Stably A^1-connected varieties and universal triviality of CH_0

0.16 MB·

by  Aravind Asok

A1
Stably -connected varieties
CH
and universal triviality of
6 0
1
0
2 Aravind Asok∗

n
a
J

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Abstract

] WestudytherelationshipbetweenseveralnotionsofconnectednessarisinginA1-homotopy
G
theoryofsmoothschemesoverafieldk: A1-connectedness,stableA1-connectednessandmo-
A
tivic connectedness, and we discuss the relationship between these notations and rationality
h. properties of algebraic varieties. Motivically connected smooth proper k-varieties are pre-
t cisely those with universally trivial CH0. We show that stable A1-connectedness coincides
a
m withmotivicconnectedness,undersuitablehypothesesonk. Then,weobservethatthereexist
stably A1-connected smooth proper varieties over the field of complex numbers that are not
[ A1-connected.

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v ToChuckWeibelontheoccasionofhis65thbirthday.
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1 Introduction
1
0
1. Suppose k isafield, and X isaconnected smooth projective k-variety ofdimension d. Recall that
0 X iscalledstablyk-rationalifthereexistsanintegerm ≥ 0suchthatk(X)(t ,...,t )isapurely
1 m
6
transcendental extension ofk. Determining whethera“nearly rational” (e.g.,rationally connected)
1
: varietyis(not)stablyk-rational isarathersubtleproblemingeneral.
v
Ithas been know forsometimethat ifX isastably k-rational variety, then the Chowgroup of
i
X 0-cycles is universally trivial: for every extension L/k, the degree map CH (X ) → Z is an iso-
0 L
r
a morphism(see,e.g.,[Mer08]). TheconditionofuniversaltrivialityofCH0 forasmoothprojective
variety X is equivalent to the existence of a Chow-theoretic decomposition of the diagonal (see,
e.g., [Voi14b]forgeneral context or[CTP14, Proposition 1.4]for thisstatement). Recently, Voisin
has shown, using degeneration techniques, that studying universal triviality of CH is fruitful for
0
disproving (stable) rationality of smooth proper varieties [Voi15b]. Her method was refined and
extended by a number of authors [CTP14, Tot15] and we now know much more about, e.g., the
failureofstablerationality ofhypersurfaces inprojective spaces.
Thispaper,inspiredbysomerelationships betweenrationalityquestions andA1-homotopythe-
ory explored in[AM11], iscentered around the following question: are there invariants of“homo-
topic nature” intermediate between universal triviality of CH and stable rationality? To explain
0
the source of this question we observe that, using duality in Voevodsky’s derived category of mo-
tives in DM (see, e.g., [MVW06]), universal triviality of CH of X is equivalent to the motive
k 0
∗AravindAsokwaspartiallysupportedbyNationalScienceFoundationAwardDMS-1254892.
1
2 1Introduction
M(X) having trivial zeroth homology sheaf (see Lemma 2.1.3 for a proof). Thus, universal trivi-
ality of CH for a smooth projective variety X is equivalent to the condition of being motivically
0
connected (see Definition 2.1.1). On the other hand, in [AM11, Theorem 2.3.6], it was observed
that stably k-rational varieties, over fields k having characteristic zero, are necessarily connected
fromthestandpoint oftheMorel-Voevodsky unstable A1-homotopy category[MV99].
If we write H (k) for the Morel-Voevodsky A1-homotopy category, sending a smooth scheme
X to its motive M(X) factors through a Hurewicz style functor Sm → H (k) → DM . How-
k k
ever, as in topology, the Hurewicz homomorphism factors through a stabilization with respect to
suspension:
Sm −→ H (k) −→ SH −→ DM ;
k k k
hereSH isthestableA1-homotopycategory ofP1-spectra (see,e.g.,[Jar00]).
k
It makes sense to ask whether the object in SH determined by a smooth scheme k (given by
k
a P1-suspension spectrum) is “connected” and we will call a smooth projective variety X stably
A1-connected ifit is“connected” in this sense (again, see Definition 2.1.1). Onehas the following
sequence ofimplications relating these various notions ofconnectedness forsmooth projective va-
rietiesoverafieldkhavingcharacteristic 0(seeLemma2.1.2andthediscussionthatprecedesitfor
moreprecisestatements):
{stablyrational} =⇒{A1-connected} =⇒{stably A1-connected} =⇒{motivically connected}.
We can refine the vague question posed above to the following: are these implications strict? The
maingoalofthisnoteistoanswerthisquestion.
Theorem 1 (See Theorem 2.2.1). If k is an infinite perfect field having finite 2-cohomological
dimension, then a smooth proper k-variety X is motivically connected if and only if it is stably
A1-connected.
The key tools in the proof are (i) the description of the zeroth stable A1-homotopy sheaf of a
smoothproperk-varietyobtainedin[AH11a]intermsofthe“Chow-Wittgroupofzerocycles”and
(ii) a spectral sequence relating a certain filtration on Chow-Witt groups from their description in
terms of cohomology of Milnor-Witt K-theory sheaves and the filtration by powers of the funda-
mental ideal in Milnor-Witt K-theory; this spectral sequence, which isaslight modification of one
consideredbyPardonandTotaro,wasdescribedin[AF15]. WereferthereadertoRemark2.2.2for
an“explanation” oftheresultinvolving Voevodsky’s slicetower.
Proposition 2 (See Proposition 2.3.2). There exist stably A1-connected smooth proper complex
varieties ofanydimension ≥ 2thatfailtobeA1-connected.
Granted Theorem 1, one knows that there exist stably A1-connected smooth proper varieties
oversomenon-algebraicallyclosedfields(examplesarisingintheworkofParimalaarediscussedin
Section 2.3). Proposition 2refines thisobservation byconstructing minimaldimensional examples
over C. It seems reasonable to expect that examples of stably A1-connected varieties that are not
A1-connected existoveralgebraically closedfieldshaving positivecharacteristic aswell.
3 2A1-homotopicnotionsofconnectedness
Over fields k having characteristic 0 there exist examples of A1-connected smooth proper k-
varieties that are not stably k-rational (we explain the examples, which can be found in work of
Colliot-The´le`ne–Sansuc on rationality problems for tori, in Example 2.3.3). We do not know of
suchanexampleoveranalgebraically closedfield.
Acknowledgements
The results below were established in the course of the Fall2015 edition of the Los Angeles alge-
braic geometry seminar, which focused onstable rationality. Inparticular, theauthor wouldlike to
thankBurtTotaroforhelpfuldiscussionsandcommentsonanearlyversionofthispaperandSasha
Merkurjevforpointing outExample2.3.3.
Notation/Conventions
Throughout this paper the letter k will be used to denote a field. By a smooth k-scheme, we will
mean a scheme that is separated, smooth and has finite type over Speck. We write Sm for the
k
category of such objects. By a sheaf on Sm , we will always mean a sheaf for the Nisnevich
k
topology.
2 A1-homotopic notions of connectedness
Section 2.1 recalls key points of homotopy theory of algebraic varieties in order to make defini-
tions regarding connectivity, i.e., Definition 2.1.1. Here, we also explain the precise link between
“motivic connectedness” and universal trivality of CH . Section 2.2 contains the proof of the re-
0
sultthatmotivically connected smoothpropervarietiesarestablyA1-connected; thisresultappears
as Theorem 2.2.1. Section 2.3 contains an analysis of stably A1-connected varieties that are not
A1-connected, whichappears inProposition 2.3.2.
2.1 Homotopycategories inbrief
Write Spc for the category of simplicial presheaves on Sm . Sending a smooth scheme X to
k k
the corresponding representable presheaf on Sm and viewing presheaves on Sm as simplicially
k k
constant simplicial presheaves, the Yoneda lemma shows that Sm → Spc is a fully-faithful em-
k k
bedding. There is an A1-local model structure on Spc [MV99, §3], and we write H (k) for the
k
corresponding homotopy category. We write πA1(X) for the Nisnevich sheaf associated with the
0
presheaf onSmk givenbyU 7→ HomH(k)(U,X)and∗fortherepresentable presheaf correspond-
ingtoSpeck.
We can consider the category of symmetric spectra valued in Spc ; this has a stable model
k
structure andwewriteSH forthecorresponding homotopy category. Asmooth schemeX deter-
k
mines a suspension symmetric P1-spectrum, denoted Σ∞X , and sending X 7→ Σ∞X extends
P1 + P1 +
to a functor H (k) −→ SH . We set S0 := Σ∞Speck ; this is the unit object for a symmetric
k k P1 +
monoidal structure ∧ on SH . As usual, one may discuss notions of connectivity for spectra, and
k
∞
it is a theorem of F. Morel that, at least if k is an infinite field, then Σ X is a (−1)-connected
P1 +
spectrum (see [Mor05, Theorem 6.1.8] and [Mor04a, 5.2.1-5.2.2]). We define the sheaf of stable
4 2.1 Homotopycategoriesinbrief
A1-connected components, denoted πsA1(X), as the Nisnevich sheaf associated with the presheaf
0
U 7→ HomSHk(S0k ∧Σ∞P1U+,Σ∞P1X+). Weset π0sA1(S0k) := π0sA1(Speck). Acelebrated result of
Morel[Mor12,Theorem6.43]showsthatπsA1(Speck) = GW,theunramifiedGrothendieck-Witt
0
sheaf(seealso[Mor04b,§6]).
Voevodsky’s category DM is discussed in, e.g., [MVW06]. The functor sending a smooth
k
scheme X to its corresponding representable presheaf with transfers extends to a functor SH →
k
DM .1 We write M(X)–the motive of X–for the object in DM corresponding to a smooth
k k
scheme X. The object M(X) has an explicit model as a bounded below complex of presheaves
with transfers. By definition, the homology sheaves of this complex (viewed simply as sheaves of
abelian groups) are concentrated in degrees > −1. The zeroth homology sheaf H (M(X)) is, by
0
definition, thezerothSuslinhomologysheafHS(X).
0
With this notation and terminology, we may introduce more precisely the various notions of
connectedness weconsider.
Definition2.1.1. SupposeX isasmoothk-scheme. WesaythatX is
1. A1-connected ifπA1(X) ∼= ∗;
0
2. stablyA1-connected ifπsA1(X) → πsA1(S0)isanisomorphism; or
0 0 k
3. motivically connected ifthecanonical mapHS(X) → Zisanisomorphism.
0
It was observed in [AM11, Theorem 2.3.6] that if k has characteristic 0, then stably k-rational
smoothproperk-varietiesareA1-connected. Notethattheconclusion of[AM11,Theorem2.3.6]is
alsotrueif“weakfactorization” holdsoverk. Thus,forexample,oneconcludesthatsmoothproper
k-rational surfaces over an arbitrary field are A1-connected. With this in mind, the relationship
betweenthenotions inDefinition2.1.1issummarizedinthefollowingresult.
Lemma2.1.2. SupposeX isasmoothproper k-varietyoveraninfinitefieldk.
1. IfX isA1-connected, thenX isstablyA1-connected.
2. IfX isstablyA1-connected, thenX ismotivically connected.
Proof. These results are essentially consequences of Morel’s stable A1-connectivity theorem (it
is appeal to this result that forces us to impose the hypothesis that k be infinite). Indeed, using
this result, one equips SH and DM with t-structures whose heart can be described explicitly:
k k
the heart of SH is the category of “homotopy modules” [Mor04a, Definition 5.2.4 and Theorem
k
5.2.6], while the heart of DM is the category of strictly A1-invariant sheaves with transfers. The
k
sheaf πsA1(X) is initial among homotopy modules admitting a map from X. With this in mind,
0
Point(1)followsfromanevidentmodificationof[Aso13,Proposition 3.5].
For Point (2), assume X is stably A1-connected, so πsA1(X) ∼= GW. Now, by assumption,
0
for any object M in the homotopy t-structure on SH , the canonical map M(k) → M(X) is a
k
bijection. Inparticular,thisistrueforMarisingfromstrictlyA1-invariantsheaveswithtransfers. In
thatcase,anymapπsA1(X) → MfactorsuniquelythroughHS(X)andweconcludebyappealing
0 0
to[Aso13,Lemma3.3]andtheYonedalemma.
1Strictlyspeaking,thisisalie,andweshouldconsiderananalogofDM constructedwithunboundedcomplexes.
k
However,alltheobjectsweconsiderwillbehighlyconnectedsonoharmisdone.
5 2.2 StableA1-connectednessvs.motivicconnectedness
The following easy (and well-known) lemma provides the connection between connectivity in
thesensejustdefinedanduniversaltrivialityofCH inthesensementionedintheintroduction; we
0
includeitheresimplyforcompleteness.
Lemma2.1.3. IfX isasmoothproperk-scheme,thenX ismotivicallyconnected ifandonlyiffor
everyextension fieldL/k,CH (X )istrivial.
0 L
Proof. The sheaf HS(X) is strictly A1-invariant, i.e., all of its cohomology presheaves are A1-
0
invariant. Asaconsequenceofthis,onededucesthatthesheafHS(X)isdeterminedbyitssections
0
onfinitelygenerated extensions ofthebasefield.
If X is an irreducible smooth proper k-scheme of dimension d, then one deduces that the fol-
lowingsequence ofisomorphism holdusingdualityandthecancellation theorem [Voe10]:
HS0(X)(SpecL):= HomDML(Z,M(XL))
∼= HomDM (M(XL)(−d)[−2d],Z)
L
∼= HomDM (M(XL),Z(d)[2d])
L
∼= CHd(X )= CH (X ).
L 0 L
Now, if HS(X)(SpecL) is trivial, then CH (X ) → Z is an isomorphism. Since we can write
0 0 L
any extension as an increasing union of its finitely generated sub-extensions, we conclude that
CH (X ) → Z is an isomorphism for extensions L/k that are not necessarily finitely generated.
0 L
Conversely, if CH (X ) → Z is an isomorphism for all finitely generated extensions, then by the
0 L
observationofthefirstparagraph,weconcludethatHS(X) → Zisanisomorphismofsheaves.
0
2.2 Stable A1-connectedness vs. motivicconnectedness
Theorem 2.2.1. Suppose k isaninfinite perfect fieldthathas finite2-cohomological dimension. If
X isamotivicallyconnected smoothprojective k-variety, thenX isstablyA1-connected.
Proof. Assume that HS(X) → Z is an isomorphism. Wewant to show that HsA1(X) → GW is
0 0
anisomorphism. Generalizing thefirstpart oftheproof ofLemma2.1.3,itwasshownin[AH11a,
Theorem 5.3.1] that πsA1(X)(L) →∼ CgHn(X ,ω ) = Hn(X ,KMW(ω )) in a fashion com-
0 L XL L n XL gn
patiblewithfieldextensions andfunctorial withrespecttopushforwards.2 HereCH (X ,ω )is
L XL
thetwistedChow-Wittgroupintroduced in[Fas08,§10].
F. Morel showed that there is a short exact sequence of sheaves on the small Nisnevich (or
Zariski)siteofX oftheform
L
0−→ In+1(ω ) −→ KMW(ω ) −→ KM −→ 0.
XL n XL n
Moreover, In+1(ω ) admits a decreasing filtration In+j(ω ) ⊃ In+j+1(ω ) with successive
XL XL XL
subquotients identified, via the Milnor conjecture on quadratic forms, with KM /2; see [AF15,
n+2
§2.1]forallofthisnotation.3 Wenowanalyzethepushfoward alongX → SpecLinmoredetail.
L
2Thestatementof[AH11a,Theorem5.3.1]alsoimposestheadditionalhypothesisthatkhascharacteristicunequal
to2. ThishypothesiswasimposedtheretoappealtoresultsaboutMilnor-Wittcohomology;thenecessaryresultswere
establishedin[Mor12]inthecasekhascharacteristic2aswell,atleastifkisperfect.
3Thecarefulreadermaywanttoassumethatkhascharacteristicunequalto2. Thereisasmallprobleminthefiber
productpresentationoftheMilnor-WittK-theorysheafdescribedin[Mor04c]thatisfixedin[GSZ16]. Asexplainedin
[Mor12,p.54],theseresultscanbeextendedtocharacteristic2aswell,replacingreferenceto[Mor04c]by[GSZ16].
6 2.2 StableA1-connectednessvs.motivicconnectedness
Webegin by analyzing the induced push-forward at the level of the graded pieces of the filtra-
tion. Theassumption HS(X) → Zisanisomorphism impliesthat CH (X ) ∼= Hn(X ,KM) =
0 0 L L n
Z. Theedgemapinthehypercohomology spectralsequenceforthemotiviccomplexesZ(n)yields
a map H2n+1,,n+1(X,Z/2) → Hn(X ,KM /2); this map is an isomorphism by connectivity
L n+j
estimates[Voe03,Lemma4.11]. Again,byduality
H2n+j,,n+j(X,Z/2) = HomDM (M(X),Z/2(n+j)[2n+j]) ∼= HomDM(Z,M(X)⊗Z/2(j)[j]).
k
Because M(X) is (−1)-connected, the weak Ku¨nneth formula allows us to identify the first non-
zerohomologysheafofM(X)⊗Z/2(j)[j] intermsofVoevodsky’stensorproductofthefirstnon-
zerohomologysheavesoftheconstituents;thisresultfollowsbytransposingtheproofof[AWW15,
Proposition3.3.9]toDM . Thus,weconcludeH (M(X)⊗Z/2(j)[j]) ∼= HS(X)⊗A1KM/2. In
k 0 0 j
particular, evaluating onLweconcludethatH (M(X)⊗Z/2(1)[1])(L) ∼= KM/2(L);oncemore
0 j
observethatthisidentification isfunctorial withrespect tofieldextensions. Thus,weconclude that
thepushfoward mapHn(X ,KM /2) →∼ KM/2(L)isfunctorial withrespecttofieldextensions.
L n+j j
Next,oneseesthatpushforward induces thefollowingmorphismofexactsequences
Hn(X ,In+1(ω )) // Hn(X ,KMW(ω )) // Hn(X ,KM) // 0
L XL L n XL L n
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
I1(L) ////GW(L) // Z //0.
Note that the morphism one term to the left is simply the map Hn−1(X ,KM) → 0 since the
L n
group H−1(Speck,KM)istrivial byconstruction. Thetworightvertical mapsareisomorphisms.
0
Therefore, ifthe mapHn(X ,In+1(ω )) → I1(L)isan isomorphism, then bythe fivelemmait
L XL
followsthatthemapHn(X ,KMW(ω )) → GW(L)isanisomorphism.
L n XL
Finally, we analyze the pushforward Hn(X ,In+1(ω )) → I1(L) using the (truncated) Par-
L XL
don spectral sequence from [AF15, §2.2]; this is the spectral sequence induced by the filtration of
the Gersten-Witt complex by sub-complexes corresponding to higher powers of the fundamental
ideal.4 We observe only that the d -differential has bidegree (1,r − 1) for r ≥ 2, and En,∗ =
r 2
Hn(X ,KM /2). Since k is assumed to have finite 2-cohomological dimension, the truncated
L n+∗
Pardon spectral sequence is bounded. Note that since X has dimension n, there are no outgoing
L
differentials from the column En,∗ so Hn(X ,In+1(ω )) admits a filtration whose associated
L XL
gradedisdescribed intermsofquotients ofHn(X ,KM /2).
L n+∗
Now, recall that pushfoward map Hn(X ,In+1(ω )) → I1(L) comes from a pushforward
L XL
defined atthelevelofGersten-Witt complexes [Fas08, The´ore`me8.3.4and Corollaire 10.4.5]; this
complex-level pushforward respects the filtration by powers of the fundamental ideal and thus in-
duces a morphism of truncated Pardon spectral sequences. Since we already saw that the maps
Hn(X ,KM /2) → KM/2(L) are isomorphisms for every j ≥ 0, the morphism of Pardon
L n+j j
spectral sequences induced by pushforward thus identifies the n-th column of the E -page of the
2
truncated Pardon spectral sequence for Hn(X ,In+1(ω )) with the 0-th column of the E -page
L XL 2
of the truncated Pardon spectral sequence for H0(SpecL,I1). In particular, we conclude that all
4Whenkhascharacteristic2,oneappealstotheRost-Schmidcomplexdescribedin[Mor12,Remark5.13]insteadof
theGersten-Wittcomplex.
7 2.3 A1-connectednessvs.stableA1-connectedness
incoming differentials on this column are trivial, and the isomorphism wehoped to construct is an
immediateconsequence.
Remark 2.2.2. The aforementioned result should not be viewed as surprising. Indeed, the proof
givenabovecanbeviewedasanexplicitformofanargumentinvolving Voevodsky’sslicefiltration
[Voe02]. Indeed, the zero slice ofthe sphere spectrum isVoevodsky’s motivic Eilenberg-MacLane
spectrumHZandeachsliceisamoduleoverHZ[Lev08]. Ifkhasfinitecohomologicaldimension,
theslicefiltrationofΣ∞X yieldsaconvergentspectralsequenceallowingustocomputeπsA1(X)
P1 + 0
in terms of information from motivic cohomology [Lev13]. That the filtration by powers of the
fundamental ideal coincides with the slice filtration is established in [Lev11]. We leave it to the
interested reader to recast the result in those terms. Note, however, that our assumption that k has
finite2-cohomological dimensionusesslightlyweakerhypotheses thantheargumentjustsketched.
Remark2.2.3. ItwouldbeinterestingtoknowwhetherTheorem2.2.1holdswithouttheassumption
thatk hasfinite2-cohomological dimension.
In line with Remark 2.2.2, Theorem 2.2.1 admits an interpretation in the theory of birational
motives developed in [KS15b, KS15c]. More precisely, combining [KS15b, Proposition 3.1.1],
Lemma2.1.3andTheorem2.2.1,wededucethefollowingresult(seealso[Tot14,Theorem2.1]for
otherequivalentcharacterizations).
Corollary 2.2.4. Assume k is a field having finite 2-cohomological dimension. If X is a smooth
properk-variety, thenX isstablyA1-connected ifandonlyifthebirational motiveofX istrivial.
2.3 A1-connectedness vs. stableA1-connectedness
If k is not algebraically closed, then it is known that one can find smooth proper k-varieties that
are motivically connected but not A1-connected since there are varieties that have a 0-cycle of
degree 1 but no k-rational point (see [Aso13, Example 4.19] where an example due to Parimala is
discussed). Wenow improvethis observation byproducing minimal dimensional counterexamples
over an algebraically closed field. The moral of the story is that the idea of “becoming connected
after P1-suspension is rather subtle”. We begin with the following lemma, which goes back to
Bloch-Srinivas [BS83,Remark2p. 1252].
Lemma2.3.1. IfX isasmoothcomplex surface such thatCH (X) ∼= ZandsuchthatH∗(X,Z)
0
istorsion free,thenHS(X) = Z.
0
Proof. In light of Lemma 2.1.3, this is an immediate consequence of [Voi15a, Corollary 2.2] or
[ACTP13,Proposition 1.9].
Recall that if X ∈ Sm , then two points x ,x ∈ X(k) are said to be elementarily R-
k 0 1
equivalent if, setting k[t] to be the semi-localization of k[x] at 0and 1, there exists an element
(0,1)
of x(t) ∈ X(k[t] ) such that x(0) = x and x(1) = x . Following Manin, we say that two
(0,1) 0 1
elements of X(k) are R-equivalent if they are equivalent for the equivalence relation generated
by elementary R-equivalence; wewrite X(k)/R for the set of R-equivalence classes of k-rational
points. IfX,Y ∈ Sm ,then(X ×Y)(k)/R ∼= X(k)/R×Y(k)/R.
k
Proposition 2.3.2. Supposek isaperfectfield.
8 2.3 A1-connectednessvs.stableA1-connectedness
1. EverystablyA1-connected smoothpropercurveisA1-connected.
2. For every integer d ≥ 2, there exist stably A1-connected smooth projective C-varieties of
dimension dthatarenotA1-connected.
Proof. For Point (1) suppose C is a smooth projective curve. By [SV96, Theorem 3.1], we know
that HS(C) = Pic(C). In particular, if C has genus > 0 then Pic(C)(k¯) = Pic(C ), which is
0 k¯
notisomorphic toZ. Thus,wecanassumewithoutlossofgenerality thatC hasgenus0,i.e.,itisa
conic. Inthatcase,sinceHS(C)∼= Z,weconcludethatC containsa0-cycleofdegree1,i.e.,ithas
0
rationalpointsofextensions ofcoprimedegree. Inparticular, ithasarationalpointaftermakingan
extension of odd degree. Now any conic that has a rational point over an extension of odd degree
is necessarily isomorphic to P1 (this follows, e.g., from a theorem of Springer [Lam05, Theorem
VII.2.7],butismucheasier). ThefactthatP1 isA1-connected canbeseeninmanyways;weleave
thistothereader.
ByLemma2.3.1,ifY isanysmooth proper surface withtorsion free integral cohomology and
CH (S) ∼= Z,thenY ismotivically connected. Now,letS beanysmooth proper surface asinthe
0
previoussentence withKodairadimension> −∞;theclassofsuchsurfaces isnon-emptyas,after
the work of Voisin [Voi14a, Corollary 2.5], for example, it contains Barlow surfaces [Bar85]. By
Theorem2.2.1,anysuchS isstablyA1-connected aswell.
On the other hand, a smooth proper variety S is A1-connected if and only if it is universally
R-trivialby[AM11,Theorem2.4.3](i.e.,foreveryfinitelygenerated separable extension L/k,the
setX(L)/R = ∗). Inparticular,A1-connectedvarietiesarerationallyconnected(moreweakly,they
haveKodairadimension−∞). However,theS fromthepreviousparagraphhasKodairadimension
> 0byassumption.
Finally, we produce examples of dimension > 2. Since S is not A1-connected, there exists a
finitely generated separable extension L/k such that S(L)/R 6= ∗. Then, for any A1-connected
varietyX,sincethemapS×X → S inducesabijection(S×X)(L)/R → S(L)/R,weconclude
thatS×X isnotA1-connected. However,oneagainapplyingtheweakKunnethformula(asinthe
proofofTheorem2.2.1)weconcludethatHS(S ×X) ∼= HS(S)aswell.
0 0
Example 2.3.3. If k is not algebraically closed, then there exist examples ofA1-connected smooth
projective varieties that are not stably k-rational. Suppose T is a torus that is retract k-rational
but not stably k-rational. By [CTS77, Proposition 13], any smooth compactification X of T is
universally R-trivial. Therefore by [AM11, Corollary 2.4.4] we conclude that X is A1-connected
and,byassumption, notstablyrational. Ontheotherhand,suchtoriactuallyexist;seealso[HY15,
Theorems1.3-1.7]andthereferences therein.
Remark 2.3.4. Write ΩP1 for the derived P1-loop space functor on H (k). For any pointed space
(X,x),therearecanonicalmorphismsX → ΩP1ΣP1X → Ω2P1Σ2P1X → ···. WetakeX = X+
forasmoothproperk-schemeX. TosaythatX isstablyA1-connected isequivalent tosaying that
themapcolim Ωn Σn X → colim Ωn Σn Speck isanisomorphism afterapplying πA1(−).
n P1 P1 + n P1 P1 + 0
If X = Speck, then the value of this colimit is actually achieved for n = 2 [Mor12, Theorem
6.43]. Thus, given a stably A1-connected smooth proper k-variety, it makes sense to ask whether
there exists a finite integer n such that Ωn Σn X → Ωn Σn Speck becomes an isomorphism
P1 P1 + P1 P1 +
afterapplying πA1(−)forallN ≥ n(cf. [AH11b,Remark2]forcloselyrelated discussion).
0
9 2.4 BirationalinvarianceofsomehigherChowgroups
Remark2.3.5. Asacounter-pointtoProposition2.3.2,werecallthatasmoothproperk-varietyX is
A1-connectedifandonlyifX isconnected,X(k)isnon-empty,andthegenericpointη ∈X(k(X))
is R-equivalent to x. This result follows immediately by combining [AM11, Theorem 2.4.3] and
[KS15a, Theorem 8.5.1]; the proof is a straightforward consequence of existence of specialization
maps for R-equivalence classes (see, e.g., D. Madore [Mad05, Proposition 3.1], J. Kolla´r [Kol04])
or [AM11, Lemma 6.2.3]). We view this characterization of A1-connectedness as analogous to
the characterization of motivic connectedness in terms of a Chow-theoretic decomposition of the
diagonal.
2.4 Birational invarianceofsomehigherChow groups
Eventhough Theorem 2.2.1showsthatπsA1(X)isnomorerefinedofaninvariant thanHS(X),it
0 0
nevertheless highlights otherwitnessestofailureofuniversaltriviality ofCH . Indeed, thefunctor
0
L 7→ H2n+j,n+j(X ,Z/2), which appears in the proof of Theorem 2.2.1 extends to a strictly A1-
L
invariant sheaf (with transfers) on Sm (this extension can be defined as the sheafification for the
k
Nisnevich topology of U 7→ H2n+j,n+j(U ×X,Z/2)); wewillwrite H2n+j,n+j(X,Z/2) for this
sheaf. Weobserve, thatthesesheavesprovideadditional stablebirational invariants ofX.
One knows that motivic cohomology groups Hp,q(X,Z) of a smooth variety X vanish for
p > q + dimX. The line of groups Hp,q with p = q + dimX need not vanish and appear in
the proof of Theorem 2.2.1. We now observe that these groups are, quite generally, stable bira-
tional invariants of smooth proper varieties; we rephrase this in homological terms to eliminate
dependence ondimension.
Proposition 2.4.1. For any integer j ≥ 0 and any commutative ring R, the motivic Borel-Moore
homology group H−j,−j(X,R) := HomDM(R(−j)[−j],M(X)) is a stable birational invariant
ofsmoothproperk-varieties.
Proof. Weuse various standard properties ofmotivic cohomology; see [MVW06, §16] for details.
Bydefinition: H−j,−j(X ×Pn) = HomDM (R(−j)[−j]),M(X ×Pn). Bytheprojective bundle
k
formulaHomDMk(R(−j)[−j]),M(X×Pn) ∼=HomDMk(R(−j)[−j],⊕ni=0M(X)(i)[2i]). Again
byVoevodsy’scancellationtheorem[Voe10],weknowthatHomDM (R(−j)[−j],M(X)(i)[2i]) ∼=
k
HomDM (R,M(X)(i+j)[2i+j]) andbyduality,itiseasytoseethatthisgroupvanishesifi > 0.
k
Toconclude, itsuffices toobserve that H−j,−j(X,R) isabirational invariant, and this follows
by an evident modification of the usual proof by the action of correspondences ([Ful98, Lemma
16.1.11]).
We can sheafify these groups for the Nisnevich topology: simply consider H−j,−j(X) :=
H (M(X)(j)[j]); these groups can also be defined as the sheafification for the Nisnevich topol-
0
ogyofthepresheaf U 7→ H2dimX+j,dimX+j(U ×X,Z).
Proposition 2.4.2. Suppose X is a smooth proper variety. If HS(X) → Z, then for every j ≥ 0,
0
themapH−j,−j(X) → H−j,−j(Speck) = KMj isanisomorphism.
Proof. SinceH−j,−j(X) := H0(M(X)(j)[j]), itfollowsfromtheweakKu¨nneththeorem(seethe
proofofTheorem2.2.1;oneadaptstheproofin[AWW15,Proposition3.3.9])thatH (M(X)(j)[j]) ∼=
0
HS(X)⊗A1 H (Z(j)[j]) ∼= HS(X)⊗A1 KM.
0 0 0 j
10 REFERENCES
Remark 2.4.3. Assume one is working over an algebraically closed base field k. There is a cycle
class map H2d+j,d+j(X,Z) is H2d+j(X,Zℓ(d + j)). If j > 0, this map is zero for dimensional
´et
reasons. Therefore, if H−j,−j(X) is a non-trivial invariant, it detects arithmetic as opposed to
topological information.
Weconclude withthefollowingexample, whichshowsthatthegroups H−1,−1(X)actually do
naturally arise.
Example 2.4.4. The groups H−1,−1(X) can also be identified as Hd(X,KQd+1) = Hd(X,KMd+1)
forasmoothvarietyofdimensiond(hereKQ isthesheafificationoftheQuillenK-theorypresheaf
i
on Sm for the Nisnevich topology). In this guise, they appear in the very definition of norm
k
varietiesandthustheproofoftheBloch-Katoconjecture [SJ06,Theorem0.1].
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