Log canonical thresholds on smooth varieties: the Ascending Chain Condition

0.15 MB·English

by  Lawrence Ein

LOG CANONICAL THRESHOLDS ON SMOOTH VARIETIES: THE
ASCENDING CHAIN CONDITION
LAWRENCE EIN AND MIRCEA MUSTAT¸A˘
9
0
0
2 Abstract. Building on results of Koll´ar, we prove Shokurov’s ACC Conjecture for log

n canonical thresholds on smooth varieties, and more generally, on varieties with quotient
a singularities.
J

9

]
G
1. Introduction
A
.
h Let X be a smooth variety over an algebraically closed field k, of characteristic
t
a zero. If f ∈ O(X) is a nonzero regular function on X, and p ∈ X is a point such that
m
f(p) = 0, then the log canonical threshold lct (f) is an invariant of the singularity at p
p
[ of the hypersurface defined by f. This invariant plays a fundamental role in birational

2 geometry. For an overview of the many contexts in which this invariant appears, and for
v
some applications, see for example [EM]. The following is our main result.
4
4 Theorem 1.1. For every n ≥ 1, the set
4
4 T := {lct (f) | f ∈ O(X), f(p) = 0, Xsmooth, dim(X) = n}
n p
.
1
satisfies the Ascending Chain Condition, that is, it contains no infinite strictly increasing
1
8 sequences.
0
:
v The above statement has been conjectured by Shokurov in [Sho]. In fact, Shokurov’s
i
X conjecture is in a more general setting, allowing the ambient variety to have mild singular-
r ities. The interest in this conjecture comes from the fact that its general singular version
a
is related to the Termination of Flips Conjecture (see [Bir] for a statement in this direc-
tion). While the above theorem does not have such strong consequences, we believe that
it offers strong evidence for the general case of the conjecture. Moreover, it suggests that
this general case might not be out of reach.
As we will see in Proposition 3.4 below, every log canonical threshold on a variety
with quotient singularities can be also written as a log canonical threshold on a smooth
variety of the same dimension. Therefore the above theorem also implies Shokurov’s Con-
jecture for log canonical thresholds on varieties with quotient singularities.
Thefirstunconditionalresultsonlimitpointsoflogcanonicalthresholds inarbitrary
dimension havebeenproved in[dFM],using ultrafilterconstructions. Thekeypointwasto
Key words and phrases. Log canonical threshold, log resolution.
2000Mathematics Subject Classification. Primary 14E15; Secondary 14B05.
ThefirstauthorwaspartiallysupportedbyNSFgrantDMS-0700774,andthesecondauthorwaspartially
supported by NSF grant DMS-0758454,and by a PackardFellowship.
1
2 L. Ein and M. Musta¸t˘a
showthatgivenasequence ofpolynomialsf ∈ k[x ,...,x ]suchthatlim lct (f ) =
m 1 n m→∞ 0 m
c, one can construct a formal power series f ∈ K[[x ,...,x ]] (for a suitable field extension
1 n
K ofk)such thatlct (f) = c(onecandefinelogcanonicalthresholdsalsoforformalpower
0
series, and it seems that for questions involving limit points of log canonical thresholds,
this is the right framework).
Theresultsinloc. cit.havebeenreprovedbyKoll´arin[Kol1],replacingtheultrafilter
construction by a purely algebro-geometric one. His approach turned out to be better
suited for ruling out increasing sequences of log canonical thresholds. In fact, using a deep
recent result in the Minimal Model Program from [BCHM], he showed that if (with the
notationintheprevious paragraph)the formalpower series f issuch thatits logcanonical
threshold is computed by a divisor with center at the origin, then lct (f ) ≥ c for all m
0 m
large enough. In this note we show that elementary arguments on log canonical thresholds
allow us to reduce to this special case, and therefore prove Theorem 1.1 above.
1.1. Acknowledgment. We are grateful to Shihoko Ishii and Angelo Vistoli for useful
discussions and correspondence, and to J´anos Koll´ar for his comments and suggestions on
a preliminary version of this note.
2. Getting isolated log canonical centers
Let k be an algebraically closed field of characteristic zero. Fix n ≥ 1, and let
R = k[[x ,...,x ]]. We put X = Spec(R). Suppose that f ∈ R is a nonzero element
1 n
in the maximal ideal m = (x ,...,x ). Our goal in this section is to show that after
1 n
possibly replacing f by a suitable fr, we can find a polynomial g ∈ k[x ,...,x ] such that
1 n
lct (f) = lct (fg) and such that the log canonical threshold of fg can be computed by
0 0
a divisor E over X with center at the closed point. Our main reference for log canonical
thresholds in the formal power series setting is [dFM] (see also [Kol2], §8 for the basic
facts in the more familiar setting of varieties over k).
Let us fix a log resolution π: Y → X for the pair (R,f ·m). Note that by the main
resultin[Tem]suchresolutionsexist,andthisiswhatallowedin[dFM]theextensionofthe
usual results on log canonical thresholds to our setting. We write f ·O = O(− a E ),
Y i i i
m·OY = OY(− ibiEi), and KY/X = OY(− ikiEi). Note that the center ofPEi on X
is the closed poinPt (that we denote by 0) if anPd only if bi > 0.
The log canonical threshold of f is given by
k +1
i
lct (f) = min .
0
i ai
We say that a divisor E computes the log canonical threshold lct (f) if lct (f) = kj+1.
j 0 0 aj
Let I denote the set of those j for which E has center equal to the closed point. If there
j
is j ∈ I computing lct (f), then there is nothing we need to do. We now assume that this
0
is not the case. The following is our key observation.
Log canonical thresholds on smooth varieties 3
Proposition 2.1. With the above notation, suppose that c := lct (f) < ki+1 for every
0 ai
i ∈ I. If
1 k +1
i
(1) q := min −a | i ∈ I ,
i
(cid:26)b (cid:18) c (cid:19) (cid:27)
i
then lct (f ·mq) = lct (f), and there is a divisor E over X computing lct (f ·mq), with
0 0 0
center equal to the closed point.
Proof. Note that our assumption implies q > 0. By construction π is also a log resolution
of (X,f ·mq). Using the above notation for f ·O , m·O , and K , we see that
Y Y Y/X
k +1
lct (f ·mq) = min i .
0
i (cid:26)ai +qbi(cid:27)
Note that if i 6∈ I, then b = 0 and ki+1 = ki+1 ≥ c.
i ai+qbi ai
On the other hand, if i ∈ I then by the definition of q we have
k +1
i
(2) ≥ c.
a +qb
i i
This shows that lct (f · mq) ≥ c. Moreover, if i ∈ I is such that the minimum in (1) is
0
achieved, then we have equality in (2). This shows that lct (f ·mq) = lct (f), and there
0 0
is a divisor E with center equal to the closed point, that computes lct (f ·mq). (cid:3)
i 0
Our next goal is to replace the ideal mq by the suitable power of a product of very
general linear forms. Suppose that a is an ideal in k[x ,...,x ] generated by h ,...,h ,
1 n 1 d
and such that lct (a) = c. It is well-known that if N ≥ c and g ,...,g are general linear
0 1 N
combinations of h ,...,h with coefficients in k, then lct (g) = c/N, where g = g ·...·g
1 d 0 1 N
(see, for example, Prop. 9.2.26 in [Laz] for a related statement). The proof of this result,
however, relies onBertini’s Theorem, so itdoes notsimply carryover to oursetting. When
dealing with formal power series, we will argue by taking truncations. Let us start with a
small variation on the above-mentioned result.
Lemma 2.2. Let a and b be nonzero ideals in the polynomial ring k[x ,...,x ], and α,
1 n
β > 0 such that lct (aα · bβ) = c . Let N be a positive integer such that N/β ≥ c .
0 0 0
If h ,...,h generate b, and if g ,...,g are general linear combinations of the h with
1 d 1 N i
coefficients in k, then lct (aα ·gβ/N) = c , where g = g ·...·g .
0 0 1 N
Proof. The argument is the same one as in loc. cit., but we recall it for completeness.
Let µ: W → An be a log resolution of a · b. Let us write a · O = O(− u E ),
W i i i
b·OW = O(− iwiEi), and KW/An = ikiEi. Therefore P
P P k +1
i
c = min ,
0
i∈J (cid:26)αui+βwi(cid:27)
where J is the set of those i such that 0 ∈ µ(E ).
i
Since each g is a general combination of the generators of b, it follows by Bertini’s
j
Theoremthatg ·O = O(−F − w E ),whereF ,...,F aredivisors withnocommon
j W j i i i 1 N
components amongst them or wiPth the Ei, such that iEi + jFj has simple normal
P P
4 L. Ein and M. Musta¸t˘a
crossings.Inparticular,µisalogresolutionofa·g.Sinceg·O = O(− Nw E − F ),
W i i i j j
and P P
k +1 1
i
min min , = c
0
(cid:26)i∈J αui +βwi β/N(cid:27)
(we use the hypothesis that N/β ≥ c ), it follows that lct (aα ·gβ/N) = c . (cid:3)
0 0 0
In the following proposition we assume that the ground field k is uncountable. In
this case, we follow the standard terminology by saying that a very general point on an
irreducible algebraic variety over k is a point that lies outside a countable union of proper
subvarieties.
Proposition 2.3. With the notation in Proposition 2.1, let N > qc be an integer. If
g ,...,g are very general linear linear forms with coefficients in k, and g = g ·...·g ,
1 N 1 N
then lct (f · gq/N) = c, and there is a divisor E over X computing lct (f · gq/N), with
0 0
center equal to the closed point.
Proof. For every ℓ, let f denote the truncation of f of degree ≤ ℓ. Let us put c =
≤ℓ ℓ
lct (f ·mq). It follows from Proposition 2.1 and Prop. 2.5 in [dFM] that lim c = c.
0 ≤ℓ ℓ→∞ ℓ
In particular, if ℓ ≫ 0, then N > qc . For such ℓ we may apply the lemma to deduce that
ℓ
lct (f · gq/N) = c (note the since we are allowed to take the g as very general linear
0 ≤ℓ ℓ i
combinations of x ,...,x , we can simultaneously apply the lemma for all ℓ as above).
1 n
Another application of Prop. 2.5 in [dFM] now gives lct (f ·gq/N) = c.
0
Suppose that E is the divisor given by Proposition 2.1, so it computes lct (f ·mq),
0
and its center is equal to the closed point. We denote by ord the valuation of the fraction
E
field of R corresponding to E. Since g ∈ mN, we deduce ord (f · gq/N) ≥ ord (f · mq).
E E
Therefore we have
ord (K )+1 ord (K )+1
E Y/X E Y/X
(3) c = ≥ ≥ c,
ord (f ·mq) ord (f ·gq/N)
E E
where the second equality follows from the fact that lct (f · gq/N) = c. Therefore both
0
inequalities in (3) are equalities, and we see that E computes lct (f ·gq/N). (cid:3)
0
3. The proof of the ACC Conjecture
Before giving the proof of Theorem 1.1, let us describe the key construction from
[Kol1]. For every nonnegative integer m, let Pol denote the affine space ANm (with
≤m k
N = n+m ), such that for every field extension K of k, the K-rational points of Pol
m m ≤m
parame(cid:0)trize(cid:1)the polynomials in K[x ,...,x ] of degree ≤ m. We have obvious maps
1 n
π : Pol → Pol that correspond to truncation of polynomials. Given a field
m ≤m ≤(m−1)
extension K of k, a formal power series f ∈ K[[x ,...,x ]] corresponds to a sequence of
1 n
morphisms SpecK → Pol over Speck, compatible via the truncation maps. We denote
≤m
by t (f) the corresponding element of Pol (K).
m ≤m
Suppose now that (f ) is a sequence of formal power series in k[[x ,...,x ]], all
q q 1 n
being nonzero, and of positive order. We consider sequences of irreducible closed subsets
Z ⊆ Pol with the following properties:
m ≤m
Log canonical thresholds on smooth varieties 5
i) For every m, there are infinitely many q such that t (f ) ∈ Z .
m q m
ii) Z is the Zariski closure of those t (f) ∈ Z .
m m m
iii) Each truncation morphism π induces a dominant morphism Z → Z .
m m m−1
Such sequences can be constructed by induction. We start the induction by taking
Z = Pol = Spec(k).IfZ isconstructed,thenwetakeZ tobeaminimalirreducible
0 ≤0 m m+1
closet subset of π−1 (Z ) with the property that it contains t (f ) for infinitely many
m+1 m m+1 q
q. Properties i)-iii) are clear: note that the minimality assumption in the definition of
Z implies that the induced morphism Z → Z is dominant.
m+1 m+1 m
Suppose that (Z ) is a sequence satisfying i)-iii) above. Let η denote the generic
m m m
point of Z , so the truncation maps induce embeddings k(η ) ֒→ k(η ). If K is an
m m m+1
algebraically closed field containing k(η ), we get corresponding maps Spec(K) →
m m
Pol≤m that are compatible via the trSuncation morphisms. This corresponds to a formal
power series f ∈ K[[x ,...,x ]] such that t (f) gives η . Of course, this construction is
1 n m m
not unique. However, whenever f is obtained from the sequence (f ) via such a sequence
q q
(Z ) , we say that f is a generic limit of (f ) . A trivial example is when f = h for
m m q q m
every m, in which case each Z is a point, and for every field extension K of k, a generic
m
limit of this sequence is given by the image of f in K[[x ,...,x ]].
1 n
A key property of generic limits that follows from construction is that for every m,
there are infinitely many q such that lct (t (f)) = lct (t (f )). It is easy to deduce from
0 m 0 m q
this that if lim lct (f ) = c, then lct (f) = c (see Thm. 29 in [Kol1] for details).
q→∞ 0 q 0
We start with an easy lemma describing the behavior of generic limits under multi-
plication.
Lemma 3.1. Let (f ) and (h ) be two sequences of formal power series in k[[x ,...,x ]].
q q q q 1 n
If f ∈ K′[[x ,...,x ]], and h ∈ K′′[[x ,...,x ]] are generic limits of (f ) , and respectively
1 n 1 n q q
(h ) , then there is a field extension K of both K′ and K′′, such that fh ∈ K[[x ,...,x ]]
q q 1 n
is a generic limit of the sequence (f h ) .
q q q
Proof. SupposethatZ′ ,Z′′ ⊆ Pol aresequences ofsubsetssatisfyingi)-iii)above,with
m m ≤m
respect to (f ) , and respectively (h ) . If η′ and η′′ are the generic points of respectively
q q q q m m
Z′ and Z′′, then K′ contains all k(η′ ), and similarly, K′′ contains all k(η′′). For every
m m m m
m we have a morphism ϕ : Pol ×Pol → Pol that corresponds to multiplication,
m ≤m ≤m ≤m
followedbytruncationuptodegreem.Itisclearthatwehaveπ ◦ϕ = ϕ ◦(π ,π )for
m m m−1 m m
every positive m. If we take Z := ϕ (Z′ ×Z′′), then (Z ) is a sequence of irreducible
m m m m m m
closed sets that satisfies i)-iii) with respect to the sequence (f h ) . Moreover, if η is
q q q m
the generic point of Z , then we have embeddings compatible with the maps induced by
m
truncation
k(η ) ֒→ Q(k(η′ )⊗ k(η′′)),
m m k m
where for a domain S we denote by Q(S) its fraction field. Since k is algebraically closed,
K′ ⊗ K′′ is a domain. If K is an algebraically closed extension of Q(K′ ⊗ K′′), then
k k
K contains k(η ), and it follows from the construction that fh is the generic limit
m m
correspondinSg to the sequence (Zm)m, and to the embedding mk(ηm) ֒→ K. (cid:3)
S
6 L. Ein and M. Musta¸t˘a
The following is the key result of Koll´ar that we will use. Its proof uses the difficult
finite generation result of [BCHM].
Proposition 3.2. (Prop.40, [Kol1]) Let K ⊇ k be a field extension, and suppose that
F ∈ K[[x ,...,x ]] is a formal power series. Suppose that there is a divisor computing
1 n
lct (F) with center at the closed point. If Z ⊆ Pol is the k-Zariski closure of t (F),
0 m ≤m m
then there is a positive integer m, and an open subset U ⊆ Z such that for every power
m m
series G ∈ K[[x ,...,x ]] with t (G) ∈ U , we have lct (F) = lct (G).
1 n m m 0 0
Note that if F is a generic limit of a sequence (f ) of power series in k[[x ,...,x ]],
q q 1 n
then the sets Z in the above statement are the same as the sets that come up in the
m
definition of F as a generic limit. We can now prove our main result.
Proof of Theorem 1.1. We may assume that k is uncountable, and it is enough to show
that the set
T ′ = {lct (f) | f ∈ k[[x ,...,x ]], f 6= 0, ord(f) ≥ 1}
n 0 1 n
contains no strictly increasing infinite sequences. This follows since we can extend scalars
to an uncountable algebraically closed field, and we can complete at the given point
p ∈ X (in fact, it is shown in [dFM] that T = T′, and that this set is independent of the
n n
algebraically closed field k; however, we do not need this fact).
Let us suppose that (f ) is a sequence of formal power series in k[[x ,...,x ]],
q q 1 n
nonzero and of positive order, such that if we put c = lct (f ), then the corresponding
q 0 q
sequence (c ) is strictly increasing. We put c := lim c . Let f ∈ K[[x ,...,x ]] be
q q q→∞ q 1 n
a generic limit of the sequence (f ) , with K an algebraically closed extension of k. As
q q
mentioned above, we have lct (f) = c.
0
If lct (f) is computed by some divisor E with center equal to the closed point, then
0
we are done. Indeed, we apply Proposition 3.2 above to F = f to get m and an open
subset U ⊆ Z withthepropertythatwheneverG ∈ K[[x ,...,x ]]satisfiest (G) ∈ U ,
m m 1 n m m
we have lct (f) = lct (G). On the other hand, by assumption Z is the Zariski closure
0 0 m
of those t (f ) ∈ Z . Therefore we can find q with t (f ) ∈ U , so that c = c, a
m q m m q m q
contradiction.
Suppose now that lct (f) is not computed by any divisor with center equal to the
0
closedpoint.InthiscasePropositions2.1and2.3implythatwecanfindapositiverational
number q, a positive integer N and g ∈ k[[x ,...,x ]] such that lct (f · gq/N) = c, and
1 n 0
there is a divisor E computing lct (f ·gq/N), and having center equal to the closed point.
0
Let us write q/N = s/r, for positive integers r and s. It is clear that lct (frgs) = c/r,
0
and the same divisor E computes this log canonical threshold.
On the other hand, by Lemma 3.1 the power series frgs is a generic limit of the
sequence (frgs) . Since lct (frgs) is computed by a divisor with center equal to the
q q 0
closed point, using Proposition 3.2 for F = frgs, we deduce that for some q we have
lct (frgs) = c/r. Note now that lct (frgs) ≤ lct (fr) = c /r, so we have a contradiction.
0 q 0 q 0 q q
(cid:3)
This completes the proof of the theorem.
Remark 3.3. While we stated Theorem 1.1 only for log canonical thresholds of principal
ideals, it is straightforward to extend the statement to arbitrary ideals. More precisely,
Log canonical thresholds on smooth varieties 7
the theorem implies that for every n ≥ 1, the set of all log canonical thresholds lct (X,a)
p
satisfies the Ascending Chain Condition, when X varies over the smooth n-dimensional
varieties, and a over the nonzero ideals on X containing p in their support. Indeed, we
may assume that X is affine, in which case if f is a general linear combination of the
i
generators of a, for 1 ≤ i ≤ n, and if f = f ·...·f , then 1 ·lct (X,a) = lct (X,f) (note
1 n n p p
that lct (X,a) ≤ n, and apply for example Lemma 2.2). Therefore each such lct (X,a)
p p
lies in n·T , and this set satisfies the Ascending Chain Condition by Theorem 1.1.
n
The following proposition allows us to reduce log canonical thresholds on varieties
with quotient singularities to log canonical thresholds on smooth varieties. We say that a
variety X has quotient singularities at p ∈ X if there is a smooth variety U, a finite group
[
G acting onU, anda pointq ∈ V = U/G such that the two completions O andO are
X,p V,q
isomorphic as k-algebras. One can assume that U is an affine space and that the action of
d
G is linear. Furthermore, one can assume that G acts with no fixed points in codimension
one (otherwise, we may replace G by G/H and U by U/H, where H is generated by all
pseudoreflections in G, and by Chevalley’s theorem [Che], the quotient U/H is again an
affine space). Using Artin’s approximation results (see Corollary 2.6 in [Art]), it follows
that there is an ´etale neighborhood of p that is also an ´etale neighborhood of q. In other
words, there is a variety W, a point r ∈ W, and ´etale maps ϕ: W → X and ψ: W → V,
such that p = ϕ(r) and q = ψ(r). After replacing ϕ by the compositon
ϕ
W × U → W → X,
V
we may assume that in fact we have an ´etale map U/G → X containing p in its image,
with U smooth, and such that G acts on U without fixed points in codimension one.
This reinterpretation of the definition of quotient singularities seems to be well-known to
experts, but we could not find an explicit reference in the literature.
We say that X has quotient singularities if it has quotient singularities at every
point.
Proposition 3.4. Let X be a variety with quotient singularities, and let a be a proper
nonzero ideal on X. For every p in the zero-locus V(a) of a, there is a smooth variety U,
a nonzero ideal b on U, and a point q in V(b) such that lct (X,a) = lct (U,b).
p q
Proof. Let us choose an ´etale map ϕ: U/G → X with p ∈ Im(ϕ), where U is a smooth
variety, and G is a finite group acting on U without fixed points in codimension one. Let
ϕ: U → X denote the composition of ϕ with the quotient map. Since G acts without fixed
points in codimension one, ϕ is ´etale in codimension one, hence K = ϕ∗(K ). It follows
U X
ferom Proposition 5.20 in [KM] that if b = a · O , then the pair (X,aq) is log canonical
U
if and only if the pair (U,bqe) is log canonical (actually the result in loec. cit. only covers
the case when a is locally principal, but one can easily reduce to this case, by taking a
suitable product of general linear combinations of the local generators of a). We conclude
that there is a point q ∈ V(b) such that lct (X,a) = lct (U,b). (cid:3)
p q
Remark 3.5. At least over the complex numbers, one usually says that X has quotient
singularities at p if the germ of analytic space (X,x) is isomorphic to M/G, where M is
8 L. Ein and M. Musta¸t˘a
a complex manifold, and G is a finite group acting on M. It is not hard to check that in
this context this definition is equivalent with the one we gave above.
Combining Proposition 3.4 and Theorem 1.1 (see also Remark 3.3), we deduce
Shokurov’s ACC Conjecture on varieties with quotient singularities.
Corollary 3.6. For every n ≥ 1, the set
{lct (X,a) | Xhas quotient singularities, dim(X) = n,(0) 6= a, p ∈ V(a)}
p
satisfies the Ascending Chain Condition.
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Department of Mathematics, University of Illinois at Chicago, 851 South Morgan
Street (M/C 249), Chicago, IL 60607-7045, USA
E-mail address: [email protected]
Department of Mathematics, University of Michigan, 530 Church Street, Ann Ar-
bor, MI 48109, USA
E-mail address: [email protected]