# A criterion for the existence of logarithmic connections on curves over a
perfect field

S. Manikandan  Indian Institute of Science Education and Research, Tirupati
C/o Sree Rama Engineering College (Transit Campus), Karakambadi Road, Mangalam
(P.O.) Tirupati 517507. Andhra Pradesh, India<EMAIL_ADDRESS>and
Anoop Singh  School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road
Mumbai 400 005
India<EMAIL_ADDRESS>
###### Abstract.

Let $k$ be a perfect field, and $X$ an irreducible smooth projective curve
over $k$. We give a criterion for a vector bundle over $X$ to admit a
logarithmic connection singular over a finite subset of $X$ with given
residues, where residues are assumed to be rigid.

###### Key words and phrases:

perfect field, logarithmic connections, Atiyah-Weil criterion

###### 2010 Mathematics Subject Classification:

53B15, 14H60

##  1\. Introduction and statements of the results

Let $X$ be a compact Riemann surface. A famous theorem due to Atiyah [3] and
Weil [11], which is known together as the Atiyah-Weil criterion, says that a
holomorphic vector bundle over a $X$ admits a holomorphic connection if and
only if the degree of each indecomposable component of the holomorphic vector
bundle is zero (see [5] for an exposition of the Atiyah-Weil criterion). In
[6] and [8], the Atiyah-Weil criterion has been generalised for the smooth
projective curve over infinite perfect field, and perfect field, respectively.
In [4], a criterion for the existence of a logarithmic connection with
prescribed residues has been established, and hence generalising the Atiyah-
Weil criterion in logarithmic set up. More precisely, let
$S=\\{x_{1},\ldots,x_{m}\\}$ be a subset of $X$ such that $x_{i}\neq x_{j}$
for all $i\neq j$, and let $E$ be a holomorphic vector bundle over $X$. Fix a
rigid endomorphism $A(x)\in\mathrm{End}(E(x))$ for every $x\in S$, where
$E(x)$ denote the fibre of $E$ over $x\in S$. Then, we have the following.

######  Theorem 1.1.

[4, Theorem 1.3] The vector bundle $E$ admits a logarithmic connection
singular over $S$ with residues $A(x)$ at every $x\in S$ if and only if for
every direct summand $F\subset E$,

$\deg F+\sum_{x\in S}\mathrm{tr}(A(x)|_{F(x)})=0,$ (1.1)

where $F(x)$ denote the fibre of $F$ over $x\in S$.

The proof in [4, Theorem 1.3] will work for vector bundles on a smooth
projective curve defined over an algebraically closed field of characteristic
zero.

Motivated by the above discussion, we have problems related to existence of
logarithmic connections when the curve is over an algebraically closed field
of characteristic $p>0$, a prime number. Also, what will be the suitable
criterion when the base field $k$ fails to be an algebraically closed field.

In case $k$ is an algebraically closed field of characteristic $p>0$, and $X$
is an irreducible smooth projective curve over $k$, we prove the following
(see section 3 Theorem 3.1).

######  Theorem 1.2.

Let $E$ be an algebraic vector bundle on $X$ defined over algebraically closed
field $k$ of characteristic $p>0$. Then $E$ admits a logarithmic connection
singular over $S$ with residue $A(x)$ for every $x\in S$ if and only if every
indecomposable component $F$ of $E$ satisfies the following condition

$\deg{F}+\sum_{x\in S}\mathrm{tr}(A(x)|_{F(x)})\equiv 0~{}(\mbox{mod}~{}p),$
(1.2)

that is, the number $\deg{F}+\sum_{x\in S}\mathrm{tr}(A(x)|_{F(x)})\in k$ is a
multiple of $p$.

By the abuse of notation, we denote the image of $\deg{E}\in\mathbb{Z}$ under
the morphism

$\mathbb{Z}\longrightarrow\mathbb{Z}/{p\mathbb{Z}}\hookrightarrow k$

by $\deg{E}$ itself, and this is used throughout the paper.

We also show the following result (see section 4, Theorem 4.1), and this will
generalise [4, Theorem 1.1] in the logarithmic framework.

######  Theorem 1.3.

Let $k$ be a perfect field of characteristic $p$. Let $E$ be a vector bundle
on an irreducible smooth projective curve $X$ over $k$. Then, we have

  1. (1)

Assume that $p>0$, and suppose that rank of each indecomposable components of
$E$ is not divisible by $p$. Then $E$ admits a logarithmic connection singular
over $S$ with residue $A(x)$ for every $x\in S$ if and only if for every
indecomposable component $F$ of $E$ satisfies

$\deg{F}+\sum_{x\in S}\mathrm{tr}(A(x)|_{F(x)})\equiv 0~{}(\mbox{mod}~{}p),$
(1.3)

  2. (2)

If for every indecomposable component $F$ of $E$ satisfies

$\deg{F}+\sum_{x\in S}\mathrm{tr}(A(x)|_{F(x)})=0,$ (1.4)

then $E$ admits a logarithmic connection singular over $S$ with residue $A(x)$
for every $x\in S$.

##  2\. Logarithmic connection and residues over a field

Let $k$ be a field. Let $X$ be an irreducible smooth projective curve over
$k$. Let

$S:=\\{x_{1},\ldots,x_{m}\\}$

be a finite subset of closed points of $X$ such that $x_{i}\neq x_{j}$ for
$i\neq j$, and let

$Z:=x_{1}+\ldots+x_{m}$

denote the reduce effective divisor associated with $S$. Let $\Omega^{1}_{X}$
denote the cotangent bundle of $X$. Let $E$ be a vector bundle over $X$. A
logarithmic connection on $E$ singular over $S$ is a $k$-linear map

$D:E\to E\otimes\Omega^{1}_{X}\otimes\mathcal{O}_{X}(Z)$ (2.1)

which satisfies the Leibniz identity

$D(fs)=fD(s)+df\otimes s,$ (2.2)

where $f$ is a local section of $\mathcal{O}_{X}$ and $s$ is a local section
of $E$.

We shall give an equivalent definition of a logarithmic connection in terms of
splitting of logarithmic Atiyah exact sequence.

Let $\mathrm{Diff}^{1}_{X}(E,E)$ be the vector bundle over $X$ whose sections
over any open subset $U\subset X$ are the differential operators on $E|_{U}$
of order at most one. Let

$\sigma_{1}:\mathrm{Diff}^{1}_{X}(E,E)\longrightarrow
TX\otimes\mathrm{End}(E)$

be the symbol operator, where $TX$ be the tangent bundle of $X$. The symbol
operator $\sigma_{1}$ is surjective. Consider the symbol exact sequence

$0\to\mathrm{End}(E)\xrightarrow{\iota}\mathrm{Diff}^{1}_{X}(E,E)\xrightarrow{\sigma_{1}}TX\otimes\mathrm{End}(E)\to
0.$ (2.3)

Let $\mathcal{O}_{X}(-Z)\subset\mathcal{O}_{X}$ be the line bundle associated
to the divisor $-Z$. Then

$\mathcal{O}_{X}(-Z)\otimes TX\otimes\mathrm{End}(E)\subset
TX\otimes\mathrm{End}(E).$

Define a vector bundle on $X$ as follows.

$\mathrm{At}(E)(-\log Z):=\sigma_{1}^{-1}(\mathbf{1}_{E}\otimes
TX\otimes\mathcal{O}_{X}(-Z)).$

From the symbol exact sequence (2.3), we get a short exact sequence

$0\to\mathrm{End}(E)\xrightarrow{\iota}\mathrm{At}(E)(-\log
Z)\xrightarrow{\widetilde{\sigma_{1}}}TX(-\log Z)\to 0,$ (2.4)

called the logarithmic Atiyah exact sequence, where $\widetilde{\sigma_{1}}$
is the restriction of $\sigma_{1}$ and $TX(-\log
Z):=TX\otimes\mathcal{O}_{X}(-Z).$

Now, $E$ admits a logarithmic connection singular over $S$ if and only if the
logarithmic Atiyah exact sequence (2.4) splits algebraically, that is, there
exists an $\mathcal{O}_{X}$-linear homomorphism

$\alpha:TX(-\log Z)\to\mathrm{At}(E)(-\log Z)$ (2.5)

such that

$\widetilde{\sigma_{1}}\circ\alpha=\mathbf{1}_{TX(-\log Z)}.$

Next we will define residue of a logarithmic connection in $E$ at $x\in S$.
Fix $x\in S$, and let $U$ be an open subset of $X$ such that $U\cap
S=\\{x\\}$. Let $s$ be an algebraic section of $\mathrm{At}(E)(-\log Z)$ over
$U$. Then $\widetilde{\sigma_{1}}(s)$ will be a section of $TX(-\log Z)$ over
$U$. Now, evaluating $\widetilde{\sigma_{1}}(s)$ at $x$, we get

$\widetilde{\sigma_{1}}(s)(x)=\widetilde{\sigma_{1}}_{x}(s(x))=0.$

Since $\mathrm{Ker}(\widetilde{\sigma_{1}})=\mathrm{End}(E)$, we get a
$k$-linear map

$j_{x}:\mathrm{At}(E)(-\log Z)(x)\to\mathrm{End}(E)(x)=\mathrm{End}(E(x)).$
(2.6)

Note that for any $x\in S$, the fiber
$\Omega^{1}_{X}\otimes\mathcal{O}_{X}(Z)(x)$ is canonically identified with
$k$ by sending a rational $1$-form to its residue at $x$.

Given a logarithmic connection $D$ on $E$ singular over $S$, we get a unique
algebraic splitting $\alpha$ of (2.4) corresponding to $D$, and thus we define
the residue of $D$ at $x\in S$ by

$Res(D,x):=j_{x}(\alpha_{x}(1))\in\mathrm{End}(E(x)),$ (2.7)

where $1\in k$ and the fiber $TX(-\log Z)(x)$ is canonically identified with
$k$ as described above.

Next, we describe logarithmic connections with prescribed residues. For that
first notice the following.

######  Lemma 2.1.

For every $x\in S$, the fibre $\mathrm{At}(E)(-\log Z)(x)$ has the canonical
decomposition

$\mathrm{At}(E)(-\log Z)(x)=\mathrm{End}(E(x))\oplus k$ (2.8)

###### Proof.

See [4, Lemma 2.1]. ∎

Fix $A(x)\in\mathrm{End}(E(x))$ for every $x\in S$. Consider the one
dimensional vector space over $k$ generated by the vector $(A(x),1)$ in
$\mathrm{End}(E(x))\oplus k$, that is,

$l_{x}:=k.(A(x),1)\subset\mathrm{End}(E(x))\oplus k=\mathrm{At}(E)(-\log
Z)(x).$ (2.9)

Let $\mathcal{A}(E)\to X$ be the vector bundle that fits in the following
short exact sequence

$0\to\mathcal{A}(E)\to\mathrm{At}(E)(-\log Z)\to\bigoplus_{x\in
S}\frac{\mathrm{At}(E)(-\log Z)(x)}{l_{x}}\to 0.$ (2.10)

where $l_{x}$ is constructed above.

From logarithmic Atiyah exact sequence (2.4) we have the short exact sequence

$0\to\mathrm{End}(E)\otimes\mathcal{O}_{X}(-Z)\xrightarrow{\iota}\mathcal{A}(E)\xrightarrow{\widehat{\sigma_{1}}}TX(-\log
Z)\to 0,$ (2.11)

where $\widehat{\sigma_{1}}$ is the restriction of $\widetilde{\sigma_{1}}$.

######  Lemma 2.2.

A logarithmic connection in $E$ with given residues
$A(x)\in\mathrm{End}(E(x))$, for every $x\in S$ is an algebraic splitting of
the short exact sequence (2.11), that is, there exists a morphism

$h:TX(-\log Z)\to\mathcal{A}(E)$

such that $\widehat{\sigma_{1}}\circ h=\mathbf{1}_{TX(-\log Z)}$.

We recall the notion of rigid endomorphism. Let $x\in X$. An endomorphism
$\beta\in\mathrm{End}(E(x))$ is said to be a rigid if

$\beta\circ\phi(x)=\phi(x)\circ\beta$

for all $\phi\in\mathrm{H}^{0}(X,\mathrm{End}(E))$. Now onwards we shall
assume that the endomorphism $A(x)$ is rigid for every $x\in S$.

Recall that a vector bundle $E$ over $X$ is said to be decomposable if there
are vector bundles $F$ and $F^{\prime}$ such that $\mathrm{rk}(F)>0$,
$\mathrm{rk}(F)^{\prime}>0$ and

$E\cong F\oplus F^{\prime}.$

A vector bundle is called indecomposable if it is not decomposable.

From [2, p.315, Theorem 2], any vector bundle $E$ over $X$ is isomorphic to a
unique, up to reordering, direct sum of indecomposable vector bundles, and we
call it Krull-Remak-Schmidt decomposition of $E$. Let

$E=\bigoplus_{i=1}^{n}E^{i}$

be the Krull-Remak-Schmidt decomposition of $E$. Since
$A(x)\in\mathrm{End}(E(x))$ is rigid for every $x\in S$,

$A(x)(E^{i}{(x)})\subset E^{i}{(x)}.$

The restriction of $A(x)$ on $E^{i}(x)$ is denoted by $A^{i}(x)$ for every
$x\in S$ and for every $i=1,\ldots,n$. We use the notations as above for the
following.

######  Lemma 2.3.

$E$ admits a logarithmic connection singular over $S$ with residues $A(x)$ at
every $x\in S$ if and only if each indecomposable component $E^{i}$ admits a
logarithmic connection $D^{i}$ with residue $A^{i}(x)$ at every $x\in S$.

###### Proof.

Let $\iota:E^{i}\to E$ be the inclusion map, and $q^{i}:E\to E^{i}$ the
quotient map. Let

$D:E\to E\otimes\Omega^{1}_{X}(\log Z)$

be a logarithmic connection singular over $S$ with residue $A(x)$ at every
$x\in S$. Consider the composition

$(q^{i}\otimes\mathbf{1}_{\Omega^{1}_{X}(\log Z)})\circ D\circ\iota:E^{i}\to
E^{i}\otimes\Omega^{1}_{X}(\log Z),$

which satisfies the Leibniz rule and singular over $S$ with residue $A^{i}(x)$
at every $x\in S$. Conversely, given logarithmic connection $D^{i}$ on $E^{i}$
for every $1\leq i\leq n$, singular over $S$ with residue $A^{i}(x)$ at every
$x\in S$. Then $\bigoplus_{i=1}^{n}D^{i}$ gives a logarithmic connection on
$E$ singular over $S$ with residue $A(x)$ at every $x\in S$. ∎

##  3\. Criterion over an algebraically closed field of characteristic $p>0$

In this section, we assume that $k$ is an algebraically closed filed of
characteristic $p>0$, and $X$ an irreducible smooth projective curve over $k$.

######  Theorem 3.1.

Let $E$ be an algebraic vector bundle on $X$ defined over the algebraically
closed field $k$ of characteristic $p>0$. Then $E$ admits a logarithmic
connection singular over $S$ with residue $A(x)$ for every $x\in S$ if and
only if every indecomposable component $F$ of $E$ satisfies the following
condition

$\deg{F}+\sum_{x\in S}\mathrm{tr}(A(x)|_{F(x)})\equiv 0~{}(\mbox{mod}~{}p),$
(3.1)

that is, the number $\deg{F}+\sum_{x\in S}\mathrm{tr}(A(x)|_{F(x)})\in k$ is a
multiple of $p$.

###### Proof.

In view of Lemma 2.3, it is enough to prove the theorem for indecomposable
vector bundles. Without loss of generality assume that $E$ is an
indecomposable vector bundle. Consider the short exact sequence (2.11)
associated with $E$ and its extension class (called logarithmic Atiyah class)

$\phi^{A}_{E}\in\mathrm{H}^{1}(X,\mathrm{End}(E)\otimes\Omega^{1}_{X}).$ (3.2)

By Serre duality, the logarithmic Atiyah class $\phi^{A}_{E}$ corresponds to
an element

$\widetilde{\phi^{A}_{E}}\in\mathrm{H}^{0}(X,\mathrm{End}(E))^{*}.$ (3.3)

Now, we shall construct a $k$-linear morphism

$\delta_{x}:\mathrm{End}(E(x))\to\mathrm{H}^{1}(X,\mathrm{End}(E)\otimes\Omega^{1}_{X})$
(3.4)

of vector spaces for every $x\in S$. In fact, this morphism $\delta_{x}$ can
be constructed for any $x\in X$. Consider the short exact sequence

$0\to\mathrm{End}(E)\otimes\Omega^{1}_{X}\rightarrow\mathcal{O}_{X}(x)\otimes\mathrm{End}(E)\otimes\Omega^{1}_{X}\xrightarrow{Res_{x}}\mathrm{End}(E(x))\to
0,$ (3.5)

where the last map is given by sending logarithmic $1$-form with values in
$\mathrm{End}(E)$ to its residue at $x$ with values in $\mathrm{End}(E(x))$.
The morphism $\delta_{x}$ in (3.4) is the coboundary operator in the long
exact sequence of cohomology groups induced from the short exact sequence
$\eqref{eq:19}$. From Serre duality, and from $\delta_{x}$, we get an induced
morphism

$\widetilde{\delta_{x}}:\mathrm{End}(E(x))\to\mathrm{H}^{0}(X,\mathrm{End}(E))^{*},$
(3.6)

such that

$\widetilde{\delta_{x}}(\beta)(\gamma)=\mathrm{tr}(\beta\circ\gamma(x)).$

Now, consider the Atiyah exact sequence (see [3] and [7, Proposition 4.2] for
the proof)

$0\to\mathrm{End}(E)\xrightarrow{\iota}\mathrm{At}(E)\xrightarrow{\sigma_{1}}TX\to
0,$ (3.7)

and let
$\mathrm{at}(E)\in\mathrm{H}^{1}(X,\mathrm{End}(E)\otimes\Omega^{1}_{X})$
denote the extension class, known as Atiyah class. Again using Serre duality,
we have $\widetilde{\mathrm{at}(E)}\in\mathrm{H}^{0}(X,\mathrm{End}(E))^{*}$.
We claim that

$\widetilde{\phi^{A}_{E}}=\widetilde{\mathrm{at}(E)}+\sum_{x\in
S}\widetilde{\delta_{x}}(A(x))$ (3.8)

Since $k$ is an algebraically closed field and $E$ is indecomposable, any
element $\theta\in\mathrm{H}^{0}(X,\mathrm{End}(E))$ is of the form

$\theta=\nu\mathbf{1}_{E}+N,$

where $\nu\in k$ and $N$ is a nilpotent endomorphism of $E$. To prove (3.8),
it is enough to verify the formula (3.8) for $\mathbf{1}_{E}$ and $N$
separately. From [3, Proposition 18(ii)], it is know that
$\widetilde{\mathrm{at}(E)}(N)=0$. Next, since $N\in\mathrm{End}(E)$ is
nilpotent, $N_{x}\in\mathrm{End}(E(x))$ is nilpotent. Since $A(x)$ is a rigid
endomorphism, $N(x)$ commutes with $N(x)$ and hence $A(x)\circ N(x)$ is
nilpotent. Thus,

$\widetilde{\delta_{x}}(A(x))(N)=\mathrm{tr}(A(x)\circ N(x))=0.$

Let $E_{\bullet}$ be a flag

$0=E_{0}\subset E_{1}\subset\ldots\subset E_{l}=E$

of subbundles of $E$ such that

$A(x)(E_{i}(x))\subset E_{i}(x),$

for every $x\in S$ and for every $1\leq i\leq l$. Let $\mathrm{End}(E)^{0}$ be
the $\mathcal{O}_{X}$-submodule of $\mathrm{End}(E)$ consisting of
endomorphisms which preserves the flag $E_{\bullet}$. Then
$\mathrm{End}(E)^{0}$ is a subbundle of $\mathrm{End}(E)$. The inclusion
morphism $\iota:\mathrm{End}(E)^{0}\hookrightarrow\mathrm{End}(E)$, induces a
morphism

$\iota^{*}:\mathrm{H}^{1}(X,\mathrm{End}(E)^{0}\otimes\Omega^{1}_{X})\to\mathrm{H}^{1}(X,\mathrm{End}(E)\otimes\Omega^{1}_{X}).$

It can be shown that the logarithmic Atiyah class $\phi^{A}_{E}$ is in the
image of $\iota^{*}$.

Moreover, for a nilpotent endomorphism $N$, the subbundles
$\mathrm{Ker}(N^{j})$ of $E$ for $j\geq 1$, form a flag of $E$ such that

$A(x)(\mathrm{Ker}(N^{j})(x))\subset\mathrm{Ker}(N^{j})(x),$

because $A(x)\circ N^{j}(x)=N^{j}(x)\circ A(x)$ for every $j\geq 1$. We take
$\mathrm{End}(E)^{0}$ to be associated with the flag $\mathrm{Ker}(N^{j})$,
$j\geq 1$. Then, from above observation $\widetilde{\phi^{A}_{E}}(N)=0$. Thus,
(3.8) satisfies for $N$.

Consider the following composition of morphisms

$\displaystyle\mathrm{H}^{0}(X,\mathrm{End}(E))\otimes\mathrm{H}^{1}(X,\mathrm{End}(E)\otimes\Omega^{1}_{X})\xrightarrow{\cup}\mathrm{H}^{1}(X,\mathrm{End}(E)\otimes\mathrm{End}(E)\otimes\Omega^{1}_{X})$
$\displaystyle\xrightarrow{m^{*}}\mathrm{H}^{1}(X,\mathrm{End}(E)\otimes\Omega^{1}_{X})\xrightarrow{\mbox{trace}}\mathrm{H}^{1}(X,\Omega^{1}_{X})=k,$

and denote it by $\Psi$, where the first morphism is the cup product, second
morphism $m^{*}$ is induced from

$m:\mathrm{End}(E)\otimes\mathrm{End}(E)\to\mathrm{End}(E)$

which is composition of endomorphisms of $E$.

Under this composition $\Psi$, the image of
$\mathbf{1}_{E}\otimes\phi^{A}_{E}$ coincides with $\deg{E}+\sum_{x\in
S}\mathrm{tr}(A(x))\in k$, that is,

$\widetilde{\phi^{A}_{E}}(\mathbf{1}_{E})=\deg{E}+\sum_{x\in
S}\mathrm{tr}(A(x)).$

Also, under the same morphism $\Psi$, the image of $\mathrm{at}(E)$ coincides
with $\deg{E}$(see [6, Proposition 3.1]), and hence

$\widetilde{\mathrm{at}(E)}(\mathbf{1}_{E})=\deg{E}.$

As observed above,
$\widetilde{\delta_{x}}(A(x))(\mathbf{1}_{E})=\mathrm{tr}(A(x)\circ\mathbf{1}_{E})=\mathrm{tr}(A(x))$.
Thus, (3.8) verifies for $\mathbf{1}_{E}$, and hence completes the proof of
the claim.

Thus, $E$ admits a logarithmic connection singular over $S$ with given residue
$A(x)$ for every $x\in S$ if and only if the obstruction class $\phi^{A}_{E}$
vanishes which is equivalent to $\widetilde{\phi^{A}_{E}}(\mathbf{1}_{E})=0$,
that is, $\deg{E}+\sum_{x\in S}\mathrm{tr}(A(x))$ vanishes in $k$, which is
nothing but

$\deg{E}+\sum_{x\in S}\mathrm{tr}(A(x))\equiv 0~{}(\mbox{mod}~{}p).$

This completes the proof of the theorem. ∎

######  Remark 3.2.

In the proof of the Theorem 3.1, we observed that
$\widetilde{\phi^{A}_{E}}(\mathbf{1}_{E})\in k$ corresponds to
$\deg{E}+\sum_{x\in S}\mathrm{tr}(A(x))\in k$. In fact, this is true for every
field $k$. Therefore, if a vector bundle $E$ on an irreducible smooth
projective curve $X$ defined over any field $k$ admits a logarithmic
connection singular over $S$ with residue $A(x)$ for every $x\in S$, then the
number

$\deg{E}+\sum_{x\in S}\mathrm{tr}(A(x))$

is a multiple of the characteristic of $k$.

##  4\. Criterion over a perfect field

In this section, we prove an analogous result, when the base field fails to be
algebraically closed. In this section, we will assume that $k$ is a perfect
field of arbitrary characteristic, and $X$ is an irreducible smooth projective
curve defined over $k$.

######  Theorem 4.1.

Let $k$ be a perfect field of characteristic $p$. Let $E$ be a vector bundle
on an irreducible smooth projective curve $X$ over $k$. Then, we have

  1. (1)

Assume that $p>0$, and suppose that rank of each indecomposable components of
$E$ is not divisible by $p$. Then $E$ admits a logarithmic connection singular
over $S$ with residue $A(x)$ for every $x\in S$ if and only if for every
indecomposable component $F$ of $E$ satisfies

$\deg{F}+\sum_{x\in S}\mathrm{tr}(A(x)|_{F(x)})\equiv 0~{}(\mbox{mod}~{}p),$
(4.1)

  2. (2)

If for every indecomposable component $F$ of $E$ satisfies

$\deg{F}+\sum_{x\in S}\mathrm{tr}(A(x)|_{F(x)})=0,$ (4.2)

then $E$ admits a logarithmic connection singular over $S$ with residue $A(x)$
for every $x\in S$.

In view of the Remark 3.2, in the first part (1) of the above Theorem 4.1, one
direction is obvious, that is, if $E$ admits a logarithmic connection singular
over $S$ with residue $A(x)$ for every $x\in S$, then

$\deg{F}+\sum_{x\in S}\mathrm{tr}(A(x)|_{F(x)})\equiv 0~{}(\mbox{mod}~{}p).$

The hard part is to prove the converse of it. Also, notice that the converse
of the second part (2) of the Theorem 4.1 is not true. In order to prove above
Theorem 4.1, we need the notion of absolutely indecomposable vector bundles
over $X$ and some key properties of finite extensions of perfect fields.

Let $K$ be a finite extension of $k$. Let

$X_{K}=X\times_{\mbox{spec(k)}}\mbox{spec(K)}$

be the curve obtained from base change and let

$\pi:X_{K}\longrightarrow X$ (4.3)

be the natural projection. Consider the subset $S=\\{x_{1},\ldots,x_{m}\\}$ of
$X$ as above. Then

$\widetilde{S}:=\pi^{-1}(S)=\\{\pi^{-1}(x_{1}),\ldots,\pi^{-1}(x_{m})\\}$

will be a subset consists of distinct closed points of $X_{K}$. For simplicity
we set $y_{j}=\pi^{-1}(x_{j})$ for every $1\leq j\leq m$. Let

$\widetilde{Z}:=y_{1}+\cdots+y_{m}$

be the reduced effective divisor associated with $\widetilde{S}$.

Let $\mathcal{F}$ be a vector bundle over $X_{K}$, and let

$F:=\pi_{*}\mathcal{F}$ (4.4)

be the direct image of the vector bundle $\mathcal{F}$ under $\pi$. Then $F$
is a vector bundle over $X$. Let

$d:=\deg{\pi}=[K:k]$

denote the degree of the extension $K|k$. Then

$\deg{F}=d\cdot\deg{\mathcal{F}},$ (4.5)

and we get a relation between their ranks

$\mathrm{rk}(F)=d\cdot\mathrm{rk}(\mathcal{F}).$ (4.6)

Under above notations we have

######  Lemma 4.2.

If the vector bundle $\mathcal{F}$ over $X_{K}$ admits a logarithmic
connection singular over $\widetilde{S}$, then the vector bundle $F$
$(\mbox{defined}~{}in~{}\eqref{eq:26})$ over $X$ admits a logarithmic
connection singular over $S$.

###### Proof.

Let

$D:\mathcal{F}\to\mathcal{F}\otimes\Omega^{1}_{X_{K}}\otimes\mathcal{O}_{X_{K}}(-\widetilde{Z})=\mathcal{F}\otimes\Omega^{1}_{X_{K}}(\log\widetilde{Z})$

be a logarithmic connection on $\mathcal{F}$ singular over $\widetilde{S}$.

Let $U\subset X$ be an open subset, and $s\in\mathcal{F}(\pi^{-1}(U))$. Then
$D(s)$ be a local section of
$\mathcal{F}\otimes\Omega^{1}_{X_{K}}(\log\widetilde{Z})$ over $\pi^{-1}(U)$.
This implies that $D(s)$ is a local section of
$\pi_{*}(\mathcal{F}\otimes\Omega^{1}_{X_{K}}(\log\widetilde{Z}))$ over $U$.
Since

$\pi^{*}(\Omega^{1}_{X}(\log Z))=\Omega^{1}_{X_{K}}(\log\widetilde{Z}),$

from the projection formula, we get

$\displaystyle\pi_{*}(\mathcal{F}\otimes\Omega^{1}_{X_{K}}(\log\widetilde{Z}))$
$\displaystyle=\pi_{*}(\mathcal{F}\otimes\pi^{*}(\Omega^{1}_{X}(\log Z))$
$\displaystyle=\pi_{*}(\mathcal{F})\otimes\Omega^{1}_{X}(\log Z)$
$\displaystyle=F\otimes\Omega^{1}_{X}(\log Z).$

Thus, $D(s)$ gives a local section of $F\otimes\Omega^{1}_{X}(\log Z)$ over
$U$.

We define an operator

$\nabla:F\to F\otimes\Omega^{1}_{X}(\log Z)$

by

$\nabla(\hat{s}):=D(s),$

where $\hat{s}\in F(U)$, a section given by
$s\in(\pi_{*}\mathcal{F})(U)=F(U)$, for every open subset $U\subset X$.

For any $f\in\mathcal{O}_{X}(U)$,
$d(f\circ\pi)=\pi^{*}df\in\Omega^{1}_{X_{K}}(\log\widetilde{Z})(U)$, where
$d:\mathcal{O}_{X}\to\Omega^{1}_{X}(\log Z)$ is the universal derivation. The
operator $\nabla$ satisfies the following property (Leibniz rule).

$\displaystyle\nabla(f\cdot\hat{s})$ $\displaystyle=D((f\circ\pi)\cdot s)$
$\displaystyle=(f\circ\pi)D(s)+d(f\circ\pi)\otimes s$
$\displaystyle=f\nabla(\hat{s})+df\otimes\hat{s}.$

Thus, $\nabla$ is a logarithmic connection on $F$ singular over $S$. ∎

######  Remark 4.3.

In the above Lemma 4.2, we can choose residue $B(y)$ for every
$y\in\widetilde{S}$, for the logarithmic connection $D$ on $\mathcal{F}$
singular over $\widetilde{S}$, such that it maps to the given residue $A(x)$
for every $x\in S$, for the induced logarithmic connection $\nabla$ on $F$
singular over $S$, under $\pi$.

Let $Y$ be a geometrically irreducible projective curve defined over a field
$l$. For the following definition of absolutely indecomposable vector bundle,
see [1] and [8, Definition 2.2].

A vector bundle $\mathcal{E}$ over $Y$ is called absolutely indecomposable if
there is an algebraic closure $\overline{l}$ of $l$ such that the
corresponding vector bundle $\mathcal{E}\otimes_{l}\overline{l}$ over
$Y\times_{l}\overline{l}$ is indecomposable.

In fact, the condition that a vector bundle over $Y$ is absolutely
indecomposable does not depend on the choice of the algebraic closure
$\overline{l}$ (see the first paragraph on [8, p.n. 86]).

###### Proof of Theorem 4.1.

In view of Lemma 2.3, it is enough to show the theorem for indecomposable
vector bundles. To prove the first part (1), let $k$ be a perfect field of
characteristic $p>0$, and $E$ be an indecomposable vector bundle over $X$ such
that

$\deg{E}+\sum_{x\in S}\mathrm{tr}(A(x))\equiv 0~{}(\mbox{mod}~{}p),$ (4.7)

where $A(x)$ is given rigid endomorphism on $E(x)$, for every $x\in S$.

Since $k$ is a perfect field, from [1, Theorem 1.8, 4] there is a finite field
extension $K$ of $k$ with the following property.

There is an absolutely indecomposable vector bundle $\\\ cat{E}$ over

$X_{K}:=X\times_{k}K$

such that

$\pi_{*}\mathcal{E}\cong E,$ (4.8)

where $\pi:X_{K}\to X$ is the natural projection. The above result is due to
A. Tillmann[10] as mentioned in [1].

In view of (4.6), for every $y\in\widetilde{S}$, choose endomorphism
$B(y)\in\mathcal{E}(y)\cong K^{\mathrm{rk}(\mathcal{E})}$ such that

$d\cdot\mathrm{tr}(B(y))=\mathrm{tr}(A(x)),$ (4.9)

where $\pi(y)=x$, and $d$ the degree of extension $K|k$.

Note that $\mathrm{tr}(B(y))\in K$ and $\mathrm{tr}(A(x))\in k$, so we
consider $K$ as a vector space over $k$, and we take trace of
$\mathrm{tr}(B(x))\in K$ to get (4.9).

From (4.7), and (4.5), we have

$d\cdot\deg{\mathcal{E}}+\sum_{y\in\widetilde{S}}d\cdot\mathrm{tr}(B(y))\equiv
0~{}(\mbox{mod}~{}p).$ (4.10)

Since $\mathrm{rk}(E)$ is not divisible by $p$, from (4.6), $d$ is coprime to
$p$, and hence (4.10) becomes,

$\deg{\mathcal{E}}+\sum_{y\in\widetilde{S}}\mathrm{tr}(B(y))\equiv
0~{}(\mbox{mod}~{}p).$ (4.11)

Since $\mathcal{E}$ is an absolutely indecomposable vector bundle over
$X_{K}$, by above definition

$\mathcal{E}_{\overline{K}}:=\mathcal{E}\otimes_{K}{\overline{K}}$

is indecomposable vector bundle over
$X_{\overline{K}}:=X_{K}\times_{K}\overline{K}$, where $\overline{K}$ is an
algebraic closure of $K$.

Let $\widehat{\pi}:X_{\overline{K}}\longrightarrow X_{K}$ be the natural
projection. Choose a subset

$\widehat{S}=\\{\hat{y_{1}},\ldots,\hat{y_{m}}\\}\subset X_{\overline{K}}$

such that $\widehat{\pi}(\hat{y_{j}})=y_{j}$ for every $j=1,\ldots,m$. Notice
that $\deg{\mathcal{E}_{\overline{K}}}=\deg{\mathcal{E}}$. From above
observation, for every $\hat{y}\in\widehat{S}$, choose endomorphisms
$\widehat{B(\hat{y})}$ on $\mathcal{E}_{\overline{K}}(\hat{y})$ such that
$\mathrm{tr}(\widehat{B(\hat{y})})=\mathrm{tr}(B(y))$. Thus, the equation
(4.11) becomes

$\deg{\mathcal{E}_{\overline{K}}}+\sum_{\hat{y}\in\widehat{S}}\mathrm{tr}(\widehat{B(\hat{y})})\equiv
0~{}(\mbox{mod}~{}p),$ (4.12)

and from Theorem 3.1, $\mathcal{E}_{\overline{K}}$ admits a logarithmic
connection singular over $\widehat{S}$ with residue $\widehat{B(\hat{y})}$ for
every $\hat{y}\in\widehat{S}$.

Consider the short exact sequence (2.11) for the vector bundle $\mathcal{E}$
over $X_{K}$ and for the given endomorphism
$B(y)\in\mathrm{End}(\mathcal{E}(y))$ for every $y\in\widetilde{S}$, that is,

$0\to\mathrm{End}(\mathcal{E})\otimes\mathcal{O}_{X_{K}}(-\widetilde{Z})\xrightarrow{\iota}\mathcal{A}(\mathcal{E})\xrightarrow{\widehat{\sigma_{1}}}TX_{K}(-\log\widetilde{Z})\to
0.$ (4.13)

Let
$\phi^{B}_{\mathcal{E}}\in\mathrm{H}^{1}(X_{K},\mathrm{End}(\mathcal{E})\otimes\Omega^{1}_{X_{K}})$
be the extension class of (4.13).

Changing the base from $K$ to $\overline{K}$, and using the base change
formula, we get a short exact sequence

$0\to\mathrm{End}(\mathcal{E})\otimes\mathcal{O}_{X_{K}}(-\widetilde{Z})\otimes_{K}\overline{K}\xrightarrow{\iota}\mathcal{A}(\mathcal{E})\otimes_{K}\overline{K}\xrightarrow{\widehat{\sigma_{1}}}TX_{K}(-\log\widetilde{Z})\otimes_{K}\overline{K}\to
0$ (4.14)

of vector bundles over $X_{\overline{K}}$. Let
$\widehat{\phi^{B}_{\mathcal{E}}}$ be the extension class of (4.14), and note
that

$\mathrm{H}^{1}(X_{\overline{K}},\mathrm{End}(\mathcal{E})\otimes\Omega^{1}_{X_{K}}\otimes\overline{K})=\mathrm{H}^{1}(X_{K},\mathrm{End}(\mathcal{E})\otimes\Omega^{1}_{X_{K}})\otimes\overline{K},$

and

$\widehat{\phi^{B}_{\mathcal{E}}}=\phi^{B}_{\mathcal{E}}\otimes 1.$ (4.15)

Next, the short exact sequence (4.14) coincides with the short exact sequence
(2.11) associated with the vector bundle $\mathcal{E}_{\overline{K}}$ over
$X_{\overline{K}}$ with endomorphism $\widehat{B(\hat{y})}$ on
$\mathcal{E}_{\overline{K}}(\hat{y})$, for every $\hat{y}\in\widehat{S}$, that
is,

$0\to\mathrm{End}(\mathcal{E}_{\overline{K}})\otimes\mathcal{O}_{X_{\overline{K}}}(-\widehat{Z})\xrightarrow{\iota}\mathcal{A}(\mathcal{E}_{\overline{K}})\xrightarrow{\widehat{\sigma_{1}}}TX_{\overline{K}}(-\log\widehat{Z})\to
0.$

Therefore, the extension class
$\phi^{\widehat{B(\hat{y})}}_{\mathcal{E}_{\overline{K}}}$ coincides with
$\widehat{\phi^{B}_{\mathcal{E}}}$.

Since $\mathcal{E}_{\overline{K}}$ admits a logarithmic connection singular
over $\widehat{S}$ with residue $\widehat{B(\hat{y})}$ for every
$\hat{y}\in\widehat{S}$, we have

$0=\phi^{\widehat{B(\hat{y})}}_{\mathcal{E}_{\overline{K}}}=\widehat{\phi^{B}_{\mathcal{E}}}.$

Thus, from (4.15),

$\phi^{B}_{\mathcal{E}}\otimes 1=0,$

and hence $\phi^{B}_{\mathcal{E}}=0$. In view of Lemma 4.2, and Remark 4.3,
proof of $\eqref{a}$ is complete.

To prove the second part (2), let $E$ be an indecomposable vector bundle over
$X$ which satisfies (4.2). Using the same technique as above, the vector
bundle $\mathcal{E}$ over $X_{K}$ in (4.8), satisfies (4.2). Now, repeating
the above argument, we conclude that $\mathcal{E}$ admits a logarithmic
connection singular over $\widetilde{S}$ with residue $B(y)$, for every
$y\in\widetilde{S}$. Again, from Lemma 4.2, and Remark 4.3, proof of the
second part (2) of the theorem is complete. ∎

##  5\. Conclusions

The above theorems have been proved over any perfect field and they will be
very useful when we study the algebro-geometric invariants for the moduli
space of logarithmic connections over a perfect field, with fixed rigid
residues.

In case residues are not rigid, finding a suitable criterion for the existence
of a logarithmic connection is still an open problem. Our guess is the same
criterion should work but we do not know how to prove it.

## References

  * [1] J. K. Arason, R. Elman, and B. Jacob, On indecomposable vector bundles, Comm. Alg. 20 (1992), 1323–1351.
  * [2] M. F. Atiyah, On the Krull–Schmidt theorem with application to sheaves. Bull. Soc. Math. Fr. 84, 307–317 (1956).
  * [3] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.
  * [4] I. Biswas, A. Dan, A. Paul, Criterion for logarithmic connections with prescribed residues. manuscripta math.155:77–88. https://doi.org/10.1007/ s00229-017-0935-6, (2018).
  * [5] I. Biswas, N. Raghavendra, The Atiyah-Weil criterion for holomorphic connections. Indian J. Pure Appl. Math. (2008)39: 3–47.
  * [6] I. Biswas and S. Subramanian, Vector bundles on curves admitting a connection. Q. J. Math. 57 (2006), no. 2, 143–150.
  * [7] I. Biswas and A. Singh, On the relative connections Comm. Algebra 48 (2020), no. 4, 1452–1475.
  * [8] I. Biswas, A criterion for a vector bundle over a curve to admit a connection. K-Theory 36 (2005), no. 1-2, 83–90 (2006).
  * [9] P. Deligne, Equations différentielles á points singuliers réguliers. Lecture Notes in Mathematics, vol. 163. Springer, Berlin(1970).
  * [10] A. Tillmann, Unzerlegbare vectorbündel über algebraischen kurven, Thesis at the Fernuniveruniversität in Hagen, 1983.
  * [11] A. Weil, Généralization de fonctions abélienes, J. Math. Pures Appl. vol.17(1938), 47-87.


# Neutral Stellar Winds Toward the High-Mass Star-Forming Region G176.51+00.20

Yingjie Li  Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing
210008, China  Ye Xu  Purple Mountain Observatory, Chinese Academy of
Sciences, Nanjing 210008, China  Jin-Long Xu  National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China  CAS Key
Laboratory of FAST, National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, China  Dejian Liu  Purple Mountain Observatory,
Chinese Academy of Sciences, Nanjing 210008, China  University of Science and
Technology of China, Hefei, Anhui 230026, China  Jingjing Li  Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210008, China  Zehao Lin
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008,
China  University of Science and Technology of China, Hefei, Anhui 230026,
China  Peng Jiang  National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, China  CAS Key Laboratory of FAST, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
Shuaibo Bian  Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210008, China  University of Science and Technology of China, Hefei,
Anhui 230026, China  Chaojie Hao  Purple Mountain Observatory, Chinese Academy
of Sciences, Nanjing 210008, China  University of Science and Technology of
China, Hefei, Anhui 230026, China  Xiuhui Chen  College of Mathematics and
Physics, Hunan University of Arts and Science, Changde, Hunan 415300, China

###### Abstract

We observed the high-mass star-forming region G176.51+00.20 using the Five-
hundred-meter Aperture Spherical radio Telescope (FAST) with the 19-beam
tracking observational mode. This is a pilot work of searching for neutral
stellar winds traced by atomic hydrogen (i.e., H i winds) using the high
sensitivity H i line toward high-mass star-forming regions where bipolar
molecular outflows have been detected with high sensitivity by Liu et al. H i
wind was detected in this work only in Beam 1. We find here that, similar to
low-mass star formation, no matter how large the inclination is, the H i wind
is likely sufficiently strong to drive a molecular outflow. We also find that
the abundance of H i in the H i wind is consistent with that of the H i
narrow-line self-absorption (HINSA) in the same beam (i.e., Beam 1). This
implies that there is probably an internal relationship between H i winds and
HINSA. This result also reinforces the assertion that H i winds and detected
molecular outflows are associated with each other.

ISM: jets and outflows - ISM: Molecules - stars: formation – ISM: abundances –
ISM: kinematics and dynamics

††software: Astropy (Astropy Collaboration et al., 2013, 2018), Matplotlib
(Hunter, 2007), Numpy (Harris et al., 2020), Pandas (Wes McKinney, 2010;
Reback et al., 2021), Scipy(Virtanen et al., 2020), Emcee (Foreman-Mackey et
al., 2013)

##  1 Introduction

One of the important goals of contemporary astrophysics is to understand star
formation (e.g., Tan et al., 2014; Bally, 2016). Relative to low-mass star
formation, the formation mechanism(s) of high-mass stars remain poorly
understood (Shu et al., 1987; McKee & Ostriker, 2007; Tan et al., 2014), as
too are the driving scenario(s) of massive molecular outflows, an essential
phase of early high-mass star formation (Arce et al., 2007).

Early studies, especially in the 1980s and 1990s, attempted to ascertain the
driving source and mechanism of molecular outflows, mostly for low-mass stars.
Snell et al. (1985) and Strom et al. (1986) found that the mass-loss rate of
ionized gas is too low to drive CO lobes, and suggested that the stellar winds
which drive the CO lobes are largely comprised of neutral gas. Lizano et al.
(1988) carefully compared the neutral stellar wind traced by atomic hydrogen
(H i wind hereafter) detected in HH 7–11 and the corresponding CO/HCO+
outflows, and found that the H i wind was strong enough to drive the molecular
outflows. A similar conclusion was drawn by carefully comparing the H i winds
in both HH 7–11 and L1551 with the corresponding molecular outflows (see
Giovanardi et al., 1992). In such studies, which were performed with the 305 m
radio telescope of the Arecibo Observatory, the root mean square (rms) noise
was typically $\sim$ 3.5–8.5 mK @ 3.9 km s-1 with beam size of
$3^{\prime}.2$–$3^{\prime}.5$. A model based on the detected H i wind in HH
7–11 was developed by Lizano et al. (1988) to explain the entrainment of
ambient molecular gas (i.e., driving the molecular outflow) by the neutral
wind. These studies suggested that H i winds are a promising driving source of
molecular outflows. However, is such an assertion also true for massive stars?

To answer this question, samples of the molecular outflows have been
assembled. For example, 12CO, 13CO, HCO+, and CS outflowing gases have been
recently detected with high sensitivity (with main beam rms noise of dozens of
mK) toward nine high-mass star-forming regions using the 13.7 m millimeter
telescope of the Purple Mountain Observatory in Delingha (see more details in
Liu et al., 2021). Accordingly, an rms noise of $\sim$ 3 mK @ 1.0 km s-1 can
be achieved with about six hour integrations and a beam size of $\sim
2^{\prime}.9$ by using the powerful 21 cm H i line detected by the most
sensitive ground-based, single-dish Five-hundred-meter Aperture Spherical
radio Telescope 111https://fast.bao.ac.cn/(FAST; Nan, 2006; Nan et al., 2011;
Jiang et al., 2020). The use of FAST will enabled us to conduct broad
investigations of stellar winds toward star-forming regions where molecular
outflows have been detected. As such, we have carried out a pilot survey of
the H i wind toward detected molecular outflows with highly sensitive
observations. The selected object in this work is one of the nine high-mass
star-forming regions in Liu et al. (2021), G176.51+00.20.

G176.51+00.20, also known as AFGL 5157 and IRAS 05345+3157, is located 1.8 kpc
from Earth (Moffat et al., 1979; Snell et al., 1988). It is a complex region
containing multi-generational star formation (e.g., Chen et al., 2003;
Dewangan, 2019). The main research object in this work is a high-mass star-
forming region at a relatively young phase, where H2O masers and an H ii
region have been detected (Torrelles et al., 1992a). This region is marked as
an “H ii region” in figure 1(a) in Dewangan (2019). There is a dense NH3 core
in the center of this region, which has become synonymous with this region
(e.g., Torrelles et al., 1992a; Chen et al., 2003; Jiang et al., 2013). By
constructing a 3.6 cm map with the Very Large Array, the excitation source of
this region was identified as a zero-age main-sequence B3 star (Torrelles et
al., 1992b). The bolometric luminosity of this excitation source was
determined as 1.7 $\times$ 103 $L_{\odot}$ via an 8–1200 $\mu$m SED fit (see
Molinari et al., 2008). The age of this star-forming region was suggested to
be $\sim$ 2 $\times$ 105 yr by investigating the near-infrared H2 line
emission using the 1.88 m telescope of Okayama Astronomical Observatory, Japan
(see Chen et al., 2003).

The famous bipolar outflow (traced by 12CO) detected by Snell et al. (1988) is
centered on the dense NH3 core (Molinari et al., 2002; Zhang et al., 2005;
Dewangan, 2019), suggesting that the excitation source of this bipolar outflow
is probably the zero-age main-sequence B3 star (see Torrelles et al., 1992b).
Liu et al. (2021) confirmed this bipolar outflow with high-sensitivity 12CO,
13CO, C18O, HCO+, and CS line emission observations, and broadened the blue
lobe of this outflow (traced by 12CO) by a factor of $\sim$ 2\. This makes the
dense NH3 core more suitable to investigate whether its H i wind is strong
enough to drive the molecular outflows.

The remainder of this paper is organized as follows. In Section 2, we describe
the data used in this work. Section 3 analyses the H i wind associated with
the molecular outflows. In Section 4, discussion of whether the H i wind is
strong enough to drive the molecular outflows is presented. Finally, Section 5
gives a summary and the main conclusions of this work.

##  2 H i Observations with FAST

FAST is located in Guizhou Province of southwest China (Nan et al., 2011). It
is equipped with a 19-beam receiver (the frequency range is 1.0–1.5 GHz with a
bandwidth of 500 MHz) and has dual linear polarizations (i.e., XX and YY; see
Li et al., 2018; Jiang et al., 2019, 2020). The spectral resolution is $\sim$
477 Hz, corresponding to a velocity resolution of $\sim$ 0.1 km s-1 at 1.4
GHz. The half-power beam width (HPBW) is $\sim$ 2.9′ at 1.4 GHz, and the
pointing error is $\sim$ 0.2′ (see Jiang et al., 2019, 2020).

Observations toward G176.51+00.20 were conducted on August 19th and 20th,
2021. The observational mode was 19-beam tracking, and the total integration
time was 335 minutes with a sampling rate of one second. A new algorithm is
used to obtain a flat baseline. In this algorithm, there are three main steps:
(1) a polynomial fitting over the velocity range of [$-7000$, 7000] km s-1;
(2) removing the standing waves by using fast Fourier transforms; (3)
calibrating the unsmooth parts in the baseline by extreme envelope curves (for
more details of the pipeline used to obtain the flat baseline, see Liu et al.,
2022). For Beams 2, 3, 5, 6, and 8–19, the spectra of two linear polarizations
(i.e., XX and YY) were dealt with separately and then averaged. However,
because of the bad baseline, only the YY polarization spectrum from Beam 1 and
the XX spectra from Beams 4 and 7 were retained. The mean rms noise of the
nineteen spectra is $\sim$ 7 mK @ 0.1 km s-1 (see the rms noise of each
spectrum in Table 1). Figure 1 presents the positions of the central seven
beams superposed on CO molecular maps, and Figure 2 shows the spectrum of the
central seven beams and the H i narrow-line self-absorption lines (HINSA,
$T_{\mathrm{ab}}$) in each beam (see Li et al., 2022).

Table 1: List of the Positions and Rms Noise of the Nineteen Beams Index | R.A. | Decl. | Rms | Index | R.A. | Decl. | Rms
---|---|---|---|---|---|---|---
| (J2000) | (J2000) | (mK) |  | (J2000) | (J2000) | (mK)
1 | 05h37m53s | 31∘59′58′′ | 8.0 | 2 | 05h38m20s | 32∘00′00′′ | 6.1
3 | 05h38m07s | 31∘55′01′′ | 6.8 | 4 | 05h37m39s | 31∘54′59′′ | 11.2
5 | 05h37m26s | 31∘59′57′′ | 6.1 | 6 | 05h37m39s | 32∘04′56′′ | 6.2
7 | 05h38m06s | 32∘04′57′′ | 7.7 | 8 | 05h38m47s | 32∘00′01′′ | 8.8
9 | 05h38m34s | 31∘55′02′′ | 6.5 | 10 | 05h38m20s | 31∘50′03′′ | 7.7
11 | 05h37m53s | 31∘50′02′′ | 6.3 | 12 | 05h37m26s | 31∘50′00′′ | 5.9
13 | 05h37m12s | 31∘54′58′′ | 6.6 | 14 | 05h36m59s | 31∘59′55′′ | 6.9
15 | 05h37m12s | 32∘04′54′′ | 6.3 | 16 | 05h37m26s | 32∘09′53′′ | 6.9
17 | 05h37m53s | 32∘09′55′′ | 6.1 | 18 | 05h38m20s | 32∘09′56′′ | 6.9
19 | 05h38m33s | 32∘04′58′′ | 6.7 |  |  |  |
Figure 1: The central seven beams superposed on the integrated intensity map
of C18O in the range of [$-$30, $-$10] km s-1 with the molecular data from Liu
et al. (2021). The blue and lime contours show the integrated intensity map of
12CO and 13CO in the same velocity range as the C18O map.

(a) Beam 1 (b) Beam 2 (c) Beam 3 (d) Beam 4 (e) Beam 5 (f) Beam 6 (g) Beam 7

Figure 2: Top: spectra of the central seven beams (green lines) and the HINSA
($T_{\mathrm{ab}}$, red lines) obtained in Li et al. (2022). Bottom: the flat
baseline. $v_{\mathrm{{c,w}}}$ represents the central velocity of the HINSA
feature (i.e., at $-$18.42 km s-1) associated with the H i wind (see below),
and $v_{\mathrm{{c,p}}}$ represents the central velocity of another HINSA
feature (i.e., at $-$3.47 km s-1) in Beam 1.

##  3 Data Analysis and Results

###  3.1 Identification of the H i Wind

We defined the excess brightness temperature (EBT, hereafter) as the spectrum
resulting from the subtraction of the average spectrum of the off-positions
from the spectrum of the on-position (i.e., the target beam). Similar to the
work of Lizano et al. (1988), the six beams located immediately around the
target beam (i.e., which acted as the on-position) were used as the off-
positions. The baseline of the EBT was flat, i.e., the rms noise was $\sim$
1.8–2.3 mK @ 2.0 km s-1 (i.e., smooth over twenty channels), except for Beams
3 and 4, whose rms noises were, respectively, 3.3 and 6.1 mK @ 2.0 km s-1. We
also defined the standard deviation (STD) of EBT (SEBT) to evaluate the
variation among the spectrum from different off-positions. For instance, for
Beam 1, we first separately calculated the EBT of Beam 1 relative to its six
surrounding beams, and then computed the STD of these six EBTs, which is the
SEBT of Beam 1.

(a) Beam 1 (b) Beam 2 (c) Beam 3 (d) Beam 4 (e) Beam 5 (f) Beam 6 (g) Beam 7

Figure 3: EBT (blue lines) and SEBT (green lines) of Beams 1–7 (smoothed over
twenty channels, blue lines). For the spectrum in which the high-velocity line
wing was present, the “triangular fit” is plotted as a red oblique line, where
the line center (red vertical line) is taken as the velocity of the HINSA (see
Figure 2 and Li et al., 2022).

####  3.1.1 H i Wind in Beam 1

From Figure 3, it can be seen that Beam 1 presents an evident high-velocity
red wing whose maximum velocity exceeds $\sim$ 120 km s-1. The SEBT is also
small in the high-velocity red wing, indicating this wing may well be true,
rather than the changes or fluctuations of the emission from one beam area to
another. This high-velocity red wing was crudely fitted by a “triangular fit”
(see e.g., Lizano et al., 1988). Assuming this high-velocity line wing is
bipolar, the symmetric line of the high-velocity red wing with respect to the
line center, $v_{\mathrm{{c,w}}}$, is also presented as a red oblique line
(see the red isosceles triangle of the high-velocity bipolar wing in Figure
3(a)). Therein, $v_{\mathrm{{c,w}}}$ is taken as the velocity of the
corresponding HINSA (see Figure 2 and Li et al., 2022), which is indicated as
a red vertical line in Figure 3(a). This symmetric line is referred to as the
presumed high-velocity blue wing, although there is no blue wing present in
Figure 3(a). However, we find that the velocity of the bulge in Beam 7 (see
Figure 3(g)) overlaps with that of the presumed high-velocity blue wing in
Beam 1. Therefore, a further analysis of the high-velocity line wing in Beam 1
was conducted where we took different beams as the off-positions (see Figure
4).

When using Beams 2–7 as off-positions (see Figures 3), we also took beams
orientated along West–East (i.e., Beams 2 and 5), Northeast–Southwest (i.e.,
Beams 3 and 6), Northwest–Southeast (i.e., Beams 4 and 7), and also using
Beams 2–6 as off-positions (see Figures 4(a)–(d), respectively). These five
cases are referred to as Normal, W–E, NE–SW, NW–SE, and Lack Beam 7. The high-
velocity red wing remains basically unchanged no matter which beams are used
as off-positions (see Figure 4 and quantitative comparison in Section 3.2),
and the high-velocity blue wing is also present as long as the off-positions
do not contain Beam 7. Therefore, Beam 1 probably presents a high-velocity
bipolar wing, but the high-velocity blue wing is contaminated by other
components. In addition, there is a small pit between the “triangular fit” of
the EBT for the cases of W–E, NE–SW, and NW–SE (see the left green oblique
line in Figures 4(a)–(c)) and the presumed high velocity blue wing (see red
oblique line). This small pit likely causes the green oblique line to deviate
from the red oblique line, which implies that this high-velocity bipolar wing
is probably symmetrical; therefore, we identify it as a bipolar H i wind.
Because the blue lobe is contaminated by other components, and we presume that
this H i wind is a symmetrical bipolar H i wind, we applied the fitting result
of the red lobe to the blue lobe.

(a) Lack Beam 7 (b) W-E (c) NE-SW (d) NW-SE

Figure 4: EBT obtained from using different beams as off-positions. The red
lines (“Fit 1”) are the same as that in Figure 3(a). The green lines (“Fit 2”)
are the results of the “triangular fit” of the EBTs (fit separately for the
blue and red lobes). Lack Beam 7: using Beams 2–6 as off-positions; W–E:
taking Beams 2 and 5 as off-positions; NE–SW: using Beams 3 and 6 as off-
positions; NW–SE: using Beams 4 and 7 as off-positions.

####  3.1.2 The EBT in Other Beams

The bulge in the blue line wing of Beam 2 is, in fact, present in only one
polarization spectrum (i.e., YY). Therefore, this blue line wing is highly
likely a fake high-velocity line wing. Figure 3 also presents other high-
velocity features, e.g., wings with negative intensities in right side of EBTs
in Beams 2 and 7, and wings with positive intensities in right side of EBTs in
Beam 6, etc. These high-velocity features are more likely to stem from the
changes or fluctuations of the emission from one beam area to another, because
these features are located where SEBTs are large.

###  3.2 Physical Parameters of the H i Wind

We calculated the mass, $M_{\mathrm{HI,wind}}$, momentum,
$P_{\mathrm{HI,wind}}$, maximum velocity, $V_{\mathrm{max,wind}}$, and average
velocity, $V_{\mathrm{mean,wind}}$, of the red lobe of the H i wind
(“triangular fit”) corresponding to different off-positions, i.e., the case of
Normal (see red line in Figure 3(a)), Lack Beam 7, W–E, NE–SW, and NW–SE (see
green lines in Figures 4(a)–(d)). The specific calculations without
inclination correction are presented in Section A. The results show that all
the values of $M_{\mathrm{HI,wind}}$, $P_{\mathrm{HI,wind}}$,
$V_{\mathrm{max,wind}}$, and $V_{\mathrm{mean,wind}}$ are consistent with each
other within the errors (all the errors presented in this work are 1$\sigma$
uncertainties) for all the five cases (see the physical values in Table 2).
This reinforces the conclusion that the red lobe of the H i wind is real, and
the case of Normal can reflect the physical properties of the red lobe of the
H i wind. Therefore, the calculations below are based on the case of Normal.

Table 2: Comparison of the Physical Parameters Determined with Different Off-positions Classification |  $M_{\mathrm{HI,wind}}$ |  $P_{\mathrm{HI,wind}}$ |  $V_{\mathrm{max,wind}}$ |  $V_{\mathrm{mean,wind}}$
---|---|---|---|---
|  ($M_{\odot}$) |  ($M_{\odot}$ km s-1) |  (km s-1) |  (km s-1)
Normal |  0.11 $\pm$ 0.02 |  4.03 $\pm$ 1.76 |  120.6 $\pm$ 16.3 |  40.2 $\pm$ 5.4
Lack Beam 7 |  0.10 $\pm$ 0.02 |  3.48 $\pm$ 2.22 |  124.7 $\pm$ 20.7 |  41.6 $\pm$ 6.9
W-E |  0.09 $\pm$ 0.02 |  3.16 $\pm$ 3.56 |  129.7 $\pm$ 25.7 |  42.2 $\pm$ 8.6
NE-SW |  0.10 $\pm$ 0.02 |  3.33 $\pm$ 1.65 |  112.6 $\pm$ 16.2 |  37.5 $\pm$ 5.4
NW-SE |  0.14 $\pm$ 0.02 |  5.20 $\pm$ 2.26 |  122.2 $\pm$ 16.0 |  40.7 $\pm$ 5.3

The “triangular fit” of the red lobe of the H i wind produced
$T_{\mathrm{wind}}(v)=(14.94\pm 1.32)+(-0.15\pm 0.02)\cdot v$. The maximum
velocity, $V_{\mathrm{max,wind}}$, and the average velocity,
$V_{\mathrm{mean,wind}}$, without inclination correction are, respectively,
$120.6\pm 16.3$ and $40.2\pm 5.4$ km s-1, where
$V_{\mathrm{mean,wind}}=V_{\mathrm{max,wind}}/3$ for the triangular profile.
$V_{\mathrm{max,wind}}$ (and also $V_{\mathrm{mean,wind}}$) are 0.7 and 0.8
times that in HH 7–11 and in L1551 (both are located in low-mass star-forming
regions; see Lizano et al., 1988; Giovanardi et al., 1992), respectively. The
column density of the red lobe of the H i wind, $N_{\mathrm{HI,wind}}$, is
(1.94 $\pm$ 0.30) $\times$ 1018 cm-2 (see Table 3). Assuming that the H i wind
occupies the same spatial location as the molecular outflows, and both of them
are angularly unresolved (e.g., Lizano et al., 1988), $N_{\mathrm{HI,wind}}$
is two orders of magnitude smaller than the column density of the outflow
traced by 12CO, and three orders of magnitude smaller than that traced by
13CO, HCO+, and CS (see the column density of the molecular outflows in Liu et
al., 2021).

13CO is proved to be a great tracer of the molecular column density, and the
fractional HINSA abundance (where the molecular column density is traced by
13CO) is $\sim$ 1.1 $\times$ 10-3 (Li et al., 2022). We also calculated the
abundance of H i in the H i wind, $X_{\mathrm{HI,wind}}$, as:

$X_{\mathrm{HI,wind}}=\frac{N_{\mathrm{HI,wind}}}{N_{\mathrm{HI,wind}}+2N_{\mathrm{H_{2},outflow}}},$
(1)

where $N_{\mathrm{H_{2},outflow}}$ is the column density of the molecular
outflow (that is traced by 12CO, 13CO, HCO+, or CS; see Liu et al., 2021). The
results of $X_{\mathrm{HI,wind}}$ are listed in Table 3.
$X_{\mathrm{HI,wind}}$, determined by the molecular outflow traced by 13CO, is
$\sim$ 1.0 $\times$ 10-3 (only for the red lobe), which is consistent with the
fractional HINSA abundance.

This consistency reveals an internal correlation between the H i wind and
HINSA. It also supports the conclusion of the mixture of the H i wind and the
molecular outflow because HINSA is probably mixed with the gas in the cold,
well-shielded regions of the molecular clouds (see Li & Goldsmith, 2003; Li et
al., 2022). However, observations with higher resolution are required to
confirm the correlation between the H i wind and HINSA, and a larger sample is
essential to rule out that the observed consistency here is only a
coincidence.

Table 3: Physical Properties of the H i Wind without Inclination Correction Quantity | Value (1$\sigma$ uncertainty) | Noting
---|---|---
$N_{\mathrm{HI,wind}}$ (cm-2) | (1.94 $\pm$ 0.30) $\times$ 1018 | See Equation (A2)
$X_{\mathrm{HI,wind}}$ |  $\sim$ (4.6, 1.0, 0.4, 0.8) $\times$ 10-3 | For 12CO, 13CO, HCO+ and CS, See Equation (1)
$M_{\mathrm{HI,wind}}$ ($M_{\odot}$) | 0.22 $\pm$ 0.03 | Combined two lobes, see Equation (A4)
$P_{\mathrm{HI,wind}}$ ($M_{\odot}$ km s-1) | 8.05 $\pm$ 3.52 | Combined two lobes, see Equation (A5)
$V_{\mathrm{max,wind}}$ (km s-1) | 120.6 $\pm$ 16.3 | See Equation (A3)
$V_{\mathrm{mean,wind}}$ (km s-1) | 40.2 $\pm$ 5.4 | $V_{\mathrm{max,wind}}/3$
$E_{\mathrm{HI,wind}}$ (erg) | (3.4 $\pm$ 1.5) $\times$ 1046 | See Equation (A7)
$\Delta M_{\ast,\mathrm{m}}$ ($M_{\odot}$) |  $\sim$ 0.43, $\sim$ 0.61, $\sim$ 3.68, $\sim$ 1.53 | For 12CO, 13CO, HCO+ and CS
$\Delta M_{\ast,\mathrm{ame}}$ ($M_{\odot}$) | 1.3, $-$5.8, 10.5, 6.5, 9.0, 2.7 | Respectively relative to Beams 2–7
$l_{\mathrm{HI,wind}}$ (pc) | 0.8 | See $l_{\mathrm{outflow}}$ in Liu et al. (2021)
$t_{\mathrm{a}}$ (yr) | (1.9 $\pm$ 0.2) $\times$ 104 | $l_{\mathrm{HI,wind}}/V_{\mathrm{mean,wind}}$
$\dot{M}_{\mathrm{a}}$ ($M_{\odot}$ yr-1) | (12.2 $\pm$ 3.6) $\times$ 10-6 | $M_{\mathrm{HI,wind}}/t_{\mathrm{a}}$
$t$ (yr) | (5.3 $\pm$ 3.5) $\times$ 105 | $\Delta M_{\ast,\mathrm{a}}/\dot{M}_{\mathrm{a}}$
$L_{\mathrm{HI,wind}}$ ($L_{\odot}$) | 16.0 $\pm$ 10.1 | $\displaystyle\frac{1}{2}\dot{M}_{\mathrm{a}}V^{2}_{\mathrm{max,wind}}$

We also calculated the mass, $M_{\mathrm{HI,wind}}$, momentum,
$P_{\mathrm{HI,wind}}$, and kinetic energy, $E_{\mathrm{HI,wind}}$, of the H i
wind (combine the values of the two lobes) without inclination correction (see
the calculations in Section A and the values in Table 3). The mass and
momentum over the overall flow history are larger than the values estimated
above (see Lizano et al., 1988). The values of $M_{\mathrm{HI,wind}}$,
$P_{\mathrm{HI,wind}}$, and $E_{\mathrm{HI,wind}}$ are larger than that in HH
7–11 (see Lizano et al., 1988; Giovanardi et al., 1992) by a factor of $\sim$
7, $\sim$ 5, and $\sim 4$, respectively, and larger than that in L1551 (see
Giovanardi et al., 1992) by a factor of $\sim$ 26, $\sim$ 20, and $\sim$ 6,
respectively.

##  4 Discussion

###  4.1 Dynamical Comparison Between the H i Wind and the Molecular Outflow

To determine whether the H i wind is strong enough to drive the molecular
outflow, analyses of two dimensions were conducted and are presented here;
i.e., the mass loss of the excitation star and the lifetime of the H i wind.

####  4.1.1 Mass Loss of the Excitation Star

If we assume that the molecular outflow is entrained by the H i wind
originally ejected from the star, then the mass loss, $\Delta
M_{\ast,\mathrm{m}}$, from the excitation star over the entire history of mass
loss is $M_{\ast,\mathrm{m}}=P_{\mathrm{m,outflow}}/V_{\mathrm{max,wind}}$
(see Lizano et al., 1988), where $P_{\mathrm{m,outflow}}$ is the momentum of
the molecular outflow (see Liu et al., 2021). Both the blue and red lobes of
the molecular outflow are included in the calculation. The total $\Delta
M_{\ast,\mathrm{m}}$, which combines contributions from the 12CO, 13CO, HCO+,
and CS outflows, is 6.26 $\pm$ 0.80 $M_{\odot}$ (see the contribution from
each molecular outflow in Table 3). Because Liu et al. (2021) did not give the
error of $P_{\mathrm{m,outflow}}$, the error of $\Delta M_{\ast,\mathrm{m}}$
is only from $V_{\mathrm{max,wind}}$ and no error is presented in Table 3. It
is worth noting that the total $\Delta M_{\ast,\mathrm{m}}$ is probably
overestimated because the values of $P_{\mathrm{m,outflow}}$ are corrected for
the abundance of the corresponding tracer (see Liu et al., 2021).

Similar to the work in Lizano et al. (1988), we calculated the atomic mass
excesses, $\Delta M_{\ast,\mathrm{ame}}$, of the line core of Beam 1 relative
to the line core of the six beams tightly around Beam 1, assuming that the
low-velocity material is associated with the source and fills the beams of
FAST. We tested different velocity spans, $v_{\mathrm{s}}$, as line core (see
Figure 5), where $v_{\mathrm{s}}=v_{\mathrm{c,w}}-v$ and only
$v<v_{\mathrm{c,w}}$ is considered because there is a strong component at
$\sim$ $-$3.5 km s-1. The increase of $\Delta M_{\ast,\mathrm{ame}}$ at
$v_{\mathrm{s}}\sim$ 30–45 km s-1 may be the results of the influence of the
component at $\sim-57$ km s-1 (see Figure 2). $\Delta M_{\ast,\mathrm{m}}$ is
flat at $v_{\mathrm{s}}\sim$ 15–23 km s-1, indicating that the influence of
the component at $\sim$ $-$3.5 km s-1 may be small. We set the velocity range
of $v_{\mathrm{{c,w}}}-v\leq 20$ km s-1 as the line core, corresponding to
$v_{\mathrm{s}}\sim$ 20 km s-1 (i.e., at the middle of the flat region). The
atomic mass excesses of Beams 1 to its surrounding beams are systematically
positive (except for those between Beams 1 and 3). In addition, such
systematically positive of the atomic mass excesses is invalid for the case of
Beams 2 and 4–7 relative to their surrounding beams. 222For Beams 2 and 4–7,
the mass excess relative to their six surrounding beams is not systematically
positive. For instance, the atomic mass excesses of Beam 2 relative to its six
surrounding beams are, respectively, $-0.4$, $-4.6$, $-3.2$, $-2.3$, 3.2, and
0.6 $M_{\odot}$ (starting from the West beam and counting counterclockwise);
and for Beam 5, the corresponding values are $-2.7$, 0.7, 1.8, $-3.6$, 0.8,
and $-4.9$ $M_{\odot}$. For Beam 3, the corresponding values are 10.9, 0.4,
$-0.9$, 2.3, 5.2, and 4.4 $M_{\odot}$.These facts indicate that atomic gas
flows out from Beam 1, and these atomic mass excesses may be an indicator of
accumulated atomic mass loss from the excitation star, $\Delta
M_{\ast,\mathrm{a}}$ (e.g., Lizano et al., 1988).

Figure 5: Atomic mass excesses, $\Delta M_{\ast,\mathrm{ame}}$, as a function
of velocity spans, $v_{\mathrm{s}}$. “Beam 1-$i$” represent the atomic mass
excess of Beam 1 relative to Beam $i$, “Mean” and “Std” present the mean and
STD values of the five atomic mass excesses (i.e., except for that between
Beams 1 and 3), and the red dashed line that labeled as “3.0 $M_{\odot}$”
indicates $\Delta M_{\ast,\mathrm{ame}}=3.0$ $M_{\odot}$.

The atomic mass excesses of Beam 1 relative to its surrounding beams with a
velocity range of $|v_{\mathrm{{c,w}}}-v|\leq 20$ km s-1 are listed in Table
3, which are set to be two times of the corresponding atomic mass excesses
with a velocity range of $v_{\mathrm{{c,w}}}-v\leq 20$ km s-1. The average
value of $\Delta M_{\ast,\mathrm{ame}}$ over the five atomic mass excesses
(i.e., except for that between Beams 1 and 3, where the atomic mass excesses
are significantly different from the other atomic mass excesses) is 6.0 $\pm$
3.5 $M_{\odot}$ (i.e., $\Delta M_{\ast,\mathrm{a}}\sim$ 6.0 $\pm$ 3.5
$M_{\odot}$), where 3.5 $M_{\odot}$ is the STD of the five atomic mass
excesses. The error of atomic mass excess inherited from the original spectrum
is far less than 3.5 $M_{\odot}$, and therefore no error of atomic mass excess
is presented in Table 3. $\Delta M_{\ast,\mathrm{a}}\sim\Delta
M_{\ast,\mathrm{m}}$, and the outflowing atomic gas would undergo deceleration
when it entrained molecular outflow gas, indicating that the velocity
associated with $\Delta M_{\ast,\mathrm{a}}$ may be not less than that
associated with $\Delta M_{\ast,\mathrm{m}}$ and H i wind is likely strong
enough to drive the molecular outflow.

####  4.1.2 Lifetime of the H i Wind

The lifetime of the H i wind flow, i.e., the total duration of the flow, $t$,
without inclination correction was estimated according to Lizano et al.
(1988). The distance from the H i wind to the central excitation star,
$l_{\mathrm{HI,wind}}$, was set as the outflow length, $l_{\mathrm{outflow}}$,
of the molecular outflow (which, here, corresponds to the red lobe of 12CO;
see Liu et al., 2021), i.e., $l_{\mathrm{HI,wind}}$ $\sim$ 0.8 pc. The
crossing time of the H i wind, $t_{\mathrm{a}}$, is then
$t_{\mathrm{a}}=l_{\mathrm{HI,wind}}/V_{\mathrm{mean,wind}}$ = (1.9 $\pm$ 0.2)
$\times 10^{4}$ yr, where the uncertainty comes only from
$V_{\mathrm{mean,wind}}$ because no error of $l_{\mathrm{outflow}}$ was given
by Liu et al. (2021). Therefore, the stellar mass-loss rate,
$\dot{M}_{\mathrm{a}}$, is
$\dot{M}_{\mathrm{a}}=M_{\mathrm{HI,wind}}/t_{\mathrm{a}}$ = (12.2 $\pm$ 3.6)
$\times$ 10-6 $M_{\odot}$ yr-1. This value (without inclination correction) is
larger than that in HH 7–11 by a factor of $\sim$ 2 and larger than that in
L1551 by a factor of $\sim$ 10 (see Lizano et al., 1988; Giovanardi et al.,
1992). The lifetime of the H i wind is $t=\Delta
M_{\ast,\mathrm{a}}/\dot{M}_{\mathrm{a}}$ = (5.3 $\pm$ 3.5) $\times$ 105 yr,
where the accumulated atomic mass, $\Delta M_{\ast,\mathrm{a}}$, is set to 6.0
$\pm$ 3.5 $M_{\odot}$ (see above). All the physical properties above are
listed in Table 3. The dynamical timescale of the red lobe of the molecular
outflow, $t_{\mathrm{outflow}}$, as an indicator of the age of the flow,
ranges from 1–5 $\times 10^{4}$ yr traced by four species (i.e., 12CO, 13CO,
HCO+ and CS; see Liu et al., 2021). $t$ is about one order of magnitude larger
than $t_{\mathrm{outflow}}$, indicating that the H i wind is likely strong
enough to drive the molecular outflow. Observations with higher spatial
resolution of both H i and molecules are required to further confirm this
conclusion.

To further investigate the influence of the inclination, Figure 6 presents
both $t$ and $t_{\mathrm{outflow}}$ (including those for the 12CO, 13CO, HCO+,
and CS outflows) under different inclinations. It can be seen that the
lifetime of the H i wind is much larger than the timescale of the outflow
traced by all the four molecules, no matter what inclination is considered.
This suggests that the inclination does not change the inequality of $t\gg
t_{\mathrm{outflow}}$.

Figure 6: $t$ (red points, where the error bars indicate the 1$\sigma$
uncertainty) and $t_{\mathrm{outflow}}$ (blue solid line for 12CO, green for
13CO, black for HCO+, and yellow for CS) as functions of the inclination. The
dashed line is the dynamical timescale of the corresponding star formation
(i.e., $\sim$ 2 $\times$ 105 yr; see Chen et al., 2003).

Moreover, Chen et al. (2003) derived the timescale of the star formation
corresponding to this H i wind or molecular outflow as $\sim$ 2 $\times$ 105
yr by researching the H2 knots surrounding the excitation source. The
inclination of this H i wind or molecular outflow, $\theta_{0}$, may be $\sim
69^{\circ}\,{}^{+8^{\circ}}_{-28^{\circ}}$, assuming that the lifetime of the
H i wind equals the timescale of star formation, i.e., $t\sim 2\times 10^{5}$
yr.

###  4.2 Physical Properties with an Inclination of $69^{\circ}$

We next calculated the physical properties of the H i wind assuming that an
inclination of $69^{\circ}$ (see Table 4). The maximum velocity of the H i
wind changes to $\sim 336.2\pm 44.6$ km s-1. This value is larger than that
for HH 7–11 (assuming an inclination of $22^{\circ}$; see Herbig & Jones,
1983; Lizano et al., 1988) by a factor of $\sim$ 2, and is $\sim$ 60% of that
for L1551 (assuming an inclination of 75∘; see Snell & Schloerb, 1985;
Giovanardi et al., 1992). The value of the momentum of the H i wind after
inclination correction is 22.66 $M_{\odot}$ km s-1, which is larger than that
in HH 7–11 by a factor of $\sim 12$ (see Lizano et al., 1988; Giovanardi et
al., 1992), and a factor of $\sim 15$ for that in L1551 (see Giovanardi et
al., 1992). The values of the wind’s kinetic energy, i.e., $\sim 2.6\times
10^{47}$ erg, and the stellar mass-loss rate, i.e., $\sim 3.2\times 10^{-5}$
$M_{\odot}$ yr-1, are larger than those in HH 7–11 by factors of $\sim 28$ and
$\sim 10$, and larger than those in L1551 by factors of $\sim 3$ and $\sim 7$,
respectively (see these values in HH 7–11 and L1551 in Lizano et al., 1988;
Giovanardi et al., 1992).

The kinetic luminosity of the H i wind,
$L_{\mathrm{HI,wind}}=\displaystyle\frac{1}{2}\dot{M}_{\mathrm{a}}V^{2}_{\mathrm{max,wind}}$,
is 16.0 $\pm$ 10.1 $L_{\odot}$ without inclination correction, which is much
less than the bolometric luminosity, $L_{\ast}$, of the excitation source
(i.e., 1.7 $\times$ 103 $L_{\odot}$; see Molinari et al., 2008). However, the
kinetic luminosity of the H i wind after inclination correction is 322.0
$L_{\odot}$, i.e., $\sim$ 19% of $L_{\ast}$. The ratio of
$L_{\mathrm{HI,wind}}/L_{\ast}$ after inclination correction is comparable to
that in HH 7–11 (i.e., 20%) and in L1551 (i.e., 127%, see Giovanardi et al.,
1992), while the ratio of $L_{\mathrm{HI,wind}}/L_{\ast}$ before inclination
correction is only $\sim$ 1%. This result implies that an inclination
correction is necessary when we calculate the physical properties of the H i
wind.

Table 4: Physical Properties of the H i Wind for an Inclination of
$69^{\circ}$

Quantity |  Value (1$\sigma$ uncertainty)
---|---
$P_{\mathrm{HI,wind}}$ ($M_{\odot}$ km s-1) |  22.66 $\pm$ 9.63
$V_{\mathrm{max,wind}}$ (km s-1) |  336.2 $\pm$ 44.6
$V_{\mathrm{mean,wind}}$ (km s-1) |  112.1 $\pm$ 14.9
$E_{\mathrm{HI,wind}}$ (erg) |  (2.6 $\pm$ 1.2) $\times$ 1047
$t_{\mathrm{a}}$ (yr) |  (7.2 $\pm$ 0.9) $\times$ 103
$\dot{M}_{\mathrm{a}}$ ($M_{\odot}$ yr-1) |  (3.2 $\pm$ 0.9) $\times$ 10-5
$t$ (yr) |  (2.0 $\pm$ 1.4) $\times$ 105
$L_{\mathrm{HI,wind}}$ ($L_{\odot}$) |  322.0 $\pm$ 202.8

Note. — See description of each quantity in Table 3.

##  5 Summary and Conclusions

We obtained high-sensitivity H i spectra with an average rms noise of $\sim$ 7
mK @ 0.1 km s-1 toward the high-mass star-forming region G176.51+00.20 using
FAST with a 19-beam tracking observational mode. We searched for H i wind in
the central seven beams (i.e., Beams 1–7), but detected it only in Beam 1. The
main results are as follows:

  1. 1.

The abundance of H i in the H i wind is consistent with that of the HINSA
(where the molecular column density is traced by 13CO). This indicates that
there probably exists an internal correlation between the H i wind and HINSA;
such a correlation would enhance the argument of the association between the H
i wind and the molecular outflows.

  2. 2.

The H i wind is likely strong enough to drive the molecular outflows. This
conclusion is not affected by the value of the inclination.

  3. 3.

The mass, momentum, and kinetic energy of the H i wind and the associated
mass-loss rate of the excitation star in G176.51+00.20, a high-mass star-
forming region, are about one orders of magnitude larger than those in low-
mass star-forming regions (i.e., HH 7–11 and L1551).

##  Appendix A Mass and Momentum of the H i Wind

A crude “triangular fit” was adopted to obtain the relationship between the
brightness temperature and the radial velocity (e.g., Lizano et al., 1988) as

$T_{\mathrm{wind}}(v)=a+b\cdot v,$ (A1)

where $a$ and $b$ are constants with units of K and K (km s-1)-1,
respectively. The result of the “triangular fit” is plotted as the red solid
line in Figure 3(a). Considering that the H i wind may be contaminated or
affected by the ambient environment, and manifesting as an asymmetry between
the blue and red lobes, the blue and red lobes were fitted separately with the
line center, $v_{\mathrm{{c,w}}}$, as the velocity of HINSA (see Li et al.,
2022, see also Figure 2 and the red vertical line in Figure 3(a)).

$T_{\mathrm{wind}}$ can be used to calculate the column density of the H i
wind as:

$N_{\mathrm{HI,wind}}=1.82\times 10^{18}\int T_{\mathrm{wind}}dv,$ (A2)

where H i emission is assumed to be optically thin (Dickey & Benson, 1982;
Lizano et al., 1988; Saha et al., 2018).

The maximum velocity of the H i wind, $V_{\mathrm{{max,flow}}}$, reads:

$V_{\mathrm{{max,flow}}}=-a/b,$ (A3)

and the mean velocity of the H i wind is
$V_{\mathrm{{mean,flow}}}=V_{\mathrm{{max,flow}}}/3$ for the triangular
profile. Considering that the beam size of FAST is $2^{\prime}.9$ and the
distance to the H i wind is 1.8 kpc, the mass, $M_{\mathrm{HI,wind}}$, and
momentum, $P_{\mathrm{HI,wind}}$, of the H i wind without inclination
correction are:

$M_{\mathrm{HI,wind}}=C_{0}\int T_{\mathrm{wind}}dv,$ (A4)

and

$P_{\mathrm{HI,wind}}=C_{0}\int|v-v_{\mathrm{{c,w}}}|T_{\mathrm{wind}}dv,$
(A5)

respectively, where the constant $C_{0}$ is:

$C_{0}=0.103\;\mathrm{K^{-1}\;}M_{\odot}\;\mathrm{km^{-1}\;s},$ (A6)

where we assume that the hydrogen mass abundance is 0.74 (see Garden et al.,
1991) and the H i wind emission is from a point source (e.g., Lizano et al.,
1988). This is reasonable because the value of $V_{\mathrm{{max,flow}}}$ is
120.6 $\pm$ 16.3 km s-1 (see Lizano et al., 1988; Frank et al., 2014, and
references therein). The value of $C_{0}$ is larger than that in Equation (A6)
by a factor of $2\ln 2$ for an extended source with a uniform brightness
temperature over the beam. The kinetic energy, $E_{\mathrm{HI,wind}}$, of the
H i wind is (see Giovanardi et al., 1992):

$E_{\mathrm{HI,wind}}=\frac{1}{2}M_{\mathrm{HI,wind}}V^{2}_{\mathrm{{max,flow}}}.$
(A7)

This work made use of the data from FAST. FAST is a Chinese national mega-
science facility, operated by National Astronomical Observatories, Chinese
Academy of Sciences. We would like to thank the anonymous referee for the
helpful comments and suggestions that helped to improve the paper. This work
was sponsored by the Natural Science Foundation of Jiangsu Province (grant No.
BK20210999), the Entrepreneurship and Innovation Program of Jiangsu Province,
NSFC grants Nos. 11933011 and 11873019, and the Key Laboratory for Radio
Astronomy, Chinese Academy of Sciences. FAST, PMO 13.7m

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