Relative log Poincare lemma and
relative log de Rham theory
Fumiharu Kato
Abstract
In this paper we will generalize the classical relative Poincare lemma in the framework of log
geometry. Like the classical Poincare lemma directly implies the de Rham theorem, the comparison
between de Rham and Betti cohomologies, our log Poincare 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 Introduction
1.1
De Rham-Hodge theory and log geometry
Let f : X ! Y be a smooth morphism of complex manifolds, and X=Y the relative de Rham complex.
Then the famous classical Poincare lemma asserts that the natural morphism
f 01 OY 0!
X=Y
of complexes is a quasi-isomorphism. The Poincare lemma implies, providing f is proper, the comparison of the de Rham cohomology and the Betti cohomology (de Rham theorem):
Rf3Z Z OY = R f3 :
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 dierentials 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 ber is isomorphic to the Betti cohomology of a general
ber, 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 Poincare lemma, the relative log
Poincare 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 Poincare 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
dierent 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 Poincare 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 briey 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 (denition of the log smoothness will be presented in the
next section). We assume that the induced morphism of characteristics f 01 CY ! CX is injective
and, for every x 2 X , the relative characteristic CX=Y;x is torsion-free (see 2.1.1 for notation and
terminologies). The rst main result is generalization of the Poincare lemma:
Theorem 1.2.1 (Theorem 3.4.2) Let f : X ! Y be as above. Then the natural morphism
) 0! !;log
(f log )01 OYlog 01 f 01 O X01 H0(!X=Y
X=Y
Y
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 dierentials where the contribution
\logarithms" to OY (cf. 2.3.4), and !X=Y
)(= Ker(d: OX ! ! 1 )) may
of \logarithms" is involved (cf. 3.4.1). The f 01 OY -module H0 (!X=Y
X=Y
0
1
not be f OY ; for example, if f is log etale, such as the morphism between toric varieties arised
1 = 0. However, if the family f : X ! Y reasonably
from subdivision of fan, it is OX since !X=Y
degenerates, it equals to f 01 OY :
Proposition 1.2.2 (Proposition 3.2.5) Let f : X ! Y satisfy the conditions as in Theorem 1.2.1.
) = f 01 OY if and only if f is exact (cf. 3.2.4).
Then H0 (!X=Y
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
Rf log Z O log :
01 Olog 0!
Y01 R f3 !X=Y
Z Y
3
Y
O
Y
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
dierent way from ours.
Finally we restrict ourselves to semistable degeneration with canonically dened 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
Hq (X ; !X = ) = Hq (Xe 3 ; C)
for all q .
Here Y has been taken to be a unit disk and X denotes the ber over the origin , and Xe 3
denotes the topological generic ber.
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 Poincare lemma. After proving a
technical lemma in the rst 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 rst
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 xed in this nal 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 specied.
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 = Z0 : additive monoid of non-negative integers.
Let P be a monoid.
| R[P ]: Monoid ring with coecients 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 dened and discussed on arbitrary ringed
spaces in [3, 1.1, 1.2].) We will therefore deal with log analytic space which is, by denition, 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 =O 2 ).
X
We denote the homomorphism MX ! OX dening the log structure by X in case we need.
(3) For a morphism f : X ! Y of log analytic spaces we dene CX=Y : = Coker(f 01 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 specied.
Note that, in particular, the analytic space (Spec C)an is equipped with the trivial log structure.
Denition 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 3 N , which is the associated log structure of the
pre-log structure f 01 N ! f 01 OY ! OX .
(2) A morphism f : X ! Y of log analytic spaces is called strict if f 3 MY ! MX is an isomorphism
of log structures on X .
Denition 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 2 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 2 X )
modeled on a homomorphism h: Q ! P of monoids is a commutative diagram of log analytic
spaces
X 0! (Spec C[P ])an
??
??
y
fy
Y 0! (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 2 X and f (x) 2 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 dene local charts of morphisms.
In what follows we will discuss in the category of fs log analytic spaces dened as follows:
Denition 2.1.5 Fs log analytic space.
(1) A monoid P is said to be ne if it is nitely generated and the natural homomorphism P ! P gp
is injective.
(2) A monoid P is said to be fs if it is ne and enjoys the following property: for x 2 P gp , xr 2 P
for some r > 0 implies x 2 P .
(3) A log analytic space X is called ne (resp. fs) if it has locally a chart modeled on a ne (resp.
an fs) monoid; i.e., X has an open covering fUg23 (where every member is endowed with the
pull-back log structure) such that, for every 2 3, there exists a chart U ! (Spec C[P ])an
with P a ne (resp. an fs) monoid.
Note that, for a ne (resp. an fs) log analytic space X and a point x 2 X , the characteristic CX;x
at x is a torsion-free ne (resp. fs) monoid; this follows from the general fact that the properties ne
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 ne monoid and : M ! M=M 2 the natural projection, where M 2
denotes the submonoid of invertible elements in M . Assume M=M 2 is fs. Then there exists a cross
section s of by an exact homomorphism M=M 2 ! M . In particular, the monoid M is isomorphic
to (M=M 2 ) 8 M 2.
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 ne monoid.
As for morphisms, nding a good local chart of a morphism is a tough business even in the
fs category; we will discuss this point in xx3.1. If we give up the goodness, we can always nd
a local chart of any morphism of ne log analytic spaces; the construction is as follows: Given a
morphism f : X ! Y of ne log analytic spaces, we rst choose local charts X ! (Spec C[P ])an and
Y ! (Spec C[Q])an of X and Y modeled on ne monoids P and Q, respectively. We consider the
!
induced homomorphism P 8 Q ! MX . It is easy to see that the composite (P 8 Q)gp ! Mgp
2 , denoted by g, maps (P 8 Q)gp surjectively onto Mgp =O 2 . Due to [7, (2.10)] we seeXthat
=
O
Mgp
X
X
X X
H : = g01 (M=OX2 ) ! 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 8 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 ane 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 6 we can equip
the associated toric variety T6 with the unique log structure such that the log structure restricted
to each ane toric patch is the canonical log structure. Since every ane 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 dened by
M: = OX \ j3 OX2 nD ,0! OX ;
where j : X n 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 dened by z1 1 11 zl (l 0) where (z1 ; . . . ; zl ) is a part of local coordinate system
2 . If l = 1, elements
(z1 ; . . . ; zn) around a point x 2 X . In case l = 0 we immediately have Mx = OX;x
in Mx may divisible by z1 ; dividing an element in Mx by z1 as many times as possible, we shall
2 1 z N. In general we
eventually get an invertible element. We have therefore the equality Mx = OX;x
1
similarly have
2 1 zN 11 1 z N;
Mx = OX;x
1
l
and hence CX;x = Nl . We have moreover seen that xing the local parameter z1 ; . . . ; zl amounts
to take a cross section Nl ! Mx which can extend to a homomorphism Nl ! MjU of sheaves
of monoids on a suciently 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
2 1 zN 1 1 1 zN
by a smaller neighborhood of x if necessary, the monoid MX;y at any y 2 U equals to OX;x
i1
i
where fi1; . . . ; im g = fi j zi (y) = 0g; on the other hand, the j -th factor for zj (y ) 6= 0 in Nl is taken
over by OX2 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 MjU 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 T6 (with the canonical log structure) from a non-singular fan 6 is
nothing but the fs log analytic space associated to the pair (T6; D6), where D6 is the union of all
codimension 1 torus orbits.
m
2.2
Log smoothness
Denition 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 01OY ! OX ).
6
Denition 2.2.2 (cf. [7, (3.3)].) Log smoothness. Let f : X ! Y be a morphism of ne log analytic
spaces. Then f is said to be log smooth if the following condition is satised: (Innitesimal lifting
property) For any commutative diagram
0
T?0
s
0!
X
?
T
0!
Y
s
?
ty
?yf
of ne 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 ne log analytic spaces.
Lemma 2.2.3 Let f : X ! Y be a strict morphism of ne 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 ne (resp. fs) log analytic spaces and Y !
(Spec C[Q])an a chart of Y modeled on a ne (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 )i2I with Pi a ne (resp. torsion-free
fs) monoid for each i such that:
(1) Each member Ui ! (Spec C[Pi ])an extends to a chart of f jU modeled on a homomorphism
hi : Q ! Pi .
(2) The homomorphism hi gp is injective and Coker(hi gp )tor is a nite group.
(3) The induced strict morphism
i
Ui 0! Y 2(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, x6]) and [13, (A.2)]. The following fact is
an immediate implication from this theorem:
Corollary 2.2.5 A morphism '3 : T60 ! T6 of toric varieties (endowed with the canonical log structures as in Example 2.1.7) induced by a morphism of fan ': (N 0 ; 60 ) ! (N; 6) (cf. [15, x1.5]) is log
smooth if and only if the homomorphism ': N 0 ! N of abelian groups has nite cokernel. 2
Setting N = f0g 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 at; for example, the morphism of toric
varieties determined by subdivision of fan, like blow-up, is log smooth, even log etale (cf. [7, (3.3)]).
However, it is known that the underlying morphism of a log smooth and integral morphism is at
(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 at 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 01 (C n E ) is smooth,
7
(2) D = f 3 E is a reduced divisor with normal crossings.
The log structures on X and C are, by denition, 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 01 OC ! OX maps f 01 MC to MX .
To see this morphism is log smooth we take local charts of X and C . Take a point x 2 X and
let y = f (x). The morphism f is locally dened by z1 1 1 1 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 2(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 fUi gi2I of X and a reduced divisor D on X
such that
(1) there exists a strict and smooth morphism hi : Ui 0! (Spec C[Pi ])an to an ane toric variety
for each i 2 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 2 I ,
(3) the log structure MX is isomorphic to the log structure given by
OX \ j3OX2 nD ,0! OX ;
where j : X n 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, x7]. 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 modied 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 ts
in with the so-called manifolds with corners (cf. [14, (x10)] or [15, (x1.3)]).
Construction 2.3.1 ([9, x1]) 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 dened as follows:
0(T ; MT ) = R0 2 S1 0! C by (; ) 7! (here we set, once for all, S1 = f 2 C j j j = 1g); As a set, X log is the set of all morphisms T
of log analytic spaces.
8
!X
We endow X log with a reasonable topology as follows: The underlying analytic space of X has
locally a closed immersion into an ane 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 2 HomZ (P gp ; S1 ) in the
natural way. Then we can dene a topology locally on X log as a closed subset in Cn 2 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 X : 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 X : X log !X simply by X : 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 ane toric variety X = (Spec C[P ])an .
There is a splitting
X log = Hom(P; R0 ) 2 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 rst factor is
a manifold with corners which can be expressed more intrinsically (cf. [14, x10] or [15, x1.3]). Let 6
be a fan in N and T6 the associated toric variety. Then the above splitting can be globalized as
T6 log = Mc(N; 6) 2 (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; 6) has a big advantage over T6 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 ! 1 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 = f(z1 ; . . . ; zn) 2 Cn j jzi j < 1g and 1 = ft 2 C j jtj < 1g. The morphism f is
dened by t = z1 1 11 zl . Then X log is isomorphic as a topological space to
f(1 ; 1 ; . . . ; l ; l ; zl+1 ; . . . ; zn) j
0 i < 1; i 2 S1 (1 i l); jzj j < 1 (l + 1 j n)g
1n0l 2 [0; 1)l 2 (S1 )l :
Similarly, 1log is isomorphic to [0; 1) 2 S1 . Note that the map X : X log ! X maps
(1 ; 1; . . . ; l ; l ; zl+1; . . . ; zn) 7! (11 ; . . . ; l l ; zl+1 ; . . . ; zn)
and 1: 1log ! 1 maps (; ) to . The induced morphism f log : X log ! 1log is given by equations
1 1 1 1 l = and 1 1 11 l = :
Let us denote the ber of f log over (0; ) by Xlog . Let X; : Xlog ! X denote the restriction of X .
01 (x) is isomorphic to (S1 )l01 .
Then the inverse image of the origin of X by X;
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 ber
Xlog which is described by
f(1 ; 1 ; 2 ; 2 ) j 1 2 = 0; 0 i < 1; 1 2 = g:
9
This is the union of
(Xlog )+ = f(1 ; 1 ; 2 ; 2 ) 2 Xlog j 1 = 0g and (Xlog )0 = f(1; 1 ; 2 ; 2) 2 Xlog j 2 = 0g
both of which are cylinders with boundaries on one sides. Here we see that the vanishing cycle has
been replaced by f1 = 2 = 0g S1.
The phenomenon which is perhaps more interesting arises from the fact that the gluing of these
two cylinders depends on the parameter , i.e., the gluing is done by the relation 12 = . As the walks along S1 f(0; )g 1log , the gluing of (Xlog )+ and (Xlog )0 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 ber Xlog looks
like a nearby ber (see [16, Chap.II], where Xlog can be visually understood by gures at p.32 and
p.34 in n = 3 case), and it is twisted by moving along non-trivial paths on 101 (0). It is therefore
natural to ask whether the family f log : X log ! 1log arised from semistable degeneration is locally
topologically trivial; this has been armatively answered by S. Usui [19, (3.4)].
Construction 2.3.4 Returning to the general situation, we impose a sheaf OXlog of rings (not necessarily local) upon X log dened as follows ([9]): First we consider the following commutative diagram
of abelian sheaves on X log with exact rows:
p
exp
0 0! 2 01Z 0!
01 OX
0!
01 O 2
0! 1
X?
?
ay
X? X
?yb
k
p
p
exp Cont({; S1 ) 0! 1,
0 0! 2 01Z 0! Cont({; 01R) 0!
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 dened by
a(v) = v 0 Re(v ) and b(u) = juj01 u:
Dene
p
LX : = Cont({; 01R) 2Cont({;S1) X01 Mgp
X;
1
canonical morphism. The sheaf LX should be called the sheaf
where X01 Mgp
X ! Cont({; S ) is the
gp
0
1
of logarithms of sections in X MX because it sits in the following commutative diagram of abelian
sheaves with exact rows:
p
0! 2 01Z 0! X01?OX
?
k
hy
p
0 0! 2 01Z 0! LX
0
exp 01 O 2 0! 1
0!
X? X
?y\
exp 01 Mgp 0! 1.
0!
X
X
Now dene
OXlog = (X01OX Z SymZ LX )=a
where a is an ideal locally generated by local sections of form v 1 0 1 h(v) for all u 2 X01 OX . The
log at a point 2 X log is isomorphic to the polynomial ring over O
stalk OX;
X; () of n variables where
gp
n = rankZ (CX ) () ([9, (3.3)]); these variables correspond to logarithms of a Z-basis of (CXgp ) ( ) .
Remark 2.3.5 The topological space X log has been considered also in [10, x4], but the structure
X
X
X
sheaf is dierent.
10
3 Relative log Poincare lemma
3.1
Taking good chart
We have seen below Denition 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 sucient 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 2 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 : 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?
0! (Spec C? [P ])an
?
V
0! (Spec C[Q])an
?y
y
of f jU at x. If f is log smooth, the induced morphism
U
0! V 2(Spec C[Q])an (Spec C[P ])an
is strict and smooth.
For the proof we need to know some auxiliary facts:
Denition 3.1.2 ([7, (4.6)]) Exact homomorphism. A homomorphism f : Q ! P of monoids is called
exact if the induced homomorphism Q ! Qgp 2P gp P is an isomorphism.
Lemma 3.1.3 If f : Q ! P is a surjective homomorphism of ne monoids and Ker(f gp ) Q, then
f is exact.
Proof. Consider the natural homomorphism Q ! Qgp 2P gp P denoted by '. The injectivity of '
is clear. For the surjectivity let us take an element (; y) 2 Qgp 2P gp P . Write = u=v by u; v 2 Q.
Then we have f (u) = f (v )y . Take w 2 Q such that y = f (w). Set n = u=(vw ) 2 Ker(f gp ) Q.
Then we have = nw 2 Q and '( ) = (; y ). 2
Proposition 3.1.4 Let f : Q ! P be a surjective and exact homomorphism of ne monoids, and
assume that P gp is a nitely 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 2 P . Since b = f gp (s(b)) 2 N
and f is exact, we have s(b) 2 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 2 P gp , s(b) 2 Q implies
b = f (s(b)) 2 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 ne monoids. Assume P is fs
and torsion-free. Set K = Ker(f gp ). Then the submonoid Qe : = (f gp )01 (P ) of Qgp allows a splitting
Qe = P 8 K . More precisely, the homomorphism fe: Qe ! P induced by f gp has a cross section
s: P ! Qe which gives the splitting as above. 2
11
The proof is easy. We note that the monoid Qe is generated by Q and a nite number of elements
in Qgp since it is generated by Q and K which is a nitely generated abelian group.
In terms of toric varieties, Corollary 3.1.5 shows that, for a given equivariant closed immersion
Y ! X of ane 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 jU 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 satises p h0 = h. Write the homomorphism P 0 ! MX;x by
0. Set Pe = (pgp )01 (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
section s of Pe such that eh = s h, where eh: Q ! Pe is the composite of h0 : Q ! P 0 and P 0 ,! Pe .
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 0! (Spec C[P ])an
?
?
?
fy
Y
?y
0! (Spec C[Q])an
of f .
In case f is log smooth the chart of f jU modeled on h0 : Q ! P 0 which we have taken above should
be chosen as in Theorem 2.2.4 so that X ! Y 2(Spec C[Q])an (Spec C[P 0])an is smooth. We claim that
the morphism
X 0! Y 2(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 rst claim that the
induced morphism
(3.1.7)
X 0! Y 2(Spec C[Q])an (Spec C[Pe ])an
is smooth; indeed, since Pe is generated by P 0 and nite number of elements, we easily see that the
morphism
Y 2(Spec C[Q])an (Spec C[Pe ])an 0! Y 2(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 2(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 2(Spec C[Q])an (Spec C[Pe ])an 0! Y 2(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
Denition 3.2.1 (cf. [7, (1.7)], [3, 1.5].) Log dierentials. Let f : X ! Y be a morphism of ne log
analytic spaces. The sheaf of log dierentials of X over Y is an OX -module dened by
i
h
!1 = 1 8 (OX Z Mgp ) =K;
X=Y
X=Y
X
where 1X=Y denotes the usual sheaf of dierentials and K is the OX -submodule generated by
(d(a); 0) 0 (0; (a) a) and (0; 1 (b));
for all a 2 MX and b 2 f 01 MY .
1 should be written by dloga; it denes a natural additive
The sections of form [0; 1 a] of !X=Y
morphism
1
(3.2.2)
dlog: Mgp
X 0! !X=Y :
1 is a nite locally free OX -module; but we will not
It is known that, if f is log smooth, the !X=Y
; d) of
need this fact. Similarly to the usual case the sheaf of log dierentials yields a complex (!X=Y
gp
f 01 OY -modules by imposing d 1 dloga = 0 for a 2 MX .
:
The subject of this subsection is calculating the cohomologies of the complex !X=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
) Z ^ C gp
)
Hq (!X=Y
= H0 (!X=Y
X=Y
q
of f 01 OY -modules for all q .
Proof. Consider the morphism (3.2.2). Since its image consists of germs of closed forms, it induces
1 1 3 gp
Mgp
X ! H (!X=Y ). The image of f MY in H (!X=Y ) is obviously zero. Hence we have
gp gp
1 Mgp
X =MY = CX=Y 0! H (!X=Y ):
By cup-product we obtain
^ gp
):
f 01 OY Z CX=Y
0! Hq (!X=Y
We need to show that this morphism is an isomorphism. This can be seen stalkwise as follows: Let
x 2 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 0! (Spec C[P ])an
?
?
q
?
?y
fy
Y 0! (Spec C[Q])an
gp
of f at x. (Note that CX=Y;x = 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 : = CfQ; w1 ; . . . ; wsg 0! A: = CfP ; w1 ; . . . ; ws ; z1 ; . . . ; zr g;
13
where Cf{g means the analytic completion of C[{].
Now we remark that the ring A = CfP g for a fs monoid P has the following description: Choose a
surjective homomorphism Nr ! P . Then, in the formal completion C[[P ]], the ring CfP g coincides
with the image of the usual convergent power series CfX1 ; . . . ; Xr gP
= CfNr g under the induced
r
morphism CfN g ! C[[P ]] ofPC-algebras. In particular, for any p2P apX p 2 CfP g and any
subset P 0 P , the partial sum p2P 0 ap X p belongs to CfP g.
splits into the toric part and the usually
Therefore, returning to our situation, the complex !A=B
smooth part. Then by the usual Poincare lemma we may assume B = CfQg and A = CfP g without
loss of generality. In this case we have
^q
q
!A=B
= CfP g Z (P gp =Qgp )
^
= CfP g C (VP =VQ );
q
where we set VP = C Z P gp , etc. Let us denote the monomial in CfP g corresponding to p 2 P
1 , i.e., the image of 1 p under
by X p . Let ep be the image of p 2 P under dlog: P ! !A=B
1 . Then the exterior derivative d: !q ! ! q+1 is given by
VP ! VP =VQ ! !A=B
A=B
A=B
0
1
X
X p
d @ ap X p ep1 ^ 1 11 ^ ep A =
ap X ep ^ ep1 ^ 1 11 ^ ep :
q
p2P
q
p2P
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 2 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 ; . . . ; vi01 ; vi+1 ; . . . ; vr for 1 i r . Set Pi = Pei \ P . Then we obviously
T
have 1ir Pi = Qe .
We choose an injective homomorphism : 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 fi g and the natural
1 ! A, i.e., a log derivation, which
projection P gp =Qgp ! P gp =Pei induce the A-linear morphisms !A=B
that the interior
we also write by @i . Then it is not dicult to see from the above description of !A=B
3 , and that i : = d@i + @i d is just the
multiplication by @i induces a derivation of degree 01 on 8!A=B
multiplication by @i (p) 2 R in degree p 2 P . Dene for any subset S f1; . . . ; rg the subcomplex
by
CS !A=B
q
j i! = 0 for all i 2 S g;
CSq : = f! 2 !A=B
which is actually a subcomplex since each i commutes with d. Note that we obviously have CSq =
V
V
T
q
q
e
Cf j 2S Pj g C (VP =VQ ); in particular, we have Cf1;...;rg = CfQg C (VP =VQ ). What we need
. To do this, we claim that any C ,! !
to prove is the quasi-isomorphy of Cf1;...;rg ,! !A=B
S
A=B
is quasi-isomorphism; which we will prove by induction with respect to card(S ). Dene the second
complex ZS by the exact sequence of complexes:
0 0! C 0! ! 0! Z 0! 0:
S
A=B
S
Each element of ZSq is uniquely represented by a nite sum of formal power series of form
X
p2P n\i2S Pi
ap X p ep1 ^ 11 1 ^ ep ;
q
14
q . Hence it follows that the
and due to what we remarked above, this representation belongs to !A=B
exact sequence splits, i.e., we have !A=B
= CS 8 ZS . In case S = fj g, we have a homotopy operator
P
01 on the complex Z . Hence we have shown the claim in the case card(S ) = 1.
p2P nP (@ (p)) @i `
S
For general S = S 0 f@j g we only need to show that CS ,! CS0 is a quasi-isomorphism. But we can
prove it similarly as above. we therefore conclude
i
^
)
Hq (!A=B
= CfQe g C (VP =VQ):
)
In case q = 0 we get H0 (!A=B
= CfQe g which implies the lemma. 2
Denition 3.2.4 ([7, (4.6)]) Exact morphism. A morphism f : X ! Y of ne log analytic spaces is
said to be exact if the induced homomorphism (f 3 MY )x ! MX;x is exact (Denition 3.1.2) for any
x 2 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 2 X . Semistable degeneration is, for example, exact since the
diagonal homomorphism N ! Nl is obviously exact (cf. Example 2.2.6).
q
The next proposition is a by-product of the above proof:
) = f 01 OY
Proposition 3.2.5 Let f : X ! Y be as in Proposition 3.2.3. Then the equality H0 (!X=Y
holds if and only if f is exact.
) = f 01 OY is equivalent to Q = Qe for any point x 2 X , which is
Proof. The equality H0 (!X=Y
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 X : X log ! X the real blow-up (2.3.1). For
x 2 X we have a canonical homeomorphism
01 (x) HomZ (C gp ; S1 ):
X;x
X
Proof. By denition, X01 (x) is the set of monoid homomorphisms MX;x ! R0 2 S1 which make
the following diagram commutative
(3.3.2)
OxX;x 0!
X;x
??
C
x
??
T;
0
MX;x 0! R0 2 S1,
where the morphism in the rst row is the residue map at x. The morphism OX;x ! C induces
2 ! R>0 2 S1 . Hence, given such a morphism MX;x ! R0 2 S1 ,
the group homomorphism OX;x
we get a monoid homomorphism CX;x ! (R0 2 S1 )=(R>0 2 S1 ) = S1 . Taking the Grothendieck
gp ! S1 .
gp
1
group we have CX;x ! S . Conversely, suppose we are given a group homomorphism CX;x
Set P = CX;x, which is a torsion-free fs monoid. By characteristic splitting lemma (Lemma 2.1.6)
2 . Then we get a monoid homomorphism
we know that there exists a splitting MX;x = P 8 OX;x
MX;x ! (f0g 2 S1 ) 8 (R>0 2 S1 ) = R0 2 S1 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 X : X log
real blow-up. We dene the topological space XYlog by
X log : = X 2Y Y log :
! X be the
Y
We x notation by the following commutative diagram of topological spaces once for all:
X?log
?
f log y
X
0! X?Ylog 0!
??
?
log
X= Y
XY
fY
Y log
=
y
Y log
yf
0!
Y;
Y
with the square in the right hand side Cartesian, so that X = X=Y X .
Y
Lemma 3.3.4 Let x0 2 XYlog and set x = X (x0). Then there exists a canonical homeomorphism
01 (x0 ) HomZ (C gp ; S1 ):
Y
X=Y
X=Y;x
Proof. Let y = f (x). We rst note the obvious equality
01 (x0 ) f 2 01 (x) j f log ( ) = f log (x0 )g:
Y
X
X=Y
By Lemma 3.3.1 we have
gp ; S1 ) and 01 (y ) HomZ (C gp ; S1 ):
X01 (x) HomZ (CX;x
Y;y
Y
The morphism f log : X01 (x) ! Y01 (y ) commutes with the group homomorphism
HomZ (C gp ; S1 ) ! HomZ (C gp ; S1 )
X;x
Y;y
gp ! C gp . Then 01 (x0) is obviously homeomorphic to the kernel of this group
induced by CY;y
X;x
X=Y
gp ! C gp ); S1). 2
homomorphism, and hence is homeomorphic to HomZ (Coker(CY;y
X;x
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
#
"^q
gp
0
1
q
0
1
X CX=Y
R X=Y 3 X=Y F = F (0q ) Z
Y
p
of abelian sheaves on XYlog for all q, where F (p) = F Z (2 01)p Z.
Proof. The case q = 0 is trivial due to Lemma 3.3.4; i.e., we have
01 F X=Y 3 X=Y
(3.3.6)
=F
For the general case we consider the exact sequence constructed in (2.3.4):
p
0 0! 2 01Z 0! L 0! 01 Mgp 0! 1:
(3.3.7)
Y
Y
Y
We rst claim that this exact sequence induces exact sequences
p
(3.3.8)
0 0! 2 01Z 0! (f log )01 LY 8(f log )01 01 O X01 OX 0! X01 f 3 Mgp
Y 0! 1
Y
16
Y
and
(3.3.9)
0 0! 2
p
01Z 0! (fYlog)01LY 8(f log )01 01O X01 OX 0! X01 f 3Mgp
Y 0! 1
Y
Y
Y
Y
Y
of abelian sheaves on X log and XYlog , respectively. Indeed, pulling back the sequence (3.3.7) by f log
we obtain the exact sequence
p
(3.3.10)
0 0! 2 01Z 0! (f log )01 LY 0! X01 f 01 Mgp
Y 0! 1
which sits in the following commutative diagram with exact rows:
p
exp 01 f 01 O 2 0! 1
0 0! 2 01Z 0! X01 f 01 OY 0!
X ? Y
??
?y
(3.3.11)
k
y
p
exp 01 f 01 Mgp 0! 1.
0 0! 2 01Z 0! (f log )01 LY 0!
Y
X
We further consider the following commutative diagram with exact rows:
p
exp 01 f 01 O 2 0! 1
0 0! 2 01Z 0! X01 f 01 OY 0!
X ? Y
??
?y
(3.3.12)
k
y
p
exp 01 O2 0! 1
0 0! 2 01Z 0! X01 OX 0!
X X
log
We take push-out, in the category of abelian sheaves on X , of (3.3.11) and (3.3.12) termwise; since
f 01 O2 0! O2
??
y
??X
y
Y
f 01 MY gp 0! f 3 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:
p
0 0! 2 01Z 0! (f log )01 LY 8O X01 OX 0! X01 f 3 Mgp
Y 0! 1
??
y
k
p
0 0! 2 01Z 0!
??
y
Y
0! X01Mgp
0! 1.
X
LX
By this and applying X=Y 3 we obtain a commutative diagram
p
01 f 3Mgp 0! R1 X=Y 3 2 01Z
(3.3.13)
??
y
XY
Y
k
0! R1 X=Y 3 2p01Z.
X01 Mgp
X
(Here we have used (3.3.6).) Applying the right-exact functor X=Y 3 to (3.3.8) we get nothing but
the exact sequence (3.3.9). Hence we deduce that the morphism in the rst row is zero. We therefore
get
p
gp 0! R1
X01 CX=Y
X=Y 32 01Z:
Taking cup-product we obtain a canonical morphism
Y
Y
(3.3.14)
^q 01 gp
01 F
F (0q) Z X CX=Y 0! Rq X=Y 3 X=Y
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 Poincare lemma. The essential
object is the following; the construction is similar to [9, (3.5)]:
Construction 3.4.1 (cf. [9, (3.5)].) Dene
q;log : = Olog 01 q
!X=Y
X 01 O X !X=Y
for all q. By the construction of OXlog in Construction 2.3.4, there exists a natural arrow d: OXlog !
log 01 !1 . We can easily see that this d extends naturally to
!1;log extending LX exp
! 01Mgp d!
X
X=Y
X
X
X
X
X=Y
q;log 0! ! q+1;log
d: !X=Y
X=Y
3;log into a complex (!;log ; d) of (f log )01 Olog -modules.
for all q which make !X=Y
Y
X=Y
Now we state the main theorem:
Theorem 3.4.2 (Relative logarithmic Poincare 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 01 CY ! CX is injective and, for every x 2 X , the relative characteristic CX=Y;x
is torsion-free. Then the natural morphism
) 0! !;log
(f log )01 OYlog 01 f 01 O X01 H0(!X=Y
X=Y
is an isomorphism in D+ (X log ; (f log )01 O log ).
Y
X
Y
Proof. (The idea of the following proof is similar to that in [9, (4.7)].) We will work stalkwise. Let
2 X log and set x = X ( ), = f log ( ) 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
2
2
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
j =1
aij S j ; aij 2 Z
(1 i s) dening the inclusion Qgp ,! P gp (here we write the monoid structures of P and Q
log log additively). We have OY;
= OX;x [S 1 ; . . . ; S r ] (cf. (2.3.4)). Moreover,
= OY;y [T 1 ; . . . ; T s] and OX;
log
log
the morphism OY; ! OX; is isomorphic to that induced by (3.4.3). Let us denote OY;y simply by
O. Set
R = O[T 1 ; . . . ; T s][S 1; . . . ; S r ]=(relations (3.4.3))
(3.4.4)
log to be the obvious O[T 1 ; . . . ; T s]-algebra morphism. Let us abbreviate O [T ] =
and u: R ! OX;
O[T 1 ; . . . ; T s]. Note that, since (3.4.3) is a linear relation, O[T ] ! R is a smooth homomorphism of
;log of complexes given by
rings. Then there exists a canonical morphism R=O[T ] ! !X=Y;
7! u(h)dlogsi1 ^ 11 1 ^ dlogsi ;
1
where dlogsi denotes the image of Si under dlog: Mgp
X;x ! !X=Y;x . Since O[T ] ! R=O[T ] is quasih 1 dS i1 ^ 11 1 ^ dS i
p
p
isomorphism by the classical relative Poincare lemma, it suces to show that the morphism
)x 0! !;log
R=O[T ] O H0 (!X=Y
X=Y;
18
is a quasi-isomorphism.
;log by the degree of coecient polyWe introduce increasing ltrations both on R=O[T ] and !X=Y;
nomials; this is possible since (3.4.3) is linear. Then it is easy to see that
;log :
Grq !X=Y;
= SymqZ P gp Z !X=Y;x
On the other hand, we easily nd
Hp (Grq q gp
R=O[T ] ) = SymZ P Z
^ gp gp
(P =Q ) Z O:
Hence by Proposition 3.2.3 we see that
)x 0! Grq ! ;log
Grq R=O[T ] O H0 (!X=Y
X=Y;
is a quasi-isomorphism for each q. This implies our theorem. 2
4 Applications
4.1
Integral structure of VMHS
In this subsection, applying the relative log Poincare 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
log 01 log log 01 log
log
X01 !X=Y
(f )01 ( )01 O (fY ) OY 0! RX=Y 3 (f ) OY
in D+(X log ; (f log )01 O log ).
Y
Y
Y
Y
Y
Y
Y
Proof. Due to Proposition 3.3.5 and Proposition 3.2.3, there exists a canonical isomorphism
(4.1.2)
Hq ( 01 ! ) = Rq X=Y 3 (f log)01(Y )01OY
XY X=Y
for all q . We shall construct a morphism
log 01 log
log 01 log
log
X01 !X=Y
(f )01 ( )01 O (fY ) OY 0! RX=Y 3 (f ) OY
of complexes which yields the isomorphisms (4.1.2) tensored by (fYlog )01 OYlog .
By Theorem 3.4.2 and Proposition 3.2.5 there exists a morphism of complexes
;log 0! R
log 01 log
(4.1.3)
X=Y 3!X=Y
X=Y 3(f ) OY :
Y
Y
Y
Y
On the other hand, we have a natural morphism
(4.1.4)
01 ! O (f log )01 O log 0! X=Y 3!;log :
XY X=Y
Y
Y
Y
X=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
Rf log Z O log
01 R f3 ! 01 O log 0!
Z
Y
X=Y
Y
O
Y
3
Y
Y
in D+(Y log ; OYlog ).
Proof. We apply R fYlog3 to both sides in the quasi-isomorphism in Lemma 4.1.1. On the left hand
side we have
) O Olog
O (f log )01 Olog ) R fYlog3 (X01 !X=Y
= R fYlog3 (X01 !X=Y
Y
Y
Y
= 01 R f3 ! O Olog;
Y
Y
Y
Y
Y
X=Y
Y
Y
where the rst = is due to Lemma 4.1.6 below (note that OYlog is at 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 nd
R fYlog3 RX=Y 3 (f log )01 OYlog = Rf3log (f log )01 OYlog
= Rf3logZ Z OYlog;
where the second = is again due to Lemma 4.1.6 (Olog is at over Z since it has no torsion). 2
Y
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 01 R-module and G an R-module. Then, if f is proper and G is at over R, the
natural morphism
Rf3 F R G 0! Rf3(F f 01 R f 3 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 = 1, the unit disk, with
the log structure associated to the pair (1; 0) (cf. Example 2.1.8). We dene the fs log analytic
space by = (Spec C)an together with the pull-back log structure of M1 by the closed immersion
0 = (Spec C)an ,! 1. The log structure M is given by the chart
N 0! 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 13 : = 1 n 0, the punctured disk, with the trivial log structure so that 13 ,! 1
is strict. Let us x notation by the following commutative diagram with all squares are Cartesian in
the category of log analytic spaces:
Xe?3
0! X?3 ,0! X?
?
?
?
fy
f 3y
e 3 0! 13 ,0! 1
1
fe3 y
0- X?
?yf
0- ,
e 3 is the universal covering space of 13. The left square is the diagram of log analytic spaces
where 1
with trivial log structure. The underlying morphism of f: X ! is nothing but the central ber
of f .
We begin with the topological argument. Since f log : X log ! 1log is locally a topological trivial
family (Theorem 4.2.1), the Betti cohomology groups of each ber are isomorphic to those of Xe 3 ,
the topological generic ber; in particular, for each 2 log ( S1 ) we have
H3 (X log ; Z) = H3(Xe 3 ; Z);
where Xlog = (f log )01 ( ). Moreover, we can introduce, by the topological triviality, the well-dened
monodromy operation: For any s 2 1log the fundamental group 1 (1log ) = Z acts on the cohomology
H3 (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 Z Z Olog
01 Hq (X ; !X = ) C Olog = Rq f3
of Olog -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 0(log ; Olog ) = C, we
have
Hq (X ; !X = ) (4.2.4)
= 0(log ; Rq flog 3 Z Z Olog ):
In order to describe elements in the right hand side we x a parameter t on 1. Fixing the parameter
amounts to taking a chart N ! O1 of the log structure M1 on 1 given by a 7! ta (cf. Example
2.1.8) so that we have the splitting Z 8 C2 = M gp . Considering the exact sequence
p
exp 01 M gp 0! 1
0 0! 2 01Z 0! L 0!
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) 2 Z 8 C2; the section u is conceptually a logarithm of the parameter
t restricted to log . For 2 log we take a germ u of a branch of u near . Then elements in
log Z Olog ) are polynomials of u with coecients in Hq (X log ; C). By the local topological
(Rq f3
Z
log C on log .
triviality, elements in Hq (Xlog ; C) extend uniquely to multivalued global sections of Rq f3
P
log Z Z Olog ) , where xk 2 Hm (X log ; C). Then x comes from a global
Let x = rk=0 xk uk 2 (Rq f3
log
log
q
log
section in 0( ; R f3 Z Z O ) if and only if
r
X
k=0
T xk (u + 2
p
01)k =
21
r
X
k=0
xk uk ;
for each i. This is equivalent to
T xk =
(4.2.5)
r
X
j =k
(01)j 0k
!
p
j
(2 01)j 0k xj for k = 0; . . . ; r:
k
Denote the monodromy logarithm logT by N . Then it is easily seen that (4.2.5) is equivalent to
xk = (02
(4.2.6)
p
01N )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)
Hq (X ; !X = ) = Hq (Xlog ; C):
The last cohomology group is isomorphic to Hq (Xe 3 ; C). Hence we get the following theorem which
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
Hq (X ; !X = ) = Hq (Xe 3 ; C)
for all q .
2
References
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[15] Oda, T.: Convex bodies and Algebraic Geometry, An introduction to the theory of toric varieties. Ergeb.
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Fumiharu Kato
Graduate School of mathematics
Kyusyu University 33
Fukuoka 812, Japan
23