Mixed Hodge structures and log Hodge
structures on log deformations
Taro Fujisawa
Tokyo Denki University,
2-2 Knada Nishiki-cho, Chiyoda-ku
e-mail: fujisawa@mail.dendi.ac.jp
Introduction
(0.1) The notion of log Hodge structure, which is a generalization of variation
of Hodge structures, is introduced by Kazuya Kato, and studied by Kato
himself, T. Kajiwara, T. Matsubara, C. Nakayama and S. Usui in [KMN02]
[KU09] etc.
In the geometric setting it is natural to ask the follwoing question analogus
to the classical case.
(0.2) Question. Let f : X −→ S be a proper log smooth morphism. Does
·
the higher direct image sheaf Rq f∗ ωX/S
underlies a log Hodge structure of
weight q under some Kähler conditions?
(0.3) [KMN02] gives an affirmative answer to this question, under the assumption that f is projective and that S is log smooth. In this case, the
open subset Striv , the locus on which the log structure is trivial, plays an
important role.
Then, how about for the case where S is the standard log point? In this
case Striv is empty, on the contrary. The talk which I presented in KUSATSU
2008 symposium is a trial to answer the question above for the case of log
deformations. By definition, the base space S of a log deformation is the
standard log point.
1
(0.4) As far as I understand, polarized log Hodge structure is a globalized
notion of nilpotent orbit. In particular, the data of a log Hodge structure
over the standard log point is almost the same as one dimensional nilpotent
orbit (with no polarization). A nilpotent orbit yields a “limiting” mixed
Hodge structure as proved by Schmid in [Sch73]. On the other hand, [FN03]
showed that a proper log deformation with suitable conditions yields a mixed
Hodge structure. In this sense, the problem which I will treat here concerns
the reconstruction of a nilpotent orbit from a mixed Hodge structure.
proper
log deformation
-
mixed Hodge structure
6
?
nilpotent orbit +
log Hodge structure
on the standard log point
(0.5) First, I misunderstood that the problem above is not so difficult by using
Theorem (2.9) below. However, I found that I was wrong while I prepared
the manuscript for the talk at Kusatsu. Even now, I have not succeeded to
prove Conjecture (2.7) in Section 2. Here I will explain my present situation
as I did in the talk.
1
Review on log Hodge structures
(1.1) In this section, I will introduce the notion of log Hodge structure on
a log deformation. First, I briefly recall log geometry, in particular, the
standard log point. Then I present the notion of log Hodge structure on a
log deformation. However, I will not give the precise definition here, but give
the equivalent conditions for the case on the standard log point.
(1.2) Definition (K. Kato [Kat88]). Let X be a complex analytic space. A
pre-log structure on X is a pair (MX , αX ) of a sheaf of monoids MX on X and
a morphism of monoid shaves αX : MX −→ OX , where OX is regarded as a
monoid sheaf by its multiplication. A pre-log structure (MX , αX ) is said to
−1
∗
be a log structure if the morphism αX induces an isomorphism αX
(OX
) −→
2
∗
OX
. A log structure (MX , αX ) is simply denoted by MX if there is no danger
of confusion.
(1.3) Example. Let X be a complex manifold, D an effective divisor on X
and j : X \ D −→ X an open immersion. We set
∗
∩ OX ,
MX (D) = j∗ OX\D
which is a monoid subsheaf of OX . Equipped with the inclusion MX (D) −→
OX , MX (D) turns out to be a log structure on X. This is a typical and most
important example of log structures.
(1.4) For a log complex analytic space X, the sheaf of log differential p-forms
p
p
p+1
ωX
and the differential d : ωX
−→ ωX
are defined in [Kat88]. It satisfies
2
the Leibniz rule and the equality d = 0 as usual. Thus we obtain the log de
·
Rham complex ωX
. We can treat the ralative case by the similar way. See
[Kat88] for the detail.
(1.5) The ringed space X log . For an fs log analytic space X (‘fs’ stands for
the properties ‘fine and satutrated’, which I do not mention in this article), a
log
topological space X log , a sheaf of rings OX
on X log and a morphism of ringed
log
spaces τX : X −→ X are defined in [KN99]. We call this ringed space X log
the Kato-Nakayama space associated to X. Here I do not give its definition,
but I will describe the Kato-Nakayama space associated to the standard log
point below. I remark that the construction of the Kato-Nakayama space is
p,log
log
p
·,log
functorial. We set ωX
= OX
⊗τ −1 OX τX−1 ωX
and obtain the complex ωX
X
on X log .
(1.6) The standard log point ∗. A point is regraded as a complex manifold of
dimension 0. The ring of “holomorphic functions” on a point is just C. We
consider a monoid C∗ ⊕ N and a monoid homomorphism C∗ ⊕ N −→ C which
sends (a, n) to 0 if n ̸= 0 and to a if n = 0. Then it is easy to see that this
gives a log structure on a point. A point equipped with this log structure is
called the standard log point and denoted by ∗ in this article. The C-vector
space of the log differential 1-forms on ∗ is denoted by ω∗1 . We can see that
ω∗1 is one dimensional C-vector space whose base is denoted by dlog t. Thus
we have ω∗1 = C dlog t. The log de Rham complex ω∗· is equal to the complex
0
C ⊕ C dlog t[−1] = (0 −→ C −→ C dlog t −→ 0)
3
by definition. The notion of integrable log connection on ∗ is equivalent to
the data of pairs (V, N ), where V is a finite dimensional C-vector space and
N is a C-linear endomorphism of V . The endomorphism N is nilpotent if
and only if the corresponding log connection is nilpotent.
The associated Kato-Nakayama space ∗log is nothing but S1 = {z ∈ C :
|z| = 1} with the trivial map τ . The universal covering space of ∗log = S1 is
R equipped with the exponential map
R ∋ x 7→ e2π
√
−1x
∈ S1 ,
which is denoted by π : R −→ S1 . For a locally constant C-sheaf F on S1 , the
C-vecotr space Γ(R, π −1 F) carries a natural automorphism corresponding to
the action of the fundamental group Z of S1 . The correspondence above
gives an equivalence between the category of locally constant C-sheaves on
∗log = S1 and the one of C-vector spaces equipped with automorphisms.
For O∗log , which is locally constant C-sheaf on ∗log = S1 , Γ(R, π −1 O∗log ) is
the polynomial ring of one variable C[u] equipped with the automorphism
f (u) 7→ f (u − 1). For the complex ω∗·,log , we have
π −1 ω∗·,log = (0 −→ C[u] −→ C[u] dlog t −→ 0)
d
with the differential
∂f
dlog t
2π −1 ∂u
for any f ∈ C[u]. For a nilpotent integrable log connenction (V, N ) the
complex π −1 ω∗·,log ⊗C V is equal to
df =
1
√
∇log
(0 −→ C[u] ⊗ V −→ C[u] dlog t ⊗ V −→ 0)
where
∇log (f ⊗ v) =
1
√
∂f
dlog t ⊗ v + f dlog t ⊗ N (v).
2π −1 ∂u
For the locally constant C-sheaf Ker(O∗log ⊗ V −→ ω∗1,log ⊗ V ), the corre⊗ V −→ ω∗1,log ⊗ V ))) is equal to
sponding C-vector space Γ(R, π −1 (Ker(O∗log √
V equipped with the automorphism exp(2π −1N ).
(1.7) By using the description above, the definition of log Hodge structures
on the standard log point ∗ can be easily translated into the following form.
For the precise definition of log (mixed) Hodge structure on an fs log complex
analytic space, see [KU00], [KMN02], [KU09].
4
(1.8) Definition. A Q-log Hodge structure of weight n on the standard log
point ∗ is a triple (VQ , F, N ) where VQ is a finite dimensional Q-vector space,
where F is a finite decreasing filtration on VC = VQ ⊗Q C and where N is a
nilpotent endomorphism of VC satisfying the following conditions:
√
(1.8.1) 2π −1N is defined over Q;
(1.8.2) N (F p ) ⊂ F p−1 for every p;
√
(1.8.3) The filtration e2π −1zN F defines a Hodge structure of weight n if
Im z is sufficiently large.
(1.9) According to the definition above, (V
√Q , F, N ) is a Q-log Hodge structure
of weight n on ∗ if and only if (VR , F, 2π −1N ) is a nilpotent orbit (with no
polarization), where VR = VQ ⊗Q R.
2
Results
(2.1) First, I will give the definition of log deformation. Then I will state the
main topic of this article.
(2.2) Definition. Let X be a complex manifold and ∆ be the unit disc in C.
A morphism f : X −→ ∆ of complex manifolds is said to be a semi-stable
degeneration, if the fiber Y = f −1 (0) is a reduced simple normal crossing
divisor on X and if f is smooth over the punctrued disc ∆∗ .
(2.3) Once a semi-stable degeneration f : X −→ ∆ is given, the simple
normal crossing divisor f −1 (0) give rise to a log structure MX on X as in
Example (1.3). Moreover the divisor {0} in ∆ gives a log structure M∆ on
∆. Note that the origin equipped with the ‘pull-back’ of the log structure
M∆ is nothing but the standard log point. On the other hand, the complex
analytic space Y = f −1 (0) is equipped with a log structure MY which is the
pull-back log structure of MX . Then the semi-stable degeneration f induces
a morphism of log complex analytic spaces Y −→ ∗, which we call the log
central fiber of f .
(2.4) Definition. Let Y be a reduced complex analytic space equipped with
a log structure. A morphism of log complex analytic space f : Y −→ ∗ is said
to be a log deformation (or simply Y is a log deformaiton) if locally on Y ,
5
the morphism Y −→ ∗ is isomorphic to the log central fiber of a semi-stable
degeneration. A log deformation f : Y −→ ∗ is said to be proper, if Y is
compact.
(2.5) Let f : Y −→ ∗ be a proper log deformation. Then we obtain a
commutative diagram
π
τ
Y
Y
Y∞ −−−
→ Y log −−−
→



 log
f∞ y
yf
Y

f
y
R −−−→ S1 −−−→ ∗
τ
τ
as in [FN03]. Then a Q-vector space Γ(R, Rq(f∞ )∗ QY∞ ) = Hq (Y∞ , Q) is obtained. My aim is to show that this Q-vector space underlies a log Hodge
structure of weihgt q under suitable assumption. More precisely, I will consider the following.
(2.6) In the situation above, the relative log de Rham complex of Y over
∗ is denoted by ωY· /∗ . The cohomology of relative log de Rham complex
Hq (Y, ωY· /∗ ) = Rq f∗ ωY· /∗ admits a nilpotent integrable log connection
∇ : Rq f∗ ωY· /∗ −→ Rq f∗ ωY· /∗ ⊗ ω∗1
which corresponds to the nilpotent endomorphism
N : Hq (Y, ωY· /∗ ) −→ Hq (Y, ωY· /∗ )
as described in Section 1. The stupid filtration F on ωY· /∗ induces a finite
filtration F on the relative log de Rham cohomology Hq (Y, ωY· /∗ ). On the
other hand, there exists a natural isomorphism
C[u] ⊗Q Hq (Y∞ , Q) ≅ C[u] ⊗C Hq (Y, ωY· /∗ )
by the log Riemann-Hilbert correcepondence proved by K. Kato and C.
Nakayama in [KN99]. Substituting 0 for the variable u, we obtain an isomorphism C ⊗Q Hq (Y∞ , Q) ≅ Hq (Y, ωY· /∗ ). Then what I want to prove is
summarized as follows.
(2.7) Conjecture. Let f : Y −→ ∗ be a proper
S log deformation. The
irreducible decomposition of Y is denoted by Y = i∈I Yi . We assume that
6
all the irreducible components Yi are nonsingular. Moreover Y is assumed to
be cohomologically Kähler, that is, there exits an element ℓ ∈ H2 (Y, R) such
that the restriction ℓ|Yi is represented by a Kähler form for every i. Then the
data
(Hq (Y∞ , Q), (Hq (Y, ωY· /∗ ), N, F ))
is a Q-log Hodge structure of weight q for every integer q.
(2.8) At the time when I write this article, this conjecture is still open. In
the remainder of this article, I explain my strategy toward a proof of it.
The key is the following result.
(2.9) Theorem ([CKS86],[KK87]). Let (V, W, F ) be an R-mixed Hodge structure, N a nilpotent endomorphism of V and S : V × V −→ R a bilinear form
on V . We denote the C-linear (resp. C-bilinear) extension of N (resp. S) on
VC = V ⊗R C by the same letter N (resp. S) for simplicity. Then (V, F, N )
is a nilpotent orbit of weight q if the following conditions are satisfied:
(2.9.1) N q+1 = 0;
(2.9.2) N (Wm ) ⊂ Wm−2 for every m;
W
(2.9.3) N m induces an isomorphism GrW
m+q V −→ Gr−m+q V for every positive integre m (W is the monodromy weight filtration for N centerd
at q);
(2.9.4) N (F p ) ⊂ F p−1 for every p;
(2.9.5) S(x, y) = (−1)q S(y, x) for every x, y ∈ V ;
(2.9.6) S(F p , F q−p+1 ) = 0 for every p;
(2.9.7) The Hodge structure of weight q + l on the primitive part Pq+l =
W
l
Ker(N l+1 : GrW
q+l V −→ Grq−l−2 V ) is polarized by the form S(·, N ·)
for every positive integer l.
(2.10) I try to apply this theorem to our situation. Form now on, I consider
the C-structure only for simplicity. In [FN03], the natural mixed Hodge
structure on Hq (Y, ωY· /∗ ) is constructed. Then the conditions (2.9.2) and
(2.9.4) can be easily seen by recalling the construction in [FN03]. The condition (2.9.3) is the consequence of Morihiko Saito’s result in [Sai88]. So
the main point remained is to construct an appropriate bilinear form on the
cohomology group Hq (Y, ωY· /∗ ).
7
(2.11) For a compact Kähler complex manifold X of dimension n, the polarization on H∗ (X, C) is constructed by using the cup product, Lefschetz
operator and the trace map
H2n (X, C) −→ C,
which is an isomorphism. Comparing to this procedure, I consider multiplicative structure or “cup product” on H∗ (Y, ωY· /∗ ) and the trace map. As
for the trace map, I have got the following:
(2.12) Lemma. Let f : Y −→ ∗ be a proper log deformation such that
all the irreducible components of Y is nonsingular and Kähler. We denote
the dimension of Y by n. (By definition, Y is pure dimensional.) Then the
equality
dim H2n (Y, ωY· /∗ ) = 1
holds.
Proof. Compute the weight spectral sequence associated to the complex AC
in [Ste76].
(2.13) By the lemma above, I can expect to find a natural trace isomorphism
H2n (Y, ωY· /∗ ) −→ C.
(2.14) A candidate on the multiplicative structure is the morphism
ωY· /∗ ⊗ ωY· /∗ −→ ωY· /∗
given by the wedge product.
We can easily see that the wedge product above give rise to a bilinear
form satisfying the coditions (2.9.5), (2.9.6) for any trace map chosen. Thus,
the remaining problem is the following:
(2.15) Problem. In the situation in Conjecture (2.7), find an appropriate
trace map H2n (Y, ωY· /∗ ) −→ C and prove the condition (2.9.7).
(2.16) After the talk, Prof. Usui suggested me a different approach to find a
bilinear form, which uses the canonical splitting of a mixed Hodge structure.
In this approach, the difficult point is to prove the condition (2.9.6) because
the canonical splitting does note preserve the Hodge filtration F .
8
References
[CKS86] E. Cattani, A. Kaplan, and W. Schmid, Degeneration of Hodge
structures, Ann. of Math. 123 (1986), 457–535.
[FN03]
T. Fujisawa and C. Nakayama, Mixed Hodge Structures on Log
Deformations, Rend. Sem. Mat. Univ. Padova 110 (2003), 221–
268.
[KK87]
M. Kashiwara and T. Kawai, The Poincaré lemma for variations
of polarized Hodge structures, Publ. RIMS 23 (1987), 345–407.
[Kat88]
K. Kato, Logarithmic structures of Fontaine-Illusie, Algebraic
Analysis, Geometry and Number Theory (J.-I. Igusa, ed.), Johns
Hopkins Univ., 1988, pp. 191–224.
[KMN02] K. Kato, T. Matsubara, and C. Nakayama, Log C ∞ -Functions and
Degenerations of Hodge structures, Algebraic Geometry 2000, Azumino, Advanced Studies in Pure Mathematics, vol. 36, Mathematical Society of Japan, 2002, pp. 269–320.
[KN99]
K. Kato and C. Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C, Kodai
Math. J. 22 (1999), 191–224.
[KU00]
K. Kato and S. Usui, Logarithmic Hodge structures and classifying
spaces, CRM Proceedings and Lecture Notes 24 (2000), 115–130.
[KU09]
, Classifying Spaces of Degenerating Polarized Hodge Structures, Annales of Math. Stud., vol. 169, Princeton Univ. Press,
2009.
[Sai88]
M. Saito, Modules de Hodge Polarisable, Publ. RIMS. 24 (1988),
849–921.
[Sch73]
W. Schmid, Variation of Hodge structure : the singularities of the
period mapping, Invent. Math. 22 (1973), 211–319.
[Ste76]
J. Steenbrink, Limits of Hodge Structures, Invent. Math. 31 (1976),
229–257.
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