Stringy Hodge numbers for a class of
isolated singularities and for threefolds
Jan Schepers and Willem Veys∗

Abstract
Batyrev has deﬁned the stringy E-function for complex varieties with at most log terminal
singularities. It is a rational function in two variables if the singularities are Gorenstein. Furthermore, if the variety is projective and its stringy E-function is a polynomial, Batyrev deﬁned
its stringy Hodge numbers essentially as the coeﬃcients of this E-function, generalizing theusual notion of Hodge numbers of a nonsingular projective variety. He conjectured that they
are nonnegative. We prove this for a class of ‘mild’ isolated singularities (the allowed singularities depend on the dimension). As a corollary we obtain a proof of Batyrev’s conjecture forthreefolds in full generality. In these cases, we also give an explicit description of the stringy
Hodge numbers and we suggest a possible generalized deﬁnition of stringy Hodge numbers if
the E-function is not a polynomial.

1

Introduction

1.1. The stringy E-function is an interesting singularity invariant, introducedby Batyrev. In [Ba1] it is associated to complex algebraic varieties with at mostlog terminal singularities, and it is for instance used to formulate a topologicalmirror symmetry test for Calabi-Yau varieties with singularities.
In order to deﬁne this invariant and to recall an intriguing conjecture of
Batyrev, we quickly review Hodge-Deligne polynomials and some birational geometry.
1.2. We work over the base ﬁeld of complex numbers; a variety is assumed to beirreducible. We will denote the (singular) cohomology and compactly supported
cohomology with coeﬃcients in C of an algebraic set X by H • (X) and Hc• (X),respectively.
∗
Partially supported by project G.0318.06 from the Research Foundation - Flanders (FWO).
During the preparation of the manuscript the ﬁrst author was a Research Assistant of the
Research Foundation - Flanders (FWO).

1

When X is smooth and projective, its cohomology H i (X) = Hci (X) carries apure Hodge structure of weight i. Its Hodge numbers hp,q (X) are given by thedimensions of the (p, q)-Hodge component of H p+q (X). They satisfy the following
symmetry and Serre duality:
hp,q (X) = hq,p (X) = hd−p,d−q (X),(1)

where d is the dimension of X.
For an arbitrary X we have that Hci (X) carries a mixed Hodge structure, seefor example [De1], [De2], [Sr]. We denote its (p, q)-component by H p,q (Hci (X))
and the dimension of the latter by hp,q (Hci (X)). Note that here also for otheri, besides p + q, we can have a nontrivial (p, q)-component. One deﬁnes the
Hodge-Deligne polynomial H(X; u, v) ∈ Z[u, v] of X as
#
" 2d
d
X
X
H(X) = H(X; u, v) :=(−1)i hp,q (Hci (X)) up v q ,
p,q=0

i=0

where again d = dim X. It is a generalized Euler characteristic; so for a closed
algebraic subset Y ⊂ X, H(X) = H(Y ) + H(X \ Y ), and for algebraic sets X
and X ′ , H(X × X ′ ) = H(X) · H(X ′ ). Returning to a smooth and projective X,
the symmetries and dualities (1) are translated in terms of H(X) as
H(X; u, v) = H(X; v, u) and H(X; u, v) = (uv)d H(X; u−1 , v −1 ).

(2)

The ﬁrst equality is still true for arbitrary X, but the second in general not.
1.3. A normal variety X is called Gorenstein if its canonical divisor KX is Cartier,and Q-Gorenstein if rKX is Cartier for some r ∈ Z>0 . (Note that for instance all
hypersurfaces, and more generally all complete intersections are Gorenstein.)e → X of X, i.e. ϕ is a
Let X be Q-Gorenstein. Take a log resolution ϕ : Xeproper birational morphism from a smooth variety X such that the exceptional
locus of ϕ is a divisor whose irreducible components
P Di are smooth and havenormal crossings. Then we have rKXe −ϕ∗ (rKX ) =P i bi Di , with bi ∈ Z. bUsually∗ithis is formally rewritten as KXe − ϕ (KX ) =i ai Di , where ai = r . The∗
divisor KX|X:= KXe − ϕ (KX ) is called the discrepancy divisor of ϕ, and ai theediscrepancy of Di .
The Q-Gorenstein variety X is called terminal, canonical and log terminal if
ai > 0, ai ≥ 0 and ai > −1 for all i, respectively (this is independent of thechosen resolution). Such varieties have ‘mild’ singularities and show up in the
Minimal Model Program.
1.4. Definition [Ba1]. Let X be a log terminal variety. Take a log resolution
e → X with irreducible components Di , i ∈ T, of the exceptional divisor.
ϕ : X
For I ⊂ T , denote DI := ∩i∈T Di and DI◦ := DI \ (∪j∈T \I Dj ); so in particular2◦e We have the standard stratiﬁcation X
e =`D∅ = X.
I⊂T DI into locally closed
subsets. The stringy E-function of X isX
Y uv − 1,
Est (X) = Est (X; u, v) :=H(DI◦ ; u, v)(uv)ai +1 − 1I⊂Ti∈Iwhere ai is the discrepancy of Ei . Batyrev proved that this expression is independent of the chosen log resolution by means of motivic integration. (Alternatively,
one can use the Weak Factorization Theorem of Abramovich, Karu, Matsuki and
Wlodarczyk from [AKMW].) Concrete computations of the stringy E-functionin special cases can be found in [Da] and [DR], but pay attention to the errors in
the latter, as explained in [Sch].
Remark. The following ﬁrst three statements are clear. The fourth is an easy
calculation [Ba1].
(i) If X is smooth, then Est (X) = H(X). If X admits a crepant resolution
e → X, i.e. such that K e = 0, then Est (X) = H(X).
e
ϕ:X
X|X
(ii) If X is Gorenstein we have that all ai ∈ Z>0 . Then Est (X) is a rational
function in u and v, more precisely it is an element of Z[[u, v]] ∩ Q(u, v).

(iii) Just as Hodge-Deligne polynomials, stringy E-functions are symmetric in u
and v: Est (X; u, v) = Est (X; v, u).
(iv) An alternative expression for Est (X) is
Est (X; u, v) =X

H(DI ; u, v)

I⊂TY uv − (uv)ai +1i∈I(uv)ai +1 − 1.

1.5. Theorem. Let X be a projective log terminal variety of dimension d. Then
Est (X; u, v) = (uv)d Est (X; u−1 , v −1 ).
This is an easy consequence of the alternative expression for Est (X) above,using the similar equality (2) for Hodge-Deligne polynomials of the smooth projective varieties DI [Ba1, Theorem 3.7].
1.6. From now on we consider only Gorenstein varieties X. Since then all
discrepancies are integers, we have that the conditions canonical and log terminal
are equivalent.
Assume now that for a Gorenstein canonical projectived
P X of pdimensionq
the rational function Est (X; u, v) is in fact a polynomial p,q bp,q u v . (An intuitive formulation of this assumption is perhaps that X is close to admitting a
crepant log resolution). Then Batyrev deﬁned the stringy Hodge numbers of X
p+q
as hp,qbp,q . Note as a motivation thatst (X) := (−1)
3p,q(a) when X is smooth we have hp,qst (X) = h (X) since then Est (X) = H(X),q,p(b) by Remark 1.4 (iii) we have hp,qst (X) = hst (X),(c) Theorem 1.5 yields that hp,qst (X) can be nonzero only for 0 ≤ p, q ≤ d andp,qd−p,d−qthat hst (X) = hst(X),(d) looking at the expression for Est (X) in Remark 1.4 (iv) we have clearlyh0,0st (X) = 1.
So the classical properties of Hodge numbers for smooth projective varieties
are still satisﬁed by Batyrev’s stringy Hodge numbers for Gorenstein canonicalprojective varieties with a polynomial stringy E-function. But there is one serious problem: it is not clear that they are nonnegative !
Conjecture [Ba1, Conjecture 3.10]. Let X be a Gorenstein canonical projective
variety. Assume that Est (X; u, v) is a polynomial. Then all stringy Hodge numbers hp,qst (X) are nonnegative.
1.7. The conjecture is trivially true for varieties that admit a crepant resolution(see Remark 1.4 (i)). This is the case for all canonical surfaces, which are exactly
those with two-dimensional A-D-E singularities. (In [Ve2] there is related workon Q-Gorenstein log terminal surfaces.)
In higher dimensions other results are rare. The ﬁrst author settled the caseof A-D-E singularities of arbitrary dimension. In [Sch] he was able to prove quite
elegant concrete formulae for their stringy E-function, obtaining the following asa corollary.
Theorem. Let X be a projective variety of dimension at least 3 with at most AD-E singularities. Its stringy E-function is a polynomial if and only if dim X = 3
and all singularities are of type An (n odd) and/or Dn (n even). In that case,the stringy Hodge numbers of X are nonnegative.
1.8. In this paper we prove the following theorem (see Theorem 3.1).
Theorem. Let Y be a projective variety of dimension d ≥ 3 with at most isolated Gorenstein canonical singularities. Assume that Y admits a log resolution
f : X → Y with all discrepancy coefficientsof exceptional components > ⌊ d−4
⌋.
2
P
i j
i+j
Write Est (Y ; u, v) as a power series
b
u
v
.
Then
(−1)
b
≥
0
for
i,j
i,j≥0 i,j
i + j ≤ d.
In view of 1.6 (c) this implies nonnegativity of the stringy Hodge numbers
for such varieties Y if they have a polynomial stringy E-function. As a corollary one can prove the analogous statement for arbitrary projective threefolds
4with Gorenstein canonical singularities (Corollary 3.2), and thus one obtains a
proof of Batyrev’s conjecture for threefolds in full generality. It is interestingthat the somewhat mysterious polynomial condition on the stringy E-function in
Batyrev’s conjecture can be dropped in the cases of the theorem and the corollary. This suggests to use the numbers bi,j in these cases to generalize Batyrev’s
stringy Hodge numbers (Remark 3.3 (ii)). In addition we also give an explicit
geometric description of these numbers bi,j (and thus in particular for the stringy
Hodge numbers if they are deﬁned) as dimensions or sums of dimensions of Hodge
components of certain cohomology groups (Propositions 3.4 and 3.5).
Acknowledgement: We wish to thank Joost van Hamel for helpful discussions
about the proof of Theorem 3.1.

2

Preliminaries

2.1. We will use the following result of de Cataldo and Migliorini in a crucial
way. By H • ( · ) we still mean singular cohomology with coeﬃcients in C.
Theorem. Let Y be a projective variety of dimension n with at most isolated
singularities. Let f : X → Y be a resolution of singularities with X projective,
such that f : f −1 (Yns ) → Yns is an isomorphism, where Yns is the nonsingular partof Y , and such that f −1 (y) is a divisor for every singular point y ∈ Y . Denote by
D the total inverse image of the singular points. Then the map H i (X) → H i (D),induced by inclusion, is surjective for i ≥ n.
The proof of this result for n = 3 and for one singular point is given in [dCM2,
Theorem 2.3.4]. From this proof it is clear that it works for any dimension and any
number of singular points, but for completeness we include the argument anyway.
Proof: We embed Y in a projective space Pr and we take a generic hyperplanesection Ys (so this hyperplane section is smooth and does not contain any singularpoint of Y ). Consider the inverse image Xs := f −1 (Ys ) and denote Y \ Ys by Y0
and X\Xs by X0 . Let Y0 be the normalization of Y0 . Then Y0 still has isolated singularities and the map f : X0 → Y0 factors through Y0 . Denote the map X0 → Y0
by f . Because Y0 is aﬃne, H i (Y0 ) = 0 for i > n (this was probably ﬁrst proved by
Kaup in [Ka]). The sheaves Rk f ∗ CX0 are skyscraper sheaves above the singularpoints for k > 0, so they are ﬂasque and H j (Y0 , Rk f ∗ CX0 ) = 0 for k, j > 0. The
Leray spectral sequence for f (given by E2p,q = H p (Y0 , Rq f ∗ CX0 ) and abutting
to H • (X0 , C)) gives then an isomorphism H i (X0 ) ∼= H i (D) for i > n, because
0,i
p,q
0,i
i
p,q
E2 = E∞ = H (D) and because E2 = E∞ = 0 for p > 0 and p + q = i. There0,nis also a surjection H n (X0 ) → H n (D) because E20,n = E∞
= H n (D). Proposi5

tion (8.2.6) from [De2] states that the image of H i (X) in H i (D) is equal to theimage of H i (X0 ) in H i (D), and thus the result follows.

The following corollary is immediate from the theorem, see also [dCM1, Corollary 2.1.11]. It was already proved by Steenbrink in [St, Corollary (1.12)] for thecase where D is a divisor with smooth components and normal crossings.
Corollary. The a priori mixed Hodge structure on H i (D) is pure of weight i fori ≥ n.
2.2. We also need the construction of the mixed Hodge structure on the cohomology of an algebraic set D with smooth projective irreducible components andnormal crossings. This can be found in [KK, p.149-156]. The construction is asfollows. Denote the irreducible components of D by Di , i ∈ I = {1, . . . , α} andput for j ≥ 0aD(j) :=DJ ,J⊂I|J|=j+1

where DJ = ∩i∈J Di . So all D(k) are smooth and projective, and we write Ai (D(k) )for the C ∞ -diﬀerential i-forms with values in C on D(k) . The inclusion map
D(k) ֒→ D(k−1) deﬁned by mapping Di1 ∩ . . . ∩ Dik+1 into Di1 ∩ · · · ∩ Dil−1 ∩(k)
Dil+1 ∩ . . . ∩ Dik+1 is denoted by δl . There exists a spectral sequence {Er , dr },abutting to H • (D) and degenerating at the E2 level, with E0p,q = Aq (D(p) ) andd0 : E0p,q → E0p,q+1 given by diﬀerentiation of forms. Then E1p,q is H q (D(p) ); and
Pp+2
l (p+1) ∗d1 : E1p,q → E1p+1,q is deﬁned by) . There is a pure Hodge
l=1 (−1) (δlstructure of weight q on E1p,q , and d1 is a morphism of Hodge structures, so E2p,qinherits a pure Hodge structure and this provides H • (D) with a mixed Hodge
structure.

3

Main results and corollaries

3.1. Theorem. Let Y be a projective variety of dimension d ≥ 3 with at
most isolated Gorenstein canonical singularities. Assume that Y admits a log
resolution f : X → Y with all discrepancy coefficientsof exceptional components
P
i j
i+j
> ⌊ d−4
⌋.
Write
E
(Y
;
u,
v)
as
a
power
seriesbbi,j ≥ 0sti,j≥0 i,j u v . Then (−1)
2for i + j ≤ d.
Remark.
(i) In particular, if the stringy Hodge numbers of Y are deﬁned, then they arenonnegative in this case, in view of 1.6 (c).
6

(ii) It is not very diﬃcult to see that, if satisﬁed, then the condition on thediscrepancy coeﬃcients does not depend on the particular log resolution.
(iii) For d = 3 there is no extra condition on the discrepancy coeﬃcients (since
we already assume that Y is canonical) and for d = 4 or 5 the conditionmeans that Y has (isolated Gorenstein) terminal singularities.
Proof: Let us ﬁrst handle the case where d ≥ 4. Choose a log resolution
f : X → Y that is an isomorphism when restricted to f −1 (Yns ) and denotethe irreducible exceptional components by Dl , l ∈ I = {1, . . . , α}. Denote thediscrepancy coeﬃcient of Dl by al and put D := ∪l∈I Dl . Consider the alternative
formula 1.4 (iv) for the stringy E-function:
Est (Y ) =X

H(DJ ; u, v)

J⊂I

Y uv − (uv)al +1l∈J(uv)al +1 − 1,

P
and write this as a power series i,j≥0 bi,j ui v j . It suﬃces to prove that (−1)i+j bi,j
≥ 0 for i ≥ j and i + j ≤ d. We denote the Hodge-Deligne polynomial of X byPi ji,j ci,j u v and for J ⊂ I, J 6= ∅ we denote the Hodge-Deligne polynomial of DJ
P J i jal +1
for al > 0 is equal toby i,j ci,j u v . The power series development of uv−(uv)(uv)al +1 −1(uv − (uv)al +1 )(−1 − (uv)al +1 − (uv)2al +2 − (uv)3al +3 − · · · )= −uv + (uv)al +1 − (uv)al +2 + (uv)2al +2 − (uv)2al +3 + · · · .

Since we assume the given lower bound on the al ’s, we can write bi,j for i + j ≤ dand i ≥ j as
j
X
X
(−1)k
bi,j = ci,j +cJi−k,j−k + Ri,j ,
J⊂I
|J|=k

k=1

where

 P
{l}

l=1,...,α c0,0
al =d/2−1 P l=1,...,α c{l}0,0al =(d−3)/2
Ri,j =P{l}l=1,...,α c1,0 al =(d−3)/20

if d is even and i = j = d2 ,if d is odd and i = j = d−1,
2, j = d−1,
if d is odd and i = d+122otherwise.

In any case, Ri,j has got the right sign, so it suﬃces to study the rest of theformula for bi,j . We rewrite it as follows, using the symmetry of usual Hodge
numbers:
j
X
X
(−1)k
cd−i,d−j +cJd−i,d−j .
J⊂I
|J|=k

k=1

7

Set q = 2d − i − j and look at the {E1p,q , d1 }-term of the spectral sequence from
2.2, for all p. This gives a complex:
H q (D(0) ) → H q (D(1) ) → · · · → H q (D(i−1) ) → 0.
Here H q (D(i) ) = 0 since D(i) has dimension d − i − 1. The cohomology of thiscomplex is given by E20,q to E2i−1,q . These are subquotients of H q (D) to H q+i−1 (D)respectively. By purity of weight r of H r (D) for r > q (Corollary 2.1), E21,q up toE2i−1,q must be zero ! This means that the complex is exact. The purity of weight
q of H q (D) itself implies that H q (D) = ker(H q (D(0) ) → H q (D(1) )). So we get anexact sequence where all arrows are morphisms of pure Hodge structures:
0 → H q (D) → H q (D(0) ) → · · · → H q (D(i−1) ) → 0.
We apply the exact functor H d−i,d−j and note that H d−i,d−j (H q (D(s) )) = 0 fors = j, . . . , i − 1 since D(s) has dimension d − s − 1. Counting the dimensions of
the resulting exact sequence of vector spaces gives
dim H

d−i,d−jq(H (D)) =jX(−1)k+1

X(−1)2d−i−j cJd−i,d−j .

J⊂I
|J|=k

k=1

The surjectivity of the morphism of Hodge structures H q (X) → H q (D) from
Theorem 2.1 translates by applying H d−i,d−j to2d−i−j(−1)

cd−i,d−j ≥jX(−1)k+1

X(−1)2d−i−j cJd−i,d−j ,

J⊂I
|J|=k

k=1

and thus (−1)2d−i−j bi,j = (−1)i+j bi,j ≥ 0.
For d = 3 it can happen that some al are zero. We can express the bi,j as
follows:
b0,0 = c0,0 = 1 ≥ 0,b1,0 = c1,0 ≤ 0,
 b2,0 = c2,0 ≥ 0,
b3,0 = c3,0 ≤ 0,
P
P

{l}
{l}

b1,1 = c1,1 − l=1,...,α c0,0 = c2,2 − l=1,...,α c2,2 ,P al 6=0
P ak 6=0 {l} b2,1 = c2,1 − l=1,...,α c{l}=c−l=1,...,α c2,1 .
2,1
1,0
al 6=0al 6=0An analogous reasoning as above gives here that H (D) = ⊕αl=1 H 4 (Dl ) andH 3 (D) = ⊕αl=1 H 3 (Dl ). Using the surjectivity of H i (X) → H i (D) for i = 3, 4 andapplying the functors H 2,2 , respectively H 2,1 , immediately gives that b1,1 ≥ 0 and
4

8

b2,1 ≤ 0.



Example. In particular, the above theorem and Batyrev’s conjecture are true
for fourfolds and ﬁvefolds with isolated Gorenstein terminal singularities.
3.2. The problem in the following corollary is that the singularities can be nonisolated.
Corollary. Let Y be a projective threefold with at mostcanonP Gorenstein
i jical singularities. Write Est (Y ; u, v) as a power series
b
u
v
.
Then
i,j≥0 i,j
i+j(−1) bi,j ≥ 0 for i + j ≤ 3. In particular, Batyrev’s conjecture is true forthreefolds.
Proof: The main theorem of [Re1] states that there exists a projective variety Z
with terminal singularities and a projective birational morphism g : Z → Y thatis crepant. It follows then from [Ba1, Theorem 3.12] that Est (Z) = Est (Y ). Since
terminal singularities in dimension 3 are isolated (see for example [Ma, Corollary
4-6-6]), the corollary follows immediately from Theorem 3.1.

3.3. Remark.
(i) It is not true in general that (−1)i+j bi,j ≥ 0 for 0 ≤ i, j ≤ d, as is shown by
the following example (same notation as in the theorem above). Consider
the variety Z ′ = {x31 x5 + x42 + x23 x25 + x24 x25 = 0} ⊂ P4 . The singular locus
of Z ′ consists of the line {x1 = x2 = x5 = 0} and an E6 singularity at the
point P = {(0 : 0 : 0 : 0 : 1)}. We can resolve the singular line by fourblow-ups (ﬁrst in the singular line itself, then in a surface isomorphic to
P2 and afterwards consecutively in two curves isomorphic to P1 ). We arethen left with a variety Z which has a unique E6 singularity at P . The
stringy E-function of Z can be written as H(Z \ {P }; u, v) + (contributionof the singular point). For the second term we refer to [Sch, Theorem
2
6
5 +(uv)4 −(uv)2 +2uv−2)5.1], it is equal to 1 + (uv) (2(uv) −2(uv). The ﬁrst term can(uv)7 −132
be computed to be (uv) + 7(uv) + 7uv. As a power series, Est (Z) =1 + 7uv + 9(uv)2 − (uv)3 + (uv)4 − (uv)6 + · · · and thus b3,3 < 0.
(ii) It is tempting to generalize Batyrev’s deﬁnition of stringy Hodge numbers forany variety Y of dimension d as in the theorem or the corollary as follows:i+j
for i + j ≤ d deﬁne analogously hi,jbi,j and maybe just putst (Y ) := (−1)
d−i,d−ji,jhst(Y ) := hst (Y ).
(iii) In Theorem 3.1 we proved in fact that the numbers (−1)i+j bi,j for i + j ≤ d
and d ≥ 4 are given by (with the same notations as in the theorem)(−1)i+j bi,j = dim ker(H d−i,d−j (H 2d−i−j (X)) → H d−i,d−j (H 2d−i−j (D))) + Si,j ,9whereSi,j =

 P
d−1,d−1(H 2d−2 (Dl ))l=1,...,α dim Ha=d/2−1lPd−1,d−1(H 2d−2 (Dl ))l=1,...,α dim H
al =(d−3)/2

Pdim H d−2,d−1 (H 2d−3 (Dl )) all=1,...,α=(d−3)/2
0

if d is even and i = j = d2 ,if d is odd and,i = j = d−1
2
if d is odd and, d+1},{i, j} = { d−122otherwise.

In the following proposition we give a more intrinsic explicit description forthese numbers. The 3-dimensional case is handled in 3.5.
3.4. Proposition. Let Y be as in Theorem 3.1 and of dimension d ≥ 4. Takea log resolution f : X → Y that is an isomorphism when restricted to f −1 (Yns ),with Dl (l = 1, . . . , α) the irreducible components of the exceptional
P locus andwith al the discrepancy of Dl . Write Est (Y ) as a power series i,j≥0 bi,j ui v j .
Then for i + j ≤ d and we have(−1)i+j bi,j = dim H d−i,d−j (H 2d−i−j (Y )) + Si,j ,where Si,j is as in Remark 3.3 (iii).
Proof: By Remark 3.3 (iii) above, it is suﬃcient to prove that
dim ker(H d−i,d−j (H 2d−i−j (X)) → H d−i,d−j (H 2d−i−j (D)))= dim H d−i,d−j (H 2d−i−j (Y ))for i+j ≤ d. Let y1 , . . . , ys be the (isolated) singular points of Y and take disjoint
contractible neighbourhoods Uk for the complex topology around the points yk .
From the Leray spectral sequence for f : Vk = f −1 (Uk ) → Uk it follows thatH m (Vk ) = H m (f −1 (yk )) for all m. Now we consider the long exact sequences for(X, ∪k Vk ) and (Y, ∪k Uk ). We get the following diagram with exact rows (we usethat H m (∪k Vk ) = H m (D) and H m (∪k Uk ) = 0 for m > 0, where D is the total
exceptional locus):→ H d−1 (D) → H d (X, ∪k Vk ) → H d (X) → · · · → H 2d (X) → 0
↑
↑
↑
↑
↑
d
d
2d
→
0
→ H (Y, ∪k Uk ) → H (Y ) → · · · → H (Y ) → 0.
By excision we have
H m (Y, ∪k Uk ) ∼= H m (Y \ {y1 , . . . , ys }, ∪k Uk \ {y1 , . . . , ys })∼= H m (X \ D, ∪k Vk \ D)∼= H m (X, ∪k Vk )10

for all m and then the Mayer-Vietoris construction (see [Ro, Lemma 6.6]) gives
us a long exact sequence (note that H 2d−1 (D) = H 2d (D) = 0)
→ H d−1 (D) → H d (Y ) → H d (X) → H d (D) → H d+1 (Y ) → · · ·· · · → H 2d−1 (X) → 0 → H 2d (Y ) → H 2d (X) → 0.
Thanks to Theorem 2.1 this sequence splits as follows:→ H d−1 (D) → H d (Y ) → H d (X) → H d (D) → 0,(d)0 → H d+1 (Y ) → H d+1 (X) → H d+1 (D) → 0,..
.

(d + 1)0 → H 2d−2 (Y ) → H 2d−2 (X) → H 2d−2 (D) → 0,

(2d − 2)0 → H 2d−1 (Y ) → H 2d−1 (X) → 0,

(2d − 1)0 → H 2d (Y ) → H 2d (X) → 0.

(2d)

Now we apply the exact functor H d−i,d−j to sequence (2d − i − j) and we use
Remark 3.3 (iii) to deduce the result. We must note that H d−i,d−j (H d−1 (D)) = 0
for i + j = d. This is [KK, Corollary 2 p.154], it also follows immediately fromthe discussion in 2.2.

Remark. We see from the short exact sequences (d + 1) to (2d) that H k (Y )carries a pure Hodge structure for k > d. From the above proof it is clear that
this works for any projective Y with at most isolated singularities. Steenbrinkproved this even for complete varieties with only isolated singularities [St, Theorem (1.13)].
Example. For a projective Y of dimension d = 4 or 5 and with at most isolated
Gorenstein terminal singularities we can write for i + j ≤ d (notations as in the
proposition)X
H d−i+1,d−j+1 (H 2d−i−j+2 (Dl )).
(−1)i+j bi,j = dim H d−i,d−j (H 2d−i−j (Y )) +
1≤l≤α
al =1

Indeed, for d = 4 the second term can only be nonzero for i = j = 2 and for
d = 5 this is the case for i = j = 2 or {i, j} = {2, 3}.
3.5. Proposition. Let Y be a projective threefold with Gorenstein
P canonicalsingularities. Write the stringy E-function of Y as a power series i,j≥0 bi,j ui v j .
Take a partial crepant resolution g : Z → Y as in the proof of Corollary 3.2.
Then for i + j ≤ 3 we have(−1)i+j bi,j = dim H 3−i,3−j (H 6−i−j (Z)).
11

Proof: Recall from 3.2 that Z has terminal singularities (which are automatically isolated) and that Est (Z) = Est (Y ). For Z the analogue of Remark 3.3 (iii)(with Si,j always zero) is valid as well and thus we can proceed exactly as in the
proof of the previous proposition.


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[Ve1] W. Veys, Arc spaces, motivic integration and stringy invariants, to appear in Adv. Stud.
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Jan Schepers, Universiteit Leiden, Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden,
The Netherlands
E-mail : jschepers@math.leidenuniv.nl
Willem Veys, Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B,
3001 Leuven, Belgium
E-mail : wim.veys@wis.kuleuven.be13