Relative log Poincaré lemma and
relative log de Rham theory
Fumiharu Kato
Final version; to appear in Duke Math. J.
Abstract
In this paper we will generalize the classical relative Poincaré lemma in the framework of log
geometry. Like the classical Poincaré lemma directly implies the de Rham theorem, the comparison
between de Rham and Betti cohomologies, our log Poincaré lemma yields the formula which gives
integral structures of hyperdirect images of the log de Rham complexes; these integral structures
are nothing but the integral structures of degenerate VMHS in the semistable degeneration case.
We will also develop the relative log de Rham theory for semistable degeneration and recover the
well-known result of Steenbrink.
1
1.1
Introduction
De Rham-Hodge theory and log geometry
Let f : X → Y be a smooth morphism of complex manifolds, and Ω•X/Y the relative de Rham complex.
Then the famous classical Poincaré lemma asserts that the natural morphism
∼
f −1 OY −→ Ω•X/Y
of complexes is a quasi-isomorphism. The Poincaré lemma implies, providing f is proper, the comparison of the de Rham cohomology and the Betti cohomology (de Rham theorem):
Rf∗ Z ⊗Z OY ∼
= R f∗ Ω•X/Y .
It is, needless to say, essential to assume f to be smooth; even for semistable degeneration this
argument can no longer be literally applied.
On the other hand, Hodge theorists have been interested in the degenerating behavior of VMHS
(Variation of Mixed Hodge Structure). It has been noticed that differentials with logarithmic poles
take an essential role in studying limiting Hodge structures. Over the past few decades a considerable
number of studies have been made on this subject. J. H. M. Steenbrink proved in his exploring paper
[17] the famous result for semistable degeneration saying that the hypercohomology of the relative
logarithmic de Rham complex of the central fiber is isomorphic to the Betti cohomology of a general
fiber, the group of nearby cycles. Recently, the log geometry in the sense of J. -M. Fontaine, L.
Illusie, and K. Kato has thrown new light. The new machinery provided by log geometry such as
log topological space (real blow-up) strikingly enable us to discuss very elegantly integral structures
of degenerate VMHS, generalized Riemann-Hilbert correspondence, and so on; the recent progresses
have been made by C. Nakayama-K. Kato [9], S. Usui [19][20], T. Fujisawa [2], and T. Matsubara
[11].
In this paper we will prove generalization of classical relative Poincaré lemma, the relative log
Poincaré lemma, for log smooth morphisms f : X → Y of fs log analytic spaces satisfying suitable
1
conditions. Here we assume neither that the underlying complex analytic space of Y is non-singular
nor that the the underlying morphism of complex analytic spaces of f : X → Y is smooth. From this
generalized Poincaré lemma we will construct directly the integral structures of degenerate VMHS;
the formula which gives integral structures of degenerate VMHS is nothing but the literal counterpart
of the classical de Rham theorem. Thus we can generalize the classical de Rham theory; applying
this theory to semistable degeneration we will recover the well-known result of Steenbrink in a totally
different way.
Our technique is much indebted to the ingenious idea of Kazuya Kato and Chikara Nakayama
which has appeared in [9]; in that paper they introduced two basic ideas both of which are very
useful in treating log analytic spaces. One is the so-called real blow-up, and the other one is a certain
sheaf of rings on the blown-up space. They deduced, by adopting this machinery, generalization of
the classical Poincaré lemma in the absolute case and the classical Riemann-Hilbert correspondence.
Applying these ideas to a relative setting we can expect a totally new perspective of the theory of
degeneration; this is the main motivation of this paper.
1.2
Results of this paper
Let us briefly view the results of this paper. Let f : X → Y be a log smooth morphism of fs log analytic
spaces with Y log smooth over (Spec C)an (definition of the log smoothness will be presented in the
next section). We assume that the induced morphism of characteristics f −1 CY → CX is injective
and, for every x ∈ X, the relative characteristic CX/Y,x is torsion-free (see 2.1.1 for notation and
terminologies). The first main result is generalization of the Poincaré lemma:
Theorem 1.2.1 (Theorem 3.4.2) Let f : X → Y be as above. Then the natural morphism
•,log
−1 0 •
H (ωX/Y ) −→ ωX/Y
(f log )−1 OYlog ⊗τ −1 f −1 OY τX
X
is a quasi-isomorphism.
Here the superscript –log indicates the objects which are, or relate to the real blow-up’s; f log
is the resulting morphism of blown-up spaces, OYlog denotes the sheaf of rings obtained by adding
•,log
is the complex of log differentials where the contribution
“logarithms” to OY (cf. 2.3.4), and ωX/Y
•
1
of “logarithms” is involved (cf. 3.4.1). The f −1 OY -module H0 (ωX/Y
)(= Ker(d: OX → ωX/Y
)) may
−1
not be f OY ; for example, if f is log étale, such as the morphism between toric varieties arised
1
from subdivision of fan, it is OX since ωX/Y
= 0. However, if the family f : X → Y reasonably
−1
degenerates, it equals to f OY :
Proposition 1.2.2 (Proposition 3.2.5) Let f : X → Y satisfy the conditions as in Theorem 1.2.1.
•
Then H0 (ωX/Y
) = f −1 OY if and only if f is exact (cf. 3.2.4).
In particular, semistable degeneration is exact.
The analogue of the classical de Rham theorem is:
Theorem 1.2.3 (Theorem 4.1.5) Let f : X → Y be as in Theorem 1.2.1. Assume further that f is
◦
exact and the underlying morphism f is proper. Then there exists a canonical quasi-isomorphism
∼
•
τY−1 R f∗ ωX/Y
⊗τ −1 OY OYlog −→ Rf∗log Z ⊗Z OYlog .
Y
This follows from the above theorem by a few formal observations. This quasi-isomorphism gives
nothing but the integral structure of the canonical extension (in the sense of Deligne [1]) of VMHS
arised from the degenerating family f : X → Y .
2
These results have been partially obtained by T. Matsubara [11] in semistable degeneration case.
S. Usui [19] also obtained the integral structure in this framework for semistable degeneration in a
different way from ours.
Finally we restrict ourselves to semistable degeneration with canonically defined log structures
(cf. Example 2.2.6). S. Usui [19] proved that the morphism of this type has an excellent topological
property relating to real blow-up’s (cf. [19]). Then adding some topological arguments we will recover
the famous result by Steenbrink (cf. [17]):
Theorem 1.2.4 (Theorem 4.2.8) Depending on the choice of the logarithm of parameter, there is an
isomorphism
•
e ∗ , C)
)∼
Hq (X◦ , ωX
= Hq (X
◦ /◦
for all q.
e∗
Here Y has been taken to be a unit disk and X◦ denotes the fiber over the origin ◦, and X
denotes the topological generic fiber.
Warning: These statements make sense only in the analytic situation because our treatment is not
algebraic.
The contents of this paper are as follows:
Section 2 contains basic facts and notions of log geometry, particularly those on log smoothness
and real blow-up, which are necessary for the rest of this paper.
Section 3 devotes to prove our main theorem, the relative log Poincaré lemma. After proving a
technical lemma in the first subsection we will calculate the cohomology of the log de Rham complex
in the second subsection. In the third subsection we observe the topological nature of real blow-up
(in the relative setting), and then we will prove the main theorem in the last subsection.
In Section 4 we exhibit two applications: One is integral structure of degenerate VMHS and the
other is recovery of Steenbrink’s result.
The main part of this work was done when the author was in Mannheim University in winter semester 1995. The author would like to express his deep gratitude to Prof. Richard Pink in
Mannheim University for valuable discussions. Prof. Sanpei Usui in Osaka University read the first
draft of this paper and pointed out an error. He kindly showed the author his recent work [19]. Prof.
Arthur Ogus pointed out a gap in the proof of Proposition 3.2.3 in the preprint version which has
been fixed in this final version by making use of his technique. The author is very grateful to them.
1.3
Conventions and notation
• By a monoid we will always mean a set with a commutative and associative binary operation
which possesses a neutral element. A homomorphism of monoids is a map which preserves binary
operations and neutral elements.
• Whenever we regard a ring (commutative and unital) as a monoid, so do we by means of the
multiplication unless otherwise specified.
• We will always work only in the complex analytic category. In particular, if X is an algebraic
variety over C, the associated analytic space will be denoted by Xan .
• N = Z≥0 : additive monoid of non-negative integers.
• Let P be a monoid.
— R[P ]: Monoid ring with coefficients in the ring R,
— P gp : Grothendieck group of P .
• D+ (X, A): Derived category (bounded below) of category of A-modules on X.
3
2
Preliminaries
2.1
Fs log analytic space
Throughout this paper we will work in the framework of log geometry in the sense of J. -M. Fontaine,
L. Illusie and K. Kato. Our main references of log geometry is [7] and [3]; but our treatments will
always be complex analytic. (Note that log structure is defined and discussed on arbitrary ringed
spaces in [3, 1.1, 1.2].) We will therefore deal with log analytic space which is, by definition, a complex
analytic space together with a log structure.
Notation 2.1.1
(1) For simplicity we will often abbreviate the log analytic space (X, M) to X. It will, however,
sometimes be desirable to specify the underlying analytic space. In such a case we will denote
◦
the underlying analytic space of the log analytic space X by X . The same convention will be
used for morphisms of log analytic spaces.
(2) For a log analytic space X we denote:
◦
(i) OX : the structure sheaf of X ,
(ii) MX : the log structure of X,
×
(iii) CX : the characteristic of X (:= MX /OX
).
We denote the homomorphism MX → OX defining the log structure by αX in case we need.
(3) For a morphism f : X → Y of log analytic spaces we define CX/Y : = Coker(f −1 CY → CX ) and
call it the relative characteristic of f .
Note that the characteristic CX has a torsion-free stalk at every point; it is moreover easily seen
that every stalk of CX has no invertible element other than the neutral element.
Example 2.1.2 Canonical log structure. Let P be a monoid and consider the complex analytic
space (Spec C[P ])an . The monoid homomorphism P → C[P ] induces the homomorphism P →
O(Spec C[P ])an of sheaves of monoids, i.e., a pre-log structure on (Spec C[P ])an . Then we take the
associated log structure (cf. [3, 1.1]) which we call the canonical log structure on (Spec C[P ])an .
In the sequel we will always regard the analytic space (Spec C[P ])an as a log analytic space endowed
with the canonical log structure unless otherwise specified.
Note that, in particular, the analytic space (Spec C)an is equipped with the trivial log structure.
Definition 2.1.3 Strict morphism.
(1) Let f : X → Y be a morphism of analytic spaces and N a log structure on Y . Then the pull-back
of N to X is the log structure, denoted by f ∗ N , which is the associated log structure of the
pre-log structure f −1 N → f −1 OY → OX .
(2) A morphism f : X → Y of log analytic spaces is called strict if f ∗ MY → MX is an isomorphism
of log structures on X.
Definition 2.1.4 Chart.
(1) Let X be a log analytic space and P a monoid. Then a chart of X modeled on P is a strict
morphism X → (Spec C[P ])an of log analytic spaces.
(2) Let x ∈ X be a point. Then the chart X → (Spec C[P ])an is said to be good at x if the induced
homomorphism P → CX,x is an isomorphism of monoids.
4
(3) A chart (resp. good chart) of a morphism f : X → Y of log analytic spaces (resp. at x ∈ X)
modeled on a homomorphism h: Q → P of monoids is a commutative diagram of log analytic
spaces
X −→ (Spec C[P ])an


fy
Y


y
−→ (Spec C[Q])an ,
where the horizontal arrows X → (Spec C[P ])an and Y → (Spec C[Q])an give charts (resp.
good charts at x ∈ X and f (x) ∈ Y , respectively) and the right vertical arrow is induced by h.
◦
Any open subset U of X has the pull-back log structure so that the inclusion U ,→ X is strict; we
have therefore the obvious concept of local charts. We can similarly define local charts of morphisms.
In what follows we will discuss in the category of fs log analytic spaces defined as follows:
Definition 2.1.5 Fs log analytic space.
(1) A monoid P is said to be fine if it is finitely generated and the natural homomorphism P → P gp
is injective.
(2) A monoid P is said to be fs if it is fine and enjoys the following property: for x ∈ P gp , xr ∈ P
for some r > 0 implies x ∈ P .
(3) A log analytic space X is called fine (resp. fs) if it has locally a chart modeled on a fine (resp.
an fs) monoid; i.e., X has an open covering {Uλ }λ∈Λ (where every member is endowed with the
pull-back log structure) such that, for every λ ∈ Λ, there exists a chart Uλ → (Spec C[Pλ ])an
with Pλ a fine (resp. an fs) monoid.
Note that, for a fine (resp. an fs) log analytic space X and a point x ∈ X, the characteristic CX,x
at x is a torsion-free fine (resp. fs) monoid; this follows from the general fact that the properties fine
and fs are inherited on passage to quotients by any submonoids.
It is known that an fs log analytic space has a good local chart at any point; this follows from the
well-known useful lemma [7, (2.10)] and the following characteristic splitting lemma which follows
from Proposition 3.1.4 below (cf. [6, 1.2.5]):
Lemma 2.1.6 Let M be a fine monoid and π: M → M/M × the natural projection, where M ×
denotes the submonoid of invertible elements in M . Assume M/M × is fs. Then there exists a cross
section s of π by an exact homomorphism M/M × → M . In particular, the monoid M is isomorphic
to (M/M × ) ⊕ M × .
Here the assumption fs is essential due to the fact that it is true for a torsion-free fs monoid to
have free Grothendieck group but not in general for a torsion-free fine monoid.
As for morphisms, finding a good local chart of a morphism is a tough business even in the
fs category; we will discuss this point in §§3.1. If we give up the goodness, we can always find
a local chart of any morphism of fine log analytic spaces; the construction is as follows: Given a
morphism f : X → Y of fine log analytic spaces, we first choose local charts X → (Spec C[P ])an and
Y → (Spec C[Q])an of X and Y modeled on fine monoids P and Q, respectively. We consider the
induced homomorphism P ⊕ Q → MX . It is easy to see that the composite (P ⊕ Q)gp → Mgp
X →
gp
gp
×
×
Mgp
/O
,
denoted
by
g,
maps
(P
⊕
Q)
surjectively
onto
M
/O
.
Due
to
[7,
(2.10)]
we
see
that
X
X
X
X
×
−1
H: = g (M/OX ) → MX gives a chart of X, and hence we get a local chart of f modeled on the
homomorphism Q → H induced by the second inclusion Q ,→ P ⊕ Q. Note that the monoid H is fs
if both P and Q are fs.
Here is some examples of fs log analytic spaces:
5
Example 2.1.7 Toric variety. An affine toric variety is canonically an fs log analytic space X =
(Spec C[P ])an with P a torsion-free fs monoid (cf. [15, 1.1]). The notion of canonical log structures
can, by the canonicity, be globalized to those on general toric varieties; i.e., given a fan Σ we can equip
the associated toric variety TΣ with the unique log structure such that the log structure restricted
to each affine toric patch is the canonical log structure. Since every affine patch is fs, toric varieties
are fs log analytic spaces.
In what follows we always regard toric varieties as fs log analytic spaces by the canonical log
structure as above.
Example 2.1.8 (cf. [7, (1.5)].) Fs log analytic space representing a pair. Here is perhaps one of the
most important examples of fs log analytic spaces. Let X be a complex manifold and D a reduced
normal crossing divisor on X. Then the pair (X, D) canonically associates an fs log analytic space
(X, M) with the fs log structure M defined by
×
M: = OX ∩ j∗ OX\D
,−→ OX ,
where j: X \ D ,→ X is the inclusion. We call this log structure the canonical log structure associated
to the pair (X, D).
To see this log analytic space is fs we shall construct local charts modeled on fs monoids: The
divisor D is locally defined by z1 · · · zl (l ≥ 0) where (z1 , . . . , zl ) is a part of local coordinate system
×
(z1 , . . . , zn ) around a point x ∈ X. In case l = 0 we immediately have Mx = OX,x
. If l = 1, elements
in Mx may divisible by z1 ; dividing an element in Mx by z1 as many times as possible, we shall
×
· z1N . In general we
eventually get an invertible element. We have therefore the equality Mx = OX,x
similarly have
×
· z1N · · · zlN ,
Mx = OX,x
and hence CX,x ∼
= Nl . We have moreover seen that fixing the local parameter z1 , . . . , zl amounts
to take a cross section Nl → Mx which can extend to a homomorphism Nl → M|U of sheaves
of monoids on a sufficiently small neighborhood U of x. This homomorphism induces a morphism
U → (Spec C[Nl ])an of log analytic spaces which gives, in fact, a chart around x. Indeed, replacing U
×
by a smaller neighborhood of x if necessary, the monoid MX,y at any y ∈ U equals to OX,x
·ziN1 · · · ziNm
where {i1 , . . . , im } = {i | zi (y) = 0}; on the other hand, the j-th factor for zj (y) 6= 0 in Nl is taken
×
over by OX
on passage to associated log structure, and only i1 , . . . , im -th parts survive. Hence the
associate log structure of the pre-log structure Nl → OU coincides with M|U and then the morphism
U → (Spec C[Nl ])an is a chart. Since the monoid Nl is obviously fs, we conclude that the log analytic
space X is fs.
Note that a toric variety TΣ (with the canonical log structure) from a non-singular fan Σ is
nothing but the fs log analytic space associated to the pair (TΣ , DΣ ), where DΣ is the union of all
codimension ≥ 1 torus orbits.
2.2
Log smoothness
Definition 2.2.1 Nilpotent thickening. Let f : X → Y be a morphism of log analytic spaces.
(1) The morphism f is called an exact closed immersion if it is strict and the underlying morphism
◦
◦
◦
f : X →Y is a closed immersion.
(2) The morphism f is called a thickening of order ≤ n if it is an exact closed immersion and
I n+1 = 0, where I = Ker(f −1 OY → OX ).
6
Definition 2.2.2 (cf. [7, (3.3)].) Log smoothness. Let f : X → Y be a morphism of fine log analytic
spaces. Then f is said to be log smooth if the following condition is satisfied: (Infinitesimal lifting
property) For any commutative diagram
s0
T 0 −→ X


ty
T


yf
−→ Y
s
of fine log analytic spaces with t a thickening of order ≤ 1, there exists locally on T a morphism
g: T → X such that s0 = g ◦ t and s = f ◦ g.
Log smoothness is stable under base change taken in the category of fine log analytic spaces.
Lemma 2.2.3 Let f : X → Y be a strict morphism of fine log analytic spaces. Then f is log smooth
◦
if and only if the underlying morphism f is smooth in the usual sense. 2
The proof is formal.
Here is a practical criterion of log smoothness by means of charts due to Kazuya Kato [7, (3.5)]
(see also [13, (A.2)]):
Theorem 2.2.4 Let f : X → Y be a morphism of fine (resp. fs) log analytic spaces and Y →
(Spec C[Q])an a chart of Y modeled on a fine (resp. torsion-free fs) monoid Q. Then f is log smooth
if and only if X is covered by local charts (Ui → (Spec C[Pi ])an )i∈I with Pi a fine (resp. torsion-free
fs) monoid for each i such that:
(1) Each member Ui → (Spec C[Pi ])an extends to a chart of f |Ui modeled on a homomorphism
hi : Q → Pi .
(2) The homomorphism hi gp is injective and Coker(hi gp )tor is a finite group.
(3) The induced strict morphism
Ui −→ Y ×(Spec C[Q])an (Spec C[Pi ])an
is smooth in the usual sense. 2
The proof is parallel to that in [7, (3.5)] (see also [5, §6]) and [13, (A.2)]. The following fact is
an immediate implication from this theorem:
Corollary 2.2.5 A morphism ϕ∗ : TΣ0 → TΣ of toric varieties (endowed with the canonical log structures as in Example 2.1.7) induced by a morphism of fan ϕ: (N 0 , Σ0 ) → (N, Σ) (cf. [15, §1.5]) is log
smooth if and only if the homomorphism ϕ: N 0 → N of abelian groups has finite cokernel. 2
Setting N = {0} in particular, we see that any toric variety is log smooth over (Spec C)an ; note
that the underlying spaces of toric varieties are not necessarily smooth. Besides, the underlying
morphism of a log smooth morphism is not necessarily flat; for example, the morphism of toric
varieties determined by subdivision of fan, like blow-up, is log smooth, even log étale (cf. [7, (3.3)]).
However, it is known that the underlying morphism of a log smooth and integral morphism is flat
(see [7, (4.1)]).
Example 2.2.6 Semistable degeneration. An important example of log smooth morphisms is
semistable degeneration of complex manifolds; it is a surjective proper flat holomorphic mapping
f : X → C of complex manifolds with C being one dimensional, such that
(1) there is a discrete set of points E ⊂ C such that the map f restricted to f −1 (C \ E) is smooth,
7
(2) D = f ∗ E is a reduced divisor with normal crossings.
The log structures on X and C are, by definition, the canonical log structures associated to the
pairs (X, D) and (C, E), respectively. Then the morphism f : X → C of complex manifolds extend
naturally to the morphism of fs log analytic spaces since f −1 OC → OX maps f −1 MC to MX .
To see this morphism is log smooth we take local charts of X and C. Take a point x ∈ X and
let y = f (x). The morphism f is locally defined by z1 · · · zl = t around x where z1 , . . . , zn are as
in Example 2.1.8 and t is a local parameter around y. We take, replacing X and Y suitably by
neighborhoods, good local charts X → (Spec C[Nl ])an and Y → (Spec C[N])an subject to our choice
of parameters as in Example 2.1.8. Then we see that the diagonal homomorphism N → Nl induces
a local chart of f and that the natural morphism X → Y ×(Spec C[N])an (Spec C[Nl ])an is strict and
smooth. Due to the criterion Theorem 2.2.4 we therefore deduce that f is log smooth.
We have seen, below Corollary 2.2.5, that toric varieties are log smooth over (Spec C)an . The
following theorem shows that the converse is “almost” true:
Theorem 2.2.7 (cf. [5, (4.8)].) Let X be an fs log analytic space. Then X is log smooth over
◦
(Spec C)an if and only if there exists an open covering {Ui }i∈I of X and a reduced divisor D on X
such that
(1) there exists a strict and smooth morphism hi : Ui −→ (Spec C[Pi ])an to an affine toric variety
for each i ∈ I,
(2) the divisor Ui ∩ D on Ui is the pull-back of the union of the closure of codimension one torus
orbits in (Spec C[Pi ])an by hi for each i ∈ I,
(3) the log structure MX is isomorphic to the log structure given by
×
,−→ OX ,
OX ∩ j∗ OX\D
where j : X \ D ,→ X is the open immersion.
◦
◦
If X is a smooth complex manifold, then D is a reduced normal crossing divisor on X . 2
The proof is parallel to that in [5, §7]. In fancy words, the idea of log smoothness over (Spec C)an
amounts to consider certain pair of analytic spaces and their divisors which are locally described by
toric geometry. For example, the fs log analytic space associated to the pair (X, D) (cf. Example
2.1.8) is log smooth over (Spec C)an .
2.3
Real blow-up
We have not made the best use of working over C until now; the arguments in the preceding paragraphs can be applied, being modified slightly, to more general situation. However, our argument
will depend very much on extra structures of C from now on. In this subsection we recall the general
construction of X log following K. Kato and C. Nakayama [9]; the space X log is, roughly speaking,
real blow-up of an fs log analytic space X along its log structures. In the toric situation this idea fits
in with the so-called manifolds with corners (cf. [14, (§10)] or [15, (§1.3)]).
Construction 2.3.1 ([9, §1]) Here is a rough sketch of construction of the space X log . First, we
◦
consider the log analytic space T with T = (Spec C)an and the log structure defined as follows:
◦
Γ(T , MT ) = R≥0 × S1 −→ C by
(λ, ζ) 7→ λζ
(here we set, once for all, S1 = {ζ ∈ C | |ζ| = 1}); As a set, X log is the set of all morphisms T → X
of log analytic spaces.
8
We endow X log with a reasonable topology as follows: The underlying analytic space of X has
locally a closed immersion into an affine space Cn ; moreover, the log structure MX has locally a chart
P → MX by an fs monoid P so that X log is locally realized as a subset in Cn × HomZ (P gp , S1 ) in the
natural way. Then we can define a topology locally on X log as a closed subset in Cn ×HomZ (P gp , S1 ),
where the topology of the second factor is the obvious one. It can be seen that this topology does
not depend on the choice of those local data; thus we can endow a topology on X log by gluing. With
◦
respect to this topology the natural map τX : X log →X is continuous and proper (cf. [9, (1.3)(1)]).
We can also check that X 7→ X log is functorial.
◦
In the sequel we write the continuous map τX : X log →X simply by τX : X log → X (which is
probably not misreading).
Example 2.3.2 In the toric situation the topological spaces just introduced above can be described
in terms of more convenient object. Let us start with the affine toric variety X = (Spec C[P ])an .
There is a splitting
X log ∼
= Hom(P, R≥0 ) × Hom(P, S1 )
of topological spaces, where Hom means the set of monoid homomorphisms. The second factor on
the right hand side is just isomorphic to Hom(P gp , S1 ) which is a compact torus. The first factor is
a manifold with corners which can be expressed more intrinsically (cf. [14, §10] or [15, §1.3]). Let Σ
be a fan in N and TΣ the associated toric variety. Then the above splitting can be globalized as
TΣ log ∼
= Mc(N, Σ) × (compact torus),
where we use the notation as in the references cited above. As is indicated in these references the
manifold with corner Mc(N, Σ) has a big advantage over TΣ itself because we can easily draw a
concrete picture of it by putting appropriate boundaries to NR .
Example 2.3.3 We apply the functor X 7→ X log to semistable degeneration (Example 2.2.6). Suppose we are given semistable degeneration f : X → ∆ of complex manifolds equipped with the canonical log structures. Using notations as in Example 2.2.6 we see that the space X log is locally presented
as follows: We set X = {(z1 , . . . , zn ) ∈ Cn | |zi | < 1} and ∆ = {t ∈ C | |t| < 1}. The morphism f is
defined by t = z1 · · · zl . Then X log is isomorphic as a topological space to
{(λ1 , ζ1 , . . . , λl , ζl , zl+1 , . . . , zn ) |
0 ≤ λi < 1, ζi ∈ S1 (1 ≤ i ≤ l), |zj | < 1 (l + 1 ≤ j ≤ n)}
≈ ∆n−l × [0, 1)l × (S1 )l .
Similarly, ∆log is isomorphic to [0, 1) × S1 . Note that the map τX : X log → X maps
(λ1 , ζ1 , . . . , λl , ζl , zl+1 , . . . , zn ) 7→ (λ1 ζ1 , . . . , λl ζl , zl+1 , . . . , zn )
and τ∆ : ∆log → ∆ maps (λ, ζ) to λζ. The induced morphism f log : X log → ∆log is given by equations
λ1 · · · λl = λ
and ζ1 · · · ζl = ζ.
Let us denote the fiber of f log over (0, ζ) by Xζlog . Let τX,ζ : Xζlog → X denote the restriction of τX .
−1
Then the inverse image of the origin of X by τX,ζ
(x) is isomorphic to (S1 )l−1 .
It is very interesting that the space X log seems to recover vanishing cycles; we should test this
fact by calculating the case l = n = 2, i.e., semistable degeneration of a curve: We pick up the fiber
Xζlog which is described by
{(λ1 , ζ1 , λ2 , ζ2 ) | λ1 λ2 = 0, 0 ≤ λi < 1, ζ1 ζ2 = ζ}.
9
This is the union of
(Xζlog )+ = {(λ1 , ζ1 , λ2 , ζ2 ) ∈ Xζlog | λ1 = 0}
and (Xζlog )− = {(λ1 , ζ1 , λ2 , ζ2 ) ∈ Xζlog | λ2 = 0}
both of which are cylinders with boundaries on one sides. Here we see that the vanishing cycle has
been replaced by {λ1 = λ2 = 0} ≈ S1 .
The phenomenon which is perhaps more interesting arises from the fact that the gluing of these
two cylinders depends on the parameter ζ, i.e., the gluing is done by the relation ζ1 ζ2 = ζ. As the ζ
walks along S1 ≈ {(0, ζ)} ⊂ ∆log , the gluing of (Xζlog )+ and (Xζlog )− is twisted. This is, needless to
say, the very concrete model of Dehn twist.
Even in the higher dimensional case this kind of phenomena can be seen; the fiber Xζlog looks
like a nearby fiber (see [16, Chap.II], where Xζlog can be visually understood by figures at p.32 and
−1
p.34 in n = 3 case), and it is twisted by moving along non-trivial paths on τ∆
(0). It is therefore
log
log
log
natural to ask whether the family f : X → ∆ arised from semistable degeneration is locally
topologically trivial; this has been affirmatively answered by S. Usui [19, (3.4)].
log
Construction 2.3.4 Returning to the general situation, we impose a sheaf OX
of rings (not neceslog
sarily local) upon X defined as follows ([9]): First we consider the following commutative diagram
of abelian sheaves on X log with exact rows:
√
exp
−1
−1 ×
τX
OX
−→
τX
OX
−→ 1
0 −→ 2π −1Z −→




k
ay
yb
√
√
exp
0 −→ 2π −1Z −→ Cont(–, −1R) −→ Cont(–, S1 ) −→ 1,
where Cont(–, A) with a topological space A denotes the sheaf of germs of continuous functions on
X log with values in A, and morphisms a and b are defined by
a(v) = v − Re(v)
Define
and b(u) = |u|−1 u.
√
−1
LX : = Cont(–, −1R) ×Cont(–,S1 ) τX
Mgp
X,
−1
1
where τX
Mgp
X → Cont(–, S ) is the canonical morphism. The sheaf LX should be called the sheaf
−1
of logarithms of sections in τX
Mgp
X because it sits in the following commutative diagram of abelian
sheaves with exact rows:
√
exp
−1
−1 ×
0 −→ 2π −1Z −→ τX
OX −→ τX
OX −→ 1
k
√
0 −→ 2π −1Z −→


hy
LX
exp


y∩
−1
−→ τX
Mgp
−→ 1.
X
Now define
log
−1
OX
= (τX
OX ⊗Z SymZ LX )/a
−1
where a is an ideal locally generated by local sections of form v ⊗ 1 − 1 ⊗ h(v) for all u ∈ τX
OX . The
log
log
stalk OX,ξ at a point ξ ∈ X is isomorphic to the polynomial ring over OX,τX (ξ) of n variables where
gp
gp
n = rankZ (CX
)τX (ξ) ([9, (3.3)]); these variables correspond to logarithms of a Z-basis of (CX
)τX (ξ) .
Remark 2.3.5 The topological space X log has been considered also in [10, §4], but the structure
sheaf is different.
10
3
Relative log Poincaré lemma
3.1
Taking good chart
We have seen below Definition 2.1.5 that any fs log analytic space admits a good local chart at any
point. We have not known, however, whether a morphism of fs log analytic spaces has a good local
chart. In this subsection we present the lemma which tells us a sufficient condition for a morphism
of fs log analytic spaces to have good local charts:
Lemma 3.1.1 (cf. [6, 2.5.9].) Let f : X → Y be a morphism of fs log analytic spaces and x ∈ X a
point. Set P = CX,x and Q = CY,y , where y = f (x), and let h: Q → P be the induced homomorphism.
Choose a good local chart λ: V → (Spec C[Q])an of a neighborhood V of y. Then, if h is injective and
Coker(hgp )tor = 0, there exists a neighborhood U of x and a good chart
U


y
V
−→ (Spec C[P ])an


y
−→ (Spec C[Q])an
of f |U at x. If f is log smooth, the induced morphism
U −→ V ×(Spec C[Q])an (Spec C[P ])an
is strict and smooth.
For the proof we need to know some auxiliary facts:
Definition 3.1.2 ([7, (4.6)]) Exact homomorphism. A homomorphism f : Q → P of monoids is called
exact if the induced homomorphism Q → Qgp ×P gp P is an isomorphism.
Lemma 3.1.3 If f : Q → P is a surjective homomorphism of fine monoids and Ker(f gp ) ⊆ Q, then
f is exact.
Proof. Consider the natural homomorphism Q → Qgp ×P gp P denoted by ϕ. The injectivity of ϕ
is clear. For the surjectivity let us take an element (ξ, y) ∈ Qgp ×P gp P . Write ξ = u/v by u, v ∈ Q.
Then we have f (u) = f (v)y. Take w ∈ Q such that y = f (w). Set n = u/(vw) ∈ Ker(f gp ) ⊆ Q.
Then we have ξ = nw ∈ Q and ϕ(ξ) = (ξ, y). 2
Proposition 3.1.4 Let f : Q → P be a surjective and exact homomorphism of fine monoids, and
assume that P gp is a finitely generated free abelian group. Then there exists a cross section of f by
an exact homomorphism s: P → Q.
Proof. Take a cross section s: P gp → Qgp of f gp : Qgp → P gp . Let b ∈ P . Since b = f gp (s(b)) ∈ N
and f is exact, we have s(b) ∈ Q. Thus the homomorphism s maps P to Q which clearly gives
a cross section of f . The exactness of s follows from the fact that, for b ∈ P gp , s(b) ∈ Q implies
b = f (s(b)) ∈ Q. 2
The characteristic splitting lemma mentioned in the previous section is an immediate consequence
of this proposition (the existence of the splitting has to be shown even though the cross section exists;
but, in this case, it is easy). There is one more useful corollary:
Corollary 3.1.5 Let f : Q → P be a surjective homomorphism of fine monoids. Assume P is fs
e = (f gp )−1 (P ) of Qgp allows a splitting
and torsion-free. Set K = Ker(f gp ). Then the submonoid Q:
e
e
e
∼
Q = P ⊕ K. More precisely, the homomorphism f : Q → P induced by f gp has a cross section
e which gives the splitting as above. 2
s: P → Q
11
e is generated by Q and a finite number of elements
The proof is easy. We note that the monoid Q
in Qgp since it is generated by Q and K which is a finitely generated abelian group.
In terms of toric varieties, Corollary 3.1.5 shows that, for a given equivariant closed immersion
Y → X of affine toric varieties, there exists a Zariski open set U of X which is isomorphic to a
product of Y and a torus T
Proof of Lemma 3.1.1. The argument is local and we will tacitly replace X and Y by smaller
open sets as we need. We may assume Y = V . Take a neighborhood U of x with a chart U →
(Spec C[P 0 ])an of f |U modeled on a homomorphism h0 : Q → P 0 . (We have seen below Lemma 2.1.6
that this is possible.) We may assume X = U .
Since X → (Spec C[P 0 ])an is a chart, the composite P 0 → MX,x → P , denoted by p, maps P 0
surjectively onto P . Obviously, p satisfies p ◦ h0 = h. Write the homomorphism P 0 → MX,x by
λ0 . Set Pe = (pgp )−1 (P ) ⊆ P 0 gp as in Corollary 3.1.5 and denote the morphism Pe → P induced by
pgp by pe. Note that the morphism pe is exact due to Lemma 3.1.3. We need to construct a cross
e = s ◦ h, where h:
e Q → Pe is the composite of h0 : Q → P 0 and P 0 ,→ Pe .
section s of Pe such that h
Once we obtain such s, then by [7, (2.10)], it extends to a local chart X → (Spec C[P ])an . Since pe is
exact, for the existence of s as above we only need to show the existence of sgp . But we see, by the
standard argument, that the obstruction for the existence of sgp lies in Ext1Z (Coker(hgp ), Ker(pgp ))
which vanishes by our assumption.
Thus we get a chart
X −→ (Spec C[P ])an


fy
Y


y
−→ (Spec C[Q])an
of f .
In case f is log smooth the chart of f |U modeled on h0 : Q → P 0 which we have taken above should
be chosen as in Theorem 2.2.4 so that X → Y ×(Spec C[Q])an (Spec C[P 0 ])an is smooth. We claim that
the morphism
X −→ Y ×(Spec C[Q])an (Spec C[P ])an
(3.1.6)
is strict and smooth; strictness is trivial. In order to see the smoothness we first claim that the
induced morphism
X −→ Y ×(Spec C[Q])an (Spec C[Pe ])an
(3.1.7)
is smooth; indeed, since Pe is generated by P 0 and finite number of elements, we easily see that the
morphism
Y ×(Spec C[Q])an (Spec C[Pe ])an −→ Y ×(Spec C[Q])an (Spec C[P 0 ])an
(3.1.8)
is an open immersion. Since the composite of (3.1.7) and (3.1.8) is X → Y ×(Spec C[Q])an (Spec C[P 0 ])an
which is smooth, the smoothness of (3.1.7) follows. By the remark made below Corollary 3.1.5 we
know that the morphism
(3.1.9)
Y ×(Spec C[Q])an (Spec C[Pe ])an −→ Y ×(Spec C[Q])an (Spec C[P ])an
is a “projection” from the product with torus, and hence is smooth. Since the morphism (3.1.6) in
question is the composite of (3.1.7) and (3.1.9), we conclude that (3.1.6) is smooth as desired. 2
12
3.2
Relative log de Rham complex
Definition 3.2.1 (cf. [7, (1.7)], [3, 1.5].) Log differentials. Let f : X → Y be a morphism of fine log
analytic spaces. The sheaf of log differentials of X over Y is an OX -module defined by
h
i
1
ωX/Y
= Ω1X/Y ⊕ (OX ⊗Z Mgp
X ) /K,
where Ω1X/Y denotes the usual sheaf of differentials and K is the OX -submodule generated by
(dα(a), 0) − (0, α(a) ⊗ a)
and (0, 1 ⊗ φ(b)),
for all a ∈ MX and b ∈ f −1 MY .
1
The sections of form [0, 1 ⊗ a] of ωX/Y
should be written by dloga; it defines a natural additive
morphism
1
(3.2.2)
dlog: Mgp
X −→ ωX/Y .
1
It is known that, if f is log smooth, the ωX/Y
is a finite locally free OX -module; but we will not
•
need this fact. Similarly to the usual case the sheaf of log differentials yields a complex (ωX/Y
, d) of
gp
−1
f OY -modules by imposing d · dloga = 0 for a ∈ MX .
•
The subject of this subsection is calculating the cohomologies of the complex ωX/Y
:
Proposition 3.2.3 Let f : X → Y be a log smooth morphism of fs log analytic spaces with Y log
smooth over (Spec C)an such that the induced morphism CY → CX is injective and CX/Y has torsionfree stalks. Then there exists a canonical isomorphism
•
•
Hq (ωX/Y
)∼
) ⊗Z
= H0 (ωX/Y
q
^
gp
CX/Y
of f −1 OY -modules for all q.
Proof. Consider the morphism (3.2.2). Since its image consists of germs of closed forms, it induces
gp
1 •
1 •
∗
Mgp
X → H (ωX/Y ). The image of f MY in H (ωX/Y ) is obviously zero. Hence we have
gp ∼ gp
1 •
Mgp
X /MY = CX/Y −→ H (ωX/Y ).
By cup-product we obtain
f −1 OY ⊗Z
q
^
gp
•
−→ Hq (ωX/Y
).
CX/Y
We need to show that this morphism is an isomorphism. This can be seen stalkwise as follows: Let
x ∈ X be a point and set y = f (X). Applying Lemma 3.1.1 to Y → (Spec C)an (recall that CYgp has
torsion-free stalks because stalks of the characteristic CY are fs and torsion-free) we get a good local
chart Y → (Spec C[Q])an at y which is strict and smooth. Again applying Lemma 3.1.1 to X → Y
we get a good local chart
X −→ (Spec C[P ])an


fy
Y


y
−→ (Spec C[Q])an
gp
CX/Y,x
of f at x. (Note that
= P gp /Qgp is torsion-free since CX/Y,x has been assumed to be torsionfree and fs.) Since the horizontal arrows in this diagram are strict and smooth, we may therefore
restrict ourselves to the situation
B: = C{Q; w1 , . . . , ws } −→ A: = C{P ; w1 , . . . , ws ; z1 , . . . , zr },
13
where C{–} means the analytic completion of C[–].
Now we remark that the ring A = C{P } for a fs monoid P has the following description: Choose a
surjective homomorphism Nr → P . Then, in the formal completion C[[P ]], the ring C{P } coincides
with the image of the usual convergent power series C{X1 , . . . , Xr } ∼
= C{Nr } under the induced
P
morphism C{Nr } → C[[P ]] of C-algebras. In particular, for any p∈P ap X p ∈ C{P } and any
P
subset P 0 ⊆ P , the partial sum p∈P 0 ap X p belongs to C{P }.
•
Therefore, returning to our situation, the complex ωA/B
splits into the toric part and the usually
smooth part. Then by the usual Poincaré lemma we may assume B = C{Q} and A = C{P } without
loss of generality. In this case we have
q
∼
ωA/B
= C{P } ⊗Z
q
^
∼
= C{P } ⊗C
q
^
(P gp /Qgp )
(VP /VQ ),
where we set VP = C ⊗Z P gp , etc. Let us denote the monomial in C{P } corresponding to p ∈ P
1
by X p . Let ep be the image of p ∈ P under dlog: P → ωA/B
, i.e., the image of 1 ⊗ p under
q
q+1
1
VP → VP /VQ → ωA/B
. Then the exterior derivative d: ωA/B
→ ωA/B
is given by

d
X
p∈P

ap X p ep1 ∧ · · · ∧ epq  =
X
ap X p ep ∧ ep1 ∧ · · · ∧ epq .
p∈P
Hence the operation d is homogeneous.
We shall prove the proposition in this situation by constructing a certain homotopy operator; the
following argument is due to the communication with A. Ogus. Set r = rank P gp /Qgp , and choose
elements v1 , . . . , vr ∈ P of which the images in P gp /Qgp form a Z-basis. Let Pei be the subgroup of
P gp generated by Qgp and v1 , . . . , vi−1 , vi+1 , . . . , vr for 1 ≤ i ≤ r. Set Pi = Pei ∩ P . Then we obviously
T
e
have 1≤i≤r Pi = Q.
We choose an injective homomorphism ι: P gp /Pei ,→ R where R is now regarded as an additive
group. Let us denote the composite P → P gp /Pei ,→ R by ∂i . The composite of {ιi } and the natural
1
projection P gp /Qgp → P gp /Pei induce the A-linear morphisms ωA/B
→ A, i.e., a log derivation, which
•
we also write by ∂i . Then it is not difficult to see from the above description of ωA/B
that the interior
∗
multiplication by ∂i induces a derivation of degree −1 on ⊕ωA/B , and that κi : = d∂i + ∂i d is just the
multiplication by ∂i (p) ∈ R in degree p ∈ P . Define for any subset S ⊆ {1, . . . , r} the subcomplex
•
CS• ⊆ ωA/B
by
q
CSq : = {ω ∈ ωA/B
| κi ω = 0 for all i ∈ S},
which is actually a subcomplex since each κi commutes with d. Note that we obviously have CSq ∼
=
T
Vq
Vq
•
e
∼
C{ j∈S Pj } ⊗C (VP /VQ ); in particular, we have C{1,...,r} = C{Q} ⊗C (VP /VQ ). What we need
•
•
•
to prove is the quasi-isomorphy of C{1,...,r}
,→ ωA/B
. To do this, we claim that any CS• ,→ ωA/B
is quasi-isomorphism; which we will prove by induction with respect to card(S). Define the second
complex ZS• by the exact sequence of complexes:
•
0 −→ CS• −→ ωA/B
−→ ZS• −→ 0.
Each element of ZSq is uniquely represented by a finite sum of formal power series of form
X
ap X p ep1 ∧ · · · ∧ epq ,
p∈P \∩i∈S Pi
14
q
and due to what we remarked above, this representation belongs to ωA/B
. Hence it follows that the
•
•
•
∼
exact sequence splits, i.e., we have ωA/B = CS ⊕ ZS . In case S = {j}, we have a homotopy operator
P
−1
on the complex ZS• . Hence we have shown the claim in the case card(S) = 1.
p∈P \Pi (∂(p)) ∂i `
0
For general S = S {∂j } we only need to show that CS ,→ CS 0 is a quasi-isomorphism. But we can
prove it similarly as above. we therefore conclude
•
e ⊗C
Hq (ωA/B
)∼
= C{Q}
q
^
(VP /VQ ).
•
e which implies the lemma. 2
In case q = 0 we get H0 (ωA/B
)∼
= C{Q}
Definition 3.2.4 ([7, (4.6)]) Exact morphism. A morphism f : X → Y of fine log analytic spaces is
said to be exact if the induced homomorphism (f ∗ MY )x → MX,x is exact (Definition 3.1.2) for any
x ∈ X.
It is easy to see that f is exact if and only if the induced homomorphism CY,f (x) → CX,x of
characteristics is exact for any x ∈ X. Semistable degeneration is, for example, exact since the
diagonal homomorphism N → Nl is obviously exact (cf. Example 2.2.6).
The next proposition is a by-product of the above proof:
•
Proposition 3.2.5 Let f : X → Y be as in Proposition 3.2.3. Then the equality H0 (ωX/Y
) = f −1 OY
holds if and only if f is exact.
•
e for any point x ∈ X, which is
Proof. The equality H0 (ωX/Y
) = f −1 OY is equivalent to Q = Q
nothing but the exactness condition. 2
3.3
Topological feature
In this subsection we observe the real blow-up from a topological viewpoint. We begin with a review
of the lemma [9, (1.3)]:
Lemma 3.3.1 Let X be an fs log analytic space and τX : X log → X the real blow-up (2.3.1). For
x ∈ X we have a canonical homeomorphism
gp
−1
τX
(x) ≈ HomZ (CX,x
, S1 ).
−1
Proof. By definition, τX
(x) is the set of monoid homomorphisms MX,x → R≥0 × S1 which make
the following diagram commutative
OX,x
(3.3.2)
−→
x

αX,x 
C
x

αT,0
MX,x −→ R≥0 × S1 ,
where the morphism in the first row is the residue map at x. The morphism OX,x → C induces
×
the group homomorphism OX,x
→ R>0 × S1 . Hence, given such a morphism MX,x → R≥0 × S1 ,
we get a monoid homomorphism CX,x → (R≥0 × S1 )/(R>0 × S1 ) ∼
= S1 . Taking the Grothendieck
gp
gp
1
group we have CX,x → S . Conversely, suppose we are given a group homomorphism CX,x
→ S1 .
Set P = CX,x , which is a torsion-free fs monoid. By characteristic splitting lemma (Lemma 2.1.6)
×
we know that there exists a splitting MX,x = P ⊕ OX,x
. Then we get a monoid homomorphism
1
1 ∼
1
MX,x → ({0} × S ) ⊕ (R>0 × S ) = R≥0 × S which makes the diagram (3.3.2) commute. 2
We are going to see the relative version of this lemma.
15
Notation 3.3.3 Let f : X → Y be a morphism of fs log analytic spaces and τX : X log → X be the
real blow-up. We define the topological space XYlog by
XYlog : = X ×Y Y log .
We fix notation by the following commutative diagram of topological spaces once for all:
X log


f log y
Y log
τX/Y
−→
=
τX
Y
X
XYlog −→


fYlog y
Y log


yf
−→ Y,
τY
with the square in the right hand side Cartesian, so that τX = τX/Y ◦ τXY .
Lemma 3.3.4 Let x0 ∈ XYlog and set x = τXY (x0 ). Then there exists a canonical homeomorphism
gp
−1
τX/Y
(x0 ) ≈ HomZ (CX/Y,x
, S1 ).
Proof. Let y = f (x). We first note the obvious equality
−1
−1
τX/Y
(x0 ) ≈ {ξ ∈ τX
(x) | f log (ξ) = fYlog (x0 )}.
By Lemma 3.3.1 we have
gp
−1
τX
(x) ≈ HomZ (CX,x
, S1 )
gp
and τY−1 (y) ≈ HomZ (CY,y
, S1 ).
−1
(x) → τY−1 (y) commutes with the group homomorphism
The morphism f log : τX
gp
gp
HomZ (CX,x
, S1 ) → HomZ (CY,y
, S1 )
gp
gp
−1
(x0 ) is obviously homeomorphic to the kernel of this group
. Then τX/Y
→ CX,x
induced by CY,y
gp
gp
), S1 ). 2
→ CX,x
homomorphism, and hence is homeomorphic to HomZ (Coker(CY,y
Next, we prove the relative version of [9, (1.5)]:
Proposition 3.3.5 Let F be a sheaf of abelian groups on XYlog . Then there exists a canonical
isomorphism
"
#
−1
Rq τX/Y ∗ τX/Y
F∼
= F(−q) ⊗Z
q
^
−1 gp
C
τX
Y X/Y
√
of abelian sheaves on XYlog for all q, where F(p) = F ⊗Z (2π −1)p Z.
Proof. The case q = 0 is trivial due to Lemma 3.3.4; i.e., we have
(3.3.6)
−1
τX/Y ∗ τX/Y
F∼
=F
For the general case we consider the exact sequence constructed in (2.3.4):
√
(3.3.7)
0 −→ 2π −1Z −→ LY −→ τY−1 Mgp
Y −→ 1.
We first claim that this exact sequence induces exact sequences
√
−1
−1 ∗
(3.3.8)
0 −→ 2π −1Z −→ (f log )−1 LY ⊕(f log )−1 τ −1 OY τX
OX −→ τX
f Mgp
Y −→ 1
Y
16
and
(3.3.9)
√
−1
−1 ∗
0 −→ 2π −1Z −→ (fYlog )−1 LY ⊕(f log )−1 τ −1 O τX
OX −→ τX
f Mgp
Y −→ 1
Y
Y
Y
X log
of abelian sheaves on
and
we obtain the exact sequence
(3.3.10)
XYlog ,
Y
Y
respectively. Indeed, pulling back the sequence (3.3.7) by f log
√
−1 −1
0 −→ 2π −1Z −→ (f log )−1 LY −→ τX
f Mgp
Y −→ 1
which sits in the following commutative diagram with exact rows:
√
exp
−1 −1
−1 −1 ×
0 −→ 2π −1Z −→ τX
f OY −→ τX
f OY
(3.3.11)


y
k
√
0 −→ 2π −1Z −→ (f log )−1 LY
−→ 1


y
exp
−1 −1
−→ τX
f Mgp
−→ 1.
Y
We further consider the following commutative diagram with exact rows:
√
exp
−1 −1
−1 −1 ×
0 −→ 2π −1Z −→ τX
f OY −→ τX
f OY −→ 1
(3.3.12)


y
k
√
0 −→ 2π −1Z −→
−1
τX
OX
exp
−→


y
−1 ×
τX
OX
−→ 1
We take push-out, in the category of abelian sheaves on X log , of (3.3.11) and (3.3.12) termwise; since
f −1 OY×
−→


y
×
OX


y
f −1 MY gp −→ f ∗ Mgp
Y
is a co-Cartesian diagram of abelian sheaves on X, we obtain (3.3.8). The construction of (3.3.9) is
similar.
The sequence (3.3.8) sits in the following commutative diagram of exact rows:
√
−1
−1 ∗
0 −→ 2π −1Z −→ (f log )−1 LY ⊕OY τX
OX −→ τX
f Mgp
−→ 1
Y


y
k
√
0 −→ 2π −1Z −→
LX
−→


y
−1
τX
Mgp
X
−→ 1.
By this and applying τX/Y ∗ we obtain a commutative diagram
√
−1 ∗
τX
f Mgp
−→ R1 τX/Y ∗ 2π −1Z
Y
Y
(3.3.13)


y
k
−1
τX
Mgp
X
Y
√
−→ R1 τX/Y ∗ 2π −1Z.
(Here we have used (3.3.6).) Applying the right-exact functor τX/Y ∗ to (3.3.8) we get nothing but
the exact sequence (3.3.9). Hence we deduce that the morphism in the first row is zero. We therefore
get
√
−1 gp
τX
C
−→ R1 τX/Y ∗ 2π −1Z.
Y X/Y
Taking cup-product we obtain a canonical morphism
(3.3.14)
F(−q) ⊗Z
q
^
−1 gp
−1
τX
C
−→ Rq τX/Y ∗ τX/Y
F
Y X/Y
for each q. One can check stalkwise that (3.3.14) is in fact an isomorphism by local calculations given
by Lemma 3.3.4. 2
17
3.4
Main theorem
In this subsection we prove generalization of well-known relative Poincaré lemma. The essential
object is the following; the construction is similar to [9, (3.5)]:
Construction 3.4.1 (cf. [9, (3.5)].) Define
q,log
log
−1 q
ωX/Y
: = OX
⊗τ −1 OX τX
ωX/Y
X
log
log
for all q. By the construction of OX
in Construction 2.3.4, there exists a natural arrow d: OX
→
exp
dlog
1,log
−1
−1 1
ωX/Y
extending LX → τX
Mgp
X → τX ωX/Y . We can easily see that this d extends naturally to
q,log
q+1,log
d: ωX/Y
−→ ωX/Y
∗,log
•,log
for all q which make ωX/Y
into a complex (ωX/Y
, d) of (f log )−1 OYlog -modules.
Now we state the main theorem:
Theorem 3.4.2 (Relative logarithmic Poincaré lemma.) Let f : X → Y be a log smooth morphism of fs log analytic spaces with Y log smooth over (Spec C)an . Assume that the induced morphism
of characteristics f −1 CY → CX is injective and, for every x ∈ X, the relative characteristic CX/Y,x
is torsion-free. Then the natural morphism
•,log
−1 0 •
H (ωX/Y ) −→ ωX/Y
(f log )−1 OYlog ⊗τ −1 f −1 OY τX
X
is an isomorphism in D+ (X log , (f log )−1 OYlog ).
Proof. (The idea of the following proof is similar to that in [9, (4.7)].) We will work stalkwise. Let
ξ ∈ X log and set x = τX (ξ), η = f log (ξ) and y = f (x). Take a neighborhood U (resp. V ) around x
(resp. y) in X (resp. Y ), and set P = CX,x and Q = CY,y . Take elements S1 , . . . , Sr (resp. T1 , . . . , Ts )
gp
×
×
gp
in Mgp
X,x (resp. MY,y ) such that S i = Si mod OX,x (resp. T j = Tj mod OY,y ) give a Z-basis of P
gp
(resp. Q ). Then there exist linear relations
(3.4.3)
Ti =
r
X
aij S j ,
aij ∈ Z
j=1
(1 ≤ i ≤ s) defining the inclusion Qgp ,→ P gp (here we write the monoid structures of P and Q
log ∼
log
additively). We have OY,η
= OY,y [T 1 , . . . , T s ] and OX,ξ ∼
= OX,x [S 1 , . . . , S r ] (cf. (2.3.4)). Moreover,
log
log
the morphism OY,η → OX,ξ is isomorphic to that induced by (3.4.3). Let us denote OY,y simply by
O. Set
(3.4.4)
R = O[T 1 , . . . , T s ][S 1 , . . . , S r ]/(relations (3.4.3))
log
and u: R → OX,ξ
to be the obvious O[T 1 , . . . , T s ]-algebra morphism. Let us abbreviate O[T ] =
O[T 1 , . . . , T s ]. Note that, since (3.4.3) is a linear relation, O[T ] → R is a smooth homomorphism of
•,log
rings. Then there exists a canonical morphism Ω•R/O[T ] → ωX/Y,ξ
of complexes given by
h · dS i1 ∧ · · · ∧ dS ip 7→ u(h)dlogsi1 ∧ · · · ∧ dlogsip ,
1
•
where dlogsi denotes the image of Si under dlog: Mgp
X,x → ωX/Y,x . Since O[T ] → ΩR/O[T ] is quasiisomorphism by the classical relative Poincaré lemma, it suffices to show that the morphism
•,log
•
Ω•R/O[T ] ⊗O H0 (ωX/Y
)x −→ ωX/Y,ξ
18
is a quasi-isomorphism.
•,log
by the degree of coefficient polyWe introduce increasing filtrations both on Ω•R/O[T ] and ωX/Y,ξ
nomials; this is possible since (3.4.3) is linear. Then it is easy to see that
•,log ∼
q
•
Grq ωX/Y,ξ
.
= SymZ P gp ⊗Z ωX/Y,x
On the other hand, we easily find
q
Hp (Grq Ω•R/O[T ] ) ∼
= SymZ P gp ⊗Z
^
(P gp /Qgp ) ⊗Z O.
Hence by Proposition 3.2.3 we see that
•,log
•
Grq Ω•R/O[T ] ⊗O H0 (ωX/Y
)x −→ Grq ωX/Y,ξ
is a quasi-isomorphism for each q. This implies our theorem. 2
4
4.1
Applications
Integral structure of VMHS
In this subsection, applying the relative log Poincaré lemma to exact f : X → Y , we will deduce the
formula which gives integral structure of hyperdirect images of relative log complexes. This formula
gives, in semistable degeneration case, the integral structure of degenerate VMHS studied by S. Usui
[19], T. Fujisawa [2], and T. Matsubara [11].
Lemma 4.1.1 Let f : X → Y be as in Theorem 3.4.2. Assume further that f is exact. Then there
exists a canonical isomorphism
−1 •
τX
ωX/Y ⊗
Y
log
(fY )−1 (τY )−1 OY
∼
(fYlog )−1 OYlog −→ RτX/Y ∗ (f log )−1 OYlog
in D+ (XYlog , (fYlog )−1 OYlog ).
Proof. Due to Proposition 3.3.5 and Proposition 3.2.3, there exists a canonical isomorphism
(4.1.2)
−1 •
Hq (τX
ωX/Y ) ∼
= Rq τX/Y ∗ (f log )−1 (τY )−1 OY
Y
for all q. We shall construct a morphism
−1 •
τX
ωX/Y ⊗
Y
log
(fY )−1 (τY )−1 OY
(fYlog )−1 OYlog −→ RτX/Y ∗ (f log )−1 OYlog
of complexes which yields the isomorphisms (4.1.2) tensored by (fYlog )−1 OYlog .
By Theorem 3.4.2 and Proposition 3.2.5 there exists a morphism of complexes
(4.1.3)
•,log
τX/Y ∗ ωX/Y
−→ RτX/Y ∗ (f log )−1 OYlog .
On the other hand, we have a natural morphism
(4.1.4)
•,log
−1 •
.
ωX/Y ⊗OY (fYlog )−1 OYlog −→ τX/Y ∗ ωX/Y
τX
Y
Then by the combination of (4.1.3) and (4.1.4) we get desired morphism of complexes which is easily
checked to be a quasi-isomorphism by (4.1.2). 2
19
Theorem 4.1.5 Let f : X → Y be as in Theorem 3.4.2. Assume further that f is exact and the
◦
underlying morphism f is proper. Then there exists a canonical isomorphism
∼
•
τY−1 R f∗ ωX/Y
⊗τ −1 OY OYlog −→ Rf∗log Z ⊗Z OYlog
Y
in D+ (Y log , OYlog ).
Proof. We apply R fYlog∗ to both sides in the quasi-isomorphism in Lemma 4.1.1. On the left hand
side we have
log −1 •
−1 •
ωX/Y ) ⊗OY OYlog
R fYlog∗ (τX
ωX/Y ⊗OY (fYlog )−1 OYlog ) ∼
= R fY ∗ (τX
Y
Y
•
∼
⊗OY OYlog ,
= τY−1 R f∗ ωX/Y
log
where the first ∼
= is due to Lemma 4.1.6 below (note that OY is flat over OY since every stalk of
OYlog is a polynomial ring over the stalk of OY (see Construction 2.3.4)), and the second one is a
natural isomorphism. As for the right hand side, we find
log
R fYlog∗ RτX/Y ∗ (f log )−1 OYlog ∼
= Rf∗log (f log )−1 OY
log
∼
= Rf∗log Z ⊗Z OY ,
log
where the second ∼
= is again due to Lemma 4.1.6 (OY is flat over Z since it has no torsion). 2
Here is an auxiliary lemma which has been used in the above proof:
Lemma 4.1.6 Let f : X → Y be a continuous map of topological spaces, and R a sheaf of rings on
Y . Let F be an f −1 R-module and G an R-module. Then, if f is proper and G is flat over R, the
natural morphism
Rf∗ F ⊗R G −→ Rf∗ (F ⊗f −1 R f ∗ G)
is an isomorphism in D+ (Y, R).
Proof. This is a special case of [4, (II.2.6.6)]. 2
4.2
Result of Steenbrink
In this subsection we restrict ourselves to semistable degeneration (Example 2.2.6) which has, due
to the local description given in Example 2.2.6, the properties assumed in Theorem 4.1.5. In dealing
with this kind of morphisms topologically, the real blow-up will be very helpful: the following theorem
is due to S. Usui [19, (3.4)]:
Theorem 4.2.1 ([19, (3.4)]) Let f : X → Y be a semistable degeneration. Then f log : X log → Y log is
locally a topologically trivial family. 2
We will work from now on in the local setting; i.e., we suppose that Y = ∆, the unit disk, with
the log structure associated to the pair (∆, 0) (cf. Example 2.1.8). We define the fs log analytic
◦
space ◦ by ◦= (Spec C)an together with the pull-back log structure of M∆ by the closed immersion
0 = (Spec C)an ,→ ∆. The log structure M◦ is given by the chart
N −→ C sending a 7→ 0a .
(The fs log analytic space ◦ is usually called the standard point.) Obviously the topological space
◦log is homeomorphic to S1 .
20
Notation 4.2.2 Let ∆∗ : = ∆ \ 0, the punctured disk, with the trivial log structure so that ∆∗ ,→ ∆
is strict. Let us fix notation by the following commutative diagram with all squares are Cartesian in
the category of log analytic spaces:
e ∗ −→ X ∗ ,−→ X ←−- X◦
X


fe∗ y




f ∗y
fy
e ∗ −→ ∆∗ ,−→ ∆ ←−∆


yf◦
◦,
e ∗ is the universal covering space of ∆∗ . The left square is the diagram of log analytic spaces
where ∆
with trivial log structure. The underlying morphism of f◦ : X◦ → ◦ is nothing but the central fiber
◦
of f .
We begin with the topological argument. Since f log : X log → ∆log is locally a topological trivial
e ∗,
family (Theorem 4.2.1), the Betti cohomology groups of each fiber are isomorphic to those of X
1
log
the topological generic fiber; in particular, for each ζ ∈ ◦ (≈ S ) we have
e ∗ , Z),
H∗ (Xζlog , Z) ∼
= H∗ (X
where Xζlog = (f log )−1 (ζ). Moreover, we can introduce, by the topological triviality, the well-defined
monodromy operation: For any s ∈ ∆log the fundamental group π1 (∆log ) ∼
= Z acts on the cohomology
H∗ (Xslog , Z). We denote the generators of this action by T , the operation induced by rounding ◦log
once counter-clockwise. The operation T is well-known to be unipotent.
Lemma 4.2.3 There exists a canonical isomorphism
•
log
τ◦−1 Hq (X◦ , ωX
) ⊗C O◦log ∼
Z ⊗Z O◦log
= Rq f◦∗
◦ /◦
of O◦log -modules for all q.
Proof. Just take the pull-back of the quasi-isomorphism in Theorem 4.1.5 onto ◦log . 2
We take global sections over ◦log ≈ S1 of both sides in Lemma 4.2.3. Since Γ(◦log , O◦log ) = C, we
have
•
(4.2.4)
Hq (X◦ , ωX
)∼
= Γ(◦log , Rq f◦log ∗ Z ⊗Z O◦log ).
◦ /◦
In order to describe elements in the right hand side we fix a parameter t on ∆. Fixing the parameter
amounts to taking a chart N → O∆ of the log structure M∆ on ∆ given by a 7→ ta (cf. Example
2.1.8) so that we have the splitting Z ⊕ C× ∼
= M◦ gp . Considering the exact sequence
√
exp
0 −→ 2π −1Z −→ L◦ −→ τ◦−1 M◦ gp −→ 1
of abelian sheaves over ◦log (cf. Construction 2.3.4), we can choose a multivalued global section u of
L◦ such that exp(u) = (1, 1) ∈ Z ⊕ C× ; the section u is conceptually a logarithm of the parameter
t restricted to ◦log . For ζ ∈ ◦log we take a germ uζ of a branch of u near ζ. Then elements in
log
(Rq f◦∗
Z ⊗Z O◦log )ζ are polynomials of uζ with coefficients in Hq (Xζlog , C). By the local topological
log
triviality, elements in Hq (Xζlog , C) extend uniquely to multivalued global sections of Rq f◦∗
C on ◦log .
log
q log
k
k=0 xk uζ ∈ (R f◦∗ Z ⊗Z O◦ )ζ , where
log
Γ(◦log , Rq f◦∗
Z ⊗Z O◦log ) if and only if
Let x =
section in
Pr
r
X
k=0
xk ∈ Hm (Xζlog , C). Then x comes from a global
r
X
√
xk ukζ ,
T xk (uζ + 2π −1)k =
k=0
21
for each i. This is equivalent to
(4.2.5)
T xk =
r
X
j=k
j−k
(−1)
!
√
j
(2π −1)j−k xj
k
for k = 0, . . . , r.
Denote the monodromy logarithm logT by N . Then it is easily seen that (4.2.5) is equivalent to
√
(4.2.6)
xk = (−2π −1N )k k!x0 for k = 0, . . . , r.
This equation shows that x is determined by the constant term x0 . Thus we have an isomorphism
(4.2.7)
log
•
Hq (X◦ , ωX
)∼
= Hq (Xζ , C).
◦ /◦
e ∗ , C). Hence we get the following theorem which
The last cohomology group is isomorphic to Hq (X
is nothing but the famous result of Steenbrink (cf. [17, (2.16)]):
Theorem 4.2.8 Depending on the choice of t and u as above, there is an isomorphism
•
e ∗ , C)
Hq (X◦ , ωX
)∼
= Hq (X
◦ /◦
for all q. 2
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Fumiharu Kato
Graduate School of mathematics
Kyusyu University 33
Fukuoka 812, Japan
23