A log canonical threshold test

0.13 MB·English

by  Alexander Rashkovskii

A log canonical threshold test
Alexander Rashkovskii
Abstract
In terms of log canonical threshold, we characterize plurisubharmonic functions with
5 logarithmic asymptotical behaviour.
1
0
2
1 Introduction and statement of results

n
a
J Let ubea plurisubharmonicfunction on a neighborhoodof theorigin of Cn. Its log canonical

0 threshold at 0,
2 c = sup c > 0 :e−cu L2 (0) ,
u { ∈ loc }

] is an importantcharacteristic of asymptotical behavior of u at 0. Thelog canonical threshold
V
c( ) of a local ideal in can be defined as c for the function u = log F , where
C I I ⊂ O0 u | |
F = (F ,...,F ) with F generators of . (Surprisingly, the latter notion was introduced
h. 1 p { j} I
later than its plurisubharmoniccounterpart.) For general results on log canonical thresholds,
t
a including their computation and applications, we refer to [9], [16], [17].
m
A classical result due to Skoda [22] states that
[

1 cu ≥νu−1, (1)
v
1 where ν is the Lelong number of u at 0. A more recent result is due to Demailly [8]: if 0 is
u
3 an isolated point of u−1( ), then
8 −∞
4
0 cu Fn(u) := nen(u)−1/n. (2)
≥
.
1
0 Here ek(u) = (ddcu)k (ddclog z )n−k(0) are the Lelong numbers of the currents (ddcu)k at
5 0 for k = 1,...,n, and∧d = ∂+∂¯|,|dc = (∂ ∂¯)/2πi; note that e (u) = ν . This was extended
1 u
1 −
by Zeriahi [23] to all plurisubharmonic functions with well-defined Monge-Amp`ere operator
:
v near 0.
i
X In [19], inequality (2) was used to obtain the ‘intermediate’ bounds
r
a c F (u) := ke (u)−1/k, 1 k l, (3)
u k k
≥ ≤ ≤
l beingthe codimension of an analytic set Acontaining theunboundedlocus L(u) of u. None
of the bounds for different values of k can be deduced from the others.
It is worth mentioning that relation (2) was proved in [8] on the base of a corresponding
result for ideals1 obtained in [6]:
c( ) ne( )−1/n, (4)
I ≥ I
wheree( ) is the Hilbert-Samuel multiplicity of the(zero-dimensional) ideal . Furthermore,
I I
it was shown in [6] that an equality in (4) holds if and only if the integral closure of is a
I
1A direct proof was given later in [2].
1
power of the maximal ideal m . Accordingly, the question of equality in (2) has been raised
0
in [8] where it was conjectured that, similarly to the case of ideals, the extremal functions
would be those with logarithmic singularity at 0.
The conjecture was proved in [20] where it was shown that
c = F (u) (5)
u n
if and only if the greenification g of u has the asymptotics g (z) = e (u)log z +O(1) as
u u 1
| |
z 0. Here the function g is the uppersemicontinuous regularization of the upperenvelope
u
→
of all negative plurisubharmonic functions v on a bounded neighborhood D of 0, such that
v u+O(1) near 0, see [18]. Note that if u = log F , then g = u+O(1) [21, Prop. 5.1].
u
≤ | |
Theequality situation in(1)(i.e., in(3)withk = 1)was firsttreated in[5]and[11]forthe
dimension n = 2: thefunctions satisfying c = ν−1 were proved in that case to beof the form
u u
u = clog f +v, where f is an analytic function, regular at 0, and v is a plurisubharmonic
| |
function with zero Lelong number at 0. In a recent preprint [15], the result was extended
to any n. This was achieved by a careful slicing technique reducing the general case to
the aforementioned two-dimensional result. In addition, it used a regularization result for
plurisubharmonic functions with keeping the log canonical threshold (see Lemma 1 below).
Concerning inequalities (3), it was shown in [19] that the only multi-circled plurisub-
harmonic functions u(z) = u(z ,..., z ) satisfying c = F (u) are essentially of the form
1 n u l
| | | |
cmax log z for an l-tuple J 1,...,n . Here we address the question on equalities in
j∈J j
| | ⊂ { }
the bounds (3) in the general case.
We present an approach that is different from that of [15] and which actually works also
for the‘intermediate’ equality situations. Itis basedon a recent resultof Demailly andPham
Hoang Hiep [10]: if the complex Monge-Amp`ere operator (ddcu)n is well defined near 0 and
e (u) > 0, then
1
e (u)
j−1
cu En(u) := X ,
≥ e (u)
j
1≤j≤n
where e (u) = 1. In particular, this implies (2) and sharpens, for the case of functions with
0
well-defined Monge-Amp`ere operator, inequality (1). Moreover, it is this bound that was
used in [20] to prove the conjecture from [8] on functions satisfying (5).
Given 1 < l n, let be the collection of all plurisubharmonic functions u whose
l
≤ E
unbounded loci L(u) have zero 2(n l + 1)-dimensional Hausdorff measure. For such a
−
function u, the currents (ddcu)k are well defined for all k l [12]. In particular, u if
l
≤ ∈ E
L(u) lies in an analytic variety of codimension at least l. Furthermore, we set to be just
1
E
the collection of all plurisubharmonic functions near 0.
Let c (z) denote the log canonical threshold of u at z and, similarly, let e (u,z) denote
u k
the Lelong number of (ddcu)k at z; in our notation, c (0) = c and e (u,0) = e (u). As
u u k k
is known, the sets z : c (z) c are analytic for all c > 0. Our first result describes, in
u
{ ≤ }
particular, regularity of such a set for c = c , provided c = F (u).
u u l
For u we set
l
∈ E
e (u)
j−1
Ek(u) = X , k l.
e (u) ≤
j
1≤j≤k
2
Theorem 1 Let u for some l 1, and let e (u) > 0. Then
l 1
∈ E ≥
(i) c E (u) for all k l;
u k
≥ ≤
(ii) c F (u) for all k l;
u k
≥ ≤
(iii) if u satisfies c = F (u) for some k l, then k = l and there is a neighborhood V of
u k
≤
the origin such that the set A = z : c (z) c is an l-codimensional manifold in V.
u u
{ ≤ }
Furthermore, A = z : e (u,z) e (u) .
l l
{ ≥ }
For l = 1, assertion (iii) re-proves the aforementioned result from [15]. Let A= z = 0 ,
1
{ }
then the function u c log z is locally bounded from above near A and thus extends to
u 1
− | |
a plurisubharmonic function v; evidently, ν = 0. On the other hand, all the functions
v
u= c log z +v with ν =0 satisfy c = ν .
u 1 v u u
| |
When l > 1, there are functions u such that z : c (z) c is an l-codimensional
u u
{ ≤ }
manifold, but c > F (u). Indeed, let us take u(z ,z ,z ) = max log z ,2log z .
u l 1 2 3 1 2 2
{ | | | |} ∈ E
Then A= z Cn : c (z) c = z = z = 0 , while F (u) = √2 < 3/2 = c . (Note that
u u 1 2 2 u
{ ∈ ≤ } { }
c = E (u) in this case.)
u 2
Furthermore, the same example shows that the equality (ddcu)2 = δ2[z = z = 0] does
1 2
not imply u= δlog (z ,z ) +v with plurisubharmonic v and ν = 0.
1 2 v
| |
Therefore, in the higher dimensional situation we need to deduce a more precise informa-
tion on asymptotical behavior of u near A. By analogy with the case l = n, it is tempting to
make the following conjecture.
Let u , then
l
∈ E
c = F (u) (6)
u l
if and only if, for a choice of coordinates z = (z′,z′′) Cl Cn−l, the greenification g of u
u
∈ ×
near 0 satisfies
g = e (u)log z′ +O(1) as z 0.
u 1
| | →
The’if’directionisobviousinviewofcu = cgu [20]andthetrivialfactclog|z′| = l,however
the reverse statement might be difficult to prove even in the case l = 1 because that would
imply non-existence of a plurisubharmonic function φ with e (φ) = 0 and e (φ) > 0, which
1 n
is a known open problem. Namely, let such a function φ exist, and set u= φ+log z . Then
1
| |
1 = ν c c = 1. On the other hand, for D = Dn, g = g +log z and the relation
u ≤ u ≤ log|z1| u φ | 1|
e (φ) > 0 implies g = 0 and thus liminf(g log z ) = when z 0.
n φ u 1
6 − | | −∞ →
What we can prove is the following, slightly weaker statement.
Theorem 2 If u satisfies (6), then e (u) = e (u)k for all k l and, for a choice of
l k 1
∈ E ≤
coordinates z = (z′,z′′) Cl Cn−l, the function u satisfies u e (u)log z′ +O(1) near 0,
1
∈ × ≤ | |
while the greenification g of u = max u,N log z with any N e (u) satisfies
uN N { | |} ≥ 1
g = max e (u)log z′ ,N log z′′ +O(1), z 0. (7)
uN { 1 | | | |} →
3
Let us fix a neighborhood D V of 0 to be the product of unit balls in Cl and Cn−l
⊂
and consider the greenifications with respect to D. Then the functions g are equal to
uN
max e (u)log z′ ,N log z′′ and they converge, as N , to e (u)log z′ g .
1 1 u
{ | | | |} → ∞ | |≥
Denote, for any bounded neighborhood D of 0 and any u plurisubharmonic in D,
g˜ = lim g .
u uN
N→∞
where u =max u,N log z . Evidently, g˜ g .
N u u
{ | |} ≥
Theorem 3 Let u be such that g˜ = g . Then it satisfies (6) if and only if, for a choice
l u u
∈ E
of coordinates z = (z′,z′′) Cl Cn−l, g = e (u)log z′ +O(1) as z 0.
u 1
∈ × | | →
In particular, this is true for u = αlog F +O(1), where F is a holomorphic mapping,
| |
F(0) = 0. Moreover, in this case we also have u= e (u)log z′ +O(1).
1
| |
The statement on αlog F can be reformulated in algebraic terms as follows. Let be
| | I
an ideal of the local ring , and let V( ) be its variety: V( ) = z : f(z) = 0 f . If
0
O I I { ∀ ∈ I}
codim V( ) k, then themixed Rees’ multiplicity e ( ,m )of k copies of andn k copies
0 k 0
I ≥ I I −
of the maximal ideal m is well defined[4]. If k = n, then, as shownin [8], the Hilbert-Samuel
0
multiplicity e( ) of equals e (u), where, as before, u= log F for generators F of . By
n p
I I | | { } I
the polarization formula, e ( ,m ) = e (u) for all k; by a limit transition, this holds true for
k 0 k
I
all k l if codim V( )= l.
0
≤ I
Bounds (3) specify for this case as
c( ) ke ( ,m )−1/k, 1 k l;
k 0
I ≥ I ≤ ≤
from Theorems 1 and 3 we thus derive
Corollary 1 If codim V( ) = l and c( ) = ke ( ,m )−1/k for some k l, then k = l,
0 k 0
I I I ≤
V( ) is an l-codimensional hypersurface, regular at 0, and there exists an ideal n generated
0
I
by coordinate (smooth transversal) germs f ,...,f such that = ns for some s Z .
1 l 0 0 +
∈ O I ∈
2 Proofs
In what follows, we will use the mentioned regularization result by Qi’an Guan and Xiangyu
Zhou. Note that its proof rests on the strong openness conjecture from [9], proved in [13]
and [14], see also [3].
Lemma 1 [15, Prop. 2.1] Let u be a plurisubharmonic function near the origin, σ = 1.
u
Then there exists a plurisubharmonic function u˜ u on a neighborhood of 0 such that e−2u
≥ −
e−2u˜ is integrable on V and u˜ is locally bounded on V z : c (z) 1 .
u
\{ ≤ }
We will also refer to the following uniqueness theorem.
Lemma 2 ([18,Lem. 6.3]2 and[20,Lem. 1.1]) Ifuandv are twoplurisubharmonic functions
with isolated singularity at 0, such that u v +O(1) near 0 and e (u) = e (v), then their
n n
≤
greenifications coincide.
2For thegeneral case of non-isolated singularities, see [1,Thm. 3.7]
4
Proof of Theorem 1. Sinceallthefunctionalsu c , E (u), F (u)arepositivehomogeneous
u k k
7→
of degree 1, we can always assume c = 1.
u
−
Let u˜ be the function from Lemma 1. Its unbounded locus L(u˜) is contained in the
analytic variety A= z : c (z) 1 . Since A L(u) and u , codimA l.
u l
{ ≤ } ⊂ ∈ E ≥
For u˜, statement (i) is proved in [20, Thm. 1.4]. Note that the relation u u˜ implies
≤
e (u) e (u˜) for all k l and thus E (u) E (u˜) [10]. Since c = c , this gives us (i).
k k l l u u˜
≥ ≤ ≤
Assertion (ii) follows from (i) by the the arithmetic-geometric mean theorem.
To prove (iii), we first note that (i) implies c E (u) > E (u) F (u) for any k < l, so
u l k k
≥ ≥
we cannot have c = F (u) unless k = l.
u k
Next, if the analytic variety A has codimension m > l, then u˜ , so c = c
m u u˜
∈ E ≥
E (u˜) > E (u˜) E (u) F (u), which contradicts the assumption, so codimA= l.
m l l l
≥ ≥
Now we prove that 0 is a regular point of the variety A. By Siu’s representation formula,
(ddcu)l = Xpj[Aj]+R
on a neighborhood V of 0, where p > 0, [A ] are integration currents along l-codimensional
j j
analytic varieties containing 0, and R is a closed positive current such that for any a > 0 the
analytic variety z V : ν(R,z) a has codimension strictly greater than l. If ν(R,0) > 0,
{ ∈ ≥ }
then for almost all points z A we have e (u,z) < e (u). This implies, by (ii), c (z) > c for
l l u u
∈
all such points z, which is impossible. The same argument shows that the collection A
j
{ }
consists of at most one variety and 0 is its regular point. (cid:3)
Proof of Theorem 2. By the arithmetic-geometric mean theorem, the condition c = F (u)
u l
implies, in view of the inequality c E (u), the relations
u l
≥
e (u) e (u)
k−1 j−1
=
e (u) e (u)
k j
for any k,j l, which gives us e (u) = [e (u)]k for all k l.
k 1
≤ ≤
Sincerelation (7)for e (u) = 0isobvious (inthis caseg 0), wecan assumee (u) = 1.
1 uN ≡ 1
Note that for any z, we have e (u,z) [e (u,z)]k. As follows from the proof of (iii), the
k 1
≥
relation c = F (u) implies then, on a neighborhood V of 0,
u l
A V = z V : c (z) 1 = z V : F (u,z) 1 = z V : e (u,z) 1
u l k
∩ { ∈ ≤ } { ∈ ≤ } { ∈ ≥ }
for all k l. Moreover, we have e (u,z) = e (u,z)k = 1 for almost all z A V.
k 1
≤ ∈ ∩
Let us choose, according to Theorem 1, a coordinate system such that A V = z V :
∩ { ∈
z = 0, 1 k l . Denote v(z) = log z′ , z = (z′,z′′) Cl Cn−l, then A V = z :
k
≤ ≤ } | | ∈ × ∩ {
e (u,z) e (v,z) , with equalities almost everywhere.
k k
≥ }
In particular, we have u(z) log z (0,ζ′′) +C(ζ′′) as z (0,ζ′′) for all z Cn and
≤ | − | → ∈
ζ′′ Cn−l that are close enough to 0. Assuming u(z) 0 for all z with max z < 2, we get
k
∈ ≤ | |
u(z) log z (0,ζ′′) for all z V and ζ′′ Cn−l with (0,ζ′′) V. By choosing ζ′′ = z′′
≤ | − | ∈ ∈ ∈
this gives us u(z) v(z) on V.
≤
Let u = max u,N log z and v = max v,N log z . Then u v , while for N 1
N N N N
{ | |} { | |} ≤ ≥
we get, by Demailly’s comparison theorem for the Lelong numbers [7],
e (u ) (ddcu)l (ddcN log z )n−l(0) = Nn−le (u) = Nn−l = e (v ).
n N l n N
≤ ∧ | |
5
By Lemma 2, g = g . (cid:3)
uN vN
Proof of Theorem 3. The only part to prove is the one concerning u = αlog F +O(1); we
| |
assume α = 1. As follows from Theorem 2, one can choose coordinates such that the zero
set Z of F is z : z′ = 0 V 0 Cn−l. Observe that for such a function u we have
F
{ }∩ ⊂ { }×
e (u,z) = e (u)k for all z Z near 0.
k 1 F
∈
Let betheidealgenerated by thecomponents of themappingF. Then,as mentioned in
I
Section1,e (u)equalse ( ,m ),themixedmultiplicityoflcopiesoftheideal andn lcopies
l l 0
I I −
of the maximal ideal m . By [4, Prop. 2.9], e ( ,m ) can be computed as the multiplicity
0 l 0
I
e( ) of the ideal generated by generic functions Ψ ,...,Ψ and ξ ,...,ξ m .
1 l 1 n−l 0
J J ∈ I ∈
Since e( )= e (w), where w = log Ψ , we have e (u) = e (w).
l l l
J | |
Let now v = e (u)log z′ , w = max w,N log z′′ , and v = max v,N log z′′ . Since
1 N N
| | { | |} { | |}
w log F +O(1), we have from Theorem 2 the inequality w v +O(1) and thus w
N
≤ | | ≤ ≤
v +O(1). Note that the mappingΨ satisfies the L ojasiewicz inequality Ψ (z) z′ M near
N 0
| | ≥ | |
0 for some M > 0. Therefore, for sufficiently big N we have w = w′ = max w,N log z .
N N { | |}
Then, as in the proof of Theorem 2, we compute
e (w )= e (w′ ) (ddcw)l (ddcN log z )n−l(0) = Nn−le (w) = Nn−le (u) = e (v ),
n N n N ≤ ∧ | | l l n N
which, by Lemma 2, implies g = g for the greenifications on a bounded neighborhood
wN vN
D of 0.
We can assume D = z′ < 1 z′′ < 1 , then g = v , while g w because the
{| | }×{| | } vN N wN ≤ N
latter function is maximal on D and nonnegative on ∂D. Letting N we get w v.
→ ∞ ≥
Since w u+O(1), we have, in particular, u v+O(1), which, in view of Theorem 2,
completes th≤e proof. ≥ (cid:3)
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Tek/Nat, University of Stavanger, 4036 Stavanger, Norway
E-mail: [email protected]
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