega+\sqrt{-1}\eta\wedge\overline{\eta}:=R.$

For both sides, take the wedge product of the $(1,1)$-form with itself, we get

$\displaystyle R\wedge R$ $\displaystyle=$ $\displaystyle
a^{2}\omega^{2}+2a\sqrt{-1}\eta\wedge\overline{\eta}\wedge\omega=(a^{2}+a|\eta|^{2})\omega^{2},\
\ \ \ \mbox{while}$ $\displaystyle L\wedge L$ $\displaystyle=$
$\displaystyle-\partial\overline{\eta}\,\partial\overline{\eta}-\overline{\partial}\eta\,\overline{\partial}\eta+2\partial\overline{\eta}\,\overline{\partial}\eta$
$\displaystyle=$
$\displaystyle-\partial(\overline{\eta}\,\partial\overline{\eta})-\overline{\partial}(\eta\,\overline{\partial}\eta)+2\\{\partial(\overline{\eta}\,\partial\overline{\eta})+\overline{\partial}(\overline{\eta}\,\partial\eta)-\partial\eta\,\overline{\partial\eta}\\}$

Therefore $\,\int\\!L\wedge L=-\int\partial\eta\,\overline{\partial\eta}\,\leq
0$. On the other hand, $\int\\!R\wedge R=3a^{2}V\geq 0$, so it forces $a=0$
hence $\eta=0$. This completes the proof of Gauduchon’s theorem. ∎

Note that when $n\geq 3$, one has to wedge the above $(2,2)$-forms with
$\omega^{n-2}$, which messes up the exact forms, and further integration-by-
part will involve the differential of $\omega$, and the integral of the left
hand side is no longer clearly non-positive. So one has to seek an alternative
argument.

###### Proof of Proposition 1.

Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq 2$ satisfying
equation (1). The constant $a$ is given by $\frac{1}{nV}\int|\eta|^{2}dv$. Our
goal is to show that $a=0$. Assume the contrary, namely, $a>0$. Let us as
before move the third term of the left hand side to the right, and get

$\sqrt{-1}(\partial\overline{\eta}-\overline{\partial}\eta)=a\,\omega+\sqrt{-1}\eta\wedge\overline{\eta}:=\omega_{0}.$

Since $a>0$, $\omega_{0}\geq a\omega$ is the Kähler form of another Hermitian
metric on $M^{n}$. Let $\alpha=-\sqrt{-1}\partial\eta$ and
$\psi=\sqrt{-1}(\overline{\eta}-\eta)$. Then we have
$d\psi=\omega_{0}+\alpha+\overline{\alpha}$. Since
$\alpha\overline{\alpha}\geq 0$, we have

$(d\psi)^{n}=\omega_{0}^{n}+C_{n}^{2}C_{2}^{1}\omega_{0}^{n-2}\alpha\overline{\alpha}+C_{n}^{4}C_{4}^{2}\omega_{0}^{n-4}(\alpha\overline{\alpha})^{2}+\cdots\geq\omega_{0}^{n},$

hence $\int_{M}(d\psi)^{n}>0$. But that is a contradiction since
$(d\psi)^{n}=d\big{(}\psi\wedge(d\psi)^{n-1}\big{)}$ is exact. This means that
the assumption $a>0$ can not occur, therefore $a$ must be zero, thus $\eta=0$.
This completes the proof of Proposition 1. ∎

In other words, under the assumption that $a>0$, the above construction leads
to an exact symplectic form $d\psi$ (namely the Liouville type symplectic
manifold), which cannot occur on closed manifold, and this argument works in
all dimensions.

##  3\. Torsion-critical manifolds

In this section we will derive the Euler-Lagrange equation of the functional
${\mathcal{F}}$ given by (2) and prove Proposition 2.

Let $(M^{n},g)$ be a compact Hermitian manifold. Denote by $\nabla$ the Chern
connection and $T$ its torsion tensor. Extend $g=\langle,\rangle$ bilinearly
over ${\mathbb{C}}$. Under a local holomorphic coordinate system
$z=(z_{1},\ldots,z_{n})$, consider the natural frame
$\\{\partial_{1},\ldots,\partial_{n}\\}$ where
$\partial_{i}=\frac{\partial}{\partial z_{i}}$, $1\leq i\leq n$. Write
$g_{i\bar{j}}=\langle\partial_{i},\overline{\partial}_{j}\rangle$ for the
entries of the $n\times n$ matrix of $g$, which we will also denote by $g$,
and denote its inverse matrix by $g^{-1}=(g^{\bar{j}i})$. Under the natural
frame, $\nabla$ has components

$\nabla_{\partial_{k}}\partial_{i}=\sum_{j}\Gamma^{j}_{ik}\partial_{j}=\sum_{j}\big{(}\sum_{\ell}\partial_{k}(g_{i\bar{\ell}})\,g^{\bar{\ell}j}\big{)}\partial_{j}.$

We have $T(\partial_{i},\overline{\partial}_{j})=0$, $\
T(\partial_{i},\partial_{k})=\sum_{j}T^{j}_{ik}\partial_{j}$ with

(6)
$T^{j}_{ik}=\sum_{\ell}\big{(}\partial_{i}(g_{k\bar{\ell}})-\partial_{k}(g_{i\bar{\ell}})\big{)}g^{\bar{\ell}j}.$

Let $\omega=\sqrt{-1}\sum_{i,j}g_{i\bar{j}}dz_{i}\wedge d\bar{z}_{j}$ be the
Kähler form of $g$. Gauduchon’s torsion $1$-form $\eta$ is given by
$\eta=\sum_{i}\eta_{i}dz_{i}$ where

(7)
$\eta_{i}=\sum_{k}T^{k}_{ki}=\sum_{k,\ell}\big{(}\partial_{k}(g_{i\bar{\ell}})-\partial_{i}(g_{k\bar{\ell}})\big{)}g^{\bar{\ell}k}=\sum_{k}\big{(}\Gamma^{k}_{ik}-\Gamma^{k}_{ki}\big{)}.$

Under a local coordinate system, we have
$\,|\eta|^{2}=\sum_{i,j}\eta_{i}\overline{\eta_{j}}g^{\bar{j}i}\,$, $\
\chi=\sum_{i,j}\eta_{i,\bar{j}}g^{\bar{j}i}\,$, and

$\displaystyle\sigma_{1}=\sqrt{-1}\sum_{i,j}A_{i\bar{j}}dz_{i}\wedge
d\overline{z}_{j},\ \ \ \ \ \ \
A_{i\bar{j}}=\sum_{r,s,p,q}T^{r}_{is}\overline{T^{p}_{jq}}g_{r\bar{p}}g^{\bar{q}s},$
$\displaystyle\sigma_{2}=\sqrt{-1}\sum_{i,j}B_{i\bar{j}}dz_{i}\wedge
d\overline{z}_{j},\ \ \ \ \ \ \
B_{i\bar{j}}=\sum_{r,s,p,q,k,\ell}T^{\ell}_{rs}\overline{T^{k}_{pq}}g^{\bar{p}r}g^{\bar{q}s}g_{i\bar{k}}g_{\ell\bar{j}}\,,$
$\displaystyle\phi=\sqrt{-1}\sum_{i,j}\phi_{i}^{\ell}g_{\ell\bar{j}}\,dz_{i}\wedge
d\overline{z}_{j},\ \ \ \ \ \
\phi_{i}^{j}=\sum_{r,s}T^{j}_{ir}\overline{\eta_{s}}g^{\bar{s}r},$
$\displaystyle\xi=\sqrt{-1}\sum_{i,j}\xi_{i}^{\ell}g_{\ell\bar{j}}\,dz_{i}\wedge
d\overline{z}_{j},\ \ \ \ \ \ \ \
\xi_{i}^{j}=\sum_{r,s}T^{j}_{ir,\bar{s}}g^{\bar{s}r}.$

Now suppose that $X$ is a type $(1,0)$-vector field on $M^{n}$. In a
coordinate neighborhood, write $X=\sum^{i}X^{i}\partial_{i}$. The divergence
of $X$ is the global function on $M^{n}$ defined by
$div(X)=\sum_{i}X^{i}_{\,,i}$, where the index after comma stands for
covariant derivative with respect to $\nabla$. Denote by $\lrcorner$ the
contraction on differential forms by vector fields. As is well known, we have

$d(X\lrcorner\,\omega^{n})=\big{(}div(X)-\eta(X)\big{)}\omega^{n},$

Integrating over $M^{n}$, we get the divergence theorem on Hermitian
manifolds:

######  Lemma 1.

Let $(M^{n},g)$ be a compact Hermitian manifold, and $X$ a type $(1,0)$ vector
field on $M^{n}$. Then it holds that

$\int_{M}div(X)\omega^{n}=\int_{M}\eta(X)\omega^{n}.$

Next, we consider tensor fields on $M^{n}$. Using the Hermitian metric $g$,
tensor fields can be transformed from covariant type to contra-variant type
and vice versa. So we just need to consider tensor fields of ‘pure type’,
namely, with only holomorphic part but without anti-holomorphic part. Let us
denote by ${\mathcal{T}}^{q}_{\,p}$ the vector space of all pure type $(p,q)$
tensor fields on $M$. In a local coordinate neighborhood,
$C\in{\mathcal{T}}^{q}_{\,p}$ has the expression

$C=\sum C_{I}^{J}\,dz_{i_{1}}\otimes\cdots\otimes
dz_{i_{p}}\otimes\partial_{j_{1}}\otimes\cdots\otimes\partial_{j_{q}}.$

Here $\partial_{i}=\frac{\partial}{\partial z_{i}}$, and we write for
convenience $I=(i_{1},\ldots,i_{p})$, $J=(j_{1},\ldots,j_{q})$. Each $i_{k}$
or $j_{k}$ is summed from $1$ to $n$. The Hermitian metric $g$ on $M^{n}$
naturally induces a Hermitian inner product $(\,,\,)$ on
${\mathcal{T}}_{\,p}^{q}$ by

$(C,D)=\int_{M}\langle C,\overline{D}\rangle\,dv,$

where

$\langle C,\overline{D}\rangle=\sum
C_{I}^{J}\,\overline{D_{K}^{L}}\,g^{\bar{k}_{1}i_{1}}\cdots
g^{\bar{k}_{p}i_{p}}\,g_{j_{1}\bar{\ell}_{1}}\cdots g_{j_{q}\bar{\ell}_{q}},$

and all indices are summed from $1$ to $n$. For $C\in{\mathcal{T}}^{q}_{\,p}$,
its $(1,0)$-covariant derivative
$\nabla^{\prime}C\in{\mathcal{T}}^{q}_{p\\!+\\!1}$ is defined by

$(\nabla^{\prime}C)_{i_{1}\cdots i_{p\\!+\\!1}}^{j_{1}\cdots
j_{q}}=(\nabla_{i_{p\\!+\\!1}}C)_{i_{1}\cdots i_{p}}^{j_{1}\cdots
j_{q}}=C_{i_{1}\cdots i_{p},\,i_{p\\!+\\!1}}^{j_{1}\cdots j_{q}}.$

Applying the divergence theorem, we get the following well-known integration-
by-part formula, and we include a brief proof here for readers convenience.

######  Lemma 2.

On a compact Hermitian manifold $(M^{n},g)$, for any
$C\in{\mathcal{T}}^{q}_{\,p}$ and $D\in{\mathcal{T}}^{q}_{p\\!+\\!1}$, it
holds that

$(\nabla^{\prime}C,D)=(C,D^{1}\\!-\\!D^{0}),$

where $D^{1}$, $D^{0}\in{\mathcal{T}}^{q}_{\,p}$ are given respectively by

$(D^{1})_{I}^{J}=\sum_{i,j}D_{Ii}^{J}\overline{\eta}_{j}g^{\bar{j}i},\ \ \ \
(D^{0})_{I}^{J}=\sum_{i,j}D^{J}_{Ii,\,\bar{j}}g^{\bar{j}i}.$

###### Proof.

Let $X$ be the type $(1,0)$ vector field on $M^{n}$ given in any local
coordinate neighborhood by $X=\sum_{i}X^{i}\partial_{i}$ where

$X^{i}=\sum C_{i_{1}\cdots i_{p}}^{j_{1}\cdots
j_{q}}\,\overline{D_{k_{1}\cdots
k_{p\\!+\\!1}}^{\ell_{1}\cdots\ell_{q}}}\,g^{\bar{k}_{1}i_{1}}\cdots
g^{\bar{k}_{p}i_{p}}g^{\bar{k}_{p\\!+\\!1}i}\,g_{j_{1}\bar{\ell}_{1}}\cdots
g_{j_{q}\bar{\ell}_{q}}.$

Here and below we used Einstein’s convention that repeated indices are summed
up from $1$ to $n$. From this, we get

$\eta(X)=\langle C,\overline{D^{1}}\rangle,\ \ \ \ \
div(X)=X^{i}_{\,,i}=\langle\nabla^{\prime}C,\overline{D}\rangle+\langle
C,\overline{D^{0}}\rangle.$

Combine the two identities and use the divergence theorem (Lemma 1), we get
the integration by part formula, so Lemma 2 is proved. ∎

Now we are ready to derive the Euler-Lagrange equation for the $L^{2}$-norm of
the Chern torsion. Let $(M^{n},g)$ be a compact Hermitian manifold, and let
$h$ be a Hermitian-symmetric covariant tensor of type $(1,1)$. Using the
metric $g$ to lift the anti-holomorphic part,
$h^{j}_{i}=\sum_{\ell}h_{i\bar{\ell}}g^{\bar{\ell}j}$, we may view $h$ as a
tensor in ${\mathcal{T}}_{\,1}^{1}$ as well. For small real values
$t\in(-\varepsilon,\varepsilon)$, consider Hermitian metric $g(t)=g+th$ on
$M^{n}$, with Chern torsion $T_{ik}^{j}(t)$. For convenience, let us denote by
$\overset{\,\circ}{P}$ the quantity $\frac{d}{dt}P|_{t=0}$ for any $P$
depending on $t$. We have

$\overset{\circ}{g(t)}=h,\ \ \
\overset{\circ}{\big{(}g(t)^{-1}\big{)}}=-g^{-1}hg^{-1},\ \ \
\overset{\circ}{\big{(}dv(t)\big{)}}=\overset{\circ}{\big{(}\log\det
g(t)\big{)}}\,dv=\langle h,\bar{g}\rangle\,dv.$

By (6), we get

$\overset{\circ}{T^{j}_{ik}}=\big{(}\partial_{i}(h_{k\bar{\ell}})-\partial_{k}(h_{i\bar{\ell}})\big{)}g^{\bar{\ell}j}-T^{r}_{ik}h_{r\bar{q}}g^{\bar{q}j}=\big{(}h_{k\bar{\ell},\,i}-h_{i\bar{\ell},\,k}\big{)}g^{\bar{\ell}j},$

where indices after comma stand for covariant derivatives with respect to
$\nabla$. Since

$|T|^{2}=T^{j}_{ik}\overline{T^{b}_{ac}}\,g^{\bar{a}i}g^{\bar{c}k}g_{j\bar{b}}\,,$

we get

$\overset{\circ}{|T|^{2}}=2\mbox{Re}\langle\overset{\circ}{T},\overline{T}\rangle+\langle
h,\overline{B-2A}\rangle=-4\mbox{Re}\langle\nabla^{\prime}h,\overline{T}\rangle+\langle
h,\overline{B-2A}\rangle$

Now we can apply Lemma 2, and notice that for the torsion tensor $D=T$, the
corresponding tensors $(D^{1})_{i}^{j}=\phi_{i}^{j}$ and
$(D^{0})_{i}^{j}=\xi_{i}^{j}$, so we get the following

(8) $\int_{M}\overset{\circ}{|T|^{2}}dv=\int_{M}\langle h,\overline{P}\rangle
dv,\ \ \ \ \ \mbox{where}\ \ \ P=B-2A-2(\phi+\phi^{\ast})+2(\xi+\xi^{\ast}).$

Apply this to the functional ${\mathcal{F}}=V^{\frac{1-n}{n}}\int|T|^{2}dv$,
we get $\overset{\circ}{{\mathcal{F}}}=\int_{M}\langle h,\overline{Q}\rangle
dv$, where

(9) $Q=P+(|T|^{2}+\frac{1-n}{n}b)g,\ \ \ \ \ \ \
b=\frac{1}{V}\int_{M}|T|^{2}dv.$

So $g$ will be a critical point of ${\mathcal{F}}$ if and only if $Q=0$,
namely, $-P=(|T|^{2}+\frac{1-n}{n}b)g$. Writing it equivalently in terms of
$(1,1)$-forms, this means

$2\sigma_{1}-\sigma_{2}+2(\phi+\overline{\phi})-2(\xi+\overline{\xi})=(|T|^{2}+\frac{1-n}{n}b)\omega.$

Therefore we have completed the proof of Proposition 2.

##  4\. Existence and non-existence of torsion-critical metrics

Kähler metrics are certainly torsion-critical, namely, are critical points of
the $L^{2}$-norm ${\mathcal{F}}$ of Chern torsion, and by Gauduchon’s theorem,
when $n=2$ there are no other critical points. When $n\geq 3$, however, there
are examples of non-Kähler metrics that are torsion-critical. By Proposition
2, product of torsion-critical metrics are still torsion-critical, so we just
need to find a non-Kähler torsion-critical metric in dimension $3$, then such
examples will exist in all dimensions $n\geq 3$.

Consider a compact Chern flat manifold $(M^{3},g)$ whose universal cover is
the complex Lie group $G=SO(3,{\mathbb{C}})$, consisting of all $3\times 3$
complex matrices $X$ satisfying ${}^{t}\\!XX=I$, equipped with a metric
compatible with the complex structure. $(M^{3},g)$ is a non-Kähler, compact
Chern flat manifolds. Such manifolds are always balanced and with Chern
parallel torsion (namely, $\nabla T=0$), so we have $\eta=0$, $\phi=0$, and
$\xi=0$.

To compute $\sigma_{1}$ and $\sigma_{2}$, let us take the standard left-
invariant coframe $\varphi$ on $G$, which are left-invariant $(1,0)$-forms
satisfying

$d\varphi_{1}=\varphi_{2}\wedge\varphi_{3},\ \ \
d\varphi_{2}=\varphi_{3}\wedge\varphi_{1},\ \ \
d\varphi_{3}=\varphi_{1}\wedge\varphi_{2}.$

Using $\varphi$ as unitary coframe, the metric has Chern torsion components

$T^{1}_{23}=T^{2}_{31}=T^{3}_{12}=-1,$

while other components are zero. From these structure constants, we get

$A_{i\bar{j}}=B_{i\bar{j}}=2\delta_{ij},\ \ \ \ \forall\ 1\leq i,j\leq 3.$

So $\sigma_{1}=\sigma_{2}=2\omega$. Also, $|T|^{2}=b=6$ is a constant, so
equation (4) is satisfied, and $(M^{3},g)$ is torsion-critical.

Note that the same conclusion holds for compact quotients of
$SO(k,{\mathbb{C}})$ for any $k\geq 3$: all such manifolds are non-Kähler,
balanced, and torsion-critical.

The example was actually found by restricting our attention to Strominger
parallel manifolds, or STP in short, which means Hermitian metrics satisfying
$\nabla^{s}T=0$. Here $T$ is the Chern torsion and $\nabla^{s}$ is the
Strominger (or Bismut) connection. We have the following:

######  Lemma 3.

Let $(M^{n},g)$ be a STP manifold. Then under any unitary frame $e$ it holds

(10) $\displaystyle T^{j}_{ik,\ell}$ $\displaystyle=$ $\displaystyle
T^{j}_{rk}T^{r}_{i\ell}+T^{j}_{ir}T^{r}_{k\ell}-T^{r}_{ik}T^{j}_{r\ell},$ (11)
$\displaystyle T^{j}_{ik,\bar{\ell}}$ $\displaystyle=$
$\displaystyle-T^{j}_{rk}\overline{T^{i}_{r\ell}}-T^{j}_{ir}\overline{T^{k}_{r\ell}}+T^{r}_{ik}\overline{T^{r}_{j\ell}},$
(12) $\displaystyle 0\ \ $ $\displaystyle=$ $\displaystyle
T^{j}_{rk}T^{r}_{i\ell}+T^{j}_{ir}T^{r}_{k\ell}-T^{r}_{ik}T^{j}_{r\ell}$

for any any $1\leq i,j,k,\ell\leq n$, where $r$ is summed from $1$ to $n$, and
index after comma stand for covariant derivatives with respect to the Chern
connection $\nabla$.

###### Proof.

Let $e$ be a local unitary frame, with dual coframe $\varphi$. As is well-
known, the difference between $\nabla^{s}$ and $\nabla$ are given by

$(\nabla^{s}-\nabla)e_{i}=\sum_{j=1}^{n}\big{(}\sum_{r=1}^{n}\\{T^{j}_{ir}\varphi_{r}-\overline{T^{i}_{jr}}\overline{\varphi}_{r}\\}\big{)}e_{j}.$

See for example [30] or [29], and note that the $T^{j}_{ik}$ there are half of
ours here. So if $\nabla^{s}T=0$, then by the above formula we immediately get
formula (10) and (11). To prove the last identity (12), apply formula (17) in
[30, Lemma 2], where $R$ stands for the Riemannian curvature tensor. If we
take the sum of that identity by cyclicly permuting the three indices without
bar, the curvature part disappear by the first Bianchi identity, hence we get
a formula involving only the derivative of $T$ and the quadratic terms in $T$.
Combining this identity with (10), we know that both sides of (10) must be
zero. This completes the proof of the lemma. ∎

Now we are ready to prove Proposition 3 stated in the introduction part.

###### Proof of Proposition 3.

Let $(M^{n},g)$ be a STP manifold, namely, it satisfies $\nabla^{s}T=0$. In
(11) above, take $k=\ell$ and sum it up from $1$ to $n$, we get

$\xi^{j}_{i}=-B_{i\bar{j}}+\phi^{j}_{i}+A_{i\bar{j}}.$

Therefore $\phi-\xi=B-A$ for any STP manifold, and the left hand side of
equation (4) becomes $\sigma_{2}$. This shows that any SPT metric will be
torsion-critical if and only if $\sigma_{2}=c\,\omega$ for some constant $c$.

It remains to show that any SPT torsion-critical metric must be balanced.
Assume that it is not, then we can fix a point in $M$ and a unitary frame $e$
such that $\eta_{1}=\cdots=\eta_{n-1}=0$ but $\eta_{n}\neq 0$ at the given
point. On the other hand, by letting $j=\ell$ in (12) and sum it up, we know
that any SPT manifold always satisfies

$\sum_{r}\eta_{r}T^{r}_{ik}=0$

for any $1\leq i,k\leq n$. With the above choice of unitary frame $e$, we have
$T^{n}_{ik}=0$ for any $i,k$. This means that $B_{n\overline{n}}=0$. In other
words, the $B$ tensor of a non-balanced SPT manifold can never be positive
definite. So when $g$ is assumed to be torsion-critical, then
$B_{n\overline{n}}=0$ leads to $c=\frac{1}{n}|T|^{2}=0$, hence $T=0$ and the
metric is Kähler, contradicting with the assumption that $g$ is not balanced.
This completes the proof of Proposition 3. ∎

Finally, let us prove the non-existence result, Proposition 4, which searches
for torsion-critical metrics amongst several special classes of Hermitian
metrics.

###### Proof of Proposition 4.

Let $(M^{n},g)$ be a torsion-critical manifold. We want to show that $g$ must
be Kähler if it belongs to one of the special classes listed in the
proposition. First assume that $g$ is locally conformally Kähler. As is well-
known, the torsion tensor of such a metric is determined by its torsion
$1$-form, namely, under any unitary frame $e$ we have

$T^{j}_{ik}=\frac{1}{n-1}\big{(}\delta_{ij}\eta_{k}-\delta_{kj}\eta_{i}\big{)}$

for any $1\leq i,j,k\leq n$. From this, we compute

$\displaystyle
A_{i\bar{j}}\,=\,\frac{1}{(n-1)^{2}}\big{(}\delta_{ij}|\eta|^{2}+(n-2)\eta_{i}\overline{\eta}_{j}\big{)},\
\ \ \ \
B_{i\bar{j}}\,=\,\frac{2}{(n-1)^{2}}\big{(}\delta_{ij}|\eta|^{2}-\eta_{i}\overline{\eta}_{j}\big{)},$
$\displaystyle\phi_{i}^{j}\,=\,\frac{1}{n-1}\big{(}\delta_{ij}|\eta|^{2}-\eta_{i}\overline{\eta}_{j}\big{)},\
\ \ \ \ \ \
\xi_{i}^{j}\,=\,\frac{1}{n-1}\big{(}\delta_{ij}\chi-\eta_{i,\,\bar{j}}\big{)},\
\ \ \ \ |T|^{2}\,=\,\frac{2}{n-1}|\eta|^{2}.$

Since $g$ is assumed to be torsion-critical, by plugging the above into
equation (4) we get

$\eta_{i,\bar{j}}+\overline{\eta_{j,\bar{i}}}-\eta_{i}\overline{\eta}_{j}=\frac{n-1}{2n}b\,\delta_{ij},$

or equivalently, equation (1). So by Proposition 1, we know that $\eta=0$,
hence $T=0$.

Next, let us assume that $g$ is Strominger Kähler-like, namely, the curvature
tensor of $\nabla^{s}$ obeys all the Kähler symmetries. By [30], we know that
$\nabla^{s}T=0$, so $g$ is SPT. By Proposition 3, we know that $g$ must be
balanced. By [30] again, balanced Strominger Kähler-like metrics are always
Kähler, so we have proved that $g$ must be Kähler in this case.

Now let us assume that $(M^{n},g)$ is a complex nilmanifold with nilpotent $J$
in the sense of [9]. By the beautiful result of Salamon [18], there exists
unitary left-invariant coframe $\varphi$ such that

(13) $C^{j}_{ik}=D_{jk}^{i}=0\ \ \ \ \ \mbox{unless}\ j>i,k.$

where $C^{j}_{ik}$ and $D^{j}_{ik}$ are structural constants of the nilpotent
Lie algebra determined by

$d\varphi_{j}=-\frac{1}{2}\sum_{i,k=1}^{n}C^{j}_{ik}\varphi_{i}\wedge\varphi_{k}-\sum_{i,k=1}^{n}\overline{D^{i}_{jk}}\,\varphi_{i}\wedge\overline{\varphi}_{k}.$

By [25] or [31], we have

$\displaystyle T_{ik}^{j}$ $\displaystyle=$ $\displaystyle-
C_{ik}^{j}-D_{ik}^{j}+D_{ki}^{j}$ $\displaystyle T^{j}_{ik,\overline{\ell}}$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{n}\big{(}T^{j}_{rk}\overline{\Gamma^{i}_{r\ell}}+T^{j}_{ir}\overline{\Gamma^{k}_{r\ell}}-T^{r}_{ik}\overline{\Gamma^{r}_{j\ell}}\big{)},$

where $\Gamma_{ik}^{j}=D^{j}_{ik}$ is the coefficients of the Chern connection
$\nabla$. Note that our $T^{j}_{ik}$ equals twice of that in [25] or [31].
From the above, we get $\eta_{i}=\sum_{s}D^{s}_{is}$, and

$\xi_{i}^{j}=\sum_{r,s}\\{T^{j}_{rs}\overline{D^{i}_{rs}}-T^{r}_{is}\overline{D^{r}_{is}}\\}+\phi_{i}^{j}.$

Using the conditions (13), we get

$\phi_{n}^{n}-\xi_{n}^{n}=-\sum_{r,s}|D^{r}_{ns}|^{2},\ \ \ \
A_{n\bar{n}}=\sum_{r,s}|D^{r}_{ns}|^{2},\ \ \ \
B_{n\bar{n}}=\sum_{r,s}|C^{n}_{rs}|^{2}.$

Plug them into equation (4), we get

$-\sum_{r,s}\\{2|D^{r}_{ns}|^{2}+|C^{n}_{rs}|^{2}\\}=\frac{1}{n}|T|^{2}.$

Hence $|T|^{2}=0$ and $g$ is Kähler.

Finally, let $(M^{3},g)$ be any manifold listed in Table 1 or Table 2 of [5],
which are all the complex nilmanifolds and Calabi-Yau type complex
solvmanifolds in dimension $3$. For the nilpotent ones in Table 1, we only
need to check the type (Niii) since others are with nilpotent $J$. By direct
verification, one could see that $g$ cannot be torsion-critical, unless it is
already Kähler. We will omit the details here since it is a straight-forward
computation. This completes the proof of Proposition 4. ∎

Acknowledgments. The second named author would like to thank Haojie Chen, Ping
Li, Lei Ni, Xiaolan Nie, Kai Tang, Hongwei Xu, Bo Yang, Xiaokui Yang, and
Quanting Zhao for their interests and/or helpful discussions.

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11institutetext: School of Computer Science and Technology, Soochow
University, China 11email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# A Method of Query Graph Reranking for Knowledge Base Question Answering

Yonghui Jia  Wenliang Chen

###### Abstract

This paper presents a novel reranking method to better choose the optimal
query graph, a sub-graph of knowledge graph, to retrieve the answer for an
input question in Knowledge Base Question Answering (KBQA). Existing methods
suffer from a severe problem that there is a significant gap between top-1
performance and the oracle score of top-n results. To address this problem,
our method divides the choosing procedure into two steps: query graph ranking
and query graph reranking. In the first step, we provide top-n query graphs
for each question. Then we propose to rerank the top-n query graphs by
combining with the information of answer type. Experimental results on two
widely used datasets show that our proposed method achieves the best results
on the WebQuestions dataset and the second best on the ComplexQuestions
dataset.

###### Keywords:

Knowledge base question answering Query graph ranking Query graph reranking.

##  1 Introduction

Knowledge Base Question Answering (KBQA) is defined to be a task that answers
natural language questions over knowledge bases, such as Freebase [3] and
Wikidata [20]. A representative stream of approaches to KBQA is based on
semantic parsing [10, 17, 19] which translates textual questions into some
semantic representation. To this end, formal meaning representations such as
$\lambda-DCS$ [15] are widely adopted to represent semantics of questions.
However, such meaning representations need to be mapped to logical constants
or an ontology before they can be used to retrieve answers from the knowledge
base. The mapping between the meaning representations and an ontology is
restricted by the coverage of the ontology [12].

An alternative to formal meaning representations is to represent question
semantics with query graph [16, 24, 25], which can be regarded as a sub-graph
of knowledge base. This way, the restriction incurred by the coverage of an
ontology is overcome. With query graph as the semantic representation, the
process of KBQA can be divided into two steps: query graph generation and
query graph selection. During query graph generation, the question is parsed
into a set of candidate query graphs [24]. In query graph selection, an
optimal query graph is selected through ranking methods, and the knowledge
graph node corresponding to the optimal query graph is returned as the answer
to the question. In the process described above, query graph generation
determines the upper bound of the performance that we can achieve whereas
query graph selection determines the final performance of a KBQA system. In
this paper, we put focus on question graph selection which is thought to be
more important and challenging.

Existing approaches to query graph selection generally build on single-stage
ranking [1, 16, 24], which sorts query graphs according to the similarity
between the question and the query graph. There are two typical methods to
calculate the similarity according to whether we decompose the query graph.
One method is to decompose the query graph into different sub-paths and then
calculates the similarity between the question and each sub-path [24]. The
other is designed to measure the similarity between the question and a global
representation of the whole query graph [16]. The strategy of single-stage
ranking is widely adopted, but there is room left for further improvement of
query graph selection. The claim is inspired by the observation from our
preliminary experiments that there is a significant gap between top-1
performance and the oracle score of top-n candidates. 111By oracle score of
top-n candidates, we mean the performance that we can always choose the
optimal query graph from the top-n set. In addition, error analysis shows that
some bad cases have top-1 query graphs that retrieve answers of erroneous
answer types.

To address the above problems, we propose to use a ranking-reranking strategy
for query graph selection. Specifically, in the ranking stage we convert the
query graph into a sequence and use BERT [8] to conduct semantic matching
between the question and the query graph sequence. In the reranking stage, we
incorporate more information, including answer types to sort the top-n
candidates that are produced in the ranking stage. Extensive experiments on
benchmark data demonstrate the effectiveness of the reranking stage and the
incorporation of answer types.

Our contributions in the paper are summarized as follows,

  * $\bullet$

We propose a ranking-reranking method for query graph selection in the KBQA
system. This two-stage strategy can rerank top-n query graphs more effectively
and improve top-1 performance.

  * $\bullet$

Focusing on the answer type error in query graph ranking, we propose to
combine the answer type feature and the query graph sequence feature to rerank
top-n query graphs. Experiments demonstrate that the method can alleviate the
type error in query graph ranking and make a big improvement.

Figure 1: Query graph generation procedure of “Who is the highest president of
the US after 2000?”. Figure (a) shows the focus nodes linking result, Figure
(b) shows the main path example, Figure (c) shows the result after adding
entity constraints to the main path, and Figure (d) shows the query graph
after adding all constraints.

##  2 Our Approach

Our approach belongs to the semantic parsing method based on query graph in
the field of KBQA. The system includes two main components: query graph
generation and query graph selection. In query graph generation, we utilize
the staged query graph generation method to obtain a set of query graph
candidates. In query graph selection, we propose a two-stage method to select
the optimal query graph from the candidates. In the first stage, we rank the
candidates to get top-n query graphs, while in the second stage we rerank the
top-n query graphs to obtain the best one as the optimal query graph. Finally,
the optimal query graph is used to retrieve the final answer from Knowledge
Base (KB).

###  2.1 Query Graph Generation

The goal of query graph generation is to parse the question into the form of
query graph corresponding to the underlying KB. Here we briefly describe the
query graph generation method proposed in the previous studies and the details
can be found in the related papers [16, 24].

Given a question $q$, we first conduct focus nodes linking to recognize four
node constraints, including entities, type words, time words, and ordinal
words. For entity linking, we adopt the entity linking tool SMART [23] to
obtain $<$mention, entity$>$ pairs. For type word linking, we calculate word
vector similarity scores between consecutive sub-sequences (up to three words)
in the question and all types in KB, and select top-10 $<$mention,type$>$
pairs as type candidates. For time word linking, we use regular expressions to
extract time words from the question. For ordinal word linking, we use a
predefined ordinal number dictionary and the “number + superlative” pattern to
extract ordinal numbers. Figure 2(a) shows an example of the focus nodes
linking.

After focus nodes linking, we perform one-hop and two-hop searches to obtain
the main path based on the linked entities. Figure 2(b) shows an example.
Then, the entity constraint is added to the main path as a constraint node, as
shown in Figure 2(c). After that, we add type constraint, time constraint, and
ordinal constraint in turn, and finally get a complete query graph, as shown
in Figure 2(d).

During the above procedure, the focus nodes are not fully disambiguated.
Moreover, the one-hop and two-hop searches may have different paths. Thus
query graph generation may generate more than one, usually hundreds of
candidate query graphs. In the circumstances, we can get the candidate query
graph set $G=\\{g_{1},g_{2},...,g_{n}\\}$ for each question.

###  2.2 Query Graph Selection

The goal of query graph selection is to select the optimal query graph $g^{*}$
from the candidate query graph set $G$. In this section, we first describe how
to select top-n query graphs from the candidate set by query graph ranking.
Then we present a query graph reranking method with answer type to choose the
best one from top-n candidates.

####  2.2.1 Query Graph Ranking

In this step, we sort the candidates by calculating the matching score between
the question and each candidate query graph, and obtain the top-n query graphs
for reranking. To encode the question and a query graph, we choose to convert
the query graph into a query graph sequence, which makes the query graph and
the question both in the unified sequence form. This choice simplifies the
matching strategy of the question and the query graph, and the mature sequence
matching methods can be used.

Specifically, w e first transform the query graph into the sequence form
according to its composition. In structure, a query graph consists of up to
five sub-paths: MainPath, TypePath, EntityPath, TimePath, and OrdinalPath,
corresponding to the main path and four types of node constraints
respectively. Each sub-path can be converted into a word sequence. Taking the
main path in Figure 1 as an example, the corresponding main path sequence is
“united states governing officials office holder A”. We combine the five sub-
path sequences into a query graph sequence according to a fixed sub-path
order, denoted as $g^{s}=\\{u_{1},u_{2},...,u_{n}\\}$, where $u_{i}$ is a
word.

Figure 2: (a) The matching framework for query graph ranking. During encoding,
the question $q$ and the query graph sequence $g^{s}$ are considered. (b) The
matching framework for query graph reranking. During encoding, the question
$q$, the query graph sequence $g^{s}$ and answer type $t$ are considered.

After the query graph sequence $g^{s}$ is obtained, the task of query graph
ranking is transformed into a matching task between two sequences: the
question and the query graph sequence. We use BERT to encode the question and
the query graph sequence. Specifically, for query graph sequence
$g^{s}=\\{u_{1},u_{2},...,u_{n}\\}$, we combine it with the question
$q=\\{w_{1},w_{2},...,w_{m}\\}$ to form a sentence pair
$q{g}^{s}=\\{[CLS],w_{1},…,w_{m},[SEP],u_{1},…,u_{n},[SEP]\\}$. Then the
sentence pair $q{g}^{s}$ is fed into BERT for encoding. The encoding framework
is shown in Figure 2(a). We use the output of the $[CLS]$ node in BERT as the
semantic representation of the question and the query graph sequence, denoted
as $\bm{f}$.

$\bm{f}=BERT(qg^{s}).$ (1)

The semantic representation vector $\bm{f}$ is mapped to a specific score
through a linear layer.

$s^{{}^{\prime}}=\bm{f}W,$ (2)

where $s^{{}^{\prime}}$ is the similarity score between the question and the
query graph, and $W$ is a weight matrix. We calculate the scores of all the
candidates in this way and obtain the top-n results.

During training, we sample data according to a fixed positive and negative
ratio, as do in Luo et al. (2018) [16]. That is, a positive query graph and
$N$ negative query graphs form group examples. In the optimization process, we
use the cross-entropy loss function to optimize the score $s^{{}^{\prime}}$ of
each query graph, and the optimization objective is:

$s=\frac{1}{1+e^{-s^{{}^{\prime}}}},$ (3) $L=-(y\log(s)+(1-y)\log(1-s)),$ (4)

where $y$ is the query graph label (1 represents a positive example and 0
represents a negative example).

####  2.2.2 Query Graph Reranking

Obviously, we can rank all the candidate query graphs according to the scores
given in query graph ranking, and select the one with the highest score as the
optimal query graph. However, in the preliminary experiments, we find that
there is a significant gap between top-1 performance and the oracle score of
top-n candidates. To further improve top-1 performance, we propose a model to
rerank the top-n query graphs and obtain the best query graph.

In query graph reranking, we add the answer type feature to our matching
model, where the answer type information is from KB 222In Freebase, each
entity has one or more corresponding type. We use the property value of the
relation ‘fb:common.topic.notable_types’ as the type description of the
answer.. As shown in Figure 2(b), we adopt the same way used in query graph
ranking to calculate the matching score $s^{{}^{\prime}}$ of the question and
the query graph sequence. Besides, we utilize another BERT to encode the
question and the answer type. Given a question $q=\\{w_{1},w_{2},...,w_{m}\\}$
and answer type sequence $t=\\{v_{1},v_{2},...,v_{n}\\}$ of the query graph,
they are spliced into a sequence $qt$, and then fed into BERT for semantic
encoding. We also use the output vector of its $[CLS]$ node as the semantic
feature of the question and the answer type.

$\bm{f_{t}}=BERT(qt).$ (5)

And then, the feature is mapped to a score through a linear layer.

$s^{{}^{\prime\prime}}=\bm{f_{t}}W_{t},$ (6)

where $s^{{}^{\prime\prime}}$ is the similarity score of the question and the
answer type, and $W_{t}$ is a weight matrix. Two scores, $s^{{}^{\prime}}$ and
$s^{{}^{\prime\prime}}$, are summed together as the final matching score
$s^{*}$,

$s^{*}=s^{{}^{\prime}}+s^{{}^{\prime\prime}},$ (7)

During training, we no longer randomly select negative examples to construct
training data like query graph ranking. Instead, we use the top-n examples
generated by query graph ranking from training set as training data. After
introducing the answer type feature, the final optimization objective becomes:

$s=\frac{1}{1+e^{-s^{*}}},$ (8) $L=-(y\log(s)+(1-y)\log(1-s)),$ (9)

where $y$ is the query graph label, same as in Equation (4).

##  3 Experiments

###  3.1 Experimental Setup

####  3.1.1 Datasets.

Table 1:  The partitions of WebQ and CompQ datasets.  Dataset | Train | Validation | Test
---|---|---|---
WebQ | 3,023 | 755 | 2,032
CompQ | 1,000 | 300 | 800

We conduct experiments on two widely used datasets: WebQuestions (WebQ)
333https://nlp.stanford.edu/software/sempre/ and ComplexQuestions (CompQ)
444https://github.com/JunweiBao/MulCQA/tree/ComplexQuestions. These two
datasets are based on the form of question-answer pairs, and use Freebase
555https: //developers.google.com/freebase/ as the knowledge base. The WebQ
dataset contains 5,810 question-answer pairs, including simple questions (84%)
and complex questions (16%). The CompQ dataset is full of complex questions,
containing a total of 2,100 question-answer pairs. Following the previous
studies, both WebQ and CompQ are divided into training set, validation set and
test set. The partitions of the datasets are listed in Table 1.

####  3.1.2 Implementation Details.

In query graph ranking and query graph reranking, we both use BERT-BASE for
BERT model initialization, and use Adam as the optimizer for optimization. We
choose the settings of hyper-parameters in the systems according to the
performance of the validation set. The learning rate is set to $5\times
10^{-5}$, and the maximum training epoch is set to 5. At the end of each
epoch, we evaluate the current model on the validation set and obtain the
corresponding score. After training, we select the model with the highest
score on the validation set as our final model which is further evaluated on
the test set. For the evaluation metrics, we report the average F1-score (F1)
that has been used in the previous methods[16], and also report the average
precision (P) and average recall (R).

###  3.2 Main Results

Table 2:  The comparison results of query graph ranking and query graph reranking on the test sets. Method | WebQ(%) | CompQ(%)
---|---|---
P | R | F1 | P | R | F1
Ranking (top-1) | 51.41 | 61.64 | 52.44 | 37.12 | 49.40 | 38.35
Reranking | 54.79 | 65.98 | 56.02 | 41.29 | 55.44 | 42.94

Table 2 shows the performance of query graph ranking and query graph reranking
on the WebQ and CompQ datasets, where Ranking (top-1) refers to the system
that we only use the ranking step to obtain the answer and Reranking refers to
the full system proposed in this paper. From the table, we find that compared
to the Ranking system, the Reranking system achieves much better performance.
Specifically, we obtain an absolute improvement of 3.58% F1 on the WebQ
dataset and 4.59% F1 on the CompQ, respectively. In addition, the Reranking
system also achieves better performance on precision and recall than the
Ranking system. These facts indicate that our reranking solution is quite
effective.

Table 3:  The comparison results with the previous methods on the test sets.  Category | Method | WebQ(F1%) | CompQ(F1%)
---|---|---|---
Using Query Graph |  Yih et al. (2015) [24] | 52.5 | -
Bao et al. (2016) [1] | 52.4 | 40.9
Hu et al. (2018) [10] | 53.6 | -
Luo et al. (2018) [16] | 52.7 | 42.8
|  Lan and Jiang (2020) [13] | - | 43.3
Others |  Berant et al. (2013) [2] | 36.4 | -
Jain(2016) [11] | 55.6 | -
Chen et al. (2019) [6] | 51.8 | -
Xu et al. (2019) [22] | 54.6 | -
Ours | Ranking | 52.4 | 38.4
Reranking | 56.0 | 42.9

Table 3 shows the comparison results of our systems and the previous systems.
According to whether to use query graph, the previous systems are divided into
two categories: “Using Query Graph” and “Others”. In the previous “Using Query
Graph” systems, they utilize a single-stage ranking strategy and adopt the
predefined query graph features to improve the performance of query graph
selection. Compared with them, our method is based on a two-stage strategy and
does not rely on any predefined query graph features.

As a single-stage solution, our Ranking system achieves comparable performance
on the WebQ, but performs slightly lower on the CompQ. This shows that complex
questions are more challenging and need to be solved further. Among all the
systems, our Reranking system achieves the best result on the WebQ and the
second best result on the CompQ, which indicates our reranking solution is
quite powerful.

###  3.3 Discussion and Analysis

In this section, we perform further analysis on the results of our systems. We
first discuss the feasibility of the reranking strategy. Then we check the
effect of using different components. Finally, we study some cases of using
answer types.

Figure 3: The effect of different top-n for reranking, where “Ranking-TopN”
denotes the oracle score of top-n candidates in query graph ranking and
“Reranking-Top1” denotes the top-1 performance of query graph reranking given
different top-n results.

####  3.3.1 Feasibility of Reranking

We check the oracle score of top-n candidates in query graph ranking, as shown
in the Ranking-TopN curve of Figure 3. It is obvious that the Ranking-TopN
curve at first shows a rapid growth trend, and then tends to be stable when
$N>30$. This phenomenon means that most of the correct query graphs are the
top candidates in the ranking list, but not always the top-1 one. Therefore,
it is possible to use the reranking strategy to improve the performance based
on the top-n results.

We further check the performance of reranking with different top-n results
given by the step of query graph ranking, as shown in the Reranking-Top1 curve
of Figure 3. From the figure, we find that as top-n increases, the reranking
performance at first increases quickly and then remains stable. This means
that a reranking method with good performance can be realized under a smaller
value of top-n, such as top-10. Thus the reranking solution can efficiently
obtain the final answer. We should note that there is room for further
improving our system by better top-n ranking model and top-1 reranking model.
We leave them as future work.

####  3.3.2 Reasons for reranking promotion

Table 4:  The effect of using different components.  Method | WebQ(%) | CompQ(%)
---|---|---
P | R | F1 | P | R | F1
Ranking | 51.41 | 61.64 | 52.44 | 37.12 | 49.40 | 38.35
Reranking-Base | 53.10 | 63.16 | 53.85 | 38.32 | 50.74 | 39.65
Reranking-Full | 54.79 | 65.98 | 56.02 | 41.29 | 55.44 | 42.94

Compared to the Ranking system, our Reranking system has two changes: we first
use a different strategy to construct training data for the matching model,
and then add the information of answer type. In order to explore the influence
of the above two changes on query graph reranking, we remove the answer type
feature from the final system. The results are shown in Table 4, where
Reranking-Base refers to the reranking system without the information of
answer type and Reranking-Full refers to our final system. From the table, we
can find that after removing the answer type, the performance of the system is
significantly reduced. But compared to query graph ranking, the Reranking-Base
still performs better. This shows that the performance improvement in query
graph reranking is not only related to the answer types, but also affected by
the reranking strategy itself.

####  3.3.3 Case Study

Here we perform the case study to know whether the query graph reranking can
correct the type error in the query graph ranking with the help of answer
types. For the WebQ dataset, we randomly select 50 questions that are wrong in
query graph ranking and correct in query graph reranking. The results show
that more than half of the examples (52%) are corrected by choosing the
correct answer type instead of the wrong answer type, while others are
corrected with the same answer type. Some typical types of corrected examples
are listed in Table 5.

Table 5:  Some example cases, where italic indicates the relation in the query
graph sequence. Question1: where is the Galapagos islands located on a world
map?
---
Reranking Select: Galapagos islands _containedby_ Galapagos province, pacific
ocean.
Answer Type: administrative area, statistical region
Ranking Select: Galapagos islands _mentions book_ the far side of the world.
Answer Type: book
Question2: who is princess leia in star wars?
Reranking Select: princess leia _portrayed in films actor_ carrie fisher.
Answer Type: person sign, tv actor
Ranking Select: star wars _award nominations nominated for_ star wars.
Answer Type: award nominated work, product category

##  4 Related Work

Tracing back to the solutions for knowledge base question answering, there are
two mainstream approaches: information retrieval(IR) based methods and
semantic parsing(SP) based methods. Among them, IR-based methods [21, 26]
first recognize entities in the question, then search related candidate
answers according to entities and returns the most relevant answers. The core
of IR-based methods is the matching between questions and answers, which is
usually realized by deep learning approaches [6, 9]. For example, Bordes et
al. (2014) [4] propose calculate the similarity between the question and the
answer by learning low-dimensional embedding of the question and the subgraph
of the answer. There are also some works that focus on multi-hop reasoning by
adopting Key-Value Memory Neural Networks [22] or performing a transition-
based search strategy [7]. Besides, knowledge base embeddings are also used to
improve multi-hop question answering and achieve success [18].

Different from IR-based methods, SP-based approaches pay more attention to the
semantic information of questions, trying to parse the question into
structured representation [1, 5, 14]. For example, the question can be parsed
into $\lambda-DCS$, and then map to the knowledge base to obtain answers [2].
As for complex questions, Sun et al. (2020) [19] design a novel skeleton
grammar to express complex questions and improve the ability to parse complex
questions. Besides, query graph is also a widely-used formal meaning
representation in SP-based systems which is also used in this paper. Yih et
al. (2015) [24] are the pioneer in query graph research for KBQA. Following
this line, the encoding method of complex query graph [16] and the
construction method of multi-hop complex query graph [13] are also proposed.
In this paper, we propose a two-stage method of query graph selection, which
makes the query graph based method more effective.

##  5 Conclusions

In this paper, we propose a simple yet effective method for query graph
selection, which can be divided into two stages: query graph ranking and query
graph reranking. In query graph ranking, we calculate the similarity between
the question and the query graph sequence to obtain the top-n query graph
candidates. We further rerank the candidates with the help of the information
of answer types. The experimental results show that the method proposed
provides the best results on the WebQ dataset and the second best on the CompQ
dataset.

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