#  ${\mathbb{A}}^{1}$-connectedness of moduli of vector bundles on a curve

Amit Hogadi  Department of Mathematical Sciences, Indian Institute of Science
Education and Research Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India.
<EMAIL_ADDRESS>and  Suraj Yadav  Universität Regensburg,
Universitätsstr. 31, 93040, Regensburg, Germany<EMAIL_ADDRESS>
###### Abstract.

In this note we prove that the moduli stack of vector bundles on a curve with
a fixed determinant is ${\mathbb{A}}^{1}$-connected. We obtain this result by
classifying vector bundles on a curve up to
${\mathbb{A}}^{1}$-concordance.Consequently we classify
${\mathbb{P}}^{n}$-bundles on a curve up to ${\mathbb{A}}^{1}$-weak
equivalence, extending a result in [3] of Asok-Morel. We also give an explicit
example of a variety which is ${\mathbb{A}}^{1}$-$h$-cobordant to a projective
bundle over ${\mathbb{P}}^{2}$ but does not have the structure of a projective
bundle over ${\mathbb{P}}^{2}$, thus answering a question of Asok-Kebekus-
Wendt [2].

###### Key words and phrases:

${\mathbb{A}}^{1}$-connectedness, Moduli of vector bundles,
${\mathbb{A}}^{1}$-concordance

###### 2010 Mathematics Subject Classification:

14D20, 14D23, 14F42

##  1\. Introduction

Let $C$ be a smooth projective curve of genus $g$ over a field $k$. Fix a line
bundle $\mathcal{L}\in\mathrm{Pic}(C)$. Consider the following moduli stack
$Bun_{n,\mathcal{L}}$,

$Bun_{n,\mathcal{L}}(Y)$ = { category of rank $n$ vector bundles on $C\times
Y$, such that for any object $\mathcal{V}$, $\mathrm{det}\mathcal{V}\cong
p^{\ast}(\mathcal{L}),$ where $p:C\times Y\rightarrow C\\}$

This is a smooth algebraic stack [1, Prop 1.3]. Any stack can be regarded as a
simplicial sheaf via the nerve construction (see [6]) and thus it defines an
object in the ${\mathbb{A}}^{1}$-homotopy category. Therefore it makes sense
to talk about ${\mathbb{A}}^{1}$-connectedness of $Bun_{n,\mathcal{L}}$.
Following is the main theorem of this note:

######  Theorem 1.1.

$Bun_{n,\mathcal{L}}$ is ${\mathbb{A}}^{1}$-connected for any curve $C$ over
an infinite field $k$ and $\mathcal{L}\in{\rm Pic}(C)$.

The proof of the Theorem 1.1 relies on finding an explicit
${\mathbb{A}}^{1}$-concordance (see [2, Definition 5.1] or Definition 2.1)
between a vector bundle $\mathcal{E}$ of rank $n$ and determinant
$\mathcal{L}$ to the vector bundle $\mathcal{O}_{C}^{n-1}\oplus\mathcal{L}$.
This is achieved by induction on $n$. In the course of this proof, we also
achieve the classification of vector bundles of a given rank on the curve $C$
up to ${\mathbb{A}}^{1}$-concordance (Theorem 2.5). Once the question of
${\mathbb{A}}^{1}$-connectedness of $Bun_{n,\mathcal{L}}$ is settled, it’s
natural to wonder the same about its open substack $Bun_{n,\mathcal{L}}^{s}$,
the moduli of stable vector bundles for a curve of genus greater than 1.
Assuming that $n$ and degree of $\mathcal{L}$ are co-prime, $k$ algebraically
closed, the coarse moduli space is known to be rational ([8, Theorem 1.2]) and
one may be tempted to conclude that ${\mathbb{A}}^{1}$-connectedness of an
algebraic stack is dictated by that of its coarse moduli space (if it exists).
However in Example 2.9 we show that this simply is not the case. We give an
example of “stacky” ${\mathbb{P}}^{1}$-an orbifold with ${\mathbb{P}}^{1}$ as
a coarse moduli space-which is not ${\mathbb{A}}^{1}$-connected. It should
also be noted that the ${\mathbb{A}}^{1}$-connectedness is not preserved under
rationality of a morphism of schemes as illustrated by
$\mathbb{G}_{m}\hookrightarrow{\mathbb{A}}^{1}$.

Related to ${\mathbb{A}}^{1}$-concordance is the notion of an
${\mathbb{A}}^{1}$-$h$-cobordism ([3, Definition 3.1.1] or Definition 3.1).
Furthermore projectivizations of two ${\mathbb{A}}^{1}$-concordant vector
bundles are ${\mathbb{A}}^{1}$-$h$-cobordant. As an application of
${\mathbb{A}}^{1}$-connectedness of $Bun_{n,\mathcal{L}}$, we obtain the
following theorem which classifies ${\mathbb{P}}^{n}$-bundles over any curve
of genus $g$ up to ${\mathbb{A}}^{1}$-weak equivalence. This extends the
result on classification of ${\mathbb{P}}^{n}$-bundles over ${\mathbb{P}}^{1}$
given in [3].

######  Theorem 1.2.

Let $X={\mathbb{P}}_{C}(\mathcal{E})$ and $Y={\mathbb{P}}_{C}(\mathcal{F})$ be
${\mathbb{P}}^{n}$-bundles over $C$, where $C$ lies over an infinite field.
Then the following are equivalent :

  1. (1)

$X$ and $Y$ are ${\mathbb{A}}^{1}$\- weakly equivalent.

  2. (2)

$X$ and $Y$ are ${\mathbb{A}}^{1}$-h-cobordant.

  3. (3)

${\rm det}(\mathcal{E})\otimes{\rm det}(\mathcal{F})^{-1}=\mathcal{L}^{\otimes
n+1}$, for some $\mathcal{L}\in{\rm Pic}(C)$.

In another application of our theorem we answer a question raised in [2]:
whether a variety which is ${\mathbb{A}}^{1}$-$h$-cobordant to a
${\mathbb{P}}^{1}$-bundle over ${\mathbb{P}}^{2}$ has a structure of
${\mathbb{P}}^{1}$-bundle over ${\mathbb{P}}^{2}$. The answer is no and we
prove in the following theorem that the suggested example in op. cit. indeed
works.

######  Theorem 1.3.

Let $X:={\mathbb{P}}_{{\mathbb{P}}^{1}}(\mathcal{E})$, where
$\mathcal{E}:=\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(1)$ on
${\mathbb{P}}^{1}_{k}$ (where $k$ is an infinite field). Then $X$ is
${\mathbb{A}}^{1}$-$h$-cobordant to
${\mathbb{P}}^{1}_{k}\times{\mathbb{P}}^{2}_{k}$ but doesn’t have the
structure of a ${\mathbb{P}}^{1}_{k}$-bundle over ${\mathbb{P}}^{2}_{k}$.

##  2\. Classification of vector bundles on a curve up to
${\mathbb{A}}^{1}$-concordance

In this section we classify vector bundles on a curve up to
${\mathbb{A}}^{1}$-concordance (Theorem 2.5) and obtain the proof of Theorem
1.1 as a consequence of that. Recall the following definition from [2].

######  Definition 2.1.

[2, Definition 5.1] Let $X$ be a scheme over a field $k$. Then two given
vector bundles $\mathcal{E}_{0}$ and $\mathcal{E}_{1}$ on $X$, are said to be
directly ${\mathbb{A}}^{1}$-concordant if there exists a vector bundle
$\mathcal{E}$ on $X\times{\mathbb{A}}^{1}$ such that
$i_{0}^{\ast}\mathcal{E}\cong\mathcal{E}_{0}$ and
$i_{1}^{\ast}\mathcal{E}\cong\mathcal{E}_{1}$, where
$i_{k}:X\times\\{k\\}\hookrightarrow X\times{\mathbb{A}}^{1}$, for $k=0,1$.
$\mathcal{E}_{0}$ and $\mathcal{E}_{1}$ on $X$ are
${\mathbb{A}}^{1}$-concordant if they are equivalent under the equivalence
relation generated by direct ${\mathbb{A}}^{1}$-concordance.

######  Lemma 2.2.

Let $\mathcal{E}_{0}$ and $\mathcal{E}_{1}$ be ${\mathbb{A}}^{1}$-concordant
vector bundles on a normal variety $X$ and $\mathcal{V}$ be a vector bundle on
$X\times{\mathbb{A}}^{1}$. Then
$(i_{0}^{\ast}(\mathcal{V})\otimes\mathcal{L})\oplus\mathcal{E}_{0}$ and
$(i_{1}^{\ast}(\mathcal{V})\otimes\mathcal{L})\oplus\mathcal{E}_{1}$ are
${\mathbb{A}}^{1}$-concordant, for any $\mathcal{L}\in{\rm Pic}(X)$.

###### Proof.

It is enough to prove the lemma in the case when $\mathcal{E}_{0}$ and
$\mathcal{E}_{1}$ are directly ${\mathbb{A}}^{1}$-concordant. Let the direct
${\mathbb{A}}^{1}$-concordance be given by a vector bundle $\mathcal{E}$ on
$X\times{\mathbb{A}}^{1}$. Note that $p^{\ast}:{\rm Pic}(X)\rightarrow{\rm
Pic}(X\times{\mathbb{A}}^{1})$ (where $p:X\times{\mathbb{A}}^{1}\rightarrow
X$) is an isomorphism (see [5, II, Prop. 6.6]) with the inverse given by
$i_{0}^{\ast}=i_{1}^{\ast}$. Then the lemma immediately follows from the
definition by considering the vector bundle $(\mathcal{V}\otimes
p^{\ast}\mathcal{L})\oplus\mathcal{E}$ and the fact that the pullback functor
commutes with the direct sums. ∎

In light of the previous lemma, the following corollary is rather obvious but
we state it nevertheless, keeping in mind its direct application in the proof
of Theorem 1.1.

######  Corollary 2.3.

Let $\mathcal{E}_{0}$ and $\mathcal{E}_{1}$ are ${\mathbb{A}}^{1}$-concordant
vector bundles on a normal variety $X$ Then the following statements hold

  1. (1)

$\mathcal{O}_{X}^{n}\oplus\mathcal{E}_{0}$ and
$\mathcal{O}_{X}^{n}\oplus\mathcal{E}_{1}$ are ${\mathbb{A}}^{1}$-concordant
for any $n\geq 0$.

  2. (2)

$\mathcal{O}_{X}(m)\oplus\mathcal{E}_{0}$ and
$\mathcal{O}_{X}(m)\oplus\mathcal{E}_{1}$ are ${\mathbb{A}}^{1}$-concordant
for any $m$.

###### Proof.

For the first statement take
$\mathcal{V}=\mathcal{O}_{X\times{\mathbb{A}}^{1}}^{n}$, keeping the notation
of the previous lemma in mind.
For the second statement take
$\mathcal{V}=\mathcal{O}_{X\times{\mathbb{A}}^{1}}$ and
$\mathcal{L}=p^{\ast}\mathcal{O}_{X}(m)$. ∎

###  2.1. ${\mathbb{A}}^{1}$-concordance via Ext classes

Now we look at a way of constructing ${\mathbb{A}}^{1}$-concordance between
vector bundles.

######  Proposition 2.4.

Let
$0\rightarrow\mathcal{E}_{0}\rightarrow\mathcal{E}\rightarrow\mathcal{E}_{1}\rightarrow
0$ be any short exact sequence of vector bundles on a projective scheme $X$
over a field $k$. Then $\mathcal{E}$ is directly ${\mathbb{A}}^{1}$-concordant
to $\mathcal{E}_{0}\oplus\mathcal{E}_{1}$.

###### Proof.

Consider $\mathcal{E}$ as an element in $\rm
Ext^{1}(\mathcal{E}_{1},\mathcal{E}_{0})$. If $\mathcal{E}$ is trivial then
our claim is obvious, so assume to the contrary. Consider the moduli functor
$\boldsymbol{\rm Ext^{1}}(\mathcal{E}_{1},\mathcal{E}_{0})$ given by
$Y\mapsto{\rm Ext}^{1}(p^{\ast}\mathcal{E}_{1},p^{\ast}\mathcal{E}_{0})$,
where $p:X\times Y\rightarrow X$. It’s well known ([7, Proposition 3.1]) that
this functor is representable by ${\mathbb{A}}^{n}_{k}$, where
$n=\mathrm{dim}(\rm Ext^{1}(\mathcal{E}_{1},\mathcal{E}_{0}))$ as a vector
space over $k$ and $n>0$ by the assumption that $\mathcal{E}$ is non trivial.
Therefore by representability there is a universal class $\mathcal{V}$ (of
vector bundle) on $X\times{\mathbb{A}}^{n}_{k}$ whose pullback to
$X\times{t_{i}}\hookrightarrow X\times{\mathbb{A}}^{n}_{k}$, $i=0,1$ is
$\mathcal{E}$ and $\mathcal{E}_{0}\oplus\mathcal{E}_{1}$ respectively for some
$k$-rational points $t_{i}\in{\mathbb{A}}^{n}_{k}$. For any two given
$k$-rational points in ${\mathbb{A}}^{n}_{k}$ (in our case $t_{0}$ and
$t_{1}$) there is a closed embedding
$i:{\mathbb{A}}^{1}_{k}\hookrightarrow{\mathbb{A}}^{n}_{k}$ such that the
composition ${\rm
Spec\,}k\xrightarrow{0}{\mathbb{A}}^{1}_{k}\xrightarrow{i}{\mathbb{A}}^{n}_{k}$
is $t_{0}$ and the composition ${\rm
Spec\,}k\xrightarrow{1}{\mathbb{A}}^{1}_{k}\xrightarrow{i}{\mathbb{A}}^{n}_{k}$
is $t_{1}$. Now consider the pullback of the universal class, $(id_{X}\times
i)^{\ast}\mathcal{V}$, via the map $id_{X}\times
i:X\times{\mathbb{A}}^{1}_{k}\to X\times{\mathbb{A}}^{n}_{k}$. By construction
the vector bundle $(id_{X}\times i)^{\ast}\mathcal{V}$ on
$X\times{\mathbb{A}}^{1}_{k}$ gives a direct ${\mathbb{A}}^{1}$-concordance
between $\mathcal{E}$ and $\mathcal{E}_{0}\oplus\mathcal{E}_{1}$ ∎

###  2.2. Classification result and proof of Theorem 1.1

######  Theorem 2.5.

Let $\mathcal{E}$ and $\mathcal{F}$ be rank $n$ vector bundles on the curve
$C$. Then the following hold

  1. (1)

$\mathcal{E}$ is ${\mathbb{A}}^{1}$-concordant to
$\mathcal{O}_{C}^{n-1}\oplus\det(\mathcal{E})$.

  2. (2)

$\mathcal{E}$ is ${\mathbb{A}}^{1}$-concordant to $\mathcal{F}$ iff
$\det(\mathcal{E})\cong\det(\mathcal{F})$.

###### Proof.

We first prove (1) for the case when $n=2$. For the general case we will use
induction.
Case 1: $n=2$. First assume $\mathcal{E}$ is globally generated and denote
$\det(\mathcal{E})$ by $\mathcal{L}$. Then by [5, II, Exercise 8.2], we have
the following short exact sequence

(2.1)
$0\rightarrow\mathcal{O}_{C}\rightarrow\mathcal{E}\rightarrow\mathcal{E}^{\prime}\rightarrow
0$

where $\mathcal{E}^{\prime}$ is a line bundle. By the Whitney sum formula of
Chern classes ([4, Theorem 5.3(c)])

$c_{1}(\mathcal{L})=c_{1}(\mathcal{E})=c_{1}(\mathcal{O}_{C})+c_{1}(\mathcal{E}^{\prime})=c_{1}(\mathcal{E}^{\prime}).$

Therefore by Proposition 2.4, $\mathcal{E}$ is directly
${\mathbb{A}}^{1}$-concordant to $\mathcal{O}_{C}\oplus\mathcal{L}$. For a
general $\mathcal{E}$, choose $m>>0$ such that $\mathcal{E}(m)$,
$\mathcal{L}(m)$ are globally generated. Then again by applying [5, II,
Exercise 8.2] we get a short exact sequence for $\mathcal{E}(m)$ which we
tensor by $\mathcal{O}(-m)$ to obtain the following short exact sequence.s

(2.2)
$0\rightarrow\mathcal{O}_{C}(-m)\rightarrow\mathcal{E}\rightarrow\mathcal{L}(m)\rightarrow
0$

This proves $\mathcal{E}$ is directly ${\mathbb{A}}^{1}$-concordant to
$\mathcal{O}_{C}(-m)\oplus\mathcal{L}(m)$. As the final step we now prove that
$\mathcal{O}_{C}(-m)\oplus\mathcal{L}(m)$ is directly
${\mathbb{A}}^{1}$-concordant to $\mathcal{O}_{C}\oplus\mathcal{L}$. Note that
$m$ is chosen such that $\mathcal{L}(m)$ is globally generated, therefore
$\mathcal{O}_{C}(m)\oplus\mathcal{L}(m)$ is globally generated. Hence we have
a short exact sequence which shows $\mathcal{O}_{C}(-m)\oplus\mathcal{L}(m)$
is directly ${\mathbb{A}}^{1}$-concordant to
$\mathcal{O}_{C}\oplus\mathcal{L}$.

(2.3)
$0\to\mathcal{O}_{C}(-m)\to\mathcal{O}_{C}\oplus\mathcal{L}\to\mathcal{L}(m)\to
0$

Therefore, $\mathcal{E}$ is ${\mathbb{A}}^{1}$-concordant to
$\mathcal{O}_{C}\oplus\mathcal{L}$.
Case 2: Now we handle the general case. So assume $n>2$ and choose $m$ such
that $\mathcal{E}(m)$ and $\mathcal{L}(m)$ are globally generated. Then we
have a short exact sequence giving a direct ${\mathbb{A}}^{1}$-concordance
between $\mathcal{E}$ and $\mathcal{O}_{C}(-m)\oplus\mathcal{E}^{\prime}$,
where $\mathcal{E}^{\prime}$ is a vector bundle of rank $n-1$ with determinant
$\mathcal{L}(m)$. By induction, $\mathcal{E}^{\prime}$ is
${\mathbb{A}}^{1}$-concordant to $\mathcal{O}_{C}^{n-2}\oplus\mathcal{L}(m)$.
Therefore by the second statement of Corollary 2.3 we have an
${\mathbb{A}}^{1}$-concordance between
$\mathcal{O}_{C}(-m)\oplus\mathcal{O}_{C}^{n-2}\oplus\mathcal{L}(m)$ and
$\mathcal{O}_{C}(-m)\oplus\mathcal{E}^{\prime}$. Hence $\mathcal{E}$ is
${\mathbb{A}}^{1}$-concordant to
$\mathcal{O}(-m)\oplus\mathcal{O}_{C}^{n-2}\oplus\mathcal{L}(m)$. Now by the
short exact sequence 2.3, $\mathcal{O}_{C}\oplus\mathcal{L}$ is directly
${\mathbb{A}}^{1}$-concordant to $\mathcal{O}_{C}(-m)\oplus\mathcal{L}(m)$,
which implies – by first statement of Corollary 2.3 – that
$\mathcal{O}(-m)\oplus\mathcal{O}_{C}^{n-2}\oplus\mathcal{L}(m)$ is directly
${\mathbb{A}}^{1}$-concordant to $\mathcal{O}_{C}^{n-1}\oplus\mathcal{L}$,
thus finishing the proof of (1).
For proving (2), we first observe that if
$\det(\mathcal{E})\cong\det(\mathcal{F})$ then (1) implies that $\mathcal{E}$
is ${\mathbb{A}}^{1}$-concordant to $\mathcal{F}$. Hence it remains to show
that if $\mathcal{E}$ is directly ${\mathbb{A}}^{1}$-concordant to
$\mathcal{F}$ then $\mathrm{det}(\mathcal{E})\cong\mathrm{det}(\mathcal{F})$.
So assume that $\mathcal{E}$ is directly ${\mathbb{A}}^{1}$-concordant to
$\mathcal{F}$, which by definition gives us a vector bundle
$\mathcal{E}^{\prime}$ on $C\times{\mathbb{A}}^{1}$ such that
$i_{0}^{\ast}\mathcal{E}^{\prime}\cong\mathcal{E}$ and
$i_{1}^{\ast}\mathcal{E}^{\prime}\cong\mathcal{F}$. We have
$c_{1}(\mathrm{det}\mathcal{E})=c_{1}(i_{0}^{\ast}\mathcal{E}^{\prime})=i_{0}^{\ast}(c_{1}(\mathcal{E}^{\prime}))$,
where the first equality follows from the isomorphism
$i_{0}^{\ast}\mathcal{E}^{\prime}\cong\mathcal{E}$ and the fact that for any
vector bundle $\mathcal{V}$,
$c_{1}(\mathcal{V})=c_{1}(\mathrm{det}\mathcal{V})$, while the second equality
is the functoriality of Chern classes ([4, Theorem 5.3(d)]). Similarly we have
$c_{1}(\mathrm{det}\mathcal{F})=c_{1}(i_{1}^{\ast}\mathcal{E}^{\prime})=i_{1}^{\ast}(c_{1}(\mathcal{E}^{\prime}))$.
Moreover, $(i_{0})^{\ast}=(i_{1})^{\ast}:{\rm
CH}^{1}(C\times{\mathbb{A}}^{1})\to{\rm CH}^{1}(C)$, which implies
$c_{1}(\mathrm{det}\mathcal{E})=c_{1}(\mathrm{det}\mathcal{F})$ Therefore
$\mathrm{det}(\mathcal{E})\cong\mathrm{det}(\mathcal{F})$.

∎

Before we proceed with the proof of Theorem 1.1, we recall some standard
definitions.

######  Definition 2.6.

Let $\mathcal{X}$ be a simplicial sheaf and $U$ a scheme. Then $x$ and $y$ in
$\mathcal{X}(U)$ are said to be naively ${\mathbb{A}}^{1}$-homotopic if there
exists $f:{\mathbb{A}}^{1}_{U}\rightarrow\mathcal{X}$ such that $f_{0}=x$ and
$f_{1}=y$, where $f_{i}$ is the composition
$U\xrightarrow{i}{\mathbb{A}}^{1}_{U}\xrightarrow{f}\mathcal{X}$, for $i=0,1$.

######  Definition 2.7.

For a given simplicial sheaf $\mathcal{X}$ we define
$\mathcal{S}(\mathcal{X})$ to be the Nisnevich sheafification of the presheaf
$U\mapsto\mathcal{X}(U)/\sim$, where $\sim$ is the equivalence relation
generated by naive ${\mathbb{A}}^{1}$-homotopies.

The following standard lemma will be required in our proof. It’s essentially
[11, Section 2, Corollary 3.22] combined with [9, Lemma 6.1.3.],

######  Lemma 2.8.

A simplicial sheaf $\mathcal{X}$ is ${\mathbb{A}}^{1}$-connected if
$\mathcal{S}(\mathcal{X})(F)=\ast$ for every finitely generated field
extension $F$ over $k$.

Now we have all the ingredients in place to prove Theorem 1.1.

######  Proof of Theorem 1.1.

We regard $Bun_{n,\mathcal{L}}$ as a simplicial sheaf. By definition, any two
$F$-valued points of $Bun_{n,\mathcal{L}}$ are two rank $n$ (with determinant
condition) vector bundles, say $\mathcal{E}_{0}$ and $\mathcal{E}_{1}$ on $C$.
A morphism ${\mathbb{A}}^{1}_{F}\rightarrow Bun_{n,\mathcal{L}}$ is a vector
bundle $\mathcal{E}$ on $C\times{\mathbb{A}}^{1}$. Then $\mathcal{E}_{0}$ and
$\mathcal{E}_{1}$ are naively ${\mathbb{A}}^{1}$-homotopic if and only if they
are ${\mathbb{A}}^{1}$-concordant. By Theorem 2.5 both $\mathcal{E}_{0}$ and
$\mathcal{E}_{1}$ are ${\mathbb{A}}^{1}$-concordant to
$\mathcal{O}_{C}^{n-1}\oplus\mathcal{L}$. Hence they are
${\mathbb{A}}^{1}$-concordant to each other. Therefore by Lemma 2.8,
$Bun_{n,\mathcal{L}}$ is ${\mathbb{A}}^{1}$-connected. ∎

Motivated by the question of ${\mathbb{A}}^{1}$-connectedness of moduli stack
of stable vector bundles, we observe in the example below that there does not
seem to be an immediate way of concluding ${\mathbb{A}}^{1}$-connectedness of
a stack by looking at its coarse moduli space.

######  Example 2.9.

Let $C$ be a curve of genus $2$ over a field with characteristic not equal to
$2$. In particular it is a hyperelliptic curve (See [5, IV, Exercise 1.7(a)]).
Therefore, there is a finite morphism $f:C\rightarrow{\mathbb{P}}^{1}$ of
degree $2$ and we have an action of the finite group
$G:={\mathbb{Z}}/2{\mathbb{Z}}$ on $C$. By [5, IV, Exercise 2.2(a)], such a
morphism is unramified at all but 6 points (denoted as closed subscheme
$Z^{\prime}$) of $C$. So the action of $G$ is free on $C\setminus Z^{\prime}$.
Let $Z$ denote the closed subset in ${\mathbb{P}}^{1}$ corresponding to the 6
branched points. The quotient stack $\big{[}C/G\big{]}$ has coarse moduli
space ${\mathbb{P}}^{1}$ and the morphism
$\pi:\big{[}C/G\big{]}\rightarrow{\mathbb{P}}^{1}$ gives an isomorphism of an
open subscheme of $\big{[}C/G\big{]}$ with ${\mathbb{P}}^{1}\setminus Z$. See
[12, Example 8.1.12] for more details on Quotient stacks.
Let $E(G)$ denote the simplicially contractible, simplicial sheaf with
$E(G)_{n}=G^{n+1}$(See [11, Example 1.11, page 128]). The morphism
$C\to\big{[}C/G\big{]}$ is a $G$-torsor. Moreover $G$ acts freely on the space
$E(G)\times C$ and $(E(G)\times C)/G\simeq\big{[}C/G\big{]}$.
$\pi_{0}^{{\mathbb{A}}^{1}}(G)\cong{\mathbb{Z}}/2{\mathbb{Z}}$ being a finite
abelian group is a strictly ${\mathbb{A}}^{1}$-invariant sheaf. So all the
hypothesis of the statement of [10, Theorem 6.50] are satisfied, as a
consequence of which we obtain the following long exact sequence of
${\mathbb{A}}^{1}$-homotopy groups/pointed sets, where $\ast$ is a chosen
basepoint.

$\cdots\to\pi_{0}^{{\mathbb{A}}^{1}}(G,\ast)\to\pi_{0}^{{\mathbb{A}}^{1}}(E(G)\times
C,\ast)\to\pi_{0}^{{\mathbb{A}}^{1}}(\big{[}C/G\big{]},\ast)\to\ast$

But on the account of $E(G)$ being simplicially contractible and $C$ being
${\mathbb{A}}^{1}$-rigid (as all curves of genus $g>0$ are)
$\pi_{0}^{{\mathbb{A}}^{1}}(E(G)\times
C)\cong\pi_{0}^{{\mathbb{A}}^{1}}(C)\cong C$. So by long exact sequence,
$\big{[}C/G\big{]}$ being ${\mathbb{A}}^{1}$-connected would imply surjection
of finite group ${\mathbb{Z}}/2{\mathbb{Z}}$ on
$C\cong\pi_{0}^{{\mathbb{A}}^{1}}(C)$, which can not happen.

######  Remark 2.10.

By definition, any hyperelliptic curve of genus $g$ admits a finite map of
degree $2$ to ${\mathbb{P}}^{1}$. By Hurwitz’s theorem, such a morphism has
$2g+2$ ramified points. Therefore Example 2.9 can be generalized to an
hyperelliptic curve of any genus (which is necessarily greater than 1).

##  3\. Applications

As applications of the results in the previous section we give a proof of
Theorem 1.2 and Theorem 1.3. We first recall the following definition from
[3].

######  Definition 3.1.

[3, Definition 3.1.1] Let $X_{0}$ and $X_{1}$ be smooth and proper varieties
over $k$. They are directly ${\mathbb{A}}^{1}$-$h$\- cobordant if there exists
a smooth scheme $X$ with $f:X\rightarrow{\mathbb{A}}^{1}$ a proper surjective
morphism such that

  1. (1)

the fibers of $f$ over $0$ and $1$ are $X_{0}$ and $X_{1}$ respectively

  2. (2)

the natural maps $X_{i}\hookrightarrow X$ for $i=0,1$ are
${\mathbb{A}}^{1}$-weak equivalences.

$X_{0}$ and $X_{1}$ are ${\mathbb{A}}^{1}$-$h$-cobordant if they are
equivalent under the equivalence relation generated by direct
${\mathbb{A}}^{1}$-$h$-cobordance.

While ${\mathbb{A}}^{1}$-concordance is a relation between vector bundles,
${\mathbb{A}}^{1}$-$h$-cobordism a relation between proper schemes. Note that
by [2, Lemma 6.4], projectivizations of ${\mathbb{A}}^{1}$-concordant vector
bundles are ${\mathbb{A}}^{1}$-$h$-cobordant.
Recall that, given a locally free sheaf $\mathcal{E}$ of rank $n+1$ on a
scheme $X$, the associated ${\mathbb{P}}^{n}$-bundle, denoted
${\mathbb{P}}_{X}(\mathcal{E})$ is the scheme ${\bf{Proj}}_{X}({\rm
Sym}(\mathcal{E}))$. Here ${\bf{Proj}}$ is the relative proj construction and
${\rm Sym}(\mathcal{E})$ is the symmetric algebra of $\mathcal{E}$ as an
$\mathcal{O}_{X}$-module. See [5, II, page 162] for more details.
We now paraphrase the classification of ${\mathbb{P}}^{n}$-bundles on
${\mathbb{P}}^{1}$ up to ${\mathbb{A}}^{1}$-weak equivalence proved in [3] to
highlight that Theorem 1.2 is its direct generalization to an arbitrary smooth
projective curve.

######  Proposition 3.2.

[3, Proposition 3.2.10] Let
$X:={\mathbb{P}}(\mathcal{O}_{{\mathbb{P}}^{1}}^{n}\oplus\mathcal{O}_{{\mathbb{P}}^{1}}(a))$
and
$Y:={\mathbb{P}}(\mathcal{O}_{{\mathbb{P}}^{1}}^{n}\oplus\mathcal{O}_{{\mathbb{P}}^{1}}(b))$
be two ${\mathbb{P}}^{n}$-bundles over ${\mathbb{P}}^{1}$. Then the following
statements are equivalent:

  1. (1)

$X$ and $Y$ are ${\mathbb{A}}^{1}$\- weakly equivalent.

  2. (2)

$X$ and $Y$ are ${\mathbb{A}}^{1}$-h-cobordant.

  3. (3)

$n+1$ divides $a-b$.

Note that in case of $C={\mathbb{P}}^{1}$ the condition ${\rm
det}(\mathcal{E})\otimes{\rm det}(\mathcal{F})^{-1}=\mathcal{L}^{\otimes n+1}$
in Theorem 1.2 exactly translates to the fact that $n+1$ divides $a-b$ as
stated in the Proposition 3.2. This is due to $\mathrm{Pic}({\mathbb{P}}^{1})$
being isomorphic to ${\mathbb{Z}}$. For a general curve Picard group is much
more complicated and humongous (think of the Jacobian variety of a curve) so
one doesn’t get any further simplification. We now prove Theorem 1.2, which is
an extension of the previous proposition.

###### Proof of Theorem 1.2.

(3) $\implies$ (2): By Theorem 2.5, $\mathcal{E}$ is
${\mathbb{A}}^{1}$-concordant to $\mathcal{O}_{C}^{n}\oplus\mathcal{L}_{1}$,
where $\mathcal{L}_{1}=\mathrm{det}(\mathcal{E})$ and $\mathcal{F}$ is
${\mathbb{A}}^{1}$-concordant to $\mathcal{O}_{C}^{n}\oplus\mathcal{L}_{2}$,
where $\mathcal{L}_{2}=\mathrm{det}(\mathcal{F})$. Hence $X$ and
${\mathbb{P}}_{C}(\mathcal{O}_{C}^{n}\oplus\mathcal{L}_{1})$ are
${\mathbb{A}}^{1}$-$h$-cobordant. In the exact same manner, $Y$ and
${\mathbb{P}}_{C}(\mathcal{O}_{C}^{n}\oplus\mathcal{L}_{2})$ are
${\mathbb{A}}^{1}$-$h$-cobordant. Suppose
$\mathcal{L}_{1}\otimes\mathcal{L}_{2}^{-1}=\mathcal{L}^{\otimes n+1}$. That
implies $\mathcal{L}_{1}=\mathcal{L}^{\otimes n+1}\otimes\mathcal{L}_{2}$ for
some $\mathcal{L}\in{\rm Pic}(C)$. Let
$\mathcal{E}^{\prime}=(\mathcal{O}_{C}^{n}\oplus\mathcal{L}_{2})\otimes\mathcal{L}$.
Then $\mathrm{det}(\mathcal{E}^{\prime})=\mathcal{L}_{1}$. Therefore
${\mathbb{P}}(\mathcal{E}^{\prime})$ is ${\mathbb{A}}^{1}$-$h$-cobordant to
${\mathbb{P}}(\mathcal{O}_{C}^{n}\oplus\mathcal{L}_{1})$. Furthermore,
${\mathbb{P}}(\mathcal{E}^{\prime})$ is isomorphic (as a scheme) to
${\mathbb{P}}(\mathcal{O}_{C}^{n}\oplus\mathcal{L}_{2})$ by the general fact
that tensoring a vector bundle by a line bundle gives an isomorphism of
projectivization of the two vector bundles. This proves $X$ and $Y$ are
${\mathbb{A}}^{1}$-$h$-cobordant.

(2) $\implies$ (1): this is immediate from the definition of
${\mathbb{A}}^{1}$-$h$-cobordism.

(1) $\implies$ (3) : ${\mathbb{A}}^{1}$-invariance of the Chow rings implies
that it is enough to show that the Chow rings of
${\mathbb{P}}(\mathcal{O}_{C}^{n}\oplus\mathcal{L}_{1})$ and
${\mathbb{P}}(\mathcal{O}_{C}^{n}\oplus\mathcal{L}_{2})$ are not isomorphic if
$\mathcal{L}_{1}\otimes\mathcal{L}_{2}^{-1}\neq\mathcal{L}^{\otimes n+1}$ for
any $\mathcal{L}\in{\rm Pic}(C)$. The Chow ring of $C$ – which is simply
${\mathbb{Z}}\oplus{\rm Pic}(C)$, with product of any two line bundles under
the ring structure being zero – is denoted $R$. For simplicity of notation we
will denote $\mathcal{O}_{C}^{n}\oplus\mathcal{L}_{i}$, $i=1,2$ by
$\mathcal{E}_{i}$. Then by the projective bundle formula for the Chow rings
[4, Theorem 9.6], the Chow ring of ${\mathbb{P}}(\mathcal{E}_{1})$ is
$R_{1}:=R[\zeta]/(\zeta^{n+1}+c_{1}(\mathcal{E}_{1})\zeta^{n})$. But
$c_{1}(\mathcal{E}_{1})=c_{1}(\mathcal{L}_{1})$. In the ring $R_{1}$, $\zeta$
as well as any element $x\in{\rm Pic}(C)$ has grading $1$ with $xy=0$ for
$x,y\in{\rm Pic}(C)$. Let’s assume we have a graded ring isomorphism $\phi$
between $R_{1}$ and
$R_{2}:=R[\sigma]/(\sigma^{n+1}+c_{1}(\mathcal{L}_{2})\sigma^{n})$. Then such
an isomorphism has to respect the grading and hence $\phi(\zeta)=x+a\sigma$,
where $x\in{\rm Pic}(C)$ and $a\in{\mathbb{Z}}$. Similarly
$\phi^{-1}(\sigma)=y+b\zeta$, where $b\in{\mathbb{Z}}$ and $y\in{\rm Pic}(C)$.
We first prove that $a=\pm 1$. The condition $\phi^{-1}\circ\phi(\zeta)=\zeta$
implies that $x+ay+ab\zeta=\zeta$. Hence $ab=1$, so $a=\pm 1$.
By the graded ring structure of $R_{1}$, as discussed before, $x^{i}=0$ for
any $i>1$. Moreover $\phi(\zeta^{n+1}+c_{1}(\mathcal{L}_{1})\zeta^{n})$ has to
be divisible by $\sigma^{n+1}+c_{1}(\mathcal{L}_{2})\sigma^{n}$ in $R_{2}$.
First assume $a=1$. Proof for the case $a=-1$ is similar. We expand
$\phi(\zeta^{n+1}+c_{1}(\mathcal{L}_{1})\zeta^{n})$ as
$\sigma^{n+1}+\sigma^{n}((n+1)x+c_{1}(\mathcal{L}_{1}))$ and this expression
is divisible by $\sigma^{n+1}+c_{1}(\mathcal{L}_{2})\sigma^{n}$. Comparing
coefficients we conclude that
$c_{1}(\mathcal{L}_{1})-c_{1}(\mathcal{L}_{2})=(n+1)x$. This implies that
$\mathcal{L}_{1}\otimes\mathcal{L}_{2}^{-1}=\mathcal{L}^{\otimes n+1}$, where
$c_{1}(\mathcal{L})=x$.

∎

Now, we answer a question raised in [2], negatively.

######  Question 3.3.

[2, Question 6.9.1] If $X$ is any smooth projective variety that is
${\mathbb{A}}^{1}$-h-cobordant to a ${\mathbb{P}}^{1}$-bundle over
${\mathbb{P}}^{2}$, does $X$ have the structure of a ${\mathbb{P}}^{1}$-bundle
over ${\mathbb{P}}^{2}$?

The authors further add the answer is possibly no and non-trivial rank three
vector bundles over ${\mathbb{P}}^{1}$ deformable to the trivial one are the
likely counterexamples. We now prove Theorem 1.3 which shows that the example
alluded to above is indeed a correct counterexample.

###### Proof of Theorem 1.3.

: By Theorem 2.5,
$X:={\mathbb{P}}(\mathcal{E})\xrightarrow{\pi}{\mathbb{P}}^{1}$ is
${\mathbb{A}}^{1}$-$h$-cobordant to the trivial ${\mathbb{P}}^{2}$\- bundle on
${\mathbb{P}}^{1}$, namely, ${\mathbb{P}}^{1}\times{\mathbb{P}}^{2}$. However
$X$ and ${\mathbb{P}}^{1}\times{\mathbb{P}}^{2}$ are not isomorphic as
schemes. By [5, II, Exercise 7.9(b)] an isomorphism would imply that for some
line bundle on ${\mathbb{P}}^{1}$, say $\mathcal{O}(a)$ where
$a\in{\mathbb{Z}}$,
$\mathcal{O}(a)\otimes\mathcal{E}\simeq\mathcal{O}(a)\oplus\mathcal{O}(a-1)\oplus\mathcal{O}(a+1)\simeq\mathcal{O}^{\oplus
3}$ is an isomorphism of vector bundles on ${\mathbb{P}}^{1}$, which can not
happen.
Now suppose
$X\xrightarrow{\theta}{\mathbb{P}}_{{\mathbb{P}}^{2}}(\mathcal{E}^{\prime}):=Y\xrightarrow{\phi}{\mathbb{P}}^{2}$,
with $\theta$ an isomorphism of schemes, for some rank $2$ vector bundle
$\mathcal{E}^{\prime}$ on ${\mathbb{P}}^{2}$. We thus have the following
diagram

${X\simeq
Y}$${{\mathbb{P}}^{2}}$${{\mathbb{P}}^{1}}$$\scriptstyle{\phi}$$\scriptstyle{\pi}$

Without loss of generality we can assume (by twisting $\mathcal{E}^{\prime}$
with a suitable line bundle in ${\rm Pic}({\mathbb{P}}^{2})$ as
$c_{1}(\mathcal{E}^{\prime}\otimes\mathcal{L})=c_{1}(\mathcal{E}^{\prime})+2c_{1}(\mathcal{L})$),
$c_{1}(\mathcal{E}^{\prime})\in\\{0,1\\}$. Since $Y$ is
${\mathbb{A}}^{1}$-weakly equivalent to the trivial bundle on
${\mathbb{P}}^{2}$, their Chow rings are isomorphic. By [2, Lemma 4.5], we
have $c_{1}(\mathcal{E}^{\prime})^{2}-4c_{2}(\mathcal{E}^{\prime})=0$. So
$c_{1}(\mathcal{E}^{\prime})=0=c_{2}(\mathcal{E}^{\prime})$. It thus suffices
to show that $\mathcal{E}^{\prime}$ splits as a direct sum of line bundles as
this will prove that
$\mathcal{E}^{\prime}\simeq\mathcal{O}_{{\mathbb{P}}^{2}}\oplus\mathcal{O}_{{\mathbb{P}}^{2}}$.
By the assumption that $X\simeq Y$ this will imply that $X$ is isomorphic to
${\mathbb{P}}^{2}\times{\mathbb{P}}^{1}$, which from the discussion in the
first paragraph of this proof can not happen.

We will prove that $\phi$ has a section. Such a section will give the
following short exact sequence.

(3.1)
$0\rightarrow\mathcal{L}_{1}\rightarrow\mathcal{E}^{\prime}\rightarrow\mathcal{L}_{2}\rightarrow
0$

As both Chern classes of $\mathcal{E}^{\prime}$ vanish, by the Whitney sum
formula of Chern classes, both $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ will be
trivial. Therefore such a short exact sequence has to be a split one. This
will prove
$\mathcal{E}^{\prime}\simeq\mathcal{O}_{{\mathbb{P}}^{2}}\oplus\mathcal{O}_{{\mathbb{P}}^{2}}$.
Define $F\hookrightarrow Y$ to be $\theta\circ\pi^{-1}(z)$ for a point
$z\in{\mathbb{P}}^{1}$. By construction $F\simeq{\mathbb{P}}^{2}$. We claim
$\phi$ maps $F$ isomorphically onto ${\mathbb{P}}^{2}$. First we claim that
$\phi_{|F}$ is surjective. Suppose not, then $Z:=\phi(F)$ is either a point or
an irreducible curve (not necessarily smooth) in ${\mathbb{P}}^{2}$. Since
$\phi:Y\to{\mathbb{P}}^{2}$ is a ${\mathbb{P}}^{1}$-bundle map, the fiber of
$\phi$ over each point of ${\mathbb{P}}^{2}$ is ${\mathbb{P}}^{1}$. Therefore
$Z$ can not be a point. So assume $Z$ is an irreducible curve in
${\mathbb{P}}^{2}$. Consider smooth points $z_{1}\neq z_{2}\in Z$. Then using
flatness of $\phi$, we have $\phi^{-1}(z_{i})\simeq{\mathbb{P}}^{1}\subset F$
for $i=1,2$. However any two lines in ${\mathbb{P}}^{2}$ intersect, so
$\phi^{-1}(z_{1})$ and $\phi^{-1}(z_{i})$ intersect in $F$(which is isomorphic
to ${\mathbb{P}}^{2}$). This contradicts our assumption that $z_{1}\neq
z_{2}$. This establishes the surjectivity of $\phi_{|F}$. We also conclude
$\phi|_{F}$ is a degree $d$ morphism to ${\mathbb{P}}^{2}$ with $d\geq 1$.
We now show that $d=1$. This is achieved by comparing the graded ring
isomorphism induced on the Chow rings of $X$ and $Y$. The Chow ring of $X$ is
$R_{1}:={\mathbb{Z}}[x,y]/(x^{2},y^{3})$, where

  1. (i)

$x$ is the divisor ${\mathbb{P}}^{2}$ as a fiber over a point of
${\mathbb{P}}^{1}$

  2. (ii)

$y$ corresponds to a divisor $D^{\prime}$, such that the pushforward
$\pi_{\ast}(\mathcal{O}_{X}(D^{\prime}))$ to ${\mathbb{P}}^{1}$ is the vector
bundle $\mathcal{E}$.

Similarly the Chow ring of $Y$ is $R_{2}:={\mathbb{Z}}[s,t]/(s^{2},t^{3})$
where

  1. (i)

$t$ corresponds to fiber of ${\mathbb{P}}^{1}$ (as a degree $1$ curve in
${\mathbb{P}}^{2}$) via $\phi$

  2. (ii)

$s$ corresponds to a divisor $D$, such that the pushforward
$\phi_{\ast}(\mathcal{O}_{Y}(D))$ to ${\mathbb{P}}^{2}$ is the rank two vector
bundle $\mathcal{E}^{\prime}$.

Let $\psi:R_{1}\to R_{2}$ be an isomorphism of graded rings. Then
$\psi(x)=as+bt$, where $a,b\in{\mathbb{Z}}$. We have the condition that
$\psi(x^{2})=\psi(x)^{2}=a^{2}s^{2}+2abst+b^{2}t^{2}$ lies in the ideal
generated by $s^{2}$ and $t^{3}$. This implies $b=0$. Morever
$\psi^{-1}\circ\psi(x)=x$. Therefore $a=\pm 1$. In a similar fashion one
proves that any isomorphism between $R_{1}$ and $R_{2}$ sends $y$ to $\pm t$.
Therefore we conclude that the graded ring isomorphism between $R_{1}$ and
$R_{2}$ is given by $x\mapsto\pm s$ and $y\mapsto\pm t$. This implies $s$ is
equivalent (in the Chow ring) to the class of $F\simeq{\mathbb{P}}^{2}$.
Grauert’s theorem ([5, III, Corollary 12.9]) implies that the intersection
multiplicity of divisor $D$ corresponding to $s$ (see the description of
$R_{2}$ above) with any fiber of the map $\phi$ is $1$. As $s$ and $F$ are
equivalent in the Chow ring, the same holds for $F$. This can not happen
unless $d=1$ because if not, one can consider a point $z^{\prime}$ in
${\mathbb{P}}^{2}$ such that the set
$\phi|_{{\mathbb{P}}^{2}}^{-1}(z^{\prime})$ has more than one point. This will
force $\phi^{-1}(z^{\prime})={\mathbb{P}}^{1}$ to intersect ${\mathbb{P}}^{2}$
in more than one point, meaning an intersection multiplicity of greater than
$1$ which as we just proved can not happen. This proves $\phi|_{F}$ is an
isomorphism onto ${\mathbb{P}}^{2}$ and hence establishes the existence of a
section of $\phi$. Via the short exact sequence 3.1 this proves that
$\mathcal{E}^{\prime}$ is a trivial rank 2 vector bundle on
${\mathbb{P}}^{2}$, and thereby finishes the proof.

∎

Acknowledgements. We thank the anonymous referees for their careful reading,
comments and corrections. In particular Remark 2.10 is the result of a
question by one of the referees. The second-named author was supported by SFB
1085 Higher Invariants, University of Regensburg, NBHM fellowship of the
Department of Atomic Energy, Govt. of India and Fulbright-Nehru Doctoral
Research fellowship during this work and thanks the Department of Math,
University of Southern California for the hospitality.

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