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Orbifold generic semi-positivity: an application to families of canonically polarized manifolds

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A note on degenerations of del Pezzo surfaces

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L’INSTITUT FOURIER
Rong DU & Yun GAO
On the Griffiths numbers for higher dimensional singularities
Tome 65, no 1 (2015), p. 389-395.
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Ann. Inst. Fourier, Grenoble
65, 1 (2015) 389-395
ON THE GRIFFITHS NUMBERS FOR HIGHER
DIMENSIONAL SINGULARITIES
by Rong DU & Yun GAO (*)
Abstract. — We show that Yau’s conjecture on the inequalities for (n−1)-th
Griffiths number and (n − 1)-th Hironaka number does not hold for isolated rigid
Gorenstein singularities of dimension greater than 2. But his conjecture on the
inequality for (n − 1)-th Griffiths number is true for irregular singularities.
Résumé. — Nous montrons que la conjecture de Yau sur les inégalités concer-
nant le (n−1)-ième nombre de Griffiths et le (n−1)-ième nombre de Hironaka n’est
pas vraie en général pour les singularités de Gorenstein isolées rigides de dimension
supérieure à 2. Cependant, la première conjecture sur les inégalités concernant le
(n − 1)-ième nombre de Griffiths est vraie pour les singularités irrégulières.
1. Introduction
In singularity theory, one always wants to find invariants associated to
singularities. Let (V, o) be a Stein analytic space with o as its only singular-
ity of dimension n > 2. In [8], Yau introduced a bunch of invariants which
are naturally attached to isolated singularities. These invariants are used
to characterize the different notions of sheaves of germs of holomorphic
differential forms on analytic spaces. Various formulas which relate to all
these invariants were proved in [8]. Among these invariants the Griffiths
number g(p), the Hironaka number h(p) and δ(p) are the most interesting
invariants. In 1981, Yau conjectured that the following two inequalities of
Keywords: Griffiths number, Hironaka number, rigid Gorenstein singularity, irregular
singularity.
Math. classification: 32S05, 14B05.
(*) The first author: The Research Sponsored by National Natural Science Foundation of
China, Shanghai Pujiang Program and Scientific Research Foundation for the Returned
Overseas Chinese Scholars, State Education Ministry.
The second author: The Research Sponsored by National Natural Science Foundation
of China and Innovation Program of Shanghai Municipal Education Commission.
390 Rong DU & Yun GAO
these invariants should be true for general isolated normal singularities in
[7].
Conjecture 1.1. — Let (V, o) be a Stein analytic space with o as its
only normal singularity of dimension n > 2. Then
(i) g(n−1) > n− 1;
(ii) δ(n−1) > h(n−1) + n− 1.
In ([7]), he confirmed his conjecture for surface singularities and non-
rational singularities with “good” C∗-action of dimension greater than 2.
As an application, he showed that any Gorenstein surface singularities with
“good” C∗-action are not rigid. Although this conjecture inspired researches
in singularity theory for long time, we will show that this conjecture is
not true in general. But the first part of his conjecture holds for irregular
singularities.
2. Preliminaries
Let (V, o) be a normal isolated singularity of dimension n > 2. It is well
known that holomorphic functions defined on V − {o} can be extended
across o. However for holomorphic forms, the situation is completely dif-
ferent. Even if we assume that the holomorphic forms defined on V − {o}
are L2-integrable in a neighborhood of o in the sense of Griffiths ([2]), it is
not clear whether holomorphic forms can be extended across o. In [8], the
Griffiths number g(p) was introduced to measure how many L2-integrable
holomorphic p-forms on V − {o} cannot be extended across o. Similarly,
Yau defined another class of invariants δ(p) which measures how many holo-
morphic p-forms on V − {o} cannot be extended across o in [7].
In [8], Yau studied the relations among all kinds of sheaves of germs of
holomorphic forms which were also considered by Grauert-Grothendieck,
Noether, Ferrari and Siu.
(1) Noether: Ω̄pV := π∗Ω
p
M , where π : M −→ V is a resolution of
singularities of V .
(2) Grauert-Grothendieck: ΩpV := Ω
p
CN /K
p, where K p = {fα+dg∧β :
α ∈ ΩpCN ;β ∈ Ω
p−1
CN ; f, g ∈ I } and I is the ideal sheaf of V in C
N .
(3) Ferrari: Ω̃pV := Ω
p
CN /H
p, where H p = {ω ∈ ΩpCN : ω|V \Vsing = 0}.
(4) Siu: ¯̄ΩpV := θ∗Ω
p
V \Vsing where θ : V \Vsing −→ V is the inclusion map
and Vsing is the singular set of V .
ANNALES DE L’INSTITUT FOURIER
ON THE GRIFFITHS NUMBERS 391
If V is a normal variety, then dualizing sheaf ωV of Grothendieck is
actually the sheaf ¯̄ΩnV . Clearly Ω
p
V , Ω̃
p
V are both coherent. Ω̄
p
V is a coherent
sheaf because π is a proper map. ¯̄ΩpV is also a coherent sheaf by the theorem
of Siu (see Theorem A of [5]).
Lemma 2.1. — There are several short exact sequences:
(2.1) 0 −→ Kp −→ ΩpV −→ Ω̃
p
V −→ 0,
(2.2) 0 −→ Ω̃pV −→ Ω̄
p
V −→ H
p −→ 0,
and
(2.3) 0 −→ Ω̄pV −→
¯̄ΩpV −→ J
p −→ 0,
where Kp, Hp and Jp are all supported on the singular set.
Definition 2.2. — Let (V, o) be a Stein analytic space with o as its
only singularity. Let Kp, Hp and Jp be defined as in (2.1), (2.2), and (2.3).
Then the invariantsm(p), g(p) and s(p) at o are defined to be dimKpo , dimHpo
and dimJpo respectively. δ(p) is defined to be the dimension of the co-kernel
of the natural map ΩPV →
¯̄ΩpV . We also define the geometric genus of the
singularity pg to be s(n) and irregularity of the singularity q to be s(n−1).
The following lemma can be found as Lemma 2.7 in [8]. We will provide
a short proof here.
Lemma 2.3. — Let (V, o) be a Stein analytic space with o as its only
isolated singularity. Let π : M → V be a resolution of the singularity. Then
g(p) = dim Γ(M,ΩpM )/π
∗Γ(V,ΩpV ),
and
δ(p) = dim Γ(V − {o},ΩpV )/Γ(V,Ω
p
V ),
where we identify Γ(V,ΩpV ) with its image in Γ(V − {o},Ω
p
V ).
Proof. — Because Kp is supported on the isolated singularity o and Ω̃pV
is coherent on the Stein space V , they both have vanishing cohomology
groups of degree greater than 0. So, from (2.1) and (2.2), we can have an
exact sequence as follows
0 −→ Kpo −→ Γ(V,Ω
p
V ) −→ Γ(V, Ω̄
p
V ) −→ H
p
o −→ 0.
Therefore we get the first equality by identifying Γ(V, Ω̄pV ) with Γ(M,Ω
p
M ).
The second equality is obvious by the definition. 
TOME 65 (2015), FASCICULE 1
392 Rong DU & Yun GAO
3. Inequalities for invariants of singularities
We will show that Yau’s conjecture mentioned in the first section is not
true in general.
Theorem 3.1. — Let (V, o) be an isolated rigid Gorenstein singularity
of dimension n > 3. Then (V, o) does not satisfy the above two inequalities.
Proof. — From the exact sequences in Lemma 2.1 one immediately con-
cludes the inclusions of finite dimensional vector spaces
Ω̃pV ⊂ Ω̄
p
V ⊂
¯̄ΩpV
from which the equality δ(p) = g(p) + s(p) immediately follows (see Lem-
ma 1.3 in [7]).
A rigid Gorenstein singularity has δ(n−1) = 0 (cf. Theorem 4.6 in [7])
from which it follows that g(n−1) = 0. Hence a rigid Gorenstein singularity
provides a counterexample to the conjectures that g(n−1) > n − 1 and
δ(n−1) > pg + n− 1. 
Remark 3.2. — The isolated rigid Gorenstein singularities of dimension
n > 3 do exist. For example, the existence of an isolated Gorenstein finite
quotient singularity of dimension n > 3 is given by [3], [9] or [1]. From
Theorem 3 in [4], such singularity is rigid.
Next, we are going to show that the inequality (3.4) holds for irregular
singularities.
Definition 3.3. — A normal isolated singularity is called regular if the
irregularity q = 0. Otherwise, it is called irregular.
Theorem 3.4. — Let (V, o) be a normal isolated irregular singularity
of dimension n > 2. Then
g(n−1) > n− 1.
Proof. — We will improve the methods used in Yau’s paper ([7] Theo-
rem 3.1) to estimate the lower bound of g(n−1). Let π : (M,A) → (V, 0)
be a resolution of the singularity with A = ∪si=1Ai as the exceptional set,
where each Ai is an irreducible component. We can assume without loss of
generality that the exceptional set A is a divisor in M with normal cross-
ings. Embed (V, o) into (Cm, 0) whose coordinate functions are supposed
to be z1, · · · , zm. Since the singularity is irregular, we know that
q = dim Γ(M\A,Ωn−1M )/Γ(M,Ω
n−1
M ) > 0.
ANNALES DE L’INSTITUT FOURIER
ON THE GRIFFITHS NUMBERS 393
So there exists a holomorphic (n − 1)-form ω0 on M\A that can not be
extended across A, i.e., ω0 ∈ Γ(M\A,Ωn−1M )\Γ(M,Ω
n−1
M ), which means
that ω0 must have poles along some irreducible component A1 of A. Sup-
pose ω1 has the lowest order of poles along A1 among all ω ∈ Γ(M\A,
Ωn−1M )\Γ(M,Ω
n−1
M ). Then π∗(zj)ω1 is holomorphic along A1 for all j, 1 6
j 6 m. Otherwise, it contracts to the assumption. If π∗(zj)ω1 /∈ Γ(M,Ωn−1M )
for some j, it must have poles along another irreducible component A2 of
A. Suppose ω2 ∈ Γ(M\A,Ωn−1M ) is holomorphic along A1 and has the low-
est order of poles along A2. Then π∗(zj)ω2 must be holomorphic along
A1 and A2, for all j, 1 6 j 6 m. Similarly, we can continue such kind of
steps to generate ω3, ω4, · · · if it is available. Since the number of irre-
ducible components of A is finite, by induction, there exists a non-empty
set Wk = {ω ∈ Γ(M\A,Ωn−1M )\Γ(M,Ω
n−1
M ) : ω has poles along some ir-
reducible component Ak and holomorphic along A1, . . . , Ak−1 such that
π∗(zj)ω ∈ Γ(M,Ωn−1M ) for all j, 1 6 j 6 m}.
Suppose ωk ∈Wk. We will separate our argument into two parts accord-
ing to the order of poles of ωk.
• The order of poles of ωk is greater than 1 along some exceptional
component Ak:
Choose a point b in Ak which is a smooth point of A. Let (x1, x2,
· · · , xn) be a coordinate system centered at b such that Ak is given
locally by x1 = 0 at b. Take the power series expansion of π∗(zj)
around b:
(3.1) π∗(zj) $ x
rj
1 fj , 1 6 j 6 m,
where fj is a holomorphic function such that fj(0, x2, · · · , xn) 6= 0
and “$” means local equality around b. Without loss of generality,
we may assume r1 = min{r1, . . . , rm}. It is easy to see that the
holomorphic (n− 1)-form
(3.2) dπ∗(zi1) ∧ dπ∗(zi2) ∧ · · · ∧ dπ∗(zin−1)
has vanishing order at least (n−1)r1−1 along Ak. So the vanishing
orders of the (n − 1)-form (π∗(z1))jωk, 1 6 j 6 n − 1, along Ak
are at most (n− 1)r1− 2. These (n− 1)-forms cannot be the linear
combinations of (3.2). Therefore we have produced at least n − 1
holomorphic (n− 1)-forms on M which are not obtained by pulling
back of holomorphic (n− 1)-forms on V .
• The maximal order of poles of ωk is equal to 1:
Because
H0(M,Ωn−1M ) = H
0(M,Ωn−1M (logA))
TOME 65 (2015), FASCICULE 1
394 Rong DU & Yun GAO
from Theorem 1.3 in [6], there exists an exceptional component Ak
such that the local expression of ωk around some smooth point b′
on A contains a summand of the form
x−11 gdx2 ∧ dx3 ∧ · · · ∧ dxn,
where g is a holomorphic function such that g(0, x2, · · · , xn) 6= 0
and (x1, x2, · · · , xn) is a coordinate system centered at b′ such
that Ak is given locally by x1 = 0 at b′. Similarly, if we express
π∗(zj), where 1 6 j 6 m, locally as (3.1) and still assume r1 =
min{r1, . . . , rm}, then the coefficient before dx2∧dx3∧ · · ·∧dxn in
the expression of (3.2) has vanishing order of x1 at least (n− 1)r1.
However the coefficient before dx2∧dx3∧· · ·∧dxn in the expression
of the (n− 1)-form (π∗(z1))jωk, 1 6 j 6 n− 1, has vanishing order
of x1 at most (n− 1)r1 − 1. These (n− 1)-forms cannot be the lin-
ear combinations of (3.2). Therefore we also have produced at least
n − 1 holomorphic (n − 1)-forms on M which are not obtained by
pulling back of holomorphic (n− 1)-forms on V .

Acknowledgements. The first author would like to thank S.S.-T. Yau for
useful discussions. The first author would also like thank N. Mok for sup-
porting his research when he was in the University of Hong Kong. Finally,
both authors heartily thank the referees for their invaluable suggestions
and comments. In particular, both authors are grateful to the referee who
helped them to shorten the proof of Theorem 3.1.
BIBLIOGRAPHY
[1] R. E. Anno, “Four-dimensional terminal Gorenstein quotient singularities”, Mat.
Zametki 73 (2003), no. 6, p. 813-820.
[2] P. A. Griffiths, “Variations on a theorem of Abel”, Invent. Math. 35 (1976), p. 321-
390.
[3] G. A. Miller, H. F. Blichfeldt & L. E. Dickson, Theory and applications of
finite groups, Dover Publications, Inc., New York, 1961, xvii+390 pages.
[4] M. Schlessinger, “Rigidity of quotient singularities”, Invent. Math. 14 (1971), p. 17-
26.
[5] Y.-t. Siu, “Analytic sheaves of local cohomology”, Trans. Amer. Math. Soc. 148
(1970), p. 347-366.
[6] D. van Straten & J. Steenbrink, “Extendability of holomorphic differential forms
near isolated hypersurface singularities”,Abh. Math. Sem. Univ. Hamburg 55 (1985),
p. 97-110.
[7] S. S. T. Yau, “Existence of L2-integrable holomorphic forms and lower estimates of
T 1V ”, Duke Math. J. 48 (1981), no. 3, p. 537-547.
ANNALES DE L’INSTITUT FOURIER
ON THE GRIFFITHS NUMBERS 395
[8] ———, “Various numerical invariants for isolated singularities”, Amer. J. Math. 104
(1982), no. 5, p. 1063-1100.
[9] S. S.-T. Yau & Y. Yu, “Gorenstein quotient singularities in dimension three”, Mem.
Amer. Math. Soc. 105 (1993), no. 505, p. viii+88.
Manuscrit reçu le 1er novembre 2013,
révisé le 29 novembre 2013,
accepté le 17 juin 2014.
Rong DU
East China Normal University
Department of Mathematics
Shanghai Key Laboratory of PMMP
Rm. 312, Math. Bldg, No. 500, Dongchuan Road
Shanghai, 200241, (P. R. China)
rdu@math.ecnu.edu.cn
Yun GAO
Shanghai Jiao Tong University
Department of Mathematics
Shanghai 200240, (P. R. of China)
gaoyunmath@sjtu.edu.cn
TOME 65 (2015), FASCICULE 1


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