The Zariski-Lipman conjecture for log canonical spaces

0.17 MB·English

by  Stéphane Druel

THE ZARISKI-LIPMAN CONJECTURE FOR LOG CANONICAL SPACES
STE´PHANEDRUEL
Abstract. InthispaperweprovetheZariski-Lipmanconjectureforlogcanonical spaces.
3
1
0
2
Contents

n
1. Introduction 1
a
J 2. Preliminaries 2
3. Canonical desingularizations, and theZariski-Lipman conjecture for klt spaces 3
4
4. The Camacho-Sad formula 4
2
5. Proof of Theorem 1.1 7

] References 8
G
A
.
h
1. Introduction
t
a
m TheZariski-LipmanconjectureassertsthatacomplexvarietyX withalocallyfreetangentsheafTX isnec-
essarilysmooth([Lip65]). Theconjecturehasbeenshowninspecialcases;forhypersurfacesorhomogeneous
[
complete intersections ([Hoc75, SS72]), for local complete intersections ([K¨al11]), for isolated singularities

1 in higher-dimensional varieties [vSS85, Sect. 1.6], and more generally, for varieties whose singular locus has
v codimension at least 3 [Fle88].
0
The Minimal Model Program was initiated in the early eighties as an attempt to extend the birational
1
classification of surfaces to higher dimensions. It became clear that singularities are unavoidable in the
9
5 birational classification of higher dimensional complex projective varieties; this led to the development of a
. powerful theory of singularities of pairs (see Definition 3.4 for basic notions, such as klt and log canonical
1
0 singularities). Theclassoflogcanonicalsingularitiesisthelargestclassofsingularitieswheretheconjectures
3 of the Minimal Model Programare expected to hold.
1 The Zariski-Lipman conjecture has been shown for klt spaces in [GKKP11] (see also [AD11, Corollary
:
v 5.7]). In this paper we prove the conjecture for log canonical spaces. Notice that log canonical spaces in
i general have singularities in codimension 2.
X
r Theorem 1.1 (Zariski-Lipman conjecture for log canonical spaces). Let X be a log canonical space such
a
that the tangent sheaf T is locally free. Then X is smooth.
X
Weremarkthattheresultsholdaswellforsingularitiesofcomplexanalyticspaces,andalgebraicvarieties
defined over a field of characteristic zero.
The paper is organized as follows. In section 2 we study entire solutions of a particular system of
polynomial equations. In Section 3 we review basic definitions of singularities of pairs, and the notion of
canonical desingularization. In section 4 we recall the Camacho-Sad formula, and provide applications to
surfaces with trivial logarithmic tangent sheaf (see Proposition 4.8). The proof of Theorem 1.1 occupies
section 5.
Notation and conventions. Throughoutthispaper,weworkoverthe fieldofcomplexnumbers. Varieties
are always assumedto be irreducible and reduced. We denote by Sing(X) the singular locus of a variety X.
If X is a variety, we denote by T the tangent sheaf of X.
X
2010 Mathematics Subject Classification. 14B05,32B05.
Theauthor waspartiallysupportedbytheprojectCLASS ofAgence nationaledelarecherche, under agreementANR-10-
JCJC-0111.
1
2 STE´PHANEDRUEL
2. Preliminaries
Letrbeapositiveinteger,andlete ,...,e becomplexnumbers. WedefinetherationalfunctionΦ
1 r e1,...,er
by the formula
1
Φ (x)=
e1,...,er 1
e −
r
1
e −
r−1
1
···−
1
e − .
2 e −x
1
Notice that
1
(2.1) Φ (x)= .
e1,...,er e −Φ (x)
r e1,...,er−1
Write
a (e ,...,e )x+b (e ,...,e )
Φ (x)= r 1 r r 1 r
e1,...,er c (e ,...,e )x+d (e ,...,e )
r 1 r r 1 r
where a ,b ,c ,d ∈C[z ,...,z ].
r r r r 1 r
Suppose that r >2. Then, by (2.1), we have
a (z ,...,z ) = c (z ,...,z )
r 1 r r−1 1 r−1
(2.2)  br(z1,...,zr) = dr−1(z1,...,zr−1)
 cr(z1,...,zr) = zrcr−1(z1,...,zr−1)−ar−1(z1,...,zr−1)
d (z ,...,z ) = z d (z ,...,z )−b (z ,...,z ),
r 1 r r r−1 1 r−1 r−1 1 r−1

a (z ) = 0 a (z ,z ) = −1
1 1 2 1 2
(2.3)  b1(z1) = 1 and  b2(z1,z2) = z1
 c1(z1) = −1  c2(z1,z2) = −z2
d (z ) = z d (z ,z ) = z z −1.
1 1 1 2 1 2 2 1
Thus, for any r>3, weobtain 
b (z ,...,z ) = z b (z ,...,z )−b (z ,...,z )
r 1 r r−1 r−1 1 r−1 r−2 1 r−2
(2.4)  b (z ) = 1
1 1
 b (z ,z ) = z .
2 1 2 1
Note that b ∈C[z,...,z ].
r 1 r−1
We denote by S the symmetric group on r letters. The proof of Theorem 5.2 makes use of the following
r
elementary result.
Lemma 2.5. Let r be a positive integer, and let e ,...,e be integers. Suppose that for any s ∈ S
1 r r
Φ (x)=x as rational functions. Then e =···=e , and e ∈{−1,0,1}.
es(1),...,es(r) 1 r 1
Proof. Notice that Φ (x) = x (as rational functions) if and only if a (e ,...,e ) = d (e ,...,e ),
e1,...,er r 1 r r 1 r
b (e ,...,e )=0, and c (e ,...,e )=0. Note also that we must have r >2.
r 1 r r 1 r
If r =2, then b (e ,e )=e =0, and c (e ,e )=−e =0. Thus e =e =0.
2 1 2 1 2 1 2 2 1 2
Suppose that r >3. Then
1
=Φ (x)=x,
1 e1,...,er
e −
r e −Φ (x)
r−1 e1,...,er−2
and hence
(1−e e )x+e
Φ (x)= r−1 r r−1.
e1,...,er−2 1−e x
r
Now, by replacing (e ,...,e ,e ,e ) with (e ,...,e ,e ,e ), we obtain
1 r−2 r−1 r 1 r−2 r r−1
(1−e e )x+e
Φ (x)= r−1 r r
e1,...,er−2 1−e x
r−1
THE ZARISKI-LIPMAN CONJECTURE FOR LOG CANONICAL SPACES 3
This yields e =e . This easily implies that e =···=e .
r−1 r 1 r
Suppose that r = 3. Then b (e ,e ,e ) = e e −1 = e2−1 = 0. Therefore, either e = e = e = 1, or
3 1 2 3 1 2 1 1 2 3
e =e =e =−1.
1 2 3
Suppose that r >4. We assume from now on that e 6=0. Hence e 6=0 for all i∈{1,...,r}.
1 i
Let 2 6 k 6 r be an integer such that b (e ,...,e ) = 0, and b (e ,...,e ) 6= 0 (recall that
k 1 k k−1 1 k−1
b (z )=1).
1 1
If k =2, then e =0, yielding a contradiction. Thus k >3, and by (2.4), we must have
1
e b (e ,...,e )=b (e ,...,e ).
k−1 k−1 1 k−1 k−2 1 k−2
Thus e divides b (e ,...,e ), and since e = ··· = e , e divides b (0,...,0). On the other
k−1 k−2 1 k−2 1 k−2 k−1 k−2
hand, b (0,...,0) ∈ {−1,0,1} by (2.4) again. Thus e ∈ {−1,0,1}, completing the proof of Lemma
k−2 k
2.5. (cid:3)
Remark 2.6. Φ (x) = x if and only if r ≡ 0(mod2), Φ (x) = x if and only if r ≡ 0(mod3), and
0,...,0 1,...,1
Φ (x)=x if and only if r ≡0(mod3).
−1,...,−1
3. Canonical desingularizations, and the Zariski-Lipman conjecture for klt spaces
3.1 (Logarithmic tangent sheaf). Let Y be a nonsingular variety of dimension n>1, and ∆⊂Y a divisor
with simple normal crossings. That is, ∆ is an effective divisor and its local equation at an arbitrary point
y ∈Y decomposesinthelocalringO intoaproducty ···y ,wherey ,...,y formpartofaregularsystem
y 1 k 1 k
of parameters (y ,...,y ) of O . Let
1 n y
TY(−log∆)⊆TY =DerC(OY)
bethesubsheafconsistingofthosederivationsthatpreservetheidealsheafO (−∆). Oneeasilychecksthat
Y
the logarithmic tangent sheaf T (−log∆) is a locally free sheaf of Lie subalgebras of T , having the same
Y Y
restriction to Y \∆, and hence the same rank n.
If ∆ is defined at y by the equation y ···y = 0 as above, then a local basis of T (−log∆) (after
1 k Y
localization at y) consists of
y ∂ ,...,y ∂ ,∂ ,...,∂ ,
1 1 k k k+1 n
where (∂ ,...,∂ ) is the local basis of T dual to the local basis (dy ,...,dy ) of Ω1.
1 n Y 1 n Y
A local computation shows that T (−log∆) can be identified with the subsheaf of T containing those
Y Y
vector fields that are tangent to ∆ at smooth points of ∆.
The dual of T (−log∆) is the sheaf Ω1(log∆) of logarithmic differential 1-forms, that is, of rational
Y Y
1-forms α on Y such that α and dα have at most simple poles along ∆.
Thetopexteriorpower∧nT (−log∆)istheinvertiblesheafO (−K −∆),whereK denotesacanonical
Y Y Y Y
divisor.
We will need the following observation.
Lemma 3.2. Let Y be a smooth variety of dimension at least 2, and ∆ ⊂ Y a divisor with simple normal
crossings. If H (Y is a smooth hypersurface such that ∆∩H is smooth of pure codimension 2 in X, then
there is an exact sequence
0→N ∗ →Ω1(log∆) →Ω1 (log∆ )→0.
H/Y Y |H H |H
Proof. Considerthecompositemapα: N ∗ →Ω1 →Ω1(log∆) ,andthemorphismβ: Ω1(log∆) →
H/Y Y|H Y |H Y |H
Ω1 (log∆ )inducedbytherestrictionmap. Alocalcomputationshowsthatαandβyieldanexactsequence
H |H
as claimed. (cid:3)
3.3 (Singularities of pairs). We recall some definitions of singularities of pairs, developed in the context of
the Minimal Model Program.
Definition3.4(See[KM98,section2.3]). LetX beanormalvariety,andB = a B aneffectiveQ-divisor
i i
onX, i.e., B is a nonnegative Q-linearcombinationof distinct prime Weil divisPors Bi’s on X. Suppose that
K +B is Q-Cartier, i.e., some nonzero multiple of it is a Cartier divisor.
X
Let π : Y → X be a log resolution of the pair (X,B). This means that Y is a smooth variety, π is a
birational projective morphism whose exceptional set Exc(π) is of pure codimension one, and the divisor
4 STE´PHANEDRUEL
E +π−1B has simple normal crossings,where the E ’s are the irreducible components of Exc(π). There
i ∗ i
aPre uniquely defined rational numbers a(Ei,X,B)’s such that
K +π−1B =π∗(K +B)+ a(E ,X,B)E .
Y ∗ X i i
XEi
The a(E ,X,B)’s do not depend on the log resolution π, but only on the valuations associated to the E ’s.
i i
Let
discrep(X,B)=inf{a(E,X,B)}
E
whereE runsthroughallthe primeexceptionaldivisorsofallprojectivebirationalmorphisms. Then, either
discrep(X,B)=−∞, or −16discrep(X,B)61. If X is smooth, then discrep(X,0)=1.
We saythat(X,B)is log terminal (orklt)if alla <1,and, forsomelogresolutionπ :Y →X of(X,B),
i
a(E ,X,B)>−1 for every π-exceptionalprime divisor E . We say that (X,B) is log canonical if all a 61,
i i i
and, for some log resolution π : Y → X of (X,B), a(E ,X,B) > −1 for every π-exceptional prime divisor
i
E . If these conditions hold for some log resolution of (X,B), then they hold for every log resolution of
i
(X,B). Moreover,(X,B) is log canonical (respectively, klt) if and only if discrep(X,B)>−1 (respectively,
discrep(X,B)>−1 and all a <1).
i
We say that X is klt (respectively log canonical) if so is (X,0).
3.5 (Canonical desingularization). In the proofs of Theorems 1.1 and 3.8, we will consider a suitable reso-
lution of singularities, whose existence is guaranteed by the following theorem.
Theorem 3.6 ([Kol07, Theorems 3.35, and 3.45] and [GKK10, Corollary 4.7]). Let X be a normal variety.
Then there exists a resolution of singularities π :Y →X such that
(1) π is an isomorphism over X \Sing(X), and
(2) π T (−log∆)≃T where ∆ is the largest reduced divisor contained in π−1(Sing(X).
∗ Y X
Notice that Supp(∆) = Exc(π). In particular, ∆ has simple normal crossings. We call a resolution π as
in Theorem 3.6 a canonical desingularization of X.
3.7 (Zariski-Lipman conjecture for klt spaces). Recall from [GKKP11] that the Zariski-Lipman conjecture
holds forklt spaces(see also[AD11, Corollary5.7]for relatedresults). We reproduce the prooffrom[AD11,
Corollary 5.7] for the reader’s convenience.
Theorem 3.8 ([GKKP11]). Let X be a klt space such that the tangent sheaf T is locally free. Then X is
X
smooth.
Proof. We assume to the contrary that Sing(X)6=∅. Let π :Y →X be a canonical desingularization of X,
and let ∆ be the largest reduced divisor contained in π−1(Sing(X)). Note that ∆ 6= 0 since Sing(X) 6= ∅.
Consider the morphism of vector bundles
π∗T ≃π∗(π T (−log∆))→T (−log∆),
X ∗ Y Y
where π∗(π T (−log∆))→T (−log∆) is the evaluation map. It induces an injective map of sheaves
∗ Y Y
π∗O (−K )≃π∗det(T )֒→det(T (−log∆))≃O (−K −∆).
X X X Y Y Y
This implies that a(∆ ,X) 6 −1 for any irreducible component ∆ of ∆, yielding a contradiction and
i i
completing the proof Theorem 3.8. (cid:3)
Remark3.9. WehavethefollowingreformulationofTheorem3.8. LetX beavarietysuchthatthetangent
sheaf T is locally free. If X is not smooth, then discrep(X)∈{−∞,−1}.
X
4. The Camacho-Sad formula
4.1 (Foliations). A (singular) foliation on a smooth complexe analytic surface S is a locally free subsheaf
L (T ofrank1suchthatthecorrespondingtwistedvectorfield~v ∈H0(X,T ⊗L⊗−1)hasisolatedzeroes.
S S
Its singular locus Sing(L) is the zerolocus of~v. Consideringthe naturalperfect pairing Ω1 ⊗Ω1 →ω , we
S S S
seethat L (T givesriseto a twisted1-formwithisolatedzeroesω ∈H0(X,Ω1 ⊗M)with M =ω ⊗L.
S S S
Conversely, given a twisted 1-form ω ∈ H0(X,Ω1 ⊗M) with isolated zeroes, we define a foliation as the
S
kernel of the morphism T →L given by the contraction with ω.
S
THE ZARISKI-LIPMAN CONJECTURE FOR LOG CANONICAL SPACES 5
4.2 (Camacho-Sad formula). Let C ⊂ S be a compact L-invariant curve, and let p ∈ C ∩Sing(L). Let
ω be a local 1-form defining L in a neighborhood of p, and let f be a local equation of C at p. Then
there exist nonzero local functions g and h, and a local 1-form η such that f and h are relatively prime and
gω =hdf +fη (see [Suw95, Lemma 1.1]). Following [Suw95], we set
1
CS(L,C,p)=−Res η .
p h |C
The right hand side depends only on L and C.
Example 4.3. Let (x,y) be local coordinates at p. Suppose that L is given by the local 1-form ω =
λx(1+o(1))dy −µy(1+o(1))dx with µ 6= 0. Set p = (0,0), and let C be the invariant curve defined by
x=0. Then CS(L,C,p)= λ.
µ
We can now state the Camacho-Sadformula (see [Suw95, Theorem 2.1], see also [CS82]).
Theorem 4.4 (Camacho-Sad formula). Let L be a foliation on a smooth complex analytic surface S, and
let C ⊂S be a compact L-invariant curve. Then
C2 = CS(L,C,p).
p∈C∩XSing(L)
4.5. Here we give some applications of the Camacho-Sad formula. The following two are immediate conse-
quences of Theorem 4.4, of independent interest.
Lemma 4.6. Let S be smooth surface, Q a line bundle on S, and T ։Q a surjective map of sheaves. Let
S
C ⊂X beasmoothcompleteconnectedcurveofgenusg. Ifdeg (Q)<2−2g,theng =0,anddeg (Q)=0.
C C
Proof. Let L be the kernel of T ։ Q. Suppose that deg (Q) < 2 − 2g. Then the composite map
S C
T → T → Q is the zero map, and hence T = L ( T , and N ≃ Q . In particular, C is a
C S|C |C C |C S|C C/S |C
leaf of the regular foliation by curves L ( T . By the Camacho-Sad formula (see Theorem 4.4), we must
S
have deg (Q)=C2 =0. This completes the proof of Lemma 4.6. (cid:3)
C
Corollary 4.7. Let S be smooth surface, Q a line bundle on S, and T ։ Q a surjective map of sheaves.
S
Let C ⊂X be a smooth complete connected curve of genus g 61. Then deg (Q)>0.
C
The next result is crucial for the proof of Theorem 1.1.
Proposition 4.8. Let S be smooth surface, and let C ⊂S be a possibly reducible complete curve with simple
normal crossings. If T (−logC)≃O ⊕O , then the intersection matrix of irreducible components of C is
S S S
not negative definite.
Proof. Let C′ be a connected component of C, and set C′′ =C\C′. Then, up to replacing S by S\C′′, we
may assume that C is a connected curve.
We denote the irreducible components of C by C ,...,C (r > 1). Recall that the dual graph Γ of C is
1 r
defined as follows. The vertices of Γ are the curves C , and for i 6=j, the vertices C and C are connected
i i j
by C ·C edges.
i j
We argue by contradiction and assume that the intersection matrix {C ·C } is negative definite.
i j i,j
Notice that O (K ) ≃ O (−C) since det(T (−logC)) ≃ O (−K −C), and T (−logC) ≃ O ⊕O .
S S S S S S S S S
By the adjunction formula, for 16i6r, we have
O (K )≃O (K +C ) ≃O (− C ),
Ci Ci S S i |Ci Ci j|Ci
Xj6=i
and hence
deg (O (K )=2g(C )−2=− deg (O (C ))60.
Ci Ci Ci i Ci Ci j|Ci
Xj6=i
Thus, one of the following holds.
(1) Either C is irreducible, and g(C)=1, or
(2) r >2, C ≃P1 for all 16i6r and the dual graph of C is a cycle.
i
6 STE´PHANEDRUEL
Suppose first that C is irreducible with g(C) = 1. Recall that there is a surjective map of sheaves
T (−logC)։T . Ontheotherhand, T (−logC)≃O ⊕O . Hence,thereexistsanon-zeroglobalvector
S C S S S
field ~v ∈ H0(S,T (−logC)) ⊆ H0(S,T ) such that ~v 6= 0. Since T ≃ O , ~v(s) 6= 0 for any s ∈ C. Set
S S |C C C
L =O ~v (T . Then, C is a complete L-invariant curve, disjoint from the singular locus Sing(L). Thus,
S S
by the Camacho-Sadformula (see Theorem 4.4), we must have C2 =0, yielding a contradiction.
Suppose that r > 2, C ≃ P1 for any 1 6 i 6 r and that the dual graph of C is a cycle. If r = 2, then
i
C ∩C ={p ,p } with p 6=p . Suppose that r >3. By renumbering the C ’s if necessary,we may assume
1 2 1 2 1 2 i
that for each i ∈ {1,...,r}, C meets C\C in p ∈ C and p ∈ C , where C = C . Note that
i i i i−1 i+1 i+1 r+1 1
p =p
r+1 1
Let~v ∈H0(S,T (−logC))⊆H0(S,T )fork∈{1,2}suchthatT (−logC)≃O ~v ⊕O ~v .Letλ∈C.
k S S S S 1 S 2
Set~v =~v +λ~v , and L =O ~v ⊆T .
λ 1 2 λ S λ S
Set C = C . Fix i ∈ {1,...,r}, and let (x ,y ) be local coordinates at p such that x (respectively, y )
0 r i i i i i
is a local equation of C (respectively, C ) at p . Then x ∂ and y ∂ are local generatorsof T (−logC)
i−1 i i i xi i yi S
at p . Therefore, there exist local functions a ,b ,c ,d at p such that the matrix
i i i i i i
a b
i i
(cid:18)ci di(cid:19)
is invertible, and such that
~v = a (x ,y )x ∂ +b (x ,y )y ∂ ,
1 i i i i xi i i i i yi
(cid:26) ~v2 = ci(xi,yi)xi∂xi +di(xi,yi)yi∂yi.
Thus
~v =(a (x ,y )+λc (x ,y ))x ∂ +(b (x ,y )+λd (x ,y ))y ∂ ,
λ i i i i i i i xi i i i i i i i yi
and a local generator for L is given by the 1-form
λ
ω = (a (x ,y )+λc (x ,y ))x dy −(b (x ,y )+λd (x ,y ))y dx
λ i i i i i i i i i i i i i i i i
= (a (p )+λc (p ))x (1+o(1))dy −(b (p )+λd (p ))y (1+o(1))dx .
i i i i i i i i i i i i
This implies that for λ∈C\{−b1(p1),−a1(p1),...,−br(pr),−ar(pr)}
d1(p1) c1(p1) dr(pr) cr(pr)
(1) ~v vanishes exactly at {p ,...,p };
λ 1 r
(2) CS(L ,C ,p )= ai(pi)+λci(pi) (see Example 4.3);
λ i−1 i bi(pi)+λdi(pi)
(3) CS(L ,C ,p )= bi(pi)+λdi(pi) (see Example 4.3).
λ i i ai(pi)+λci(pi)
In particular, L ( T is a foliation by curves on S, Sing(L ) = {p ,...,p }, and CS(L ,C ,p ) =
λ S λ 1 r λ i i
1 .
CS(Lλ,Ci+1,pi)
Set e =C2 ∈Z. Notice that for each i∈{1,...,r}, we have
i i
CS(L ,C ,p ) = 1 by (3)
λ i+1 i+1 CS(Lλ,Ci,pi+1)
= 1 by the Camacho-Sadformula
ei−CS(Lλ,Ci,pi)
Set x =CS(L ,C ,p )= a1(p1)+λc1(p1). Then
λ λ 1 1 b1(p1)+λd1(p1)
1
x =CS(L ,C ,p )=CS(L ,C ,p )=
λ λ 1 1 λ r+1 r+1 e −CS(L ,C ,p )
r λ r r
1 1
= =···= =Φ (x )
e − 1 1 e1,...,er λ
r er−1−CS(Lλ,Cr−1,pr−1) e −
r
1
e −
r−1
1
···−
1
e −
2 e −CS(L ,C ,p )
1 λ 1 1
for any λ∈C\{−b1(p1),−a1(p1),...,−br(pr),−ar(pr)}.
d1(p1) c1(p1) dr(pr) cr(pr)
This implies that
Φ (x)=x
e1,...,er
THE ZARISKI-LIPMAN CONJECTURE FOR LOG CANONICAL SPACES 7
as rational functions. And similarly, if s∈S , then
r
Φ (x)=x.
es(1),...,es(r)
ByLemma2.5,wemusthavee =···=e =−1,yieldingacontradictionsince(C +C )2 =−2+2C ·C >
1 r 1 2 1 2
0. This completes the proof of Proposition 4.8. (cid:3)
5. Proof of Theorem 1.1
We will use the following theorem to reduce to the surface case.
Theorem 5.1 ([Fle88, Corollary]). Let X be a variety such that the tangent sheaf T is locally free. If
X
codim Sing(X)>3, then X is smooth.
X
We are now in positionto proveour main result. Notice that Theorem1.1 is a immediate consequence of
Theorem 5.2 below.
Theorem 5.2 (Zariski-Lipman conjecture for log canonical pairs). Let (X,B) be a log canonical pair such
that the tangent sheaf T is locally free. Then X is smooth.
X
Proof. Notice first that K is Cartier since the tangent sheaf T is locally free. This implies that X is log
X X
canonical as well.
Let us assume to the contrary that Sing(X) 6= ∅. By Theorem 5.1, we have codim Sing(X) = 3. By
X
replacingX withanaffineopendensesubset,wemayassumethatX isaffine,andthatSing(X)isirreducible
of codimension 2. We may also assume without loss of generality that T ≃O⊕dim(X).
X X
Let π : Y → X be canonical desingularization of X, and let ∆ be the largest reduced divisor contained
in π−1(Sing(X)). Note that ∆ 6= 0. As in the proof of Theorem 3.8, we consider the morphism of vector
bundles
π∗T →T (−log∆),
X Y
and the induced injective map of sheaves
π∗O (−K )≃π∗det(T )֒→det(T (−log∆))≃O (−K −∆).
X X X Y Y Y
This yields a(∆ ,X) 6 −1 for any irreducible component ∆ of ∆. Thus that a(∆ ,X) = −1 since X has
i i i
log canonical singularities, and we have an isomorphism
π∗T ≃T (−log∆).
X Y
Suppose that dim(X) > 3. Let G ⊂ X be a general hyperplane section, and set H = π−1(G ) ⊂ Y.
1 1 1
Then G is a normal affine variety (see for instance [BS95, Theorem 1.7.1]). Moreover H is smooth, and
1 1
∆∩H has simple normal crossings by Bertini’s Theorem. By Lemma 3.2, there is an exact sequence
1
0→N ∗ →Ω1(log∆) →Ω1 (log∆ )→0.
H1/Y Y |H1 H1 |H1
Note that N ∗ ≃ π∗N ∗ ≃ O . Thus, there exist regular functions g ,...,g on G such that the
H1/Y G1/X Y 1 r 1
map O ≃ N ∗ → Ω1(log∆) ≃ O⊕dim(Y) is given by g ◦π ,...,g ◦π . Let i ∈ {1,...,r}
Y H1/Y Y |H1 Y |H1 1 |H1 r |H1
such that g 6= 0. Then, by replacing X with X \{g = 0} if necessary, we may assume that
i|H1∩Sing(X) i
Ω1 (log∆ )≃O⊕dim(H1) (and∆6=0). LetG ,...,G ⊂X beageneralhyperplanesections,and
H1 |H1 H1 2 dim(X)−2
setH =π−1(G )⊂Y,S =H ∩···∩H ,C =∆∩H ···∩H ,andT =G ∩···∩G =
i i 1 dim(X)−2 1 dim(X)−2 1 dim(X)−2
π(S). Then S is smooth, and C has simple normal crossings. Proceeding by induction, we conclude that
by replacing T with an appropriate open subset, we may assume that T (−logC) ≃ O⊕2 (and C 6= 0).
S S
Observe that the induced morphism π : S → T is birational with exceptional locus C. This implies that
|S
the intersectionmatrix of irreducible components of C is negative definite. But this contradictsProposition
4.8, completing the proof of Theorem 5.2. (cid:3)
8 STE´PHANEDRUEL
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St´ephane Druel: Institut Fourier, UMR 5582 du CNRS, Universit´e Grenoble 1, BP 74, 38402 Saint Martin
d’H`eres, France
E-mail address: [email protected]