# Comment on ”Polar and antiferromagnetic order in f=1 boson systems”

C. G. Bao  Center of Theoretical Nuclear Physics, National Laboratory of Heavy
Ion Accelerator, Lanzhou, 730000, PRC and State Key Laboratory of
Optoelectronic Materials and Technologies School of Physics and Engineering,
Sun Yat-Sen University, Guangzhou, PRC

###### Abstract

An inequality for the lower bound of the average number of hyperfine component
$\mu=0$ particles in the ground state of spin-1 condensates under a magnetic
field has been derived in ref.tasa13 . It is shown in this comment that, in a
broad domain of parameters usually accessed in experiments, the lower bound
appears to be negative. Thus the applicability of the inequality is very
limited.

###### pacs:

03.75.Mn,03.75.Kk

The essential results of Tasaki’s letter tasa13  are concluded in two
theorems. In particular, an inequality (eq.(11) of tasa13  ) is derived to
constrain the number of spin-component $\mu=0$ particles in the ground state
(g.s.). The inequality has the following short-comings.

(i) Eq.(11)of tasa13  can be rewritten as

$\langle\Phi_{GS},\hat{\rho}_{0}\Phi_{GS}\rangle\geq 1-c_{0}\frac{N}{2qV},$
(1)

where $c_{0}=(2g_{2}+g_{0})/3$ is the strength of the central (spin-
independent) force. Thus, this inequality demonstrates that the constraint
arises essentially from the central force. This is misleading. Since the spin-
flip is caused by the spin-dependent force with the strength
$c_{2}=(g_{2}-g_{0})/3$, it should be $c_{2}$ to play the essential role.

(ii) There is a great difference between the ground states of Rb ($c_{2}<0$)
and Na ($c_{2}>0$) condensates (Say, when $q=0$, the spins of Rb are coupled
to total spin $S=N$, while the spins of Na are coupled to $S=M$, where $M$ is
the total magnetization). It has been shown by numerical calculation that the
variations of the $\langle\Phi_{GS},\hat{\rho}_{0}\Phi_{GS}\rangle$ versus $q$
of these two species differ from each other greatly (refer to Fig.2b and
Fig.3b of bao12 , where the scales for $q$ are greatly different). Therefore,
the constraints for the two species should be greatly different.

(iii) When a magnetic field is applied, the g.s. is not necessary to have
$M=0$ (In fact, the energy of the states with a larger $M$ may be remarkably
reduced due to the negative linear Zeeman energy). It has been shown that
$\langle\Phi_{GS},\hat{\rho}_{0}\Phi_{GS}\rangle$ in fact depends on $M$
seriously.bao12  The inequality in tasa13  is confined only for the case of
$M=0$. A better constraint should take the $M-dependence$ into account.

(iv) The constraint given by eq.(11) of tasa13  is too loose. To evaluate
numerically, it is assumed that the Na atoms are trapped by a harmonic
potential $U(r)=\frac{1}{2}m\omega^{2}r^{2}$. Then the g.s. of the single-
particle hamiltonian is just the lowest harmonic oscillator state. When
$\hbar\omega$ and $\sqrt{\hbar/m\omega}$ are used as units for energy and
length, according to the definition given in tasa13 , $1/V=(1/\pi)^{3/2}$.
When $q$ and $c_{0}$ are expressed in the new units, their values are
$q=1745(B/G)^{2}/(\omega\sec)$ and $c_{0}=6.77\times
10^{-4}(\omega\sec)^{1/2}$, where $G$ is Gauss and $B$ is the magnetic field
($B/G$ and $\omega\sec$ are both a number). Accordingly, eq.(11) of tasa13
becomes

$\langle\Phi_{GS},\hat{\rho}_{0}\Phi_{GS}\rangle\geq 1-3.48\times
10^{-8}\frac{N(\omega\sec)^{3/2}}{(B/G)^{2}}$ (2)

Since $N$ is usually large, the right side might become negative. For an
example, if $N=10^{4}$ and $\omega=300\sec^{-1}$, the right side becomes
$1-\frac{1.81}{(B/G)^{2}}$. It implies that, when $B<1.34G$, the right side is
negative. Thus, in a large domain quite often accessed, the inequality makes
no sense.

In the derivation of Theorem 2 the contribution from the term
$\langle\Phi_{GS},\hat{V}\Phi_{GS}\rangle$ has been completely neglected. In
general, during the derivation of a formula or an inequality, the neglect of a
small term is allowed. However, At the present case,
$\langle\Phi_{GS},\hat{V}\Phi_{GS}\rangle$ is not a small term. Instead, it is
essential. In order to overcome the above shortcomings, this term should be
recovered.

Incidentally, the statement stated in Theorem 1 is not new. A more general
formula for
$\langle\vartheta_{SM}^{[N]},\hat{\rho}_{\mu}\vartheta_{SM}^{[N]}\rangle$ has
been given in Eq.(10) of Ref.bao12 . When $\mu=0$, the general formula can be
rewritten as

$\langle\vartheta_{SM}^{[N]},\hat{\rho}_{0}\vartheta_{SM}^{[N]}\rangle=\frac{S(S+1)(2N+1)-N-M^{2}(2N+3)}{N(2S+3)(2S-1)}.$
(3)

When $S=M=0$, the above formula gives
$\langle\vartheta_{00}^{[N]},\hat{\rho}_{0}\vartheta_{00}^{[N]}\rangle=1/3$ as
stated in Theorem 1.

## References

  * (1) H. Tasaki, Phys. Rev. Lett. 110, 230402 (2013).
  * (2) C.G. Bao, J. Phys. A, 45, 235002 (2012).


# Obstructions to algebraizing topological vector bundles

Aravind Asok
Department of Mathematics
University of Southern California
Los Angeles, CA 90089-2532
<EMAIL_ADDRESS>Aravind Asok was partially supported by National Science
Foundation Award DMS-1254892. Jean Fasel
Institut Fourier - UMR 5582
Université Grenoble Alpes
F-38402 Saint Martin d’Hères
<EMAIL_ADDRESS>Michael J. Hopkins
Department of Mathematics
Harvard University
Cambridge, MA 02138
<EMAIL_ADDRESS>Michael J. Hopkins was partially supported by National
Science Foundation Award DMS-0906194

###### Abstract

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a
complex topological vector bundle on $X$ (viewed as a complex manifold) to be
algebraic is that all Chern classes must be algebraic cohomology classes,
i.e., lie in the image of the cycle class map. We analyze the question of
whether algebraicity of Chern classes is sufficient to guarantee
algebraizability of complex topological vector bundles. For affine varieties
of dimension $\leq 3$, it is known that algebraicity of Chern classes of a
vector bundle guarantees algebraizability of the vector bundle. In contrast,
we show in dimension $\geq 4$, algebraicity of Chern classes is insufficient
to guarantee algebraizability of vector bundles. To do this, we construct a
new obstruction to algebraizability using Steenrod operations on Chow groups.
By means of an explicit example, we observe that our obstruction is non-
trivial in general.

##  1 Introduction

Suppose $X$ is a smooth complex algebraic variety. We write $X^{an}$ for
$X({\mathbb{C}})$ viewed as a complex manifold. Write $\mathscr{V}_{n}(X)$ for
the set of isomorphism classes of rank $n$ algebraic vector bundles on $X$,
$\mathscr{V}^{an}(X)$ for the set of isomorphism classes of rank $n$ analytic
vector bundles on $X^{an}$ and $\mathscr{V}^{top}(X)$ for the set of
isomorphism classes of rank $n$ complex topological vector bundles on
$X^{an}$.

For any integer $n\geq 0$, the assignment $X\mapsto X^{an}$ gives rise to a
sequence of functions

$\mathscr{V}_{n}(X)\longrightarrow\mathscr{V}^{an}(X)\longrightarrow\mathscr{V}^{top}(X).$

An element of $\mathscr{V}^{top}(X)$ that lies in the image of the composite
map $\mathscr{V}_{n}(X)\longrightarrow\mathscr{V}^{top}(X)$ will be called an
algebraizable vector bundle. The motivating problem of this paper is:
characterize algebraizable vector bundles among topological vector bundles.
This problem is very old; it is studied explicitly for projective varieties of
small dimension, for example, in work of Schwarzenberger [Sch61], Atiyah–Rees
[AR76] and Bănică–Putinar [BP87]. Very little is known about this problem for
varieties of dimension $\geq 4$.

Suppose $\mathcal{E}^{top}\to X^{an}$ is a complex topological vector bundle.
If $\mathcal{E}^{top}$ is algebraizable, then the Chern classes
$c_{i}^{top}(\mathcal{E}^{top})\in H^{2i}(X^{an},{\mathbb{Z}})$ of
$\mathcal{E}^{top}$ are algebraic, i.e., they lie in the image of the cycle
class map. Using this observation, one can show that the composite map is, in
general, neither injective nor surjective. If $X$ is a smooth affine variety,
then by Grauert’s Oka-principle ([Gra58, §2 Satz I, II] or [For11, Theorem
7.2.1] for a textbook treatment), every topological vector bundle on $X$
admits a unique analytic structure, i.e., the map
$\mathscr{V}^{an}(X)\to\mathscr{V}^{top}(X)$ is a bijection. The following
question is a concrete form of the problem stated above.

######  Question 1.

If $X$ is a smooth complex affine variety, and $\mathcal{E}^{top}\to X$ is a
complex topological vector bundle with algebraic Chern classes, then is
$\mathcal{E}^{top}$ is algebraizable?

Question 1 has a positive answer in small dimensions. First, for any smooth
affine variety, by Serre’s splitting theorem [Ser58, Théorème 1], any
algebraic vector bundle of rank $r>\dim X$ can be written as the direct sum of
a vector bundle of rank $\leq\dim X$ and a trivial bundle. Therefore, to
answer Question 1, it suffices to establish that topological vector bundles of
rank below the dimension with algebraic Chern classes are algebraizable. In
any dimension, $Pic(X)$ is the set of isomorphism classes of line bundles on
$X$, so for bundles of rank $1$ in any dimension algebraicity of the first
Chern class guarantees algebraizability.

In dimension $2$, a positive answer to Question 1 follows from the work of
Murthy and Swan [MS76] who show that if $X$ is a smooth complex affine
surface, then for any pair $(c_{1},c_{2})\in CH^{1}(X)\times CH^{2}(X)$ there
is an algebraic vector bundle of rank $2$ with those Chern classes. Similarly,
in dimension $3$, Question 1 is true by work of Mohan Kumar and Murthy [KM82,
Theorem 2.1]. Indeed, Mohan Kumar and Murthy established existence of
algebraic vector bundles with of rank $\leq 3$ with arbitrary prescribed Chern
classes on smooth affine threefolds.

The main result of this paper is that Question 1 admits a negative answer for
smooth complex affine varieties of dimension $\geq 4$. To see this, we will
give a necessary and sufficient condition for algebraizability of rank $2$
bundles on smooth complex affine $4$-folds. We then show, by means of explicit
examples, that the necessary and sufficient condition is, in general, non-
trivial. More precisely, we establish the following results (the first result
is established just after Theorem 2.2.2 in the body of the text).

######  Theorem 2.

Suppose $X$ is a smooth complex affine variety of dimension $4$, and
$\mathcal{E}\to X$ is a rank $2$ complex topological vector bundle with Chern
classes $c_{1}^{top}$ and $c_{2}^{top}$. Assume $c_{1}^{top}$ and
$c_{2}^{top}$ are algebraic, i.e., lie in the image of the cycle class map.
The bundle $\mathcal{E}$ is algebraizable if and only if we can choose
$(c_{1},c_{2})\in CH^{1}(X)\times CH^{2}(X)$ mapping to
$(c_{1}^{top},c_{2}^{top})$ under the cycle class map such that
$Sq^{2}c_{2}+c_{1}\cup c_{2}=0\in CH^{3}(X)/2$; here $Sq^{2}$ is Voevodsky’s
motivic Steenrod squaring operation.

######  Remark 3.

Note that, even if $c_{1}^{top}$ is zero in the above statement, unless the
cycle class map $CH^{1}(X)\to H^{2}(X,{\mathbb{Z}})$ is injective in degree
$1$ we cannot guarantee that $c_{1}$ can be chosen to vanish. Moreover, a
priori it is possible that $Sq^{2}c_{2}+c_{1}\cup c_{2}$ is always zero, but
the following result shows that this is not the case.

######  Theorem 4 (See Corollary 3.1.4).

There exists a smooth hypersurface $Z$ of bidegree $(3,4)$ in
${\mathbb{P}}^{1}\times{\mathbb{P}}^{3}$, such that, setting
$X:=({\mathbb{P}}^{1}\times{\mathbb{P}}^{3})\setminus Z$,

  * (a)

$CH^{i}(X)\to H^{2i}(X,{\mathbb{Z}})$ is an isomorphism for $i\leq 2$ (in
particular, the Chern classes of every topological vector bundle are
algebraic), and

  * (b)

$X$ carries a rank $2$ topological vector bundle $\mathcal{E}$ with Chern
classes $(0,c_{2})$ such that $Sq^{2}c_{2}\neq 0$ (in particular,
$\mathcal{E}$ is not algebraizable).

Theorem 2 is established by first observing that the map
$\mathscr{V}_{n}(X)\to\mathscr{V}_{n}^{top}(X)$ factors through
$[X,BGL_{n}]_{{\mathbb{A}}^{1}}$, i.e., the set of ${\mathbb{A}}^{1}$-homotopy
classes of maps $X$ to $BGL_{n}$. Thus, to produce the obstruction, it
suffices to obstruct existence of an ${\mathbb{A}}^{1}$-homotopy class of
maps. This is accomplished by analyis of the Moore-Postnikov factorization of
the map assigning to the universal rank $2$ vector bundle over the
Grassmannian its Chern classes; these ideas are discussed in Section 2.1. The
primary obstruction to existence of a lifting yields the condition of the
statement, and we establish a vanishing theorem showing that, under suitable
hypotheses, all higher obstructions vanish. To prove necessity of the
vanishing of the obstruction, we appeal to Morel’s vector bundle
classification [AHW15, Theorem 1]: if $X$ is smooth and affine, then
$[X,BGL_{n}]_{{\mathbb{A}}^{1}}\cong\mathscr{V}_{n}(X)$.

The construction of the example in Theorem 4 is closely related with the
failure of the integral Hodge conjecture for the hypersurface $Z$. The failure
of injectivity of the cycle class map $CH^{3}(X)\to
H^{6}(X^{an},{\mathbb{Z}})$ is precisely what allows the examples above to
exist. In Section 3.2, we explain how some general conjectures on algebraic
cycles suggest examples like those above are “generic”. By considering
products of the form $X\times{\mathbb{A}}^{n}$ with $X$ as in Theorem 4, one
may construct examples of non-algebraizable topological vector bundles with
algebraic Chern classes in any dimension $\geq 4$. To place the above results
in a broader context, we propose the following.

######  Conjecture 5.

If $X$ is a smooth complex affine variety, and the cycle class map
$H^{*,*}(X,{\mathbb{Z}})\to H^{*}(X^{an},{\mathbb{Z}})$ from integral motivic
cohomology to ordinary integral cohomology is an isomorphism, then every
topological vector bundle on $X^{an}$ is algebraizable.

Theorem 2 establishes Conjecture 5 for rank $2$ bundles on smooth affine
$4$-folds. Using techniques similar to those used to establish Theorem 2, one
can show that Conjecture 5 is true for rank $3$ and rank $4$ bundles on smooth
complex affine $4$-folds. Using Serre’s splitting theorem, one can then
conclude that Conjecture 5 is true in dimension $4$. Because of the
${\mathbb{A}}^{1}$-representability of vector bundles, Conjecture 5 is
suggested by existence of suitable Adams style resolutions of the motivic
space $BGL_{n}$ by Voevodsky’s motivic Eilenberg-MacLane spaces; we will
discuss this point of view further in [AFH].

#### Acknowledgements

The first-named author would like to thank Burt Totaro for some helpful
correspondence as regards the integral Hodge conjecture on threefolds and for
pointing out the paper [Tot13]. The third-named author would like to thank
Philip Griffiths for correspondence related to the paper [Gri72], which was a
source of inspiration for the questions considered here.

##  2 Obstruction theory: proof of Theorem 2

The goal of this section is to prove Theorem 2. We begin with some
preliminaries regarding Chern classes and obstruction theory in
${\mathbb{A}}^{1}$-homotopy theory. We will use rather freely the terminology
of [MV99, Mor12]; rather than bulking up this paper with a long section of
notatoin and terminology, we have chosen to focus on the argument; we will
follow the notations and conventions of [AF15b, §2.1] and we refer the reader
there for terminology not explicitly defined here (e.g., spaces, homotopy
sheaves, strong and strict ${\mathbb{A}}^{1}$-invariance of sheaves and their
basic properties). When we consider cohomology of a sheaf on a smooth scheme,
we mean cohomology in the Nisnevich site. We also remind the reader that
strongly or strictly ${\mathbb{A}}^{1}$-invariant sheaves are unramified and
that to check a morphism of unramified sheaves is an isomorphism, it suffices
to check this on sections over finitely generated extensions of the base
field. For a general discussion of the Moore-Postnikov factorization in
${\mathbb{A}}^{1}$-homotopy theory, we refer the reader to [AF15b, §6].

###  2.1 Chern classes and the basic obstruction theory problem

Write $BGL_{n}$ for the simplicial classifying space of $GL_{n}$ (see, e.g.,
[MV99, §4.1]). The determinant map $GL_{n}\to{{\mathbf{G}}_{m}}$ induces a map
of classifying spaces $BGL_{n}\to B{{\mathbf{G}}_{m}}$. If $X\to BGL_{n}$ is a
simplicial homotopy class of maps representing a rank $n$ vector bundle on a
smooth scheme $X$, then the composite map $X\to BGL_{n}\to
B{{\mathbf{G}}_{m}}$ represents the determinant line bundle of this vector
bundle. Some ${\mathbb{A}}^{1}$-homotopy sheaves of $BGL_{r}$ were computed in
[Mor12, Theorem 7.20]; we refer the reader to [AF14a, Lemma 3.1, Theorem 3.2]
for convenient references in the form we require.

######  Lemma 2.1.1.

There are canonical isomorphisms
$\bm{\pi}_{1}^{{\mathbb{A}}^{1}}(BGL_{n}){\stackrel{{\scriptstyle{\scriptscriptstyle{\sim}}}}{{\;\rightarrow\;}}}{{\mathbf{G}}_{m}}$
induced by the determinant and
$\bm{\pi}_{2}^{{\mathbb{A}}^{1}}(BGL_{2})\cong{{\mathbf{K}}}^{MW}_{2}$.

The motivic cohomology of $BGL_{n}$ is a polynomial algebra over the motivic
cohomology of a point in variables $c_{1},\ldots,c_{n}$ where $c_{i}$ has
bidegree $(2i,i)$. By Voevodsky’s (unstable)
${\mathbb{A}}^{1}$-representability of motivic cohomology [Del09, §2.3 Theorem
2], each $c_{i}$ corresponds to an ${\mathbb{A}}^{1}$-homotopy class of maps

$c_{i}:BGL_{n}\to K({\mathbb{Z}}(i),2i).$

To be precise, we observe that there is an ${\mathbb{A}}^{1}$-weak equivalence
$Gr_{n}\to BGL_{n}$, where $Gr_{n}$ is the infinite Grassmannian [MV99,
§Proposition 3.7]. The space $Gr_{n}$ can be written as a colimit of smooth
schemes, and the motivic cohomology of $Gr_{n}$ for these finite dimensional
approximations stabilizes by the argument given by Edidin-Graham [EG98,
Definition-Proposition 1 and §2.7]. Here and henceforth, we can fix suitable
${\mathbb{A}}^{1}$-fibrant models of $BGL_{n}$ and $K({\mathbb{Z}}(i),2i)$ so
that the ${\mathbb{A}}^{1}$-homotopy classes of maps above are represented by
actual morphisms of spaces.

Morel and Voevodsky study a complex realization functor [MV99, §3.3]. Under
complex realization $Gr_{n}$ is sent to the usual complex Grassmannian and the
maps $c_{i}:BGL_{n}\to K({\mathbb{Z}}(i),2i)$ are sent to the usual Chern
class maps $c_{i}^{top}:Gr_{n}\to K({\mathbb{Z}},2n)$. Indeed, this follows
from the fact that $Gr_{n}$ is (stably) cellular, and the cycle class map from
Chow groups to (the even part of) ordinary cohomology is an isomorphism. We
use these compatibilities without mention in the sequel.

The space $K({\mathbb{Z}}(n),2n)$ is not an Eilenberg-MacLane space in the
sense that its ${\mathbb{A}}^{1}$-homotopy sheaves are not, in general,
concentrated in a single degree. The following result identifies the
${\mathbb{A}}^{1}$-homotopy sheaves. The first statement is simply a
reformulation of the ${\mathbb{A}}^{1}$-representability of motivic cohomology
mentioned above, while the second statement is a reformulation of the
Nesterenko-Suslin-Totaro theorem (see, e.g., [MVW06, Theorem 5.1]).

######  Lemma 2.1.2.

There is a canonical isomorphism
$\bm{\pi}_{i}^{{\mathbb{A}}^{1}}(K({\mathbb{Z}}(n),2n))\cong{\mathbf{H}}^{2n-i,n}$,
where $\mathbf{H}^{2n-i,n}$ is the sheafification of the presheaf $U\mapsto
H^{2n-i,n}(U,{\mathbb{Z}})$. In particular, $K({\mathbb{Z}}(n),2n)$ is
${\mathbb{A}}^{1}$-$(n-1)$-connected and its first non-vanishing
${\mathbb{A}}^{1}$-homotopy sheaf is ${{\mathbf{K}}}^{M}_{n}$, the $n$-th
unramified Milnor K-theory sheaf.

Ideally, we would study the existence of a vector bundle of rank $2$ with
given Chern classes $(c_{1},c_{2})$ by studying an obstruction theory problem
deduced from the Moore-Postnikov factorizatoin of the map
$(c_{1},c_{2}):BGL_{2}\to K({\mathbb{Z}}(1),2)\times K({\mathbb{Z}}(2),4)$.
The fact that $K({\mathbb{Z}}(n),2n)$ is not an Eilenberg-MacLane space causes
various technical complications and we use our homotopy sheaf computations to
produce a technically simpler problem.

Lemma 2.1.2 yields a canonical map $K(Z(n),2n)\to K({{\mathbf{K}}}^{M}_{n},n)$
corresponding to the first stage of the ${\mathbb{A}}^{1}$-Postnikov tower of
the former. In particular, composition of the universal $n$-th Chern with this
stage of the Postnikov tower yields a map:

(2.1) $c_{n}^{\prime}:BGL_{r}\to K({{\mathbf{K}}}^{M}_{n},n)$

for any $r\geq n$; this is a modified version of the universal $n$-th Chern
class.

######  Remark 2.1.3.

Since ${{\mathbf{K}}}^{M}_{1}={{\mathbf{G}}_{m}}$ and since the motivic
complex ${\mathbb{Z}}(1)$ is ${{\mathbf{G}}_{m}}$ concentrated in a single
degree [MVW06, Theorem 4.1], the map $K({\mathbb{Z}}(1),2)\to
K({{\mathbf{K}}}^{M}_{1},1)$ induced by the Postnikov tower is a simplicial
weak equivalence. In particular, the maps $c_{1}$ and $c_{1}^{\prime}$ lie in
the same simplicial homotopy class. Explicitly identifying the homotopy
sheaves of $\bm{\pi}_{i}^{{\mathbb{A}}^{1}}(K({\mathbb{Z}}(n),2n))$ in all
degres seems at the moment intractable: the Beilinson-Soulé vanishing
conjecture predicts that $K({\mathbb{Z}}(n),2n)$ is
${\mathbb{A}}^{1}$-$(2n-1)$-truncated, i.e., its homotopy sheaves vanish for
$i>2n-1$.

###  2.2 Homotopy sheaves and Moore-Postnikov factorizations

Consider the map

$(c_{1}^{\prime},c_{2}^{\prime}):BGL_{2}\to K({{\mathbf{K}}}^{M}_{1},1)\times
K({{\mathbf{K}}}^{M}_{2},2),$

where $c_{1}^{\prime}$ and $c_{2}^{\prime}$ are the maps mentioned in 2.1. Our
goal is to analyze the homotopy fiber of this map. Write $\mathscr{F}_{2}$ for
the ${\mathbb{A}}^{1}$-homotopy fiber of $(c_{1}^{\prime},c_{2}^{\prime})$.

By Lemma 2.1.1 there is a canonical identification
$\bm{\pi}_{1}^{{\mathbb{A}}^{1}}(BGL_{2})\cong{{\mathbf{G}}_{m}}$ and the map
$c_{1}^{\prime}$ yields an isomorphism of ${\mathbb{A}}^{1}$-fundamental
sheaves of groups (see also Remark 2.1.3). As a consequence, we will be
considering a twisted obstruction theory problem and we will have to keep
track of the action of ${{\mathbf{G}}_{m}}$ on the higher
${\mathbb{A}}^{1}$-homotopy sheaves of the ${\mathbb{A}}^{1}$-homotopy fiber.
We begin by identifying the relevant ${\mathbb{A}}^{1}$-homotopy sheaves of
$\mathscr{F}_{2}$ together with the action of ${{\mathbf{G}}_{m}}$ (for the
relevant definitions about twisted ${\mathbb{A}}^{1}$-homotopy sheaves, we
refer the reader to [AF15b, §2.4]).

######  Proposition 2.2.1.

There are canonical isomorphisms
$\bm{\pi}_{2}^{{\mathbb{A}}^{1}}(\mathscr{F}_{2})\cong\mathbf{I}^{3}$ and
$\bm{\pi}_{3}^{{\mathbb{A}}^{1}}(\mathscr{F}_{2})\cong\bm{\pi}_{3}^{{\mathbb{A}}^{1}}(BGL_{2})\cong\bm{\pi}_{2}^{{\mathbb{A}}^{1}}(SL_{2})$.
Moreover, the actions of ${{\mathbf{G}}_{m}}$ on these sheaves coincide with
the actions described in [AF14b, Propositions 6.3 and 6.5].

###### Proof.

Observe that $K({{\mathbf{K}}}^{M}_{1},1)\times K({{\mathbf{K}}}^{M}_{2},2)$
is ${\mathbb{A}}^{1}$-$2$-truncated, i.e., has ${\mathbb{A}}^{1}$-homotopy
sheaves concentrated in degrees $\leq 2$. The second assertion is therefore
immediate from the isomorphism
$\bm{\pi}_{3}^{{\mathbb{A}}^{1}}(BGL_{2})\cong\bm{\pi}_{2}^{{\mathbb{A}}^{1}}(SL_{2})$.

For the first statement, the long exact sequence in
${\mathbb{A}}^{1}$-homotopy sheaves of a fibration yields (using Lemma 2.1.1)
a short exact sequence

$0\longrightarrow\bm{\pi}_{2}^{{\mathbb{A}}^{1}}(\mathscr{F}_{2})\longrightarrow{{\mathbf{K}}}^{MW}_{2}\longrightarrow{{\mathbf{K}}}^{M}_{2}\longrightarrow
0.$

On the other hand, recall that there is an exact sequence of sheaves of the
form

$0\longrightarrow\mathbf{I}^{3}\longrightarrow{{\mathbf{K}}}^{MW}_{2}\longrightarrow{{\mathbf{K}}}^{M}_{2}\longrightarrow
0;$

this is the sheafified version of the exact sequence of [Mor04, Corollaire
5.4]. We will identify these two exact sequences.

To this end, let us unwind some of the identifications. Recall that the
${\mathbb{A}}^{1}$-fiber sequence

${{\mathbf{G}}_{m}}\longrightarrow BSL_{2}\longrightarrow BGL_{2}$

yields the identification
$\bm{\pi}_{2}^{{\mathbb{A}}^{1}}(BSL_{2})\cong\bm{\pi}_{2}^{{\mathbb{A}}^{1}}(BGL_{2})$.

Identify $SL_{2}\cong Sp_{2}$ and consider the following commutative diagram:

$\textstyle{BSp_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BGL_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BSp_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BGL_{\infty}.}$

The left vertical map induces an isomorphism on
$\bm{\pi}_{2}^{{\mathbb{A}}^{1}}$ by [AF14b, Theorem 2.6] and yields the
isomorphism
$\bm{\pi}_{2}^{{\mathbb{A}}^{1}}(BSp_{2})\cong\bm{\pi}_{2}^{{\mathbb{A}}^{1}}(BSp_{\infty})\cong\mathbf{KSp}_{2}$.
By Suslin’s theorem $\mathbf{KSp}_{2}\cong{{\mathbf{K}}}^{MW}_{2}$ [Sus87,
Theorem 5.2] (see [AF14c, Theorem 4.1.2] and the discussion there for an
explanation in the context we consider). On the other hand, the map on
$\bm{\pi}_{2}^{{\mathbb{A}}^{1}}$ induced by the right vertical map is a map
${{\mathbf{K}}}^{MW}_{2}\to{{\mathbf{K}}}^{M}_{2}$ (see [AF14a, Lemma 3.1 and
Theorem 3.2]). The bottom vertical map induces the hyperbolic map which
coincides with the standard surjection
${{\mathbf{K}}}^{MW}_{2}\to{{\mathbf{K}}}^{M}_{2}$.

The second Chern class map $BGL_{2}\to K({{\mathbf{K}}}^{M}_{2},2)$
necessarily factors through the $BGL_{2}\to BGL_{\infty}$ and the map
$BGL_{\infty}\to K({{\mathbf{K}}}^{M}_{2},2)$ is an isomorphism on
$\bm{\pi}_{2}^{{\mathbb{A}}^{1}}$ by Matsumoto’s theorem identifying Quillen’s
$K_{2}$ of a field with Milnor’s $K_{2}$. Therefore, the second Chern class
map induces a map on $\bm{\pi}_{2}^{{\mathbb{A}}^{1}}$ that, up to the
identifications described above, coincides with the hyperbolic map
${{\mathbf{K}}}^{MW}_{2}\to{{\mathbf{K}}}^{M}_{2}$, which is precisely what is
asserted above.

By construction, the actions mentioned in the statement are inherited from the
${{\mathbf{G}}_{m}}$-actions on homotopy sheaves of $BGL_{2}$ as described in
the referenced propositions. ∎

######  Theorem 2.2.2.

Suppose $X$ is a smooth affine $4$-fold over an algebraically closed field
having characteristic unequal to $2$, and fix a line bundle ${\mathcal{L}}$ on
$X$ with first Chern class $c_{1}\in CH^{1}(X)$. Given $c_{2}\in CH^{2}(X)$,
the pair $(c_{1},c_{2})$ are the first and second Chern classes of a vector
bundle of rank $2$ and determinant ${\mathcal{L}}$ if and only if
$Sq^{2}c_{2}+c_{1}\cup c_{2}=0$.

###### Proof.

Suppose $X$ is a smooth affine $4$-fold over $\operatorname{Spec}k$ and fix
$(c_{1},c_{2})\in CH^{1}(X)\times CH^{2}(X)$ as in the statement. Such a pair
determines a map $\psi:X\to
K({{\mathbf{K}}}^{M}_{1},1)\times{{\mathbf{K}}}({{\mathbf{K}}}^{M}_{2},2)$.
Fix a line bundle ${\mathcal{L}}$ on $X$ representing $c_{1}$. We now analyze
the Moore-Postnikov factorization of the map
$(c_{1},c_{2}^{\prime}):BGL_{2}\to K({{\mathbf{K}}}^{M}_{1},1)\times
K({{\mathbf{K}}}^{M}_{2},2)$.

This analysis proceeds in several steps, which we now outline. First, we
analyze the primary obstruction: we begin by constructing a “quotient” of the
primary obstruction, which is easier to identify. Then, we show that this
“quotient” of the primary obstruction actually coincides with the primary
obstruction by means of suitable cohomological vanishing statements. Next, we
analyze the secondary obstruction. We show the secondary obstruction vanishes
again by establishing a general cohomological vanishing result. Finally, we
appeal to general theory to show that no further obstructions can arise.

Step 1: Analyzing the primary obstruction. The primary obstruction to lifting
$\psi$ to a map $X\to BGL_{2}$ is, by means of Proposition 2.2.1, a class in
$H^{3}(X,\mathbf{I}^{3}({\mathcal{L}}))$.

Step 1a: A quotient of the primary obstruction. To identify this obstruction
more explicitly, we consider the exact sequence of [Fas13, §2.1 p. 423]:

$0\longrightarrow\mathbf{I}^{j+1}({\mathcal{L}})\longrightarrow\mathbf{I}^{j}({\mathcal{L}})\longrightarrow\mathbf{I}^{j}/\mathbf{I}^{j+1}\longrightarrow
0.$

By the sheafified version of the Milnor conjecture on quadratic forms [OVV07]
there is a canonical isomorphism of sheaves
$\mathbf{I}^{j}/\mathbf{I}^{j+1}\cong{{\mathbf{K}}}^{M}_{j}/2$. We consider
the composite map (via this isomorphism)

$H^{2}(X,{{\mathbf{K}}}^{M}_{2})\longrightarrow
H^{3}(X,\mathbf{I}^{3}({\mathcal{L}}))\longrightarrow
H^{3}(X,{{\mathbf{K}}}^{M}_{3}/2).$

The first map here is precisely the $k$-invariant in the Moore-Postnikov
factorization and the identifications of Proposition 2.2.1 show that it is
exactly the connecting homomorphism in the long exact sequence in cohomology
associated with the exact sequence of sheaves on $X$:

$0\longrightarrow\mathbf{I}^{3}({\mathcal{L}})\longrightarrow{{\mathbf{K}}}^{MW}_{2}({\mathcal{L}})\longrightarrow{{\mathbf{K}}}^{M}_{2}\longrightarrow
0.$

The composite of this connecting homomorphism with the reduction map is
studied in [AF15a]. Indeed, generalizing a result of Totaro [Tot03], we prove
in [AF15a, Theorem 4.1.4] that if $c_{2}\in H^{2}(X,{{\mathbf{K}}}^{M}_{2})$,
then the composite map just described

$H^{2}(X,{{\mathbf{K}}}^{M}_{2})\longrightarrow
H^{3}(X,{{\mathbf{K}}}^{M}_{3}/2)$

sends $c_{2}\mapsto Sq^{2}c_{2}+c_{1}({\mathcal{L}})\cup c_{2}$.

Step 1b: the actual primary obstruction. The relationship between the
obstruction just mentioned and the actual primary obstruction is measured by
the difference between the groups $H^{3}(X,\mathbf{I}^{3}({\mathcal{L}}))$ and
$H^{3}(X,{{\mathbf{K}}}^{M}_{3}/2)$. In this case, we have the long exact
sequence

$\cdots\longrightarrow H^{3}(X,\mathbf{I}^{4}({\mathcal{L}}))\longrightarrow
H^{3}(X,\mathbf{I}^{3}({\mathcal{L}}))\longrightarrow
H^{3}(X,{{\mathbf{K}}}^{M}_{3}/2)\longrightarrow
H^{4}(X,\mathbf{I}^{4}({\mathcal{L}}))\longrightarrow\cdots.$

Since $X$ is a smooth affine $4$-fold, by [AF14b, Proposition 5.2] the groups
$H^{3}(X,\mathbf{I}^{4}({\mathcal{L}}))$ and
$H^{4}(X,\mathbf{I}^{4}({\mathcal{L}}))$ vanish. Thus, the the primary
obstruction coincides with the “quotient” constructed in Step 1a, i.e., the
primary obstruction to lifting is precisely the vanishing of
$Sq^{2}c_{2}+c_{1}\cup c_{2}$.

Step 2: Analyzing the secondary obstruction. If the primary obstruction
vanishes, then we can choose a lift to the second stage of the Moore-Postnikov
factorization of the map $(c_{1},c_{2}^{\prime})$. Upon choosing a lift to
this stage, we obtain a secondary obstruction lying in
$H^{4}(X,\bm{\pi}_{3}^{{\mathbb{A}}^{1}}({\mathscr{F}}_{2})({\mathcal{L}}))$.
By means of Proposition 2.2.1, this obstruction is an element of
$H^{4}(X,\bm{\pi}_{3}^{{\mathbb{A}}^{1}}(BGL_{2})({\mathcal{L}}))$. A priori,
this obstruction depends on the choice of lift, but we claim
$H^{4}(X,\bm{\pi}_{3}^{{\mathbb{A}}^{1}}(BGL_{2})({\mathcal{L}}))=0$,
independent of this choice.

To see this, recall by [AF14b, Theorem 3.3] that
$\bm{\pi}_{3}^{{\mathbb{A}}^{1}}(BSp_{2})$ is an extension of
${{\mathbf{K}}}^{Sp}_{3}$ by a certain sheaf $\mathbf{T}^{\prime}_{4}$.
Observe first that $H^{4}(X,\mathbf{GW}^{2}_{3}({\mathcal{L}}))=0$ by explicit
construction of the Gersten resolution. Indeed,
$H^{4}(X,\mathbf{GW}^{2}_{3}({\mathcal{L}}))$ is simply a quotient of
$\bigoplus_{x\in
X^{(4)}}(\mathbf{GW}^{2}_{3}({\mathcal{L}}))_{-4}(\kappa_{x})$ and the latter
vanishes by [AF14b, Lemma 4.11].

By [AF14b, Corollary 4.9] the induced ${{\mathbf{G}}_{m}}$-action on
$\mathbf{T}^{\prime}_{4}$ is trivial. Again using [AF14b, Theorem 3.3],
$\mathbf{T}^{\prime}_{4}$ is a quotient of $\mathbf{S}^{\prime}_{4}$ by
$\mathbf{D}_{5}$; the former is a quotient of ${{\mathbf{K}}}^{M}_{4}/12$
while the latter is a quotient of $\mathbf{I}^{5}$. Observe that
$H^{4}(X,{{\mathbf{K}}}^{M}_{4}/12)\cong CH^{4}(X)/12$. Since $X$ is smooth
and affine over an algebraically closed field, $CH^{4}(X)$ is divisible and
therefore, $H^{4}(X,{{\mathbf{K}}}^{M}_{4}/12)$ is trivial. On the other hand,
$H^{4}(X,\mathbf{I}^{5})$ is trivial as well since the sheaf $\mathbf{I}^{5}$
is itself trivial [AF14b, Proposition 5.1].

Step 3. Lifting to $BGL_{2}$. Since the secondary obstruction vanishes, we may
choose a lift to the third stage of the Moore-Postnikov factorization of the
map. Since $X$ has Nisnevich cohomological dimension $4$, there are no further
obstructions to lifting and we may arbitrarily make a choice of lift of $X$ to
the $4$-th stage of the Moore-Postnikov factorization of the map
$(c_{1},c_{2}^{\prime})$. Moreover, again using the fact that $X$ has
Nisnevich cohomological dimension $4$ we see that beyond the $4$-th stage of
the factorization all lifts are uniquely determined. Thus, by the same
argument as [AF14b, Proposition 6.2], we obtain an element of
$[X,BGL_{2}]_{{\mathbb{A}}^{1}}$. Combining the discussion of the previous
steps we see that a necessary and sufficient condition to lift
$(c_{1},c_{2})\in CH^{1}(X)\times CH^{2}(X)$ to an element of
$[X,BGL_{2}]_{{\mathbb{A}}^{1}}$ is the vanishing of $Sq^{2}c_{2}+c_{1}\cup
c_{2}\in CH^{3}(X)/2$.

Step 4. Geometrization. Finally, we apply Morel’s
${\mathbb{A}}^{1}$-representability theorem for vector bundles [AHW15, Theorem
1] to identify $[X,BGL_{2}]_{{\mathbb{A}}^{1}}$ with $\mathscr{V}_{2}(X)$. ∎

###### Proof of Theorem 2.

Consider the map

$(c_{1}^{top},c_{2}^{top}):BGL_{2}({\mathbb{C}})\longrightarrow
K({\mathbb{Z}},2)\times K({\mathbb{Z}},4).$

The map $(c_{1}^{top},c_{2}^{top})$ is a $4$-equivalence and thus the homotopy
fiber is $5$-connected. If $X$ is a smooth affine $4$-fold, then $X$ has the
homotopy type of a CW complex of dimension $\leq 4$. A straightforward
obstruction theory argument then shows that there is a bijection
$\mathscr{V}_{2}^{top}(X){\stackrel{{\scriptstyle{\scriptscriptstyle{\sim}}}}{{\;\rightarrow\;}}}H^{2}(X,{\mathbb{Z}})\times
H^{4}(X,{\mathbb{Z}})$. Combining this observation with Theorem 2.2.2 yields
Theorem 2. ∎

######  Remark 2.2.3.

We believe it is possible to establish Theorem 2 by a direct analysis of the
Moore-Postnikov factorization of $(c_{1},c_{2}):BGL_{2}\to
K({\mathbb{Z}}(1),2)\times K({\mathbb{Z}}(2),4)$, but a first complication
arises in identifying the ${\mathbb{A}}^{1}$-homotopy sheaves of the
${\mathbb{A}}^{1}$-homotopy fiber. Moreover, explicit identification of
primary obstruction also becomes significantly more difficult: this seems to
require understanding Nisnevich cohomology of $K({\mathbb{Z}}(2),4)$ with
coefficients in ${{\mathbf{K}}}^{M}_{3}/2$.

##  3 Constructing examples: proof of Theorem 4

The goal of this section is to construct explicit examples of smooth complex
affine varieties for which (a) the cycle class map $CH^{i}(X)\to
H^{2i}(X,{\mathbb{Z}})$ is an isomorphism for $i\leq 2$ and (b) there exist
classes $\alpha\in CH^{2}(X)$ such that $Sq^{2}\alpha\neq 0$.

###  3.1 Explicit examples of non-algebraizable vector bundles

If $\dim X<3$, $CH^{3}(X)$ is trivial by definition and if $\dim X=3$ then
$CH^{3}(X)$ is divisible (which is classical); thus, if $\dim X\leq 3$, then
$CH^{3}(X)/2$ is trivial. Therefore the first dimension that can support non-
trivial examples of the kind we envision is dimension $4$.

######  Proposition 3.1.1.

Suppose $Y$ is a smooth projective variety of dimension $\geq 4$, $Z$ is an
ample hypersurface on $Y$ (i.e., ${\mathcal{O}}_{Y}(Z)$ is an ample line
bundle) and set $X:=Y\setminus Z$. If the cycle class map $CH^{i}(Y)\to
H^{2i}(Y,{\mathbb{Z}})$ is an isomorphism for $i\leq 2$, then the cycle class
maps $CH^{i}(X)\to H^{2i}(X,{\mathbb{Z}})$ is an isomorphism in the same
range.

###### Proof.

First, by the assumption on the dimension and the hypothesis that $Z$ is an
ample hypersurface, the Grothendieck-Lefschetz theorem guarantees that the
pullback map $Pic(Y)\to Pic(Z)$ is an isomorphism [Har70, Theorem 1.5]. By the
usual Lefschetz hyperplane theorem [Voi07, Theorem 1.1], the map
$H^{i}(Y,{\mathbb{Z}})\to H^{i}(Z,{\mathbb{Z}})$ is an isomorphism for $i\leq
2$. Since the cycle class map commutes with pullbacks [Voi07, Proposition
9.2.1(i)] and since the map $Pic(Y)\to H^{2}(Y,{\mathbb{Z}})$ is an
isomorphism by assumption, we conclude that $Pic(Z)\to H^{2}(Z,{\mathbb{Z}})$
is an isomorphism.

Since the cycle class map commutes with Gysin maps [Voi07, Proposition
9.2.1(ii)], there is a commutative diagram of the form

$\textstyle{CH^{i-1}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{CH^{i}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{CH^{i}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{H^{2i-2}(Z,{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2i}(Y,{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2i}(X,{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

The left vertical map is an isomorphism for $i=1$ essentially by definition,
and an isomorphism for $i=2$ by the discussion of the previous paragraph. The
middle map is an isomorphism for $i\leq 2$ by assumption. Combining these
facts, a diagram chase yields the statement we wanted. ∎

Fix an isomorphism
$CH^{*}({\mathbb{P}}^{1}\times{\mathbb{P}}^{3})\cong{\mathbb{Z}}[\xi,\tau]/\langle\xi^{2},\tau^{4}\rangle$.
If $Z$ is a smooth hypersurface of bidegree $(d_{1},d_{2})$, then under this
isomorphism $[Z]=d_{1}\xi+d_{2}\tau$. We now identify the Chow groups of the
complement of $Z$ in ${{\mathbb{P}}^{1}}\times{\mathbb{P}}^{3}$ in low
degrees; we do this since the computation is concrete and elementary (the full
strength of this statement is not necessary in Theorem 3.1.3 below).

######  Proposition 3.1.2.

Suppose $Z\subset{{\mathbb{P}}^{1}}\times{\mathbb{P}}^{3}$ is a hypersurface
of bidegree $(d_{1},d_{2})$ with $d_{1}\neq 0,d_{2}\neq 0$. Set
$g=gcd(d_{1},d_{2})$ and pick $m$ and $n$ such that $md_{1}+nd_{2}=g$. If
$X:={{\mathbb{P}}^{1}}\times{\mathbb{P}}^{3}\setminus Z$, then

$\begin{split}CH^{1}(X)&\cong{\mathbb{Z}}/d_{1}{\mathbb{Z}}\oplus{\mathbb{Z}}/d_{2}{\mathbb{Z}},\text{
and }\\\
CH^{2}(X)&\cong{\mathbb{Z}}/g{\mathbb{Z}}\oplus{\mathbb{Z}}/{\scriptstyle\frac{d_{2}^{2}}{g}}{\mathbb{Z}}.\end{split}$

Under the first isomorphism ${\mathbb{Z}}/d_{1}{\mathbb{Z}}$ is generated by
the image of $\xi$ while ${\mathbb{Z}}/d_{2}{\mathbb{Z}}$ is generated by the
image of $\tau$. Under the second isomorphism, the class $\xi\tau$ is sent to
$(1,-\frac{md_{2}}{g})$, while the class $\tau^{2}$ is sent to a generator of
${\mathbb{Z}}/{\scriptstyle\frac{d_{2}^{2}}{g}}{\mathbb{Z}}$. If $d_{1}\nmid
d_{2}$ and $d_{2}\nmid d_{1}$ (e.g., if $gcd(d_{1},d_{2})=1$), then $\xi\tau$
can be assumed to restrict non-trivially to $CH^{2}(X)$.

###### Proof.

It suffices to identify the map $CH^{j-1}(Z)\to
CH^{j}({\mathbb{P}}^{1}\times{\mathbb{P}}^{3})$. Since
$i^{*}:CH^{j-1}({\mathbb{P}}^{1}\times{\mathbb{P}}^{3})\to CH^{j-1}(Z)$ is an
isomorphism for $j\leq 2$ as observed in the proof of Proposition 3.1.1, it
remains to compute the cokernel of the map
$i^{*}i_{*}:CH^{j-1}({\mathbb{P}}^{1}\times{\mathbb{P}}^{3})\to
CH^{j}({\mathbb{P}}^{1}\times{\mathbb{P}}^{3})$, which comes from the
intersection with a divisor formula [Ful98, Proposition 2.6c].

By definition $[Z]\in CH^{1}({{\mathbb{P}}^{1}}\times{\mathbb{P}}^{3})$ is
precisely $d_{1}\xi+d_{2}\tau$. The first isomorphism of the statement then
follows immediately from this:
$CH^{1}(X)\cong{\mathbb{Z}}\xi\oplus{\mathbb{Z}}\tau/\langle
d_{1}\xi+d_{2}\tau\rangle$. For the second isomorphism we proceed as follows.
Consider the matrix representing $i^{*}i_{*}$ with respect to the bases
$\xi,\tau$ of $CH^{1}({\mathbb{P}}^{1}\times{\mathbb{P}}^{3})$ and
$\xi\tau,\tau^{2}$ of $CH^{2}({\mathbb{P}}^{1}\times{\mathbb{P}}^{3})$. This
matrix is not diagonal, but can be put in Smith normal form via the identity:

$\begin{pmatrix}1&0\\\
-\frac{md_{2}}{g}&1\end{pmatrix}\begin{pmatrix}d_{2}&d_{1}\\\
0&d_{2}\end{pmatrix}\begin{pmatrix}n&-\frac{d_{1}}{g}\\\
m&\frac{d_{2}}{g}\end{pmatrix}=\begin{pmatrix}g&0\\\
0&\frac{d_{2}^{2}}{g}\end{pmatrix}$

In particular, the cokernel of the map $i^{*}i_{*}$ can be computed from that
of $diag(g,\frac{d_{2}^{2}}{g})$, which is what was asserted in the statement.
The statements regarding the images of $\xi\tau$ and $\tau^{2}$ are immediate,
and the final statement follows from the choice of Bezout identity presenting
$gcd(d_{1},d_{2})$ that arises from the Euclidean algorithm. ∎

######  Theorem 3.1.3.

Suppose $Z\subset{\mathbb{P}}^{1}\times{\mathbb{P}}^{3}$ is a smooth complex
hypersurface of bidegree $(3,4)$ defined over $\overline{{\mathbb{Q}}}$ that
specializes modulo some prime $p$ to the singular hypersurface
$y_{0}^{3}x_{0}^{4}+y_{0}^{2}y_{1}x_{1}^{4}+y_{0}y_{1}^{2}x_{2}^{4}+y_{1}^{3}x_{3}^{4}$
over $\overline{\mathbb{F}}_{p}$. The classes $\xi\tau$ and $\xi\tau^{2}$ both
restrict non-trivially from ${\mathbb{P}}^{1}\times{\mathbb{P}}^{3}$ to
$X=({\mathbb{P}}^{1}\times{\mathbb{P}}^{3})\setminus Z$ and if $\psi$ is the
image of $\xi\tau\in CH^{2}(X)$, then $Sq^{2}\psi\neq 0\in CH^{3}(X)/2$.

###### Proof.

Observe that $Sq^{2}(\xi\tau)=\xi\tau^{2}+\xi^{2}\tau=\xi\tau^{2}$ in
$CH^{*}({\mathbb{P}}^{1}\times{\mathbb{P}}^{3})$ by the Cartan formula [Voe03,
Proposition 9.7] (note that $Sq^{1}(\xi)=Sq^{1}(\tau)=0$ since
$H^{3,1}(W,{\mathbb{Z}})$ vanishes for any smooth scheme $W$). Since Steenrod
operations are compatible with pullbacks the statement that $Sq^{2}(\psi)$ is
non-trivial will follow from the assertion that $\xi\tau$ and $\xi\tau^{2}$
restrict non-trivially to $X$. Since $gcd(3,4)=1$, the fact that $\xi\tau$
restricts non-trivially to $X$ follows immediately from Proposition 3.1.2.
Thus, it remains to show that $\xi\tau^{2}$ restricts non-trivially to $X$.

To this end, we use an idea of Totaro who showed that every curve in $Z$ has
even degree over ${\mathbb{P}}^{1}$. Indeed, see [Tot13, Proof of Theorem 3.1]
for this precise statement. Granted this statement, one deduces immediately
that $i_{*}[C]$ for a $1$-cycle $C$ on $Z$ is of the form
$2r\xi\tau^{2}+s\tau^{3}$. In particular, $\xi\tau^{2}$ does not lie in the
image of $i_{*}$ or the corresponding map for Chow groups modulo $2$ and
therefore restricts non-trivially to $CH^{3}(X)/2$. (Note: this last
observation reproves the fact that $\xi\tau$ restricts non-trivially to $X$
since it follows immediately from the previous statement that $Sq^{2}(\psi)$,
which is the restriction of $Sq^{2}(\xi\tau)=\xi\tau^{2}$ to $X$, is non-
trivial.)

∎

######  Corollary 3.1.4.

If $X$ is a variety as described in the statement of Theorem 3.1.3, then any
rank $2$ topological vector bundle on $X$ has algebraic Chern classes. For a
given $c_{2}\in H^{4}(X,{\mathbb{Z}})$, there is a unique topological vector
bundle $\mathcal{E}$ over $X$ with $c_{1}(\mathcal{E})=0$ and given
$c_{2}(\mathcal{E})=c_{2}$. Moreover, if $c_{2}$ is the image of $\xi\tau$
under restriction, then the associated topological vector bundle has algebraic
Chern classes yet fails to be algebraizable.

###### Proof.

By Proposition 3.1.1 the algebraicity of Chern classes is true for any pair
$(c_{1},c_{2})$. As discussed in the proof of Theorem 2 at the end of
Subsection 2.2 there map $(c_{1},c_{2}):\mathscr{V}_{2}^{top}(X)\to
H^{2}(X,{\mathbb{Z}})\times H^{4}(X,{\mathbb{Z}})$ is a pointed bijection.
Thus, the pair $(0,c_{2})$ determines a unique rank $2$ topological vector
bundle on $X$. In the case where $c_{2}$ is the image of $\xi\tau$ under
restriction, it follows immediately from Theorem 3.1.3 that $Sq^{2}(c_{2})\neq
0\in CH^{3}(X)/2$. Therefore, the result follows from Theorem 2.2.2. ∎

###  3.2 Conjectures on cycles and genericity of the examples

The example of the previous section seems intimately related to the failure of
the integral Hodge conjecture. We explore this connection in greater detail
now.

######  Example 3.2.1.

There are examples due to Kollár-van Geemen that show the integral Hodge
conjecture can fail for hypersurfaces in projective space [Tre92] or [SV05,
Theorem 2]. Indeed, suppose $Z\subset{\mathbb{P}}^{4}$ is a hypersurface of
degree $d$. Assume that for some integer $p$ coprime to $6$, $p^{3}$ divides
$d$. Then, for a general $Z$, any curve $C\subset Z$ has degree divisible by
$p$. Observe that, in this case, $Pic(Z)\cong{\mathbb{Z}}$ by the
Grothendieck-Lefschetz theorem. Setting $X:={\mathbb{P}}^{4}\setminus Z$, we
can compute $CH^{i}(X)$ in low degrees. Identify
$CH^{*}({\mathbb{P}}^{4})={\mathbb{Z}}[\xi]/\xi^{5}$. Then,
$CH^{1}(X)={\mathbb{Z}}/d{\mathbb{Z}}$ generated by the image of $\xi$ and
$CH^{2}(X)={\mathbb{Z}}/d{\mathbb{Z}}$ generated by the class of $\xi^{2}$.
Note that $Sq^{2}(\xi^{2})=2\xi Sq^{2}\xi=0\in CH^{3}(X)/2$. Take a
topological complex vector bundle on $X$ of rank $2$ with Chern classes of the
form $(m\xi,a\xi^{2})$. If $m$ is even, all such bundles are necessarily
algebraizable. If $m$ is odd, then $Sq^{2}(a\xi^{2})+m\xi\cup
a\xi^{2}=am\xi^{3}$. If $a$ is also odd, then $am$ is odd and so $am\xi^{3}$
could restrict non-trivially to $CH^{3}(X)/2$. However, the construction above
only shows that $CH^{3}(X)$ is a quotient of ${\mathbb{Z}}/n{\mathbb{Z}}$
where $p|n$. However, since $p$ is odd, we do not know whether $\xi^{3}$
restricts non-trivially to $CH^{3}(X)/2$.

######  Example 3.2.2.

By [Tot13, Lemma 5.1], there is a smooth hypersurface
$Z\subset{\mathbb{P}}^{4}$ of degree $48$ over $\bar{Q}$ for which the
integral Hodge conjecture fails. The construction of this example is, however,
rather involved. Set $X={\mathbb{P}}^{4}\setminus Z$. In this case,
$Pic(X)\cong{\mathbb{Z}}/48{\mathbb{Z}}$ and
$CH^{2}(X)\cong{\mathbb{Z}}/48{\mathbb{Z}}$, generated by $\xi$ and $\xi^{2}$
in the notation of Example 3.2.1. Totaro shows that every curve $C\subset X$
has even degree over $\bar{Q}$, i.e., the pushforward map $CH^{2}(Z)\to
CH^{3}(X)$ has image contained in $2\xi^{3}$. In particular, in this example
$CH^{3}(X)/2$ is necessarily non-trivial, and $\xi^{2}$ restricts non-
trivially. As before, if we fix Chern classes $(m\xi,a\xi^{2})$ with both $a$
and $m$ odd, then $ma\xi^{3}$ restricts non-trivially to $CH^{3}(X)/2$ and we
thus obtain more non-algebraizable bundles with algebraic Chern classes.

Let us recall the following conjecture of Nori, as modified by Totaro.

######  Conjecture 3.2.3 (Nori,Totaro).

If $Y$ is a smooth projective variety, and $Z\subset Y$ is a very general,
sufficiently ample hypersurface, then the restriction map $CH^{i}(Y)\to
CH^{i}(Z)$ is an isomorphism for $i<\dim Z$.

Combined with Theorem 2, Conjecture 3.2.3 suggests that non-algebraizable
topological complex vector bundles of rank $2$ should be rather common.

######  Example 3.2.4.

Under the hypotheses of Conjecture 3.2.3, set $\xi=[Z]\in Pic(Y)$ and
$X:=Y\setminus Z$, which is necessarily affine. Observe then that if $Z\subset
Y$ has dimension $3$, then $Pic(X)=Pic(Y)/\langle\xi\rangle$,
$CH^{2}(X)=CH^{2}(Y)/\langle\xi\cup Pic(Y)\rangle$, while
$CH^{3}(X)=CH^{3}(Y)/\langle\xi\cup CH^{2}(Y)\rangle$. Thus, assuming Nori’s
conjecture if $Y={\mathbb{P}}^{4}$ then for $d$ sufficiently large, and any
sufficiently general hypersurface of degree $d$, then
$Pic(X)\cong{\mathbb{Z}}/d{\mathbb{Z}}$,
$CH^{2}(X)\cong{\mathbb{Z}}/d{\mathbb{Z}}$ and
$CH^{3}(X)\cong{\mathbb{Z}}/d{\mathbb{Z}}$ generated by the image of
$\xi,\xi^{2},\xi^{3}$. If $d$ is even, then we expect only those topological
complex vector bundles of rank $2$ whose Chern classes are of the form
$(m\xi,a\xi^{2})$ with $a$ and $m$ both odd to be non-algebraizable.

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