Extension of 2-forms and symplectic varieties

26 Pages·0.28 MB·English

by  Yoshinori Namikawa

1
Extension of 2-forms and symplectic varieties
0
0
2
Yoshinori Namikawa
n
a
J

7 Introduction
1

In this paper we shall prove two theorems (Stability Theorem,Local Torelli
]
G Theorem) for symplectic varieties.
Let us recall the notion of a symplectic singularity. Let X be a good rep-
A
resentative of a normal singularity. Then the singularity is symplectic if the
.
h regular locus U of X admits an everywhere non-degenerate holomorphic closed
at 2-formω whereω extendstoaregularformonY foraresolutionofsingularities
m Y → X. Similarly we say that a normal compact Kaehler space Z is a sym-
plectic variety ifthe regularlocus V ofZ admits anon-degenerateholomorphic
[
closed 2-form ω where ω extends to a regular form on Z˜, where Z˜ → Z is a
3 resolution of singularities of Z. When Z has a resolution π :Z˜ →Z such that
v
(Z˜,π∗ω) is a symplectic manifold, we call Z has a symplectic resolution.
4
1 Examples
1
(i) This is one of examples of symplectic singularities studied in [Be 1]. For
0
detailssee[Be1]andthereferencesthere. LetQ⊂Pn−1 beageneralquadratic
1
0 hypersurface. IdentifyapointoftheGrassmannianGr(2,n)withalineinPn−1.
0 LetGr (2,n)be the subvarietyofGr(2,n) correspondingto the lines ofPn−1
iso
/ contained in Q. It is checked that dimGr (2,n)=dimGr(2,n)−3=2n−7.
h iso
t Embed Griso(2,n) into P1/2n(n−1)−1 by the Plu¨cker embedding Gr(2,n) →
a
P1/2n(n−1)−1. Nowconsider the cone X overGr (2,n). Then the germ(X,0)
m iso
at the vertexis a symplectic singularityof dimension 2n−6. The X is actually
v: obtained as the closure O¯min of the minimal nilpotent orbit Omin of the Lie
i algebra Lie(SO(n)), and O has the Kostant-Kirillov symplectic 2-form.
X min
(ii)LetA:=C2l/ΓbeanAbelianvarietyofdimension2l. Let(z ,z ,...,z ,z )
r 1 2 2l−1 2l
a be the standard coordinates of C2l. Then Z/2Z acts on A by z → −z
i i
(i=1,...,2l). The quotient Z of A by the action becomes a symplectic variety
ofdimension2l. Asymplectic2-formis,forexample,givenbyΣ dz ∧dz .
1≤i≤l i l+i
The Z has singularities, and Z has no symplectic resolution when l>1.
(iii) These are symplectic varieties studied by O’Grady [O]. Let S be a
polarized K3 surface. Let c be an even number with c ≥ 4. Denote by M
0,c
the moduli space of rank 2 semi-stable torsion free sheaves with c = 0 and
1
c = c. M becomes a projective symplectic variety of dim = 4c−6. The
2 0,c
singularlocus Σ has dimension2c. Moreover,O’Grady showedthat M has a
0,4
1
symplecticresolution,howeverM hasQ-factorialterminalsingularitieswhen
0,c
c≥6 (cf. section 3 of the e-print version of [O]: alg-geom/9708009). Therefore
M have no symplectic resolution when c≥6.
0,c
A symplectic singularity / variety will play an important role in the gener-
alized Bogomolov decomposition conjecture (cf. [Kata], [Mo]):
Conjecture: Let Y be a smooth projective variety over C with Kodaira di-
mension0. Then thereisafiniteetalecoverY′ →Y suchthatY′ isbirationally
equivalent to Y ×Y ×Y , where Y is an Abelian variety, Y is a symplectic
1 2 3 1 2
variety, and Y is a Calabi-Yau variety.
3
In this conjecture we hope that it is possible to replace Y and Y by their
2 3
birationalmodelswithonlyQ-factorialterminalsingularitiesrespectively. Main
results are these.
Theorem 7(Stability Theorem): Let (Z,ω) be a projective symplectic
variety. Let g :Z →∆ bea projective flat morphism from Z toa 1-dimensional
unit disc ∆ with g−1(0)=Z. Then ω extends sideways in theflat family sothat
it gives a symplectic 2-form ω on each fiber Z for t ∈ ∆ with a sufficiently
t t ǫ
small ǫ.
In the above, the result should also hold for a (non-projective) symplectic
variety (Z,ω) and for a proper flat morphism g. But two ingredients remained
unproved in the general case (cf. Remark below Theorem 7).
Let Z be a symplectic variety. Put Σ := Sing(Z) and U := Z \ Σ. Let
π :Z →S betheKuranishifamilyofZ,whichis,bydefinition,asemi-universal
flat deformation of Z with π−1(0) = Z for the reference point 0 ∈ S. When
codim(Σ⊂Z)≥4,S issmoothby[Na1,Theorem2.4]. Z isnotprojectiveover
S. But we canshow that everymember of the Kuranishifamily is a symplectic
variety (cf. Theorem 7’). Define U to be the locus in Z where π is a smooth
map and let π :U →S be the restriction of π to U. Then we have
Theorem 8(Local Torelli Theorem): Assume that Z is a Q-factorial
projective symplectic variety. Assume h1(Z,O ) = 0, h0(U,Ω2) = 1, dimZ =
Z U
2l≥4 and Codim(Σ⊂Z)≥4. Then the following hold.
(1) R2π (π−1O ) is a free O module of finite rank. Let H be the image of
∗ S S
the composite R2π C→R2π C→R2π (π−1O ). Then H is a local system on
∗ ∗ ∗ S
S with H =H2(U ,C) for s∈S.
s s
(2) The restriction map H2(Z,C) → H2(U,C) is an isomorphism. Take a
resolution ν : Z˜ → Z in such a way that ν−1(U) ∼= U. For α ∈ H2(U,C) we
take a lift α˜ ∈ H2(Z˜,C) by the composite H2(U,C) ∼= H2(Z,C) → H2(Z˜,C).
Choose ω ∈ H0(U,Ω2) = C. This ω extends to a holomorphic 2-form on Z˜.
U
Normalize ω in such a way that (ωω)l =1. Then one can define a quadratic
Z˜
form q :H2(U,C)→C as R
2
q(α):=l/2 (ωω)l−1α˜2+(1−l)( ωlωl−1α˜)( ωl−1ωlα˜).
Z Z Z
Z˜ Z˜ Z˜
This form is independent of the choice of ν :Z˜ →Z.
(3)Put H :=H2(U,C). Then there exists a trivialization of the local system
H: H∼=H ×S. Let D :={x∈P(H);q(x)=0,q(x+x)>0}. Then one has a
period map p:S →D and p is a local isomorphism.
Note that when Z is a symplectic variety with terminal singularities, the
condition codim(Σ⊂Z)≥4 is always satisfied ([Na 2]).
Thestabilitytheoremwillbeprovedbyusingthe followingtheoremandthe
factthataprojectivevarietywithrationalsingularitieshasDuBoissingularities
(cf. [Ko]):
Theorem 4.. Let X be a Stein open subset of a complex algebraic variety.
Assume that X has only rational Gorenstein singularities. Let Σ be the sin-
gular locus of X and let f : Y → X be a resolution of singularities such that
f|Y\f−1(Σ) : Y \f−1(Σ) ∼= X \Σ. Then f∗Ω2Y ∼= i∗Ω2U where U := X \Σ and
i:U →X is a natural injection.
The same result was obtained by D. van Straten and Steenbrink [S-S] for
an arbitrary isolated normal singularity with dim ≥ 4, and later Flenner [Fl]
proveditforarbitrarynormalsingularitywithCodim(Σ⊂X)≥4. ByTheorem
4 X is a symplectic singularity if and only if X is rational Gorenstein and the
regular part of X admits an everywhere non-degenerate 2-form (cf. Theorem
6). Thisfactisoftenusefultodeterminethatcertainkindsofsingularitiesgiven
by G.I.T. quotient are symplectic (Example 6’).
The local Torelli theorem has been proved for non-singular symplectic vari-
eties by Beauville [Be 2, Theoreme 5]. In our singular case, it is based on the
Hodge decomposition H2(U,C) = H0(U,Ω2)⊕H1(U,Ω1)⊕H2(U,O ). We
U U U
need the conditionthat codim(Σ⊂Z)≥4 to have this decomposition(cf. [Oh,
Na 1, Lemma 2.5]). We shall prove that R2π C is a constant sheaf around
∗
0 ∈ S by using the Q-factoriality of Z. When Z is not Q-factorial, the state-
ment for H in (1) does not hold as it stands because H2(Z,C) → H2(U,C) is
not surjective.
We have formulated the local Torelli theorem for a projective symplec-
tic variety, but it is possible to get similar statements for a general (non-
projective) symplectic variety. For a non-projective symplectic variety, Q-
factoriality should be replaced by a certain condition which is equivalent to
the Q-factoriality in the projective case (cf. Remark (2) on the final page).
ThefollowingproblemwouldbeofinterestinviewofGlobalTorelliProblem.
Problem. LetZ beaQ-factorial projective symplecticvariety withterminal
singularities. Assume that Z is smoothable by a suitable flat deformation. Is Z
then non-singular?
3
InthefirstsectionweshallproveTheorem4. Othertwotheoremsareproved
in the second section.
Notation. Let F be a coherent sheaf on a normal crossing variety D.
Assume that F is a locally free sheaf on the regular locus of D. Then Fˆ =
F/(torsion) by definition. Here (torsion) means the subsheaf of the sections
whose support are contained in the singular locus of D.
§1. Extension properties for Rational Gorenstein Singularities
Proposition 1. Let X be a Stein open subset of a complex algebraic va-
riety. Assume that X has only rational Gorenstein singularities. Let Σ be the
singular locus of X and let f :Y →X be a resolution of singularities such that
f|Y\f−1(Σ) :Y \f−1(Σ)∼=X\Σ and D :=f−1(Σ) is a simple normal crossing
divisor. Then f Ω2(logD) ∼= i Ω2 where U := X \Σ and i : U → X is a
∗ Y ∗ U
natural injection.
Proof. Let ω ∈ H0(U,Ω2). ω is a meromorphic 2-form on Y. In fact,
U
Coker[f Ω2 → i Ω2] is a torsion sheaf whose support is contained in Σ. Hence
∗ ∗ U
φω is an element of Γ(X,f Ω2) for a suitable holomorphic function φ on X.
∗ Y
Since(X,p)∼=(R.D.P)×(Cn−2,0)forallp∈X outsidecertaincodimension
3 (in X) locus Σ ⊂ Σ ([Re]), it is clear that f Ω2(logD) ∼= i Ω2 at such
0 ∗ Y ∗ U
points p. Let F be an irreducible component of D with f(F) ⊂ Σ . Put
0
k := dimΣ −dimf(F). We shall prove that ω has at worst log pole along F
0
by the induction on k.
(a) k =0:
(a-1): Put l := codim(Σ ⊂ X). Note that l ≥ 3. Take a general l dimen-
0
sional complete intersection H := H ∩...∩H . Let p ∈ H ∩f(F). Since
1 n−l
H is general, p ∈ f(F) is a smooth point. Replace X by a suitable small open
neighborhoodofp. ThenH∩f(F)={p}. H hasauniquedissidentpointpand
othersingularitiesarelocallyisomorphicto(R.D.P.)×(Cl−2,0). Byperturbing
H we can define a flat holomorphic map g : X →∆n−l such that the fiber X
0
over 0 ∈ ∆n−l coincides with H. We may assume that g has a section passing
through p and each fiber g−1(t) intersects f(F) only in this section. The map
f :Y →X givesasimultaneousresolutionofX (t∈∆n−l). Since H is general
t
and X is sufficiently small, D := D∩Y are normal crossing divisors of Y for
t t t
all t ∈ ∆n−l. Let D′ be the union of irreducible components of D which are
mapped in this section. D′ → ∆n−l is a proper map. We put π = g◦f. We
often write ∆ for ∆n−l.
(a-2): We shall prove the following.
Claim. By replacing ∆n−l by a smaller disc and by restricting everything
(e.g. X, Y, D, D′ ...) over the new disc, we have a subset K ⊂ Y which
contains D′ and which is proper over ∆n−l with the following property:
The ω is mapped to zero by the comoposition of the maps
4
H0(U,Ω2)∼=H0(Y\D′,Ω2 (logD))→H0(Y\K,Ω2 (logD))→H1(Y,Ω2(logD)).
U Y\D′ Y\K K Y
Ifthe claimis verified,thenω| extends to alogarithmic2-formonY. It
Y\K
is clear that its restriction to U is ω.
Proof of Claim. We shall first prove that R1πΩ2(logD) = 0. There is a
! Y
filtration π∗Ω2 ⊂G ⊂Ω2(logD) which yields exact sequences
∆ Y
0→G →Ω2(logD)→Ω2 (logD)→0
Y Y/∆
0→π∗Ω2 →G →π∗Ω1 ⊗Ω1 (logD)→0.
∆ ∆ Y/∆
BytheexactsequencesitsufficestoprovethatR1πO =R1πΩ1 (logD)=
! Y ! Y/∆
R1πΩ2 (logD)=0.
! Y/∆
We shall use the relative duality theorem due to Ramis and Ruget [R-
R] to prove these facts. Before applying the relative duality we note that
Riπ Ωl−2 (logD)(−D) = Riπ Ωl−1 (logD)(−D) = Riπ Ωl (logD)(−D) =
∗ Y/∆ ∗ Y/∆ ∗ Y/∆
0fori≥l−1. Toprovethese,weonlyhavetocheckthatRiπ Ωl−2(logD )(−D )=
∗ Yt t t
Riπ Ωl−1(logD )(−D ) = Riπ ω = 0 for i ≥ l −1 and for t ∈ ∆, for ex-
∗ Yt t t ∗ Yt
ample, by using [B-S, VI, Cor. 4.5, (i)]). Since l ≥ 3, these follow from a
vanishing theorem in [St] except for the vanishing of R2π Ω1 (logD )(−D ).
∗ Yt t t
But, by the same argument as the proof of [Na-St, Theorem (1.1)] we see that
R2π Ω1 (logD )(−D )=0.
∗ Yt t t
Now the relative duality says that
RπRHom(Y;Ωl−j (logD)(−D)⊗π∗Ωn−l,ω· )
! Y/∆ ∆ Y
∼=RHomtop(∆,Rπ (Ωl−j (logD)(−D)⊗π∗Ωn−l),ω· )
∗ Y/∆ ∆ ∆
for j =0,1,2.
WehaveR1−nπRHom(Y;Ωl−j (logD)(−D)⊗π∗Ωn−l,ω· )∼=R1πΩj (logD).
! Y/∆ ∆ Y ! Y/∆
ThereforewehavetoprovethatExtop1−n(∆,Rπ (Ωl−j (logD)(−D)⊗π∗Ωn−l),ω· )=
∗ Y/∆ ∆ ∆
0.
ChooseaboundedcomplexofFNfreeO modulesL·representingRπ Ωn−j(logD)(−D)
∆ ∗ Y/∆
(cf. [R-R]). Since Riπ Ωn−j(logD)(−D)=0 for i≥l−1, we have Hi(L·)=0
∗ Y/∆
fori≥l−1. LetQ:=Ker[Ll−2 →Ll−1]. ThenRHomtop(∆,Rπ (Ωl−j (logD)(−D)⊗
∗ Y/∆
π∗Ωn−l),ω· )isrepresentedbythecomplexHomtop(...→Ll−3 →Q→0...,Ωn−l[n−
∆ ∆ ∆
l]). HenceweknowthatExtop1−n(∆,Rπ (Ωl−j (logD)(−D)⊗π∗Ωn−l),ω· )=
∗ Y/∆ ∆ ∆
0.
5
Wearenowinapositiontojustifytheclaim. Letj :Y \D′ →Y. Theω de-
finesanelementof(π j j∗Ω2(logD)) . Byacoboundarymap(π j j∗Ω2(logD)) →
∗ ∗ Y 0 ∗ ∗ Y 0
(R1Γπ,D′Ω2Y(logD))0,itdefinestheobstructionclassob(ω)∈(R1Γπ,D′Ω2Y(logD))0.
Bytheobservationabove,ob(ω)issenttozerobythenaturalmap(R1Γπ,D′Ω2Y(logD))0 →
(R1πΩ2(logD)) . Therefore there is a small disc ∆ ⊂ ∆ and a subset
! Y 0 ǫ
K ⊂ π−1(∆ǫ)(= Yǫ) which contains (π|D′)−1(∆ǫ)(= Dǫ′) and which is proper
over∆ ,suchthatob(ω)isalreadyzeroinH1(Y ,Ω2 (logD′)). Thisisnothing
ǫ K ǫ Yǫ ǫ
but our claim.
(b) k: general
(b-1): Takeagenerall+k dimensionalcompleteintersectionH :=H ∩...∩
1
H . Let p∈H∩f(F). p∈f(F) is a smooth point. Replace X by a small
n−k−l
openneighborhoodofp. ThenH∩f(F)={p}. ByperturbingH,wecandefine
a flat holomorphic map g : X → ∆n−k−l with g−1(0) = H. We may assume
thatg hasasectionpassingthroughpandeachfiberg−1(t)intersectsf(F)only
in this section. The map f :Y →X gives a simultaneous resolution of X (t∈
t
∆n−l−k). SinceH isgeneralandX issufficientlysmall,D :=D∩Y arenormal
t t
crossing divisors of Y for all t ∈ ∆n−l−k. Let D′ be the union of irreducible
t
components of D which are mapped in the section. Then D′ → ∆n−k−l is a
proper map. We put π =g◦f. We often write ∆ for ∆n−k−l. By an induction
hypothesis we have an isomorphism H0(Y \D′,Ω2 (logD))∼=H0(U,Ω2).
Y\D′ U
(b-2): We shall prove the following
Claim. By replacing ∆n−l by a smaller disc and by restricting everything
(e.g. X, Y, D, D′ ...) over the new disc, we have a subset K ⊂ Y which
contains D′ and which is proper over ∆n−l with the following property:
The ω is mapped to zero by the comoposition of the maps
H0(U,Ω2)∼=H0(Y\D′,Ω2 (logD))→H0(Y\K,Ω2 (logD))→H1(Y,Ω2(logD)).
U Y\D′ Y\K K Y
Ifthe claimis verified,thenω| extends to alogarithmic2-formonY. It
Y\K
is clear that its restriction to U is ω.
The proof of the claim is similar to the claim in (a-2). When we ap-
ply the relative duality we need the vanishings: Riπ Ωl+k−2(logD )(−D ) =
∗ Yt t t
Riπ Ωl+k−1(logD )(−D ) = Riπ ω = 0 for i ≥ l +k − 1 and for t ∈ ∆.
∗ Yt t t ∗ Yt
Q.E.D.
Lemma2. Letp∈X beaSteinopen neighborhood ofapoint pofacomplex
algebraic variety. Assume that X is a rational singularity of dimX ≥ 3. Let
f : Y → X be a resolution of singularities of X such that E := f−1(p) is
a simple normal crossing divisor. Then H0(Y,Ωi ) → H0(Y,Ωi (logE)) are
Y Y
isomorphisms for i=1,2.
Proof. By the assumptionwe cantakea complete algebraicvarietyZ which
6
containsX asanopenset. Wemayassumethatf isobtainedfromaresolution
Z˜ →Z. Set V :=Z˜\E. Recall that the natural exact sequence
→Hj(Z˜,C)→Hj(V,C)→Hj+1(Z˜,C)→
E
is obtained from the following exact sequence of the complexes by taking hy-
percohomology
0→Ω· →Ω· (logE)→Ω· (logE)/Ω· →0.
Z˜ Z˜ Z˜ Z˜
Introduce the stupid filtrations F· (cf. [De]) on three complexes and take
Hj(Gri ) of the sequence of complexes. Then we have
F
→Hj−i(Ωi )→Hj−i(Ωi (logE))→Hj−i(Ωi (logE)/Ωi )→.
Z˜ Z˜ Z˜ Z˜
We know that this exact sequence coincides with the exact sequence
→Gri (Hj(Z˜,C)→Gri (Hj(V,C)→Gri (Hj+1(Z˜,C))→
F F F E
whichcomesfromthemixedHodgestructures. Inparticular,themapHj−i(Ωi (logE)/Ωi )→
Z˜ Z˜
Hj−i+1(Ωi ) is interpreted as the map Gri (Hj+1(Z˜,C))→Gri (Hj+1(Z˜,C)).
Z˜ F E F
WenextconsiderthenaturalmapofmixedHodgestructures: Hj+1(Z˜,C)→
Hj+1(E,C). We have Gri (Hj+1(E,C)) ∼= Hj−i+1(Ωˆi ) (cf. [Fr]), (see Intro-
F E
duction, for the notation Ωˆi ). Note that Ωˆi is isomorphic to the cokernel of
E E
the injection Ωi (logE)(−E)→Ωi . Therefore the composed map
Y Y
Hj−i(Ωi (logE)/Ωi )→Hj−i+1(Ωi )→Hj−i+1(Ωi /Ωi (logE)(−E))
Z˜ Z˜ Z˜ Y Y
is interpreted as the map
Gri (Hj+1(Z˜,C))→Gri (Hj+1(E,C).
F E F
By the isomorphisms H·(Y,C) ∼= H·(E,C), H·(Y) have natural mixed
Hodge structures. We put U′ :=Y \E.
(i=2): We shallfirstprovethatthenaturalmapβ :H3(Y,C)→H3(Y,C)
E
is an injection.
Byalocalcohomologyexactsequenceitsufficestoshowthatα:H2(Y,C)→
H2(U′,C)isasurjection. BecauseX hasonlyrationalsingularity,H1(Y,O∗)⊗
Y
C∼=H2(Y,C).
On the other hand, one can prove that H1(U′,O∗ ) ⊗ C → H2(U′,C)
U′
is a surjection; in fact, since H·(U′,C) ∼= H·(Y,Rj Ω· ) ∼= H·(Y,j Ω· ) ∼=
∗ U′ ∗ U′
H·(Y,Ω· (logE))wherej :U′ →Y istheimmersion[De],andsinceH·(U′,C)∼=
Y
H·(U′,Ω· ), we have a commutative diagram of Hodge spectral sequences
U′
7
Hq(Y,Ωp(logE)) >Hp+q(U′,C)
Y
(1)
 
 
Hq(Uy′,Ωp ) >Hp+yq(U′,C)
U′
Let F· (resp. F·) be the filtration of Hp+q(U′,C) by the first (resp. sec-
1 2
ond) spectral sequence. In particular, when p+q = 2, we have a surjection
Gr0 H2(U′,C) → Gr0 H2(U′,C). Because X has only rational singularity,
F1 F2
H2(Y,O ) = 0, hence Gr0 H2(U′,C) = 0. Therefore Gr0 H2(U′,C) = 0,
Y F1 F2
which implies that the natural map H2(U′,C) → H2(U′,OU′) is the zero
map. It is now clear that H2(U′,Z) → H2(U′,OU′) is also the zero map, and
hence H1(U′,O∗ )→H2(U′,Z) is a surjection. Therefore one has a surjection
U′
H1(U′,O∗ )⊗C→H2(U′,C).
U′
It is enough to prove that H1(Y,O∗) → H1(U′,O∗ ) is surjective in order
Y U′
to prove that α is surjective. Put f0 := f|U′ : U′ → U, where U := X \{p}.
Let D be irreducible exceptional divisors of f where f(D ) 6= {p}. Let L ∈
i i
Pic(U′). Since dimX ≥3 and U ⊂X is 1-concave at p, there exists a reflexive
coherent sheaf F on X of rank 1 such that F| ∼= (f0L)∗∗ by [S, Theorem 5],
U ∗
where ∗∗ means the double dual. By taking the double dual of both sides of
the natural map (f0)∗f0L → L we get an injection ((f0)∗f0L)∗∗ → L, hence
∗ ∗
L ∼= ((f0)∗f∗0L)∗∗ ⊗OU′(ΣaiDi) with some ai ≥ 0. On the other hand, we
have an injection ((f0)∗f∗0L)∗∗ →(f∗f∗F)∗∗|U′ and M :=(f∗f∗F)∗∗ ∈Pic(Y).
We have ((f0)∗f∗0L)∗∗ ∼= M|U′ ⊗OU′(Σ(−bi)Di) with some bi > 0. Therefore
L∼=M ⊗OY(Σ(ai−bi)Di)|U′.
We are now in a position to prove the lemma for (i = 2). Let us consider
the exact sequence
0→Ω2/Ω2(logE)(−E)→Ω2(logE)/Ω2(logE)(−E)→Ω2(logE)/Ω2 →0.
Y Y Y Y Y
Fromthissequencewehaveamapδ :H0(E,Ω2(logE)/Ω2)→H1(E,Ω2/Ω2(logE)(−E)).
Y Y Y Y
The map β is a morphism of mixed Hodge structures and δ can be interpreted
as the map Gr2(H3(Y,C) → Gr2(H3(Y,C). We already proved that β is an
F E F
injection. Hence δ is also an injection by the strict compatibility of the filtra-
tions F. Note that δ is factorized as H0(E,Ω2(logE)/Ω2) →γ H1(Y,Ω2) →
Y Y Y
H1(E,Ω2/Ω2(logE)(−E)) where γ is the last map in the following exact se-
Y Y
quence
H0(Y,Ω2)→τ H0(Y,Ω2(logE))→H0(E,Ω2(logE)/Ω2)→H1(Y,Ω2).
Y Y Y Y
Since δ is injective, γ is also injective. Hence τ is surjective by the exact
sequence.
8
(i=1): We shallfirstprovethatthenaturalmapβ :H2(Y,C)→H2(Y,C)
E
is an injection. By a local cohomology exact sequence it suffices to show that
α : H1(Y,C) → H1(U′,C) is a surjection. Since X has rational singularities,
the sequence
H0(Y,O )→H0(Y,O∗)→H1(Y,Z)→0
Y Y
is exact. By the same argument as (i = 2), H1(U′,Z) → H1(U′,OU′) is the
zero map because X has rational singularities. Hence the sequence
H0(U′,OU′)→H0(U′,OU∗′)→H1(U′,Z)→0
is also exact. The restriction map H0(Y,O∗) → H0(U′,O∗ ) is an iso-
Y U′
morphism because it is factorized as H0(Y,O∗) ∼= H0(X,O∗ ) ∼= H0(X \
Y X
{p},OX∗\{p})∼=H0(U′,OU∗′). SimilarlytherestrictionmapH0(Y,OY)→H0(U′,OU′)
is also an isomorphism. Hence the restriction H1(Y,Z) → H1(U′,Z) is an iso-
morphism by the exact sequences above, and α is an isomorphism.
Let us consider the exact sequence
0→Ω1/Ω1(logE)(−E)→Ω1(logE)/Ω1(logE)(−E)→Ω1(logE)/Ω1 →0.
Y Y Y Y Y
Fromthissequencewehaveamapδ :H0(E,Ω1(logE)/Ω1)→H1(E,Ω1/Ω1(logE)(−E)).
Y Y Y Y
The map β is a morphism of mixed Hodge structures and δ can be interpreted
as the map Gr1(H2(Y,C) → Gr1(H2(Y,C). We already proved that β is an
F E F
injection. Hence δ is also an injection by the strict compatibility of the filtra-
tions F. Note that δ is factorized as H0(E,Ω1(logE)/Ω1) →γ H1(Y,Ω1) →
Y Y Y
H1(E,Ω1/Ω1(logE)(−E)) where γ is the last map in the following exact se-
Y Y
quence
H0(Y,Ω1)→τ H0(Y,Ω1(logE))→H0(E,Ω1(logE)/Ω1)→H1(Y,Ω1).
Y Y Y Y
Since δ is injective, γ is also injective. Hence τ is surjective by the exact
sequence. Q.E.D.
Remark. In the proof of Lemma 2 the map H0(E,Ωi /Ω1(logE)(−E))→
Y
H0(E,Ωi (logE)/Ωi (logE)(−E)) is surjective for i = 1,2 because we have
Y Y
proved that δ is injective.
Proposition3. Let X beaStein open subsetof acomplex algebraic variety.
Assumethat X has only rational Gorenstein singularities. Let Σ be the singular
locus of X and let f : Y → X be a resolution of singularities such that D :=
f−1(Σ)isasimplenormalcrossingdivisor andsuchthatf| :Y \D∼=X\Σ.
Y\D
Then f Ω2 ∼=f Ω2(logD).
∗ Y ∗ Y
9
Proof. Since (X,p) ∼= (R.D.P.)×(Cn−2,0) for all p ∈ X outside certain
codimension 3 (in X) locus Σ ⊂Σ ([Re]), it is clear that f Ω2 ∼=f Ω2(logD)
0 ∗ Y ∗ Y
at such points p.
Let ω ∈ H0(Y,Ω2(logD)). Let F be an irreducible component of D with
Y
f(F)⊂Σ . Putk :=dimΣ −dimf(F). Weshallprovethatω isregularalong
0 0
F by the induction on k.
(a) k =0:
(a-1): Put l := codim(Σ ⊂ X). Note that l ≥ 3. Take a general l dimen-
0
sional complete intersection H := H ∩H ∩...∩H . Let p ∈ H ∩f(F).
1 2 n−l
p ∈ f(F) is a smooth point. Replace X by a small open neighborhood of p.
Then H ∩f(F) = {p}. Moreover, H˜ := f−1(H) is a resolution of singularities
ofH. SinceX hascanonicalsingularities,H hasalsocanonicalsingularities. H
has a unique dissident point p and other singular points are locally isomorphic
to (R.D.P.) × (Cl−2,0). By perturbing H we can define a flat holomorphic
map g : X → ∆n−l with g−1(0) = H. We may assume that g has a section
passing through p and each fiber X :=g−1(t) intersects f(F) only in this sec-
t
tion. Denote by p ∈ X this intersection point. By definition p = p. The
t t 0
map f : Y → X gives a simultaneous resolution of X for t ∈ ∆n−l. Let D′
t
be the union of irreducible components of D which are mapped in this section.
Since H is general and X is sufficiently small, every irreducible component of
D′ is mapped onto the section. D′ → ∆n−l is a proper map and every fiber
D′ is a normal crossing variety. Note that f−1(p ) = D′. We put π = g◦f.
t t t t
We often write ∆ for ∆n−l. There are filtrations (π|D′)∗Ω2∆ ⊂ F ⊂ Ωˆ2D′ and
π∗Ω2 ⊂G ⊂Ω2(logD′) which yield the following exact sequences
∆ Y
0→F →Ωˆ2 →Ωˆ2 →0,
D′ D′/∆
0→(π|D′)∗Ω2∆ →F →(π|D′)∗Ω1∆⊗Ωˆ1D′/∆ →0,
0→G →Ω2(logD′)→Ω2 (logD′)→0,
Y Y/∆
0→π∗Ω2 →G →π∗Ω1 ⊗Ω1 (logD′)→0.
∆ ∆ Y/∆
(a-2): Let us consider the exact sequence
0→Ω2/Ω2(logD′)(−D′)→Ω2(logD′)/Ω2(logD′)(−D′)→Ω2(logD′)/Ω2 →0.
Y Y Y Y Y
We shall prove the following.
Claim. ThemapH0(D′,Ω2/Ω2(logD′)(−D′))→H0(D′,Ω2(logD′)/Ω2(logD′)(−D′))
Y Y Y
is surjective.
Proof. Note that Ω2/Ω2(logD′)(−D′) ∼= Ωˆ2 . By the exact sequences
Y Y D′
above we have two commutative diagrams with exact columns
10