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ANNALES DE L’INSTITUT FOURIER
JOSEPH H. M. STEENBRINK
Du Bois invariants of isolated complete
intersection singularities
Annales de l’institut Fourier, tome 47, no 5 (1997), p. 1367-1377
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Ann. Inst. Fourier, Grenoble
47, 5 (1997), 1367-1377
DU BOIS INVARIANTS OF ISOLATED COMPLETE
INTERSECTION SINGULARITIES
by Joseph H.M. STEENBRINK (*)
1. Introduction.
In this paper we study invariants of isolated singularities which arise
from Du Bois5 work on the filtered de Rham complex [1]. Therefore we call
them Du Bois invariants. In general, these are hard to compute, and their
meaning is not very clear. The reason to consider them here is that one of
them occurs in the deformation theory of singular Calabi-Yau threefolds.
This note is the result of an attempt to understand them better, and aims
to give a complete survey of what is known about them until now. We relate
them to the Hodge numbers of the local and vanishing cohomology groups.
Our main results are Theorem 4 which expresses the Tjurina number of
certain Gorenstein singularities in terms of Du Bois invariants and Hodge
numbers of the link, and Theorem 6 which expresses the Hodge numbers
of the Milnor fibre of certain three-dimensional complete intersections in
similar terms. We also address the question whether the Du Bois invariants
are semicontinuous under deformation of the singularity.
(*) Partially supported by the University of Grenoble and the Laboratoire J. Dieudonne
of Nice.
Key words: Isolated complete intersection singularity - Hodge numbers - Du Bois
invariants.
Math. classification: 32S35 - 32S05.
1368 JOSEPH H.M. STEENBRINK
2. Du Bois invariants.
Let (X^x) be an isolated singularity of pure dimension n and let
TT : (Y,E) —> (X^x) be a good resolution of it, i.e. E is a divisor with
normal crossings on Y. We fix a representative TT : Y —> X with X a
contractible Stein space; then E c—^ Y is a homotopy equivalence. We put
U:=Y\E^X\{x}.
The Du Bois invariants W^ of (X, rr) are defined by
^(X,^) :=dimHq(Y,^(\ogE){-E))
for p > 0 and ^ > 0. Their definition does not depend on the choice of the
good resolution, as the Du Bois invariants can be defined in terms of the
filtered de Rham complex (n^,F) of X (see [1]):
^^dimJ^Gr^^).
This follows from [13, Prop. 3.3].
We list some properties of the Du Bois invariants:
THEOREM 1. — 6^(X, x)=0ifp+q>norq>n.
See [3, Thm 2] and [5, Thm 6.6.1, Thm 6.7.1] for the first case.
The second case follows from the fact that Y is an 77-dimensional complex
manifold without compact component: by Serre duality, ^(Y^F) = 0 for
all locally free sheaves F on Y.
THEOREM 2. — If{X, x) is a tone isolated singularity, then bP^ = 0
for all p > 0 and q > 0.
See [5, Exp. V Sect. 4].
PROPOSITION 1. — If (X,x) has depth > k, then &°'9 = 0 for
0 < q < k- 1.
Proof. — Suppose that (X,x) has depth > k. Then H^(X, Ox) =
0 for % < k by [4]. As X is a Stein space, this implies that ^(^7, Ou) = 0 for
0 < i < k— l.By duality of Grauert-Riemenschneider's vanishing theorem
we have H^Y, Oy) = 0 for i < n, so H^Y, Oy) ^ H\U, Ou) for i < n -1.
Hence ^(V.Oy) = 0 for 0 < i < k - 1. By [11, Lemma 2.14], the maps
H^Y, OY(-E)) -^ fT(y, Oy) are injective for all z, so H^Y, Oy(-E)) = 0
for 0 < i < k - 1.
DU BOIS INVARIANTS OF SINGULARITIES 1369
In case (X, x) is Cohen-Macaulay, i.e. has depth n, the only possibly
non-zero b0^ is fc0'71-1. We call (X,x) a Du Bois singularity if b°^ = 0
for 0 < q < n. Note that the vanishing of H^Y.Oy) in this range
characterizes rational singularities. Du Bois singularities, in particular
Gorenstein ones, have been studied by Ishii [6], [7]. She gives an example
of a small deformation of a normal isolated Du Bois singularity such that
the deformed singularity is no longer Du Bois, and shows that this cannot
happen in the Gorenstein case. It follows that the Du Bois invariants are
not upper semicontinuous under deformation in general, but might still
be semicontinuous for Gorenstein singularities. However, we will give a
counterexample to this in Section 4.
If one only considers deformations which can be simultaneously
resolved, then the invariants bP^ do depend upper semicontinously on the
parameters. This follows from general semicontmuity theorems concerning
higher direct images of coherent sheaves under flat proper mappings.
The restriction of the complex f^y (log E)(-E)) to E is acyclic, hence
HP(y,f2y(log^)(-E))) ^ iT(y,E,C) = 0.
This implies that the first spectral sequence of hypercohomology
(1) E^ = H^YWogE)(-E)) ̂  H^(y,^(logE)(-£;)))
converges to 0. It follows, that also ^l'n-l = 0 for a Du Bois singularity.
The Du Bois invariant &1'1 has a particular importance for rational
singularities:
THEOREM 3. — Let (X, x) be a rational isolated singularity and let
TT : Y —> X be a good resolution. Then the subspace generated by the
image of the map
dlog : H\Y,0^) -^ -^(r,^)
has codimension ̂ ^(X^x).
See [10, Prop. 2.1].
3. Invariants from local cohomology.
Let X,y be as in the previous section. Let L = 9X be the link of
(X, x). The cohomology groups H'1^) carry a mixed Hodge structure. (See
1370 JOSEPH H.M. STEENBRINK
[11, Sect. 1] for the proofs of the following statements.) We will focus on
its Hodge filtration, and define invariants Z^9 of (X, x) by
l^ := dimGr^H^^L.C) = dimH^E.^^ogE) 0 0^).
Poincare duality for L induces a duality of mixed Hodge structures
H^-i-i^ ^ Hom(^(L), Q(-n))
and we obtain the equalities
p,q ^ jn-p.n-q-l
If we put h^ := dimGr^Gr^H^L, C), then one has h^ = h^ and
^p'^-p=^^'9.
q
By the semipurity theorem [11, Cor. 1.12], /^'9 = 0 in the following cases:
i f z < n , p + g > % and if i > n^p + q < i.
LEMMA 1. — Ifp < n then f^ l^-1 > f; IP-^.
i=0 i=0
The proof is an easy consequence of semipurity.
LEMMA 2. — Let (X, x) be a rational singularity. Then 1°^ = r'° =
0 for all i.
Proof. — We have 1°^ = dimH^E, 0^) = 0 because H^Y, Oy) = 0
(as (X,x) is rational) and H^Y.Oy) —^ H^E.OE) is surjective by [11,
Lemma 2.14]. Moreover, 1^° < 1°^.
We recall the following invariants:
r{X,x) = dimTj^^ (the Tjurina number);
q'{X^) = dim^o(£/,^-l)/^o(y,^-l(log^)(-^));
a{X,x) := dlmHO(Y^7y)/dHO(Y^1y~l(\ogE){-E)) (compare [15,
1.8] for the case n = 2).
THEOREM 4. — Let (X, x) be an isolated Gorenstein singularity of
dimension at least three for which Z1'71-3 = 0. Then
r = y1-1'1 + 61'71-2 - ̂ n-1 + y1'71"2 - r-150 + q ' .
If moreover (X,x) is Du Bois and Z71"1'0 = 0, then
T^^^+^^+Z1'71"2.
DU BOIS INVARIANTS OF SINGULARITIES 1371
Proof.— We have Tj^ c± H\U,Qu) ^ H^U,^-1) as (X,x)
is Gorenstein. We have the exact sequence of local cohomology for
^7y~l(\ogE)(-E) which reduces to
0 -^ <y ^ HE(Y^-\\ogE)(-E)) -^ H^Y^-^logE^-E))
-^ H\U^-1) -^ H^^dogE^-E)) -^ 0
as &rl-l'2 = 0. Hence
T = q' + y1-1'1 + ̂  - ̂
where we write
h^ = dim^y.^-^log £0(-£')) = dim^^^y^^logE))
f o r z = 1,2. .
Because ^l'n-3 = 0, we have the exact sequence
0 ̂  H^^Y^^ogE^-E)) -. H^^Y^^logE)) -> Gr^-^L)
-> ̂ "-^^^(^^(-E)) ̂  ̂ -^^^(log^)) -^ Gr^W -^ 0
which implies that
1,2 LI _ Ll,n-2 l̂,n-l i 7l,n-2 7l,n-l
{IE ~ I^E ~ ° — 0 +t — t
Finally use that Z1'71-1 = Z71-1'0.
Suppose that (X, re) is moreover Du Bois. Consider the exact sequence
HO(E^1Y-l(\ogE)^OE)-^HE(Y^1Y-l(\ogE)(-E))
^HE(Y^-\\ogE)).
By duality, the third space has dimension &l'n-l = 0 as {X^x) is Du Bois,
and the first one has dimension Z71"1'0. Hence /n-1'0 = 0 implies h^ = 0,
so g' = 0.
COROLLARY. — Let (X, x) be an isolated toric Gorenstein singula-
rity of dimension at least three. Then r = Z1'71"2. Indeed, toric singularities
are rational. As &l'n-2 = 0, the formula for r holds even without the hy-
pothesis l1^-3 = 0.
The condition l1^-3 = 0 is fulfilled for isolated complete intersection
singularities, see the next section, and for rational singularities in dimension
three. More generally we have
PROPOSITION 2. — Let (X,x) be a three-dimensional Gorenstein
Du Bois singularity with l1^ = /2'0 = 0 (e.g. a rational singularity). Then
T=26 1 ' l +^ l ' l -a .
1372 JOSEPH H.M. STEENBRINK
Proof. — As (X^x) is Du Bois, &1'2 = 0. The convergence to zero of
the spectral sequence (1) implies that in this case, a = 61'1 — &2'1.
This improves [9, Theorem 1]. If moreover (X^x) is a toric singu-
larity, we get T = I1'1. The rigidity of three-dimensional cyclic quotient
singularities (for which Z151 = 0) is a direct consequence.
4. The case of isolated complete intersection singularities.
The link L of an n-dimensional isolated complete intersection sin-
gularity is (n — 2)-connected, so H2'^) -^ 0 can only occur for i €
{0, n - 1, n, 2n - 1}. Hence IP^ = 0 unless p + q € {0, n - 1, n, 2n - 1}.
Recall that for an n-dimensional isolated complete intersection sin-
gularity (X, x) and p = 0 , . . . , n, an invariant 5p(X, x) can be defined in the
following way. Choose any one parameter smoothing / : ( X ' ^ x ) —>• (C,0)
of (X, x) and define
^(X,^):=dimGr^Cx/
as a Hodge number of the Milnor fibre of (X, x). It does not depend on the
choice of the smoothing (see [12]). Moreover it is an upper semicontinuous
n
invariant under deformations of (X,.r). Of course, ^ Sp{X,x) = /^(X,a-),
p=o
the Milnor number of (X, x).
For each one-parameter smoothing with Milnor fibre F one has the
exact sequence of mixed Hodge structures
0 -, H71-1^) -^ ^{F.L) -^ ^(F) -> ̂ (L) -^ 0.
Taking into account the duality of IT^F, L) and H71^) (w.r.t. Q(-n)) we
obtain
^ - s^-p = ̂ 'n-p - ̂ n-p-l == r-P^P-1 - IP^-P-\
P P
We conclude that ^ s^ < ̂  «n-z for 0 < p < n.
1=0 %=o
Note that Sn(X^x) is the geometric genus of (X^x) [11, Prop. 2.13]
mdth8iiso(X,x)=Sn(X,x)-lo-n~l=dimHn-l(Y,OY)-dlmHn-l(E,OE)
= ^o'n-l by [11, Lemma 2.14]. In particular SQ = 0 characterizes Du Bois
singularities among isolated complete intersection singularities. In general,
there seem no other connections between Du Bois invariants and Hodge
numbers of the Milnor fiber.
DU BOIS INVARIANTS OF SINGULARITIES 1373
THEOREM 5. — Let (X,x) be an isolated complete intersection
singularity of dimension n. Then ̂ (X, x) = 0 unless p + q e [n - 1, n}.
Proof. — If n < 2 the theorem reduces to Theorem 1. So suppose
that n > 3. Then &°'1 = 0 as (X, x) is Cohen-Macaulay. So we still have to
deal with the case 2 < p + 9 <n -2. We have the exact sheaf sequences
0 -. ̂ (\ogE)(-E) -^ ^(logE) -^ WogE) 0 OE -^ 0.
Note that ^(^^(logE) 0 0^;) = 0 for n < p + q < 2n - 1 (as
HP^(L) = 0 in that range). As also ̂  (V,^ (log E) {-E)) = 0 for these
p ^ q (by Theorem 1), we obtain
H^Y^^ogE)) = 0 for n < p+ q < 2n - 1.
By duality we get
^(V, ̂ \ogE)(-E)) = 0 for 1 < p + q < n - 1.
Suppose l < j ? + < 7 < n - l . Then H^(^) = 0 by [2, Prop. 2.3]. Hence
the natural map
HW^ogE^-E)) -. ̂ (Y,^(logE)(-E))
is an isomorphism for 2 < p-\-q < n-1. Therefore H'1^ fl,^(\ogE){-E)) =
0 in this range.
The convergence of the spectral sequence (1) to zero implies
PROPOSITION 3. — For (X,x) an isolated complete intersection
singularity of dimension n, we have
n-l
Q^^^-1'71-^-^71-^.
p=l
LEMMA 3. — Let (X, x) be a Du Bois isolated complete intersection
singularity of dimension at least three with ^n-1'0 = 0. Then
^ = 671-1'! + 61'71-2 + ̂ n-2 + l0^-1 + a.
Proof. — By [8], for any isolated complete intersection singularity of
dimension at least two,
^ - T = ̂ n-1 - r-150 +01+02+03
for certain invariants 01,02,03. For a Du Bois singularity, 01 = 0, and
03 < Z71-1'0 by its definition. Moreover, ^n-1'0 = 0 implies that 02 = a. So
we have ^ — r = ^o'n-l + a. We conclude by the second case of Theorem 4.
1374 JOSEPH H.M. STEENBRINK
THEOREM 6. — Let (X, x) be a three-dimensional Du Bois isolated
complete intersection singularity with Z2'0 = 0. Then the Hodge numbers
of the Milnor fibre are given by
SQ = 0,5i = ̂ l,52 = b1-1 + l^\S3 = I0-2.
Proof. — The statements about SQ and 53 have been proved pre-
viously, as well as the identity s-z - «i = I111 - ;2'0 = ;1'1. Finally
51 + S2 + 53 = ̂  = 261'1 + I1-1 + ̂ 2.
COROLLARY. — For rational isolated complete intersection singula-
rities of dimension three, the invariant b1'1 is upper semicontinuous under
deformation.
Indeed, this follows from the semicontinuity of «i. As a consequence,
we get a generalization of [10, Theorem 2.2]: if {X,x) is a rational isolated
complete intersection singularity of dimension three, which is not an
ordinary double point, then ^^(X.x) > 0. Indeed, if (X,x) is not an
ordinary double point, it deforms into a singularity of type A^ or to the
intersection of two quadrics (type D^) which have «i = 1 and s\ = 2
respectively.
We now give our example that the invariant &151 is not upper semi-
continuous under deformations in general, even for two-dimensional hyper-
surfaces.
Example. — Let ft(x, y , z) = tx6 + x7 + x^y4 + y7 + z2 and let Xf
be the surface germ at 0 € C3 given by ft(x,y,z) = 0. Then b°^(Xt) = 3
if t ̂  0 whereas b°^(Xo) = 2. This shows that the invariant &°'1 does not
depend upper semicontinuously on deformation parameters in general.
To compute these numbers, we use the formula pg + a = dim Qf /~Kn
y>o from [14, Cor. (6.3)] which is valid for a two-dimensional isolated
hypersurface singularity. Here Qf := ^/df A ^2, V is its Kashiwara-
Malgrange filtration, pg = dimQ-^/V^o is the geometric genus and ~K is
the kernel of multiplication by / in Qf. In the example at hand, ft is semi-
quasi-homogeneous for all t, with weights (- ,- ,-) for t ^ 0 and with
weights ( - , . . ) for t = 0. For all t we have the equalities
VoQf=V^Qf=(x2^xy^2)Qf
which imply that &051 = pg = 3 so &1'1 = 3 — a.
DU BOIS INVARIANTS OF SINGULARITIES 1375
Comparing the difference between the maximal and minimal weights
of (^f with the weights of x and y , one checks that xloQf = yloQf = 0.
A simple Grobner basis calculation shows that for t -^ 0 one has ~K =
(x,y2)Qf hence a = 0 and &1'1 = 3. On the other hand, if t = 0 then
K = {x^y^Qf c V>o and a = 1 so &1'1 = 2.
5. Some consequences and remarks.
PROPOSITION 4. — Let (X,x) be an isolated tone complete inter-
section singularity. Then either [X, x) is a surface singularity of type Ak
with k >1 or an ordinary double point in dimension three.
Proof. — It is well-known that the only toric surface singularities are
the cyclic quotient singularities; these can only be Gorenstein if they are
of type Ak.
A toric isolated complete intersection singularity of dimension three
has 51 = &151 = 0 so must be the ordinary double point.
If (X, x) is a complete intersection of dimension at least four, then
^{L) = 0. For a toric isolated singularity this group has rank equal
to k — dim(X) where k is the number of one-dimensional faces of the
corresponding cone. One has k = dim(X) if and only if the cone is
simplicial, i.e. X is a quotient singularity. However, isolated quotient
singularities in dimension at least three are rigid, so cannot be complete
intersections.
We end with the following question. Let (X, x) be a three-dimensional
isolated rational complete intersection singularity (more generally: a hyper-
surface section of an isolated Gorenstein singularity X' with T^, = 0) and
let (V, E) —^ (X, x) be a good resolution. One has the exact sequence
0 -^ H^Y^logE^-E)) -^ T^ -. Hi(Y^y{\og E)(-E)) -^ 0
as Tj^ ^ H^{Y \ E,f^). Is it true that the tangent cone to the
discriminant of the versal deformation of (X,a;) (which is in a natural
way a cone inside T^ ^) is invariant under translations by the subspace
H\Y^y{\og E)(-E))7
In [3] it is shown that the tangent cone to the discriminant is invariant
under translations by mT^ where m is the maximal ideal of Ox,x- So one
1376 JOSEPH H.M. STEENBRINK
may as well ask whether H1 (V, ̂  (log E) (-£')) C mT^. This has been
proven in the hypersurface case in [10].
If the answer to the question is positive, then the first proof of
smoothability of Q-factorial Calabi-Yau threefolds from [10] goes through
if these have isolated complete intersection singularities.
In [9], the second proof of [10] is generalized even further: Namikawa
there admits smoothable isolated singularities which are either toric or have
smooth semiuniversal base space.
BIBLIOGRAPHY
[I] Ph. Du BOIS, Complexe de de Rham nitre d'une variete singuliere, Bull. Soc. Math.
Prance, 109 (1981), 41-81 .
[2] G-M. GREUEL, Dualitat in der lokalen Kohomologie isolierter Singularitaten, Math.
Ann., 250 (1980), 157-173 .
[3] G.-M. GREUEL und LE DUNG TRANG, Spitzen, Doppelpunkte und vertikale Tan-
genten in der Diskriminante verseller Deformationen von vollstandigen Durchsch-
nitten, Math. Ann., 222 (1976), 71-88.
[4] A. GROTHENDIECK, Local cohomology, Lecture Notes in Math.41, Springer-Verlag,
Berlin etc., 1967.
[5] F. GUILLEN, V. NAVARRO AZNAR, P. PASCUAL-GAINZA, F. PUERTA, Hy-
perresolutions cubiques et descente cohomologique, Lecture Notes in Math. 1335,
Springer—Verlag, Berlin etc., 1988.
[6] S. ISHII, On isolated Gorenstein singularities, Math. Ann., 270 (1985), 541-554.
[7] S. ISHII, Small deformations of normal singularities, Math. Ann., 275 (1986), 139-
148 .
[8] E.J.N. LOOIJENGA, J.H.M. STEENBRINK, Milnor number and Tjurina number for
complete intersections, Math. Ann., 251 (1985), 121-124.
[9] Y. NAMIKAWA, Deformation theory of Calabi-Yau threefolds and certain invariants
of singularities, Preprint November 1995.
[10] Y. NAMIKAWA, J.H.M. STEENBRINK, Global smoothing of Calabi-Yau threefolds,
Invent. Math., 122 (1995), 403-419.
[II] J.H.M. STEENBRINK, Mixed Hodge structures associated with isolated singulari-
ties, Proc. Symp. Pure Math., 40, Part 2 (1983), 513-536.
[12] J.H.M. STEENBRINK, Semicontinuity of the singularity spectrum, Invent. Math.,
79 (1985), 557-565.
[13] J.H.M. STEENBRINK, Vanishing theorems on singular spaces, Asterisque, 130
(1985), 330-341.
DU BOIS INVARIANTS OF SINGULARITIES 1377
[14] D. VAN STRATEN, J. STEENBRINK, Extendability of holomorphic differential forms
near isolated hypersurface singularities, Abh. Math. Sem. Univ. Hamburg, 55
(1985), 97-110.
[15] J.M. WAHL, A characterization of quasi-homogeneous Gorenstein surface singula-
rities, Compos. Math., 55 (1985), 269-288.
Manuscrit recu Ie 25 fevrier 1997,
accepte Ie 10 juillet 1997.
Joseph H.M. STEENBRINK,
University of Nijmegen
Department of Mathematics
Toernooiveld
6525 ED Nijmegen (The Netherlands).
steenbri@sci. kun. nl


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