Grothendieck--Lefschetz type theorems for the local Picard group

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by  János Kollár

GROTHENDIECK–LEFSCHETZ TYPE THEOREMS
FOR THE LOCAL PICARD GROUP
3
1 JA´NOSKOLLA´R
0
2

n Aspecialcaseofthe LefschetzhyperplanetheoremassertsthatifX is asmooth
a projectivevarietyandH ⊂X anampledivisorthen the restrictionmapPic(X)→
J
Pic(H) is an isomorphism for dimX ≥4 and an injection for dimX ≥3.
0
If X is normal, then the isomorphism part usually fails. Injectivity is proved in
3
[Kle66, p.305] and an optimal variant for the class group is established in [RS06].

] Forthelocalversionsofthesetheorems,studiedin[Gro68],theprojectivevariety
G
is replacedby the germofasingularity(p∈X)andthe ample divisorby a Cartier
A divisorp∈X ⊂X. Theusual(global)Picardgroupisreplacedbythelocal Picard
0
. group Picloc(p∈X); see Definition 7.
h
Grothendieck proves in [Gro68, XI.3.16] that if depth O ≥ 4 then the map
t p X
a
between the local Picard groups Picloc(p ∈ X) → Picloc(p ∈ X ) is an injection.
m 0
Note that this does not imply the Lefschetz version since a cone over a smooth
[
projective variety usually has only depth 2 at the vertex.

2 The aim of this note is to propose a strengthening of Grothendieck’s theorem
v that generalizes Kleiman’s variant of the global Lefschetz theorem. Then we prove
7
some special cases that have interesting applications to moduli problems.
1
3 Problem 1. Let X be a normal (or S and pure dimensional) scheme, X ⊂X a
2 0
0
Cartier divisor and x ∈ X a closed point. Assume that dim X ≥ 4. What can
. 0 x
1 one say about the kernel of the restriction map between the local Picard groups
1
2 restX :Picloc(x∈X)→Picloc(x∈X )? (1.1)
X0 0
1
We consider three conjectural answers to this question.
:
v
(2) The map (1.1) is an injection if X is S .
i 0 2
X (3) The kernel is p-torsion if X is an excellent, local F -algebra.
p
r (4) The kernel is contained in the connected subgroup of Picloc(x∈X).
a
The main result of this paper gives a positive answer to the topological variant
(1.4) in some cases. The precise conditions in Theorem 13 are technical and they
might even seem unrealistically special. Instead of stating them, I focus on three
applications first.
Myinterestinthissubjectstartedwithtryingtounderstandhigherdimensional
analogs of three theorems and examples concerning surface singularities and their
deformations; see [Ses75, Art76, Har10] for introductions. The main results of this
note imply that none of them occurs for isolated singularities in dimensions ≥3.
2 (Three phenomena in the deformation theory of surfaces).
(1) ThereisaprojectivesurfaceS withquotientsingularitiesandamplecanon-
0
ical class such that S has a smoothing {S :t∈D} where S is a rational
0 t t
surface for t6=0.
1
2 JA´NOSKOLLA´R
Explicit examples were written down only recently (see [LP07, PPS09a,
PPS09b]orthesimpler[Kol13,3.76]),butithasbeenknownforalongtime
that K2 can jump in flat families of surfaces with quotient singularities.
ThesimplestsuchexampleisclassicalandwasknowntoBertini(though
he probably did not consider K2 for a singular surface). Let C ⊂ P5 be
4
the cone overthe degree4 rationalnormalcurvein P4. Ithas twodifferent
smoothings. In one family {S : t ∈D} (where S := C ) the general fiber
t 0 4
isP1×P1 ⊂P5 embeddedbyOP1×P1(1,2). Intheotherfamily{Rt :t∈D}
(where R0 :=C4) the general fiber is P2 ⊂P5 embedded by OP2(2).
Note that K2 =9 and K2 =8 for t6=0, thus K2 jumps in one of the
Rt St
families. In fact, it is easy to compute that K2 = 9, so the jump occurs
C4
in the family {S :t∈D}.
t
(2) There are non-normal, isolated, smoothable surface singularities (0 ∈ S )
0
whose normalization is simple elliptic [Mum78].
(3) Every rational surface singularity (0 ∈ S ) has a smoothing that admits a
0
simultaneous resolution.
It is known that such smoothings form a whole component of the defor-
mationspaceDef(0∈S ),calledthe Artin component[Art74]. Generaliza-
0
tionsofthiscanbeusedtodescribeallcomponentsofthedeformationspace
of quotient singularities [KSB88], and, conjecturally, of all rational surface
singularities and many other non-rational singularities [Kol91, dJvS92].
The higher dimensional versions of these were studied with the ultimate aim
of compactifying the moduli space of varieties of general type; see [Kol12] for an
introduction. The generaltheory of [KSB88, Kol12] suggests that one should work
with log canonical singularities; see [KM98] or (9) for their definition and basic
properties. This guides our generalizations of (2.1–2).
In order to develop (2.3) further, recall that a surface singularity is rational iff
its divisor class group is finite; see [Mum61].
In this paper we state the results for normal varieties. In the applications to
moduli questions one needs these results for semi-log canonical pairs (X,∆). All
the theorems extend to this general setting using the methods of [Kol13, Chap.5];
see the forthcoming [Kol14, Chap.3] for details.
We say that a variety W is smooth (resp. normal) in codimension r if there is a
closedsubscheme Z ⊂W of codimension ≥r+1 suchthat W \Z is smooth (resp.
normal).
Theorem 3. None of the above examples (2.1–3) exists for varieties with isolated
singularities in dimension ≥3. More generally the following hold.
(1) Let X be a projective variety with log canonical singularities and ample
0
canonical class. If X is smooth in codimension 2 then every deformation
0
of X also has ample canonical class.
0
(2) Let X be a non-normal variety whose normalization is log canonical. If
0
X is normal in codimension 2 then X is not smoothable, it does not even
0 0
have normal deformations.
(3) Let X be a normal variety whose local class groups are torsion and {X :
0 t
t ∈ D} a smoothing. If X is smooth in codimension 2 then {X : t ∈ D}
0 t
does not admit a simultaneous resolution.
GROTHENDIECK–LEFSCHETZ TYPE THEOREMS 3
Our results in Sections 3–4 are even stronger; we need only some control over
the singularities in codimension 2.
SuchresultshavebeenknownifX (orits normalization)hasrationalsingulari-
0
ties. Theseessentiallyfollowfrom[Gro68,XI.3.16];see[Kol95]fordetails. Thusthe
newpartofTheorem3.1–2isthattheclaimsalsoholdforlogcanonicalsingularities
that are not rational.
Further comments and problems. 1
The theorems of SGA rarely have unnecessary assumptions, so an explanation
is needed why Problem 1 could be an exception. One reason is that while our
assumptions are weaker, the conclusions in [Gro68] are stronger.
Theorem4. [Gro68,XI.2.2]LetX beaschemeofpuredimension n+1,X ⊂X a
0
CartierdivisorandZ ⊂X aclosedsubschemesuchthatZ :=X ∩Z hasdimension
0 0
≤ n−3. Let D∗ be a Cartier divisor on X \Z such that D∗| extends to a
X0\Z0
Cartier divisor on X . Assume furthermore that X is S and depth O ≥ 3.
0 0 2 Z0 X0
Then D∗ extends to a Cartier divisor on X, in some neighborhood of X . (cid:3)
0
In Problem 1 we assume that Z is contained in X , thus it is not entirely sur-
0
prising that the depth condition depth X ≥3, could be relaxed.
Z0 0
Example12showsthatinTheorem4theconditiondepth X ≥3isnecessary.
Z0 0
Another reasonwhy Problem1 may have escaped attention is that the topolog-
ical version of it fails. In (28–29) we construct normal, projective varieties Y (of
arbitrarylarge dimension) with a single singular point y ∈Y and a smooth hyper-
planesectionH (notpassingthroughy)suchthatthe restrictionmapH2(Y,Q)→
H2(H,Q) is not injective. However, the kernel does not contain (1,1)-classes. In
the example Y even has a log canonical singularity at y.
The arguments in Section 5 show that, at least over C, a solution to Problem 1
would be implied by the following.
Problem5. LetW beanormalSteinspaceofdimension≥3andLaholomorphic
line bundle on W. Assume that there is a compact set K ⊂ W such that the
restriction of c (L) is zero in H2(W \K,Z).
1
Does this imply that c (L) is zero in H2(W,Z)?
1
Another approach would be to use intersection cohomology to restore Poincar´e
duality in (26.2). For this to work, the solution of the following is needed.
Problem 6. Let W be a normal analytic space. Is there an exact sequence
H1(W,O )→Pican(W)⊗Q→IH2(W,Q)? (6.1)
W
Note first that the sequence exists. Indeed, let g : W′ → W be a resolution of
singularities. If L is a line bundle on W, then g∗L is a line bundle on W′ hence it
has a Chern class c1(cid:0)g∗L) ∈ H2(W′,Z). By the decomposition theorem [BBD82]
IH2(W,Q) is a direct summand of H2(W′,Q) (at least for algebraic varieties).
Arapura explained to me that the sequence (6.1) should be exact for projective
varieties by weight considerations but the general complex case is not clear. For
our applications we need the case when W is Stein.
1RecentlyBhattanddeJongprovedConjecture1.3ingeneralandConjecture1.2forschemes
essentiallyoffinitetypeincharacteristic0.
4 JA´NOSKOLLA´R
Acknowledgments. I thank D. Arapura, C. Xu and R. Zong for discussions and
remarks. The formulation of Conjectures 1.2–4 owes a lot to the comments and
results of B. Bhatt and A. J. de Jong. The referees comments helped to eliminate
several inaccuracies. Partial financial support was provided by the NSF under
grant number DMS-07-58275 and by the Simons Foundation. Part of this paper
was written while the author visited the University of Utah.
1. Definitions and examples
Definition 7 (Local Picard groups). Let X be a scheme and p∈X a point. The
local Picard group Picloc(p ∈ X) is a group whose elements are S sheaves F on
2
some neighborhood p ∈ U ⊂ X such that F is locally free on U \{p}. Two such
sheavesgive the same element if they are isomorphic oversome neighborhoodof p.
The product is given by the S -hull of the tensor product.
2
One can also realize the local Picardgroupas the direct limit of Pic(U\{p})as
U runs through all open Zariski neighborhoods of p or as Pic(SpecO \{p}).
x,X
If X is normal and X \{x} is smooth then Picloc(p ∈ X) is isomorphic to the
divisor class group of O .
x,X
In many contexts it is more natural to work with the ´etale-local Picard group
Picet−loc(p ∈ X) := Pic(SpecOh \{p}) where Oh is the Henselization of the
x,X x,X
local ring O .
x,X
If X is defined over C, let W ⊂X be the intersection of X with a small (open)
ball around p. The analytic local Picard group Pican−loc(p ∈ X) can be defined as
aboveusing(analytic)S sheavesonW. By[Art69],thereisanaturalisomorphism
2
Picet−loc(p∈X)∼=Pican−loc(p∈X).
Note that Pican(W \{p}) is usually much bigger than Pican−loc(p ∈ W). (This
happens already for X = C2.) However, Pican(W \{p}) = Pican−loc(p ∈ X) if
depth O ≥3 [Siu69].
p X
(The literature does not seem to be consistent; any of the above four variants is
called the local Picard group by some authors.)
Let X be a complex space and p ∈ X a closed point. Set U := X \{p}. As
usual, Pic(U)∼=H1(U,O∗) and the exponential sequence
U
0→Z −2→πi O −ex→p O∗ →1
U U U
gives an exact sequence
H1(cid:0)U,OU(cid:1)→Pic(U)−c→1 H2(cid:0)U,Z(cid:1).
A piece of the local cohomology exact sequence is
H1(cid:0)X,OX(cid:1)→H1(cid:0)U,OU(cid:1)→Hp2(cid:0)X,OX(cid:1)→H2(cid:0)X,OX(cid:1).
Thus if X is Stein then we have an isomorphism
H1(cid:0)U,OU(cid:1)∼=Hp2(cid:0)X,OX(cid:1)
and the latter vanishes iff depth O ≥ 3; see [Gro67, Sec.3]. Combining with
p X
[Siu69] we obtain the following well known result.
Lemma 8. Let X be a Stein space and p ∈ X a closed point. If depth O ≥ 3
p X
then taking the first Chern class gives an injection
c1 :Pican−loc(p∈X)֒→H2(cid:0)X\{p},Z(cid:1). (cid:3)
GROTHENDIECK–LEFSCHETZ TYPE THEOREMS 5
Definition 9 (Log canonical singularities). (See [KM98] for an introduction and
[Kol13] for a comprehensive treatment of these singularities.)
Let X be a normal variety such that mK is Cartier for some m > 0. Let
X
f : Y → X be a resolution of singularities with exceptional divisors {E : i ∈ I}.
i
One can then write
mKY ∼f∗(mKX)+m·Pi∈Ia(Ei,X)Ei.
The number a(E ,X)∈ Q is called the discrepancy of E ; it is independent of the
i i
choice of m.
Ifmin{a(E ,X):i∈I}≥−1thentheminimumisindependentoftheresolution
i
f :Y →X and its value is called the discrepancy of X.
X is called log canonical if min{a(E ,X) : i ∈ I} ≥ −1 and log terminal if
i
min{a(E ,X) : i ∈ I} > −1. A cone over a smooth variety with trivial canonical
i
class is log canonical but not log terminal.
Let X be log canonical, g : X′ → X any resolution and E ⊂ X′ an exceptional
divisorsuch that a(E,X)=−1. The subvariety g(E)⊂X is called a log canonical
center of X. Log canonical centers hold the key to understanding log canonical
varieties, see [Kol13, Chaps.4–5].
Log terminal singularities are rational [KM98, 5.22]. Log canonical singularities
are usually not rational but they are Du Bois [KK10].
Logcanonicalsingularities(andtheirnon-normalversions,calledsemi-logcanon-
ical singularities) are precisely those that are needed to compactify the moduli of
varieties of general type.
We use only the following two theorems about log canonical singularities.
Theorem 10. [Kaw07,Ale08] Let X be a normal variety over a field of character-
istic 0, Z ⊂X a closed subscheme of codimension ≥3 and Z ⊂X ⊂X a Cartier
0
divisor such that X \Z is normal and the normalization of X is log canonical.
0 0
Assumealso that K is Q-Cartier. Then X is normal and it does not contain any
X 0
log canonical center of X. (cid:3)
Theorem 11. Let X be a normal variety over a field of characteristic 0, Z ⊂ X
a closed subscheme of codimension ≥3 and D a Weil divisor on X that is Cartier
on X \Z.
Then thereisaproper, birationalmorphism f :Y →X suchthatf issmall(that
is, its exceptional set has codimension ≥ 2) and the birational transform f−1D is
∗
Q-Cartier and f-ample if one of the following assumptions is satisfied.
(1) [KSB88] There is a Cartier divisor Z ⊂X ⊂X suchthat X \Z is normal
0 0
and the normalization of X is log canonical.
0
(2) [Bir11, HX13, OX12] X is log canonical and Z does not contain any log
canonical center of X. (cid:3)
ObservethatDisQ-Cartierifff :Y →X isanisomorphism. Inourapplications
we show that Y 6=X leads to a contradiction.
Comments on thereferences. [Kaw07]provesthatthepair(X,X )islogcanoni-
0
cal. ThisimpliesthatX doesnotcontainanylogcanonicalcenterofX byaneasy
0
monotonicity argument [KM98, 2.27]. Then [Ale08] shows that X is S , hence
0 2
normal since we assumed normality in codimension 1.
6 JA´NOSKOLLA´R
[KSB88] claimed (11.1) only for dimX =3 since the necessary results of Mori’s
program were known only for dimX ≤ 3 at that time. The proof of the general
case is the same.
The second case (11.2) is not explicitly claimed in the references but it easily
follows from them. For details on both cases see [Kol13, Sec.1.4].
The next example shows that Theorem 4 fails if depth X < 3, even if the
Z0 0
dimension is large.
Example 12. Let (A,Θ) be a principally polarized Abelian variety over a field k.
The affine cone over A with vertex v is
Ca(A,Θ):=SpeckPm≥0H0(cid:0)A,OA(mΘ)(cid:1).
Note that depth C (A,Θ)=2 since H1(A,O )6=0.
v a A
SetX :=C (A,Θ)×Pic0(A)withf :X →Pic0(A)thesecondprojection. Since
a
L(Θ) has a unique section for every L ∈ Pic0(A), there is a unique divisor D on
A
A×Pic0(A)whoserestrictiontoA×{[L]}isthe abovedivisor. Bytakingthe cone
we get a divisor D on X.
X
For L ∈ Pic0(A), let D denote the restriction of D to the fiber C (A,Θ)×
[L] X a
{[L]} of f. We see that
(1) D[L] is Cartier iff L∼=OA.
(2) mD[L] is Cartier iff Lm ∼=OA.
(3) D is not Q-Cartier for very general L∈Pic0(A).
[L]
2. The main technical theorem
The following is our main result concerning Problem 1. In the applications the
key question will be the existence of the bimeromorphic morphism f : Y → X.
This is a very hardquestion in general,but in our cases existence is guaranteedby
Theorem 11.
Theorem 13. Let f : Y → X be a proper, bimeromorphic morphism of normal
analyticspacesofdimension≥4andLalinebundleonY whoserestrictiontoevery
fiber is ample. Assume that there is a closed subvariety Z ⊂ Y of codimension
Y
≥ 2 such that Z := f(cid:0)ZY(cid:1) has dimension ≤ 1 and f induces an isomorphism
Y \ZY ∼=X\Z.
Let X ⊂ X be a Cartier divisor such that Z ∩ X is a single point p. Let
0 0
p∈U ⊂X beacontractibleopenneighborhood ofp. NotethatU\Z ∼=f−1(U)\ZY,
hence the restriction L| makes sense. Set U := X ∩U. The following are
U\Z 0 0
equivalent.
(1) The map f is an isomorphism over U.
(2) The Chern class of L|U\Z vanishes in H2(cid:0)U \Z,Q(cid:1).
(3) The Chern class of L|U0\{p} vanishes in H2(cid:0)U0\{p},Q(cid:1).
Proof. If f is an isomorphism then L is a line bundle on the contractible space
U hence c1(L) = 0 in H2(cid:0)U,Q(cid:1). Thus (2) holds and clearly (2) implies (3). The
key part is to prove that (3) ⇒ (1).
TheassumptionandtheconclusionarebothlocalnearpintheEuclideantopol-
ogy. By shrinking X we may assume that the Cartier divisor X givesa morphism
0
g : X → D to the unit disc D whose central fiber is X . Note that Y := f−1(X )
0 0 0
is a Cartier divisor in Y.
GROTHENDIECK–LEFSCHETZ TYPE THEOREMS 7
Let W ⊂ X be the intersection of X with a closed ball of radius 0 < ǫ ≪ 1
around p. Set W :=X ∩W, V :=f−1(W) and V :=f−1(W ). We may assume
0 0 0 0
that W,W are contractible and f−1(p) is a strong deformation retract of both V
0
and of V .
0
Let D¯ ⊂ D denote the closed disc of radius δ. If 0 < δ ≪ ǫ then the
δ
pair (cid:0)W0,∂W0(cid:1) is a strong deformation retract of (cid:0)W ∩g−1D¯δ,∂W ∩g−1D¯δ(cid:1) and
(cid:0)V0,∂V0(cid:1)is astrongdeformationretractof(cid:0)V ∩(gf)−1D¯δ,∂V ∩(gf)−1D¯δ(cid:1). These
retractions induce continuous maps (unique up-to homotopy)
rc :(cid:0)Vc,∂Vc(cid:1)→(cid:0)V0,∂V0(cid:1), (13.4)
where V is the fiber of g◦f :V →D over c∈D . The induced maps
c δ
rc∗ :Q∼=H2n(cid:0)V0,∂V0,Q(cid:1)→∼= H2n(cid:0)Vc,∂Vc,Q,(cid:1)∼=Q (13.5)
are isomorphisms where n = dimCV0 = dimCVc. Our aim is to study the cup
product pairing
H2(cid:0)V0,∂V0,Q(cid:1)×H2n−2(cid:0)V0,Q(cid:1)→H2n(cid:0)V0,∂V0,Q(cid:1)∼=Q. (13.6)
(See [Hat02], especially pages 209 and 240 for the relevant facts on cup and cap
products.) We prove in Lemmas 14 and 15, by arguing on V , that it is zero and
c
in Lemma 17.2, by arguing on V , that it is nonzero if V →W is not finite. Thus
0 0 0
f−1(p) is 0-dimensional, hence f is a biholomorphism. (cid:3)
For later applications, in the next lemmas we consider the more general case
when f : Y → X is a proper, bimeromorphic morphism of normal analytic spaces
and g :X →D a flat morphism of relative dimension n.
Lemma 14. Notation and assumptions as in (13). If H2n−2(cid:0)Vc,Q(cid:1) = 0 then the
cup product pairing
H2(cid:0)V0,∂V0,Q(cid:1)×H2n−2(cid:0)V0,Q(cid:1)→H2n(cid:0)V0,∂V0,Q(cid:1)∼=Q is zero.
Proof. Using r∗ and the Poincar´e duality map, the cup product pairing factors
c
throughthe followingcupandcapproductpairings,wherethe righthandsidesare
isomorphic by (13.5),
H2(cid:0)V0,∂V0,Q(cid:1) × H2n−2(cid:0)V0,Q(cid:1) → H2n(cid:0)V0,∂V0,Q(cid:1) ∼= Q
↓ ↓ ↓ ||
H2(cid:0)Vc,∂Vc,Q(cid:1) × H2n−2(cid:0)Vc,Q(cid:1) → H2n(cid:0)Vc,∂Vc,Q(cid:1) ∼= Q
↓ ↓ ↓ ||
H2n−2(cid:0)Vc,Q(cid:1) × H2n−2(cid:0)Vc,Q(cid:1) → H0(cid:0)Vc,Q(cid:1) ∼= Q
The first factor in the bottom row is zero, hence the pairing is zero. (cid:3)
We apply the next result to V →W to check the homology vanishing assump-
c c
tion in Lemma 14.
Lemma 15. Let V′ → W′ be a proper bimeromorphic map of normal complex
spaces of dimension n≥3. Assume that every fiber has complex dimension ≤n−2
and W′ is Stein. Then H2n−2(cid:0)V′,Q(cid:1)=0.
Proof. Let E′ ⊂ V′ denote the exceptional set and F′ ⊂ W′ its image. Then
dimF′ ≤n−2, hence the exact sequence
H2n−2(cid:0)F′,Q(cid:1)→H2n−2(cid:0)W′,Q(cid:1)→H2n−2(cid:0)W′,F′,Q(cid:1)→H2n−3(cid:0)F′,Q(cid:1)
8 JA´NOSKOLLA´R
showsthatH2n−2(cid:0)W′,Q(cid:1)∼=H2n−2(cid:0)W′,F′,Q(cid:1). Thelattergroupisinturnisomor-
phic to H2n−2(cid:0)V′,E′,Q(cid:1) which sits in an exact sequence
H2n−2(cid:0)E′,Q(cid:1)→H2n−2(cid:0)V′,Q(cid:1)→H2n−2(cid:0)V′,E′,Q(cid:1).
Here H2n−2(cid:0)E′,Q(cid:1) is generated by the fundamental classes of the compact irre-
ducible components of E′, but we assumed that there are no such. Thus we have
an injection
H2n−2(cid:0)V′,Q(cid:1)֒→H2n−2(cid:0)W′,Q(cid:1).
Since W′ is Stein and 2n−2 > n, Theorem 16 implies that H2n−2(cid:0)W′,Q(cid:1) = 0.
Thus we conclude that H2n−2(cid:0)V′,Q(cid:1)=0. (cid:3)
During the proof we have used the following.
Theorem 16. [Ham83, Ham86] or [GM88, p.152]. Let W be a Stein space of
dimension n. Then H (W,Z) and Hi(W,Z) both vanish for i>n. More generally,
i
W is homotopic to a CW complex of dimension ≤n. (cid:3)
Nextwedescribetwocaseswhenthe cupproductpairing(13.6)isnonzero. The
first of these is used in Proposition 22 and the second in Theorem 13.
Lemma 17. Let f : V → W be a projective, bimeromorphic morphism between
0 0 0
irreducible complex spaces. Let p ∈ W be a point. Assume that f is an isomor-
0 0
phism over W \{p} and dimf−1(p) > 0. Assume furthermore that one of the
0 0
following holds.
(1) There is a nonzero Q-Cartier divisor E ⊂V supported on f−1(p).
0 0 0
(2) There is an f -ample line bundle L such that c (L)| =0.
0 1 ∂V0
Then f−1(p) has codimension 1 in V and the cup product pairing
0 0
H2(cid:0)V0,∂V0,Q(cid:1)×H2n−2(cid:0)V0,Q(cid:1)→H2n(cid:0)V0,∂V0,Q(cid:1)∼=Q is nonzero.
Proof. ConsiderfirstCase(1). ThenE 6=0showsthatf−1(p)hascodimension
0 0
1.
Let H be a relatively veryample line bundle. We have c1(E0)∈H2(cid:0)V0,∂V0,Q(cid:1)
and c1(H)∈H2(cid:0)V0,Q(cid:1). If E0 is effective then
c1(E0)∪c1(H)n−1 =c1(cid:0)H|E0(cid:1)n−1 ∈H0(cid:0)E0,Q(cid:1)→H0(cid:0)V0,Q(cid:1) (17.3)
is positive. If E is not assumed effective then we claim that
0
c1(E0)∪c1(E0)∪c1(H)n−2 ∈H2n(cid:0)V0,∂V0,Q(cid:1) is nonzero.
Thecompleteintersectionof(n−2)generalmembersofH givesanalgebraicsurface
S, proper over W such that E ∩S is a nonzero linear combinationof exceptional
0 0
curves. Thus, by the Hodge index theorem,
c1(E0)2∪c1(H)n−2 =c1(cid:0)E0|S(cid:1)2 <0,
completing the proof of (1).
Next assume that (2) holds. By assumption we can lift c (L) to c˜ (L) ∈
1 1
H2(cid:0)V0,∂V0,Q(cid:1). (The lifting is in fact unique, but this is not important for us.)
From this we obtain a class
(cid:2)c˜1(L)(cid:3)∈H2n−2(cid:0)V0,Q(cid:1)=H2n−2(cid:0)f−1(p),Q(cid:1)=PQ[Ai] (17.4)
GROTHENDIECK–LEFSCHETZ TYPE THEOREMS 9
where A ⊂ f−1(p) are the irreducible components of dimension n−1. So far we
i
have not established that dimf−1(p) = n −1, thus the sum in (17.4) could be
empty. The key step is the following.
Claim 17.5. (cid:2)c˜1(L)(cid:3) = Pai[Ai] where ai < 0 for every i and the sum is not
empty.
Once this is shown we conclude as in (17.3) using the equality
c˜1(L)∪c1(L)n−1 =Pai·c1(cid:0)L|Ai(cid:1)n−1 <0. (17.6)
In order to prove (17.5) we aim to use [KM98, 3.39], except there it is assumed
that every A is Q-Cartier. To overcome this, take a resolution π : V′ → V
i 0 0
and write the homology class (cid:2)c˜1(π∗L)(cid:3) is a linear combination Pa′i[A′i] where
A′ ⊂ (f ◦π)−1(p) are the irreducible components of dimension n−1. Since L is
i
f-ample and dimf−1(p)>0, we see that π∗L is nef and not numerically trivial on
(f ◦π)−1(p). Apply [KM98, 3.39.2] to π∗L. We obtain that −Pa′i[A′i] is effective
and its support contains (f ◦π)−1(p). Thus a′i < 0 for every i and so (cid:2)c˜1(L)(cid:3) =
π∗Pa′i[A′i] shows (17.5) unless there are no f-exceptional divisors Ai ⊂f−1(p).
If this happens, then Pa′i[A′i] is π-exceptional and, as the homology class of
π∗L,ithaszerointersectionwitheverycurvethatis contractedbyπ. Thuswecan
apply[KM98,3.39.1]toboth±Pa′i[A′i]andconclude thatPa′i[A′i]=0. This is a
contradictionsinceLandhenceπ∗Lhavepositiveintersectionwithsomecurve. (cid:3)
3. Deformations of log canonical singularities
Here we derive stronger forms of the three assertions of Theorem 3. We start
with (3.1–2).
Theorem 18. Let X be a normal variety over C and g :X →C a flat morphism
of pure relative dimension n to a smooth curve. Let 0∈C be a point and Z ⊂X
0 0
a closed subscheme of dimension ≤n−3. Assume that
(1) K is Q-Cartier on X\Z ,
X 0
(2) the fibers X are log canonical for c6=0,
c
(3) X \Z is log canonical and
0 0
(4) the normalization of X is log canonical.
0
Then X is normal and K is Q-Cartier on X.
0 X
Proof. By localization we may assume that Z = {p} is closed point. Next we
0
use Theorem 11 to obtain f : Y →X such that f is an isomorphism over X \{p}
and f−1K is an f-ample Q-Cartier divisor.
∗ X
By the Lefschetz principle, we may assume that everything is defined over C.
We apply Theorem 13 to L := mf−1K for a suitable m > 0. Let U be the
∗ X
intersection of X with a small ball around p and set U := X ∩U. Note that U
0 0 0
is naturally a subset of X, of Y and also of the normalization of X . The latter
0
shows that L is trivial,thus the assumption (13.3) is satisfied. Hence f is an
U0\{p}
isomorphism and so K is Q-Cartier.
X
Now Theorem 10 implies that X is normal. (cid:3)
0
Theorem19. LetX bealogcanonical variety ofdimension ≥4over C andp∈X
a closed point that is not a log canonical center (9). Let p ∈X ⊂ X be a Cartier
0
10 JA´NOSKOLLA´R
divisor. Let p∈U be a Stein neighborhood such that U and U :=X ∩U are both
0 0
contractible. Then the restriction maps
Picloc(p∈X)→Picloc(p∈X0) and Picloc(p∈X)→H2(cid:0)U0\{p},Z(cid:1)
are injective.
Proof. Let D be a divisor on X such that D|X\{p} is Cartier and c1(cid:0)D|X0\{p}(cid:1)
is zero in H2(cid:0)X0\{p},Z(cid:1).
First we show that D is Q-Cartier at p. By Theorem 11 there is a proper
birationalmorphismf :Y →X suchthatf isanisomorphismoverX\{p},f−1D
∗
is f-ample and f has no exceptional divisors.
The Cartier divisor X gives a morphism X →D whose central fiber is X .
0 0
As in Theorem 13 let W ⊂ X be the intersection of X with a closed ball of
0 0 0
radiusǫ aroundp andV :=f−1(W ). Set n:=dimX . By Lemma 14 we see that
0 0 0
the cup product pairing
H2(cid:0)V0,∂V0,Q(cid:1)×H2n−2(cid:0)V0,Q(cid:1)→H2n(cid:0)V0,∂V0,Q(cid:1)∼=Q is zero.
On the other hand, by (17.2) it is nonzerounless f :Y →X is finite. Thus f is an
isomorphism and D is Q-Cartier at p.
Now we can use [Gro68, X.3.2] to show that D is Cartier at p. (cid:3)
Corollary 20. Let g : X → C and Z ⊂ X be as in Theorem 18. Assume that
0 0
the fibers X are all log canonical and K is Q-Cartier. Let D∗ be Cartier divisor
c X
on X \Z such that D∗| extends to a Cartier divisor on X .
0 X0\Z0 0
Then D∗ extends to a Cartier divisor on X.
Proof. Choose Z to be the smallest closed subset such that D∗ is Cartier on
X \Z. We need to show that Z = ∅. If not, let p ∈ Z be a generic point. By
localization we are reduced to the case when Z ={p} is a closed point of X. Note
that p is not a log canonical center of X by Theorem 10.
Thus (20) is a special case of Theorem 19. (cid:3)
21 (Proof of Theorem 3.1–2). Let g : X → C be a flat, proper morphism to a
smooth curve whose fibers are normal and log canonical. Let 0 ∈ C be a closed
point and Z ⊂ X a subscheme of codimension ≥ 3 such that K is Q-Cartier
0 0 X
on X \Z . Then K is Q-Cartier by Corollary 20 thus mK is Cartier for some
0 X X
m > 0. So O (mK ) is a line bundle on X. For a flat family of line bundles,
X X
ampleness is an open condition, proving (3.1).
The second assertion (3.2) directly follows from Theorem 18. (cid:3)
4. Stability of exceptional divisors
Weconsiderpart3ofTheorem3. Letg :X →C beaflatmorphismtoasmooth
curve. Let 0 ∈C be a closed point such that X is Q-factorial. Let Z ⊂ X be a
0 0 0
subschemeofcodimension≥3andf :Y →X beaprojective,birationalmorphism
such that f is an isomorphism over X \Z and f :Y →X is birational but not
0 0 0 0
an isomorphism.
Let H ⊂Y be an ample divisor. Since X is Q-factorial, m·f (H ) is Cartier
0 0 0 0 0
for some m>0. Thus
E0 :=f0∗(cid:0)mf0(H0)(cid:1)−mH0