[title=Index of definitions and notations,intoc]

# Betti numbers of real semistable degenerations via real logarithmic geometry

Emiliano Ambrosi and Matilde Manzaroli

###### Abstract.

Let $X\rightarrow C$ be a totally real semistable degeneration over a smooth
real curve $C$ with degenerate fiber $X_{0}$. Assuming that the irreducible
components of $X_{0}$ are simple from a cohomological point of view, we give a
bound for the individual Betti numbers of a real smooth fiber near $0$ in
terms of the complex geometry of the degeneration. This generalizes previous
work of Renaudineau-Shaw, obtained via combinatorial techniques, for tropical
degenerations of hypersurfaces in smooth toric varieties. The main new
ingredient is the use of real logarithmic geometry, which allows to work with
not necessarily toric degenerations.

###### Contents

  1. 1 Introduction
  2. 2 Notation and conventions
  3. 3 Log-varieties and analytification
  4. 4 Real log varieties and equivariant cohomology
  5. 5 Proof of Theorem 3.4.2(1)
  6. 6 Proof of Theorem 3.4.2(2)

##  1\. Introduction

Let $X$ be a real algebraic variety, let $X(\mathbb{C})$ be the set of its
complex points and $X(\mathbb{R})$ the set of its real points. For a
topological space $Y$, set
$b_{i}(Y):=\textrm{dim}_{\mathbb{F}_{2}}(H^{i}(Y,\mathbb{Z}/2\mathbb{Z}))$ for
its $i^{th}$ Betti number.

###  1.1. Individual Betti numbers

The Smith-Thom inequality

(1.1.1) $\sum_{i}b_{i}(X(\mathbb{R}))\leq\sum_{i}b_{i}(X(\mathbb{C}))$

bounds the total Betti number of the real points of $X$ with the total Betti
number of its complexification. A variety is said to be maximal, in the sense
of Smith-Thom, if (1.1.1) is an equality. The problem of finding (non trivial)
bounds for the individual Betti numbers $b_{i}(X(\mathbb{R}))$ in terms of the
geometry of $X(\mathbb{C})$ is a central topic in real algebraic geometry. For
example, it was conjectured by Viro in [Vir80] that if $X$ is a smooth
projective real surface such that $X(\mathbb{C})$ is simply connected then
$b_{1}(X(\mathbb{R}))\leq\textrm{dim}(H^{1}(X,\Omega^{1}_{X}))$. Even if the
conjecture has been disproved by Itenberg in [Ite93, Ite97], many smooth
projective real algebraic varieties constructed with the tools at our disposal
(for example, using Viro’s patchworking [Vir83, Vir84]) verify the inequality

(1.1.2) $b_{i}(X(\mathbb{R}))\leq\sum_{j}h^{i,j}(X)$

where $h^{i,j}(X):=\dim(H^{i}(X,\Omega^{j}_{X}))$. Following [BS22], if a
smooth projective real variety has no torsion in cohomology and satisfies the
inequality (1.1.2), we say that it is sub-Hodge expressive.

The main goal of this paper is to explain this phenomenon in some cases.
Roughly speaking, the general principle is that, often, the constructions of
real varieties with prescribed topology are achieved first by constructing a
degenerated version of the pair $(X(\mathbb{C}),X(\mathbb{R}))$, whose
irreducible components are simpler to deal with and then by deforming them
back in non-trivial ways. Our main results (Theorem 1.3.1, Corollary 1.3.2),
shows that if these pieces are very simple, from a cohomological point of
view, then the variety obtained gluing these pieces satisfies the inequality
(1.1.2), up to some torsion. The main novelty of our approach is to use (real)
logarithmic geometry, which allows to extend previous results for toric
degenerations to more general, non necessarily toric, families.

###  1.2. The case of tropically smooth hypersurfaces

####  1.2.1. Renaudineau-Shaw theorem

The most general result previously known in this setting is [RS22, Theorem
1.4], which proves a conjecture of Itenberg ([Ite17]) and whose proof involves
real and complex combinatorial algebraic geometry arguments. Renaudineau and
Shaw show that for a real compact hypersurface $X$ near the
$\mathbb{Q}$-regular smooth tropical limit inside a smooth toric variety (see
[IKMZ19] for the definitions involved), one has the following inequality
([RS22, Theorem 1.5]):

$b_{i}(X_{t}(\mathbb{R}))\leq\sum_{j}\dim(H_{i,j}(Trop(X),\mathbb{Z}/2\mathbb{Z})),$

where $H_{i,j}(Trop(X),\mathbb{Z}/2\mathbb{Z})$ is the $(i,j)$ tropical
homology group with $\mathbb{Z}/2\mathbb{Z}$-coefficients (which is
independent from the real structure of the family) and $X_{t}(\mathbb{R})$ are
the real points of a smooth fiber near the tropical limit. Moreover, by
[ARS21], one knows that the variant $H_{i,j}(Trop(X),\mathbb{Z})$ of tropical
homology with integer coefficients is torsion free, while by [IKMZ19] one
knows that its rank is the $(i,j)$ Hodge number of the general fiber $X_{t}$
of the family. Summing up, one gets the sub-Hodge expressive inequality
([RS22, Theorem 1.4]):

$b_{i}(X_{t}(\mathbb{R}))\leq\sum_{j}\dim(H^{i}(X_{t},\Omega^{j})).$

In this paper we generalize this result to more general, non necessarily
toric, families of varieties, avoiding the use of combinatorial arguments and
constructing, via real logarithmic geometry, a space which allows to
simultaneously relate real and complex data.

######  Remark 1.2.1.1.

Very recently, there have been other two generalizations of the results and
the techniques in [RS22]:

  * •

On the combinatorial side, in [BdMR22], Brugallé, López de Medrano and Rau
extend the main results of [RS22] to $PL$-manifolds associated to non-
necessarily convex triangulations of non-singular lattice polytopes.

  * •

On the geometric side, in a work in progress of Renaudineau, Shaw and Rau, the
authors generalize [RS22, Theorem 1.5] and the strategy therein to general
families with smooth tropicalisation near the tropical limit inside a smooth
toric variety.

####  1.2.2. Geometry of tropicalisation

In order to put our main results (Theorem 1.3.1, Corollary 1.3.2) and their
hypothesis into context, we present an overview of the algebraic geometry
behind the tropicalisation. In the setting of Section 1.2.1, one can construct
from $Trop(X)$ a toric degeneration $T\rightarrow C$ of the ambient toric
variety and an associated one dimensional toric degeneration $X\rightarrow C$
of $X$, whose intersection with every toric orbit of $T$ is the complement of
an hyperplane arrangement endowed with the standard complex conjugation.
Complements of hyperplane arrangements endowed with the standard complex
conjugation have tight properties, both from the complex and the real point of
view:

  1. (a)

the only non trivial cohomology group of their real points is the $0$-th;

  2. (b)

they are maximal varieties, in the sense of (1.1.1);

  3. (c)

the mixed Hodge structure on the $i^{th}$-cohomology group with rational
coefficients is pure of type $(i,i)$.

In the proof of the results of [IKMZ19], property (c) is used explicitly to
give a geometric interpretation to the combinatorial complex computing
$H_{i,j}(Trop(X),\mathbb{Q})$ (see [IKMZ19, Section 5.3]), while in [RS22]
properties (a) and (b) are used implicitly to compare the dimensions of
different combinatorial vector spaces associated to $Trop(X)$; see [RS22,
Corollary 3.15].

###  1.3. Main result

Assume that $C$ is a smooth real curve and $f:X\rightarrow C$ a real
projective morphism which is smooth outside a real point $0\in C(\mathbb{R})$
and strictly-semistable around $0$, in the sense that the irreducible
components of $X_{0}$ are smooth and, locally analytically around $0$, the
family $f:X(\mathbb{C})\rightarrow C(\mathbb{C})$ is isomorphic to the
standard semistable degeneration
$\mathrm{Spec}(\mathbb{C}[x_{1},\dots,x_{n},T]/(x_{1}\dots
x_{n}-T))\rightarrow\mathrm{Spec}(\mathbb{C}[T]).$ Assume furthermore that
$f:X\rightarrow C$ is totally real, i.e. that the irreducible components of
$X_{0}(\mathbb{C})$ are real. Write

$X_{0}=\bigcup_{i\in I}X_{i}$

for the decomposition of $X_{0}$ in irreducible components and for every
subset $J\subseteq I$ set

$X_{J}:=\bigcap_{i\in J}X_{i}\quad\text{and}\quad
X^{0}_{J}:=X_{J}\setminus\bigcup_{i\not\in J}X_{i}.$

Then

$X_{0}=\coprod_{J\subseteq I}X^{0}_{J}$

is a stratification $\mathfrak{I}:=\\{X^{0}_{J}\\}_{J\subseteq I}$ of $X_{0}$
by smooth real algebraic subvarieties. Fix a refinement
$\mathfrak{Z}:=\\{X^{0}_{\Delta}\\}$ of $\mathfrak{I}$, made of smooth real
algebraic varieties.

In Section 3.4 we construct, for every ring $A$ and every $q\geq 0$ a
canonical cochain complex $C_{q,\mathfrak{Z},A}^{\bullet}$ of $A$-modules
depending only on the complex geometry of the stratification $\mathfrak{Z}$.
This complex plays, in this general setting, the role of the combinatorial
complex computing $H_{i,j}(Trop(X),A)$ used in [RS22, ARS21, IKMZ19]. Inspired
by the discussion in Section 1.2.2, we consider the following condition on the
members of $\mathfrak{Z}$.

  1. (a)

$H^{i}(X^{0}_{\Delta}(\mathbb{R}),\mathbb{Z}/2\mathbb{Z})=0$ for all $i\geq 1$
and $X^{0}_{\Delta}\in\mathfrak{Z}$;

  2. (b)

$X^{0}_{\Delta}$ is a maximal variety, for all
$X^{0}_{\Delta}\in\mathfrak{Z}$;

  3. (c)

the mixed Hodge structure on $H^{i}(X^{0}_{\Delta},\mathbb{Q})$ is pure of
type $(i,i)$ and $H^{i}(X^{0}_{\Delta},\mathbb{Z})$ is torsion free, for all
$i\geq 1$ and $X^{0}_{\Delta}\in\mathfrak{Z}$.

Our main result is the following.

######  Theorem 1.3.1.

  1.   2. (1)

Assume that (a) and (b) hold. Then, for every $t\in C(\mathbb{R})$ close to
$0$ one has:

$b_{p}(X_{t}(\mathbb{R}))\leq\sum_{q}\textrm{dim}(H^{p}(C_{q,\mathfrak{Z},\mathbb{Z}/2\mathbb{Z}}^{\bullet})).$

  3. (2)

Assume that (a),(b) and (c) hold. Then for every $t\in C(\mathbb{R})$ close to
$0$, one has:

    1. (i)

$\textrm{dim}(H^{p}(C_{q,\mathfrak{Z},\mathbb{Z}}^{\bullet}\otimes\mathbb{Q}))=h^{p,q}(X_{t})$

    2. (ii)

$C_{q,\mathfrak{Z},\mathbb{Z}}^{\bullet}\otimes\mathbb{Z}/2\mathbb{Z}\simeq
C_{q,\mathfrak{Z},\mathbb{Z}/2\mathbb{Z}}^{\bullet}$

Theorem 1.3.1 directly implies the following corollary, which was the main
motivation for this paper.

######  Corollary 1.3.2.

Assume that (a), (b), (c) hold and that
$H^{p}(C_{q,\mathfrak{Z},A}^{\bullet})$ is torsion free for every
$p,q\in\mathbb{N}$. Then for every $t\in C(\mathbb{R})$ close to $0$ and every
$p\in\mathbb{N}$ one has

$b_{p}(X_{t}(\mathbb{R}))\leq\sum_{q}h^{p,q}(X_{t}).$

######  Remark 1.3.3.

We point out that even in very easy examples is necessary to refine the
standard stratification $\\{X^{0}_{J}\\}$ in order to guarantee that the
strata satisfy the hypothesis in Theorem 1.3.1. If ones take the trivial
degeneration of $\mathbb{P}^{1}$, the stratification $\\{X^{0}_{J}\\}$ is the
trivial one and hence it does not satisfy the hypothesis. So, one has to
further stratify $\mathbb{P}^{1}$, for example as

$\mathbb{P}^{1}=\Big{(}\mathbb{P}^{1}-\Big{(}\\{[0,1]\\}\coprod\\{[1,0]\\}\Big{)}\Big{)}\coprod\\{[0,1]\\}\coprod\\{[1,0]\\},$

which corresponds to the stratification of $\mathbb{P}^{1}$ into toric orbits.

In the case of degenerations constructed from a smooth $\mathbb{Q}$-regular
tropicalisation inside a smooth toric variety, the stratification
$\mathfrak{Z}:=\\{X^{0}_{\Delta}\\}$ is the stratification obtained
intersecting the special fiber with the toric orbits of the ambient toric
variety, hence we see that our results generalise those in [RS22] to non
necessarily toric degenerations.

Under the hypothesis of Theorem 1.3.1, in order to understand the link between
the Hodge numbers and the real Betti numbers, it remains to understand the
torsion in the cohomology of $C_{p,\mathfrak{Z},\mathbb{Z}}^{\bullet}$. While
torsion appears in general, this raises the following question.

######  Question 1.3.4.

Let $X$ be a smooth complete intersection having smooth $\mathbb{Q}$-regular
tropicalisation inside a smooth toric variety. Is the cohomology of
$C_{p,\mathfrak{Z},\mathbb{Z}}^{\bullet}$ torsion free?

###  1.4. Strategy

####  1.4.1. Main ideas

In order to attack this kind of problems, the general involved strategy was to
separately compute complex and real information and, in a second moment, show
that they are related. For example, in [IKMZ19] it is shown that tropical
homology is related to Hodge numbers, in [RS22] it is first shown that the
tropical interpretation of the Viro patchworking method ([Vir83, Vir84]) gives
a way to compute the real Betti numbers via the (co)homology of a tropical
sheaf on the tropical varieties ([RS22, Theorem 3.7 and Remark 3.8]) and then
it is shown how to relate the (co)homology of this sheaf to tropical homology
([RS22, Section 4]).

A different approach to similar problems has been recently proposed by
Brugallé in [Bru22]. There, complex and real invariants are shown to satisfies
the same gluing relations under totally real semistable degenerations, so
that, in order to relate the global invariants, it is enough to relate the
local invariants, which are easier to compute. Inspired by this, our basic
strategy is to relate the real Betti and the Hodge numbers via the geometry of
a common ambient space. The main innovation of this paper is the use of real
logarithmic geometry to construct and study this common ambient space, which
allows to use a more sophisticated and less combinatorial machinery. After the
construction of such a space, realised in Sections 3.2.2 and 4.1.2, the
cohomology of the general real fiber can be computed by a filtered complex;
see Section 5.4. The idea of the use of filtered complexes is inspired by
[RS22], where it was constructed via combinatorial techniques. Since these
combinatorial approach are not available in our general setting, we use a
different approach based on equivariant cohomology.

######  Remark 1.4.1.

Observe that [Bru22, Proposition 2.1] applies under the hypothesis of Theorem
1.3.1 and implies that, for all $t\in C(\mathbb{R})$ close to $0$, one has
$\chi=\sigma$, where $\chi$ is the Euler characteristic of $X_{t}(\mathbb{R})$
and $\sigma$ is the signature of $X_{t}(\mathbb{C})$.

####  1.4.2. Equivariant cohomology

To explain our strategy in more details, let us recall a modern proof of the
Smith-Thom inequality (1.1.1), as in [BBF+60, Chapter 4, IV, Pag. 55]. Let $G$
be $\mathbb{Z}/2\mathbb{Z}$ acting via the complex conjugation on $X$. Then
there is a spectral sequence

$E_{2}^{a,b}:=H^{a}(G,H^{b}(X,\mathbb{Z}/2\mathbb{Z}))\Rightarrow
H^{a+b}_{G}(X,\mathbb{Z}/2\mathbb{Z})$

where $H^{a}(G,H^{b}(X,\mathbb{Z}/2\mathbb{Z}))$ is the group cohomology of
$G$ acting on $H^{b}(X,\mathbb{Z}/2\mathbb{Z})$ and
$H^{a+b}_{G}(X,\mathbb{Z}/2\mathbb{Z})$ is the $G$-equivariant cohomology of
$X$. Since $H^{a+b}_{G}(X,\mathbb{Z}/2\mathbb{Z})\simeq
H^{*}(X(\mathbb{R}),\mathbb{Z}/2\mathbb{Z})$ for $a+b>2\textrm{dim}(X)$, for
every $m>2\textrm{dim}(X)$ one gets a filtration ${}_{m}F^{i}$ of
$H^{*}(X(\mathbb{R}),\mathbb{Z}/2\mathbb{Z})$ such that
${}_{m}F^{i}/_{m}F^{i+1}$ is a subquotient of
$H^{i}(G,H^{m-i}(X,\mathbb{Z}/2\mathbb{Z}))$. Since
$\textrm{dim}(H^{a}(G,H^{b}(X,\mathbb{Z}/2\mathbb{Z})))\leq\textrm{dim}(H^{b}(X,\mathbb{Z}/2\mathbb{Z}))$,
one gets the desired inequality. This argument shows also that $X$ is maximal
if and only if the spectral sequence degenerates at $E_{2}$ and the action of
$G$ on $H^{*}(X,\mathbb{Z}/2\mathbb{Z})$ is trivial. Since we assume
maximality for every stratum of our degeneration, we could try to apply this
argument to each stratum and then glue them together to relate complex and
real geometry of the general fiber; see Section 4.3 for more details.

####  1.4.3. Real logarithmic geometry

From a topological point of view, as remarked in [Bru22], the general real
fiber is the union of coverings of the real strata of the special fiber.
Naively, one can hope to construct a stratification of the real general fiber
from the one of the real special fiber gluing this covering. Unfortunately, it
is unclear to us how to compare this kind of constructions with the complex
geometry of the general fiber, since the gluing conditions might be quite
complicated. On the other hand, what one can do is to use real and complex
logarithmic geometry to simultaneously stratify the complex and the real
general fiber; see Sections 3 and 4.

More precisely, there is a natural structure $X^{log}$ and $C^{log}$ of real-
log variety, in the sense of [Arg21], on $X$ and $C$ making the morphism
$X^{log}\rightarrow C^{log}$ a smooth morphism of real log-varieties. Taking
the fiber in $0$, we get a morphism $X_{0}^{log}\rightarrow 0^{log}$. Then we
can take the analytification, in the sense of Kato-Nakayama [KN99b], of this
morphism to construct a morphism of $C^{\infty}$-manifolds with corners
$X_{0}^{log}(\mathbb{C})\rightarrow S^{1}$ endowed with involutions. The main
point of logarithmic geometry is that the fiber $X_{0,1}^{log}(\mathbb{C})$ at
$1$ of this morphism is homeomorphic to the general fiber of the family
([NO10]), while its real part $X_{0,1}^{log}(\mathbb{R})$ is homeomorphic to
the real part of a general (positive) fiber ([Arg21] or [Rau22]). This
$X_{0,1}^{log}(\mathbb{C})$ is the common ambient space we were looking for.

Figure 1.
$(X^{log}_{0,1}(\mathbb{C}),X^{log}_{0,1}(\mathbb{R}))\rightarrow(X_{0}(\mathbb{C}),X_{0}(\mathbb{R}))$.
On the left a maximal case and on the right a non-maximal one.

a

It remains to relate this space to the special fiber. The formalism of
logarithmic geometry gives a morphism $X_{0,1}^{log}(\mathbb{C})\rightarrow
X_{0}(\mathbb{C})$ compatible with the involutions, which is a nice locally
constructible fibration with fiber at $x$ equal to $S^{1}$ at some power
$n_{x}$, depending only on which stratum $X^{0}_{J}$ the point $x$ lies. This
allows to relate the complex geometry of the general fiber to that of the
special fiber. To understand the real part, one has just to take the fixed
points of the involutions, and thanks to the totally real assumption, one gets
a locally constructible cover $\pi:X_{0,1}^{log}(\mathbb{R})\rightarrow
X_{0}(\mathbb{R})$ with fiber at $x$ equal to $\\{\pm 1\\}$ at some power
$n_{x}$ depending only on which stratum $X^{0}_{J}$ the point $x$ lies; see
Section 5.2. The situation is explained in an example in Fig.1, in the case of
a family of real elliptic curves (endowed with two different real structures,
respectively on the left and on the right of Fig.1) degenerating to the union
of three genus $0$ curves.

####  1.4.4. Filtred complexes

In [RS22], in order to prove the main theorem, it is given an explicit complex
which computes combinatorially the real Betti numbers and which is filtered by
subcomplexes $K_{i}$, in way that the resulting graded quotients are
isomorphic to the complex computing the tropical homology. The bound is then
obtained as a direct consequence of the spectral sequence for a filtered
complex. Taking inspiration from the above argument, we now have all the
ingredients to prove the desired bound. Once $X^{log}_{0,1}(\mathbb{C})$ and
$X^{log}_{0,1}(\mathbb{R})$ are stratified in a compatible way via the
stratification of the special fiber, one can run the argument of Section
1.4.2, using equivariant cohomology on each stratum. First, since the real
part of the strata have only $H^{0}$, the spectral sequence computing the
cohomology of a stratified space reduces to a single complex, so that we can
compute the cohomology groups of $X^{log}_{0,1}(\mathbb{R})$ via a complex
(Section 5.3) whose terms are the cohomology of the strata. Then, since the
strata are maximal and have only $H^{0}$, the spectral sequence for
equivariant cohomology gives a filtration of the real cohomology of each real
stratum whose graduated complex is the cohomology of the corresponding complex
stratum. Putting these together, we get the filtration that we were looking
for and we conclude by using the spectral sequence of a filtered complex. This
ends the proof of Theorem 1.3.1(1).

####  1.4.5. Hodge numbers

We finally explain how to get Theorem 1.3.1(2). By the arguments in [IKMZ19,
Pag. 31], the theory of limiting Hodge structures and Theorem 1.3.1’s
hypothesis, one has that the $(p,q)$-Hodge numbers of the general fiber can be
computed as dimension of the weight $2q$ part in the limiting Hodge structure
of $X^{log}_{0,1}(\mathbb{C})$ of degree $p+q$. Hence it is enough to show
that the cohomology of the complex
$C_{q,\mathfrak{Z},\mathbb{Z}}^{\bullet}\otimes\mathbb{Q}$ computes the weight
filtration on the limiting Hodge structure.

While this would follows easily from a Hodge theory for the cohomology with
compact support of open logarithmic varieties, this theory seems not to be
developed. One of the problem is that the open stratum $X^{0}_{J}$ endowed
with its natural log-structure is not a log-smooth variety, so that one can
not naively appeal to some form of logarithmic Poincaré duality.

To avoid this problem, we filter
$C_{q,\mathfrak{Z},\mathbb{Z}}^{\bullet}\otimes\mathbb{Q}$ by exploiting the
Leray spectral sequence for the morphisms
$X^{log}_{\Delta,1}(\mathbb{C})\rightarrow X^{0}_{\Delta}(\mathbb{C})$, to
relate the cohomology of
$C_{q,\mathfrak{Z},\mathbb{Z}}^{\bullet}\otimes\mathbb{Q}$ to the one of the
nearby cycles sheaves on $X^{0}_{\Delta}(\mathbb{C})$. At the same time, we
compute the weight filtration on $H^{n}(X^{log}_{0,1},\mathbb{Q})$ by
exploiting the Leray Spectral sequence for the morphism
$X^{log}_{0,1}(\mathbb{C})\rightarrow X_{0}(\mathbb{C})$, also called the
vanishing cycles spectral sequence. The main point is that the assumption on
the weights allows us to show that the terms appearing in the different
spectral sequences involved coincide, since they both compute the same weight
part of a mixed Hodge structure. Since the details are somehow involved, we
refer the reader to Section 6.2, for a more precise account.

###  1.5. Organisation of the paper

The paper is organised as follows. In Section 2.1, we fix some basic notation.
Section 3 and 4 are devoted to recall and complement some properties of
complex and real log-varieties respectively. In Section 5, we prove the first
point of Theorem 1.3.1 and in Section 6 we prove the second point of Theorem
1.3.1.

###  1.6. Acknowledgement

The authors are grateful to Erwan Brugallé, Taro Fujisawa, Kris Shaw, Atsushi
Shiho for helpful discussions. A thanks to Michele Ancona, Erwan Brugallé and
Olivier Benoist for useful comments on a preliminary version of this paper.
The authors thank Athene Grant and CAS young fellows program ”Real structures
in discrete, symplectic, and tropical geometries”.

##  2\. Notation and conventions

###  2.1. Stratification

Let $X$ be a topological space and $\\{X^{0}_{\Delta}\\}$ a collection of
locally closed subspaces of $X$. We write $\\{X_{\Delta}\\}$ for the closure
of $X^{0}_{\Delta}$. We say that $\\{X^{0}_{\Delta}\\}$ is a stratification of
$X$ if

$X=\coprod X^{0}_{\Delta}\quad\text{and for every $\Delta$}\quad
X_{\Delta}=\coprod_{X^{0}_{\Delta^{\prime}}\subseteq
X_{\Delta}}X^{0}_{\Delta^{\prime}}.$

Let $\\{X^{0}_{\Delta^{\prime}}\\}$ another stratification of $X$. We say that
$\\{X^{0}_{\Delta^{\prime}}\\}$ is a refinement of $\\{X^{0}_{\Delta}\\}$ if
every $X^{0}_{\Delta^{\prime}}$ is included is some $X^{0}_{\Delta}$ and for
every $X^{0}_{\Delta}$ the collection
$\\{X^{0}_{\Delta^{\prime}}\\}_{X^{0}_{\Delta^{\prime}}\subseteq
X^{0}_{\Delta}}$ is a stratification of $X^{0}_{\Delta}$.

###  2.2. Degenerations

Let $C$ be a curve over a field $k$, a point $0\in C(k)$ and $f:X\rightarrow
C$ a projective morphism of smooth varieties. We say that $f$ is semistable
(around $0$) if it is smooth over $C-\\{0\\}$ and if, étale locally around
$0$, it is isomorphic to the standard semistable degeneration
$\mathrm{Spec}(k[x_{1},\dots,x_{n},T]/(x_{1}\dots
x_{n}))\rightarrow\mathrm{Spec}(k[T])$. We say that $f:X\rightarrow C$ is
strictly semistable (around $0$) and the irreducible components
$\\{X_{i}\\}_{i\in I}$ of the fiber $X_{0}$ in $0$ are smooth. Setting for
every subset $J\subseteq I$

$X_{J}:=\bigcap_{i\in J}X_{i}\quad\text{and}\quad
X^{0}_{J}:=X_{J}\setminus\bigcup_{i\not\in J}X_{i},$

the collection $\mathfrak{I}:=\\{X^{0}_{J}\\}_{J\subseteq I}$ is a
stratification of $X_{0}$ by smooth algebraic subvarieties. If $k=\mathbb{R}$,
we say that $f$ is totally real, if the irreducible components of
$X_{0}(\mathbb{C})$ are real.

###  2.3. Local orientations

Let $C$ be a smooth real curve and $0\in C(\mathbb{R})$ a real point. The set
$C(\mathbb{R})$ is homeomorphic to a disjoint union of open subsets of
$S^{1}$’s. Denote with $\mathcal{H}$ the real connected component of
$C(\mathbb{R})$ containing $0$. We say that $t\in C(\mathbb{R})$ is near $0$,
if $t$ belongs to $\mathcal{H}$. Let $D\subseteq\mathbb{C}$ be an open disc
centered in $(0,0)$ and endowed with the canonical involution induced by the
complex conjugation acting on $\mathbb{C}$.

An orientation around $0\in C$ is a pair $(U,\psi)$ where $U$ is an open
neighbourhood of $0$ in $C(\mathbb{C})$ stable under complex conjugation and
$\phi$ is an equivariant homeomorphism $\psi:U\rightarrow D$ sending $0$ to
$0$. An orientation induces an homeomorphism
$\phi_{\mathbb{R}}:U(\mathbb{R})\xrightarrow{\simeq}(-1,1)$, where
$U(\mathbb{R})$ is the set of points of $U$ fixed by the involution. Let
$U_{+}:=\phi_{\mathbb{R}}^{-1}((0,1))$. A point $t\in C(\mathbb{R})$ near $0$
is said to be positive with respect to an orientation $(U,\psi)$ if it lies in
$(\mathcal{H}\cap U_{+})\setminus\\{0\\}$; otherwise negative.

Assume that $X\rightarrow C$ is a morphism of real algebraic varieties. A
fiber $f^{-1}(t)$, for $t\in C(\mathbb{R})$, is said to be near $f^{-1}(0)$ if
$t$ is near $0$. Moreover, for some fixed orientation $(U,\phi)$ around $0$, a
fiber $f^{-1}(t)$ is said to be positive if $t$ is near $0$ and it is positive
with respect to $(U,\psi)$; otherwise negative.

##  3\. Log-varieties and analytification

In this section we construct the complex $C_{p,\mathfrak{Z},A}^{\bullet}$
appearing in Theorem 1.3.1 and we state Theorem 1.3.1 in a more precise way
(Theorem 3.4.2). To this end, we need to recall a few basic notions from
logarithmic geometry (Section 3.1), and from their associated analytification
(Section 3.2). Then we study the geometry of certain stratifications of these
analytifications (Section 3.3) and we finally construct the complex
$C_{p,\mathfrak{Z},A}^{\bullet}$ in Section 3.4.

###  3.1. Logarithmic varieties

####  3.1.1. Definitions and examples

We start recalling a few basic basic notions from logarithmic geometry. For
more details see for example [Ill02] and [Kat89].

Let $k$ be a field and $X$ a $k$-variety. We consider $\mathcal{O}_{X}$ as a
sheaf of (abelian) monoids with the multiplication operation.

######  Definition 3.1.1.1.

  1.   2. (1)

A pre-log structure on $X$ is a morphism of sheaf of (abelian) monoids
$\alpha_{X}:M\rightarrow\mathcal{O}_{X}$.

  3. (2)

A pre-log structure $\alpha_{X}:M\rightarrow\mathcal{O}_{X}$ is said a log
structure if the natural morphism
$\alpha_{|\alpha^{-1}(\mathcal{O}_{X}^{*})}:\alpha^{-1}(\mathcal{O}_{X}^{*})\rightarrow\mathcal{O}_{X}^{*}$
is an isomorphism.

  4. (3)

Given a pre-log structure $\alpha_{X}:M\rightarrow\mathcal{O}_{X}$, there is a
canonical associated log-structure
$\alpha^{\prime}_{X}:M\rightarrow\mathcal{O}_{X}$ (see [Kat89, Section 1.3]).

  5. (4)

A log-variety is a pair $(X,M)$ where $X$ is a $k$-variety and $M$ a log-
structure on $X$.

All the logarithmic structures in this paper will be assumed to be fine and
saturated ([Ill02, 1.3]) and all the operation between log-varieties will be
done in the category of fine and saturated log varieties. Given a log variety
$(X,M)$, we consider $\mathcal{O}_{X}^{*}$ as a submonoid of $M$ via $\alpha$.
We write $M^{gr}$ for the groupification of the sheaf of monoids $M$ and
$\overline{M}$ (resp. $\overline{M}^{gr}$) for the quotient
$M/\mathcal{O}_{X}^{*}$ (resp. $\overline{(M/\mathcal{O}_{X}^{*})}^{gr})$.
There is a natural notion of morphism between logarithmic varieties. Given a
log-variety $(X,M)$ and a morphism of $k$-varieties $f:Y\rightarrow X$, there
is an induced log-structure $f^{*}M$ on $Y$ ([Kat89, Section 1.4]) and a
morphism of log varieties $f:(Y,f^{*}M)\rightarrow(X,M)$ (morphisms of this
type are called strict). Here some examples that are fundamental for our
purposes and, therefore, that will be treated further in Examples 3.2.1.1,
4.1.1.1.

######  Example 3.1.1.2.

In the following examples anytime a morphism of monoids
$\mathbb{N}\rightarrow\mathbb{N}^{r}$ is given, it is the diagonal map which
sends $1$ to $\sum\limits_{i=1}^{r}e_{i}$, where $e_{i}$ denotes the $n$-uple
with the $i$-th component equals $1$ and the remaining $0$.

  1. (1)

The trivial log-structure on $X$ is given by $(X,\mathcal{O}_{X}^{*})$ and
$\alpha_{X}$ is the natural inclusion
$\mathcal{O}^{*}_{X}\rightarrow\mathcal{O}_{X}$. In this way any variety can
be consider as a log-variety with the trivial log-structure. If $(X,M)$ is a
log-variety, there is a canonical morphism $(X,M)\rightarrow X$.

  2. (2)

Let $X=\mathrm{Spec}(k)$ and let $\mathbb{N}\rightarrow k$ be the map sending
$1$ to $0$. The associated log-structure $\mathbb{N}\times k^{*}\rightarrow k$
sends $(1,\lambda)$ to $0$ and $(0,\lambda)$ to $\lambda$. It is called the
standard log-structure on the point and we write $\mathrm{Spec}(k)^{\log}$ for
the associated log-scheme.

  3. (3)

Let $C$ be $\mathbb{A}^{1}=Spec(k[T])$ and let $\mathbb{N}\rightarrow k[T]$ be
the map sending $1$ to $T$. The associated log structure $\mathbb{N}\times
k^{*}\rightarrow k[T]$ sends $(n,f)$ to $fT^{n}$. It is called the log
structure on $\mathbb{A}^{1}$ associated to the divisor $0$. We write
$C^{log}$ for the associated log-scheme. The restriction of such log-structure
to $0\in\mathbb{A}^{1}=Spec(k[T])$, induces the standard log-structure on the
point.

  4. (4)

Let $X=\mathrm{Spec}(k[x_{1}\dots x_{n},T]/(x_{1}\dots x_{n}-T))$ and
$\mathbb{N}^{n}\rightarrow k[x_{1}\dots x_{n},T]/(x_{1}\dots x_{n}-T)$ be the
pre-log structure sending $e_{i}$ to $x_{i}$. The associated log structure
$\mathbb{N}^{n}\times k^{*}\rightarrow k[x_{1}\dots x_{n},T]/(x_{1}\dots
x_{n}-T)$ sends $(e_{i},f)$ to $fx^{i}$. We write $X^{log}$ for the associated
log-variety. If $C=\mathbb{A}^{1}$ is endowed with the log-structure in
Example 3.1.1.2(3), i.e. the previous example, the natural morphism
$X\rightarrow C$, extends to a morphism of log varieties $X^{log}\rightarrow
C^{log}$ described by the commutative diagram

${\mathbb{N}}$${k[T]}$${\mathbb{N}^{n}}$${k[x_{1}\dots x_{n},T]/(x_{1}\dots
x_{n}-T).}$$\scriptstyle{1\mapsto T}$$\scriptstyle{1\mapsto\sum
e_{i}}$$\scriptstyle{e_{i}\mapsto x_{i}}$

  5. (5)

Retain the notation of Example 3.1.1.2(4) and take the fiber
$X^{log}_{0}\rightarrow 0^{log}$ at $0$ of the morphism $X^{log}\rightarrow
C^{log}$. Then the situation is described by the commutative diagram

${\mathbb{N}}$${k}$${\mathbb{N}^{n}}$${k[x_{1}\dots x_{n}]/(x_{1}\dots
x_{n}).}$$\scriptstyle{1\mapsto 0}$$\scriptstyle{1\mapsto\sum
e_{i}}$$\scriptstyle{e_{i}\mapsto x_{i}}$

  6. (6)

Retain the notation of Example 3.1.1.2(5), set $J=\\{1,\dots,r\\}$ and endow
the open strata $X^{0}_{J}:=\mathrm{Spec}(k[x^{\pm 1}_{r+1}\dots x^{\pm
1}_{n}])\subseteq X_{0}$, with the log structure induced by the pre-log
structure $\mathbb{N}^{r}\rightarrow k[x^{\pm 1}_{r+1}\dots x^{\pm 1}_{n}]$
sending $e_{i}$ to $0$. We write $X_{J}^{0,log}$ for the corresponding log-
variety. The natural morphism $X^{0,log}_{J}\rightarrow 0^{log}$ is described
by the commutative diagram

${\mathbb{N}}$${k}$${\mathbb{N}^{r}}$${k[x^{\pm 1}_{r+1}\dots x^{\pm
1}_{n}].}$$\scriptstyle{1\mapsto 0}$$\scriptstyle{1\mapsto\sum
e_{i}}$$\scriptstyle{e_{i}\mapsto 0}$

####  3.1.2. The case of a semistable family

More generally, let $C$ be a smooth $k$-curve and $0\in C(k)$. Let
$X\rightarrow C$ be a projective family, smooth over
$U_{C}:=C\setminus\\{0\\}$ and having strictly semistable reduction at $0$.

The natural inclusion $\mathcal{O}_{C}\cap
j_{*}\mathcal{O}_{U_{C}}^{*}\rightarrow\mathcal{O}_{C}$ is induces a log
structure on $C$, called the log-structure $C^{\log}$ associated to the
divisor $0$ in $C$. Locally around $0$, in the étale topology, the logarithmic
structure is given by the one in Example 3.1.1.2(3); see [Kat89, Example 1.5].

Let $U_{X}:=X\setminus X_{0}\subseteq X$. The natural inclusion
$\mathcal{O}_{X}\cap j_{*}\mathcal{O}_{U_{X}}^{*}\rightarrow\mathcal{O}_{X}$
is a pre-log structure on $X$, which induces a log-strucure $X^{log}$ giving
rise to a morphism of log-variety $X^{\log}\rightarrow C^{\log}$, which,
locally around $0$, is isomorphic to the one given in Example 3.1.1.2(5). The
restriction of such log structure to $X_{0}$ induces a canonical log-structure
$X^{\log}_{0}$ on $X_{0}$ and a morphism
$X^{\log}_{0}\rightarrow\mathrm{Spec}(k)^{\log}$ (see [Kat89, Example 3.7
(2)]), which, locally around $0$, is isomorphic to the one given in Example
3.1.1.2(6).

Endowing the strata $X^{0}_{J}\subseteq X_{0}$ with the induced log-structure,
we get a strict closed immersion of log-schemes $X^{0,log}_{J}\subseteq
X_{0}^{log}$, which is locally isomorphic to the one in example 3.1.1.2(7).
The situation is summarized in the following commutative diagram:

${X_{J}^{0,log}}$${X_{J}^{0}}$${X_{0}^{log}}$${X^{log}}$${X}$${X_{0}}$${\mathrm{Spec}(k)^{log}}$${C^{log}}$${C}$${\mathrm{Spec}(k)}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{f^{log}_{0}}$$\scriptstyle{\pi}$$\scriptstyle{f^{log}}$$\scriptstyle{f}$$\scriptstyle{f_{0}}$$\scriptstyle{\pi}$$\scriptstyle{0}$$\scriptstyle{\pi}$$\scriptstyle{0}$

###  3.2. Analytification

Assume now that $k=\mathbb{C}$ and $(X,M)$ is a log $\mathbb{C}$-variety. We
recall how to associate to $(X,M)$ a topological space, the Kato-Nakayama
space of $(X,M)$, which is a topological incarnation of $(X,M)$ extending the
usual analytification for $\mathbb{C}$-varieties. We refer the reader to
[KN99a] for the original construction and to [Ill02, Section 5.5] for a nice
introduction.

####  3.2.1. Definitions and examples

More precisely, there is a natural functorial way to associate to $(X,M)$ a
$C^{\infty}$-manifold with corner $(X,M)^{an}$, called the Kato-Nakayama space
of $(X,M)$. Concretely, consider the pair $(\mathrm{Spec}(\mathbb{C}),\pi)$
where $\pi:\mathbb{R}_{\geq 0}\times S^{1}\rightarrow\mathbb{C}$ is the polar
coordinates defined by $\pi(x,y)=xy$. Then, by definition

$(X,M)^{an}:=\mathrm{Hom}((\mathrm{Spec}(\mathbb{C}),\pi),(X,M))$

endowed with a natural topology. We will often write $(X,M)(\mathbb{C})$ for
$(X,M)^{an}$.

If $M$ is the trivial log-structure, then $(X,M)^{an}=X(\mathbb{C})$ is the
usual analytification. In general, for every log-variety $(X,M)$, the natural
morphism $(X,M)\rightarrow X$ induces a proper morphism $(X,M)^{an}\rightarrow
X^{an}$, whose fiber over a point $x\in X^{an}$ is
$(S^{1})^{\textrm{rank}(\overline{M}^{gp}_{x})}$. Here some examples which are
relevant for this paper.

######  Example 3.2.1.1.

  1.   2. (1)

As already mentioned, if $X$ has the trivial log structure then
$X^{an}=X(\mathbb{C})$.

  3. (2)

In the notation of Example 3.1.1.2(2), $(Spec(\mathbb{C})^{\log})^{an}=S^{1}$,
since the only morphisms $h:\mathbb{N}\rightarrow\mathbb{R}_{\geq 0}\times
S^{1}$ making the diagram

${\mathbb{N}}$${\mathbb{R}_{\geq 0}\times
S^{1}}$${\mathbb{\mathbb{C}}}$${\mathbb{C}.}$$\scriptstyle{h}$$\scriptstyle{1\mapsto
0}$$\scriptstyle{(x,y)\mapsto xy}$$\scriptstyle{id}$

commutative are those sending $1$ to $(0,y)$ for $y\in S^{1}$.

  4. (3)

In the notation of Example 3.1.1.2(3), then $C^{log,an}\simeq\mathbb{R}_{\geq
0}\times S^{1}$ is the real blow-up of $\mathbb{C}$ at $0$, as one readily
sees by looking at the set of commutative diagrams

${\mathbb{N}}$${\mathbb{R}_{\geq 0}\times
S^{1}}$${\mathbb{\mathbb{C}}[T]}$${\mathbb{C}.}$$\scriptstyle{1\mapsto
T}$$\scriptstyle{(x,y)\mapsto xy}$$\scriptstyle{t\mapsto a_{i}}$

Moreover, the natural map $C^{log,an}\simeq\mathbb{R}_{\geq 0}\times
S^{1}\rightarrow\mathbb{C}\simeq C^{an}$ is the usual real-blow-up map (the
polar coordinates).

  5. (4)

Retain the notation of Example 3.1.1.2(6). Then
$X^{0,\log}_{J}(\mathbb{C})\simeq(S^{1})^{r}\times(\mathbb{C}^{*})^{n-r}$, as
one readily sees by looking at the set of commutative diagrams

${\mathbb{N}^{r}}$${\mathbb{R}_{\geq 0}\times
S^{1}}$${\mathbb{\mathbb{C}}[x_{r+1}^{\pm 1}\dots x_{n}^{\pm
1}]}$${\mathbb{C}.}$$\scriptstyle{e_{i}\mapsto 0}$$\scriptstyle{(x,y)\mapsto
xy}$$\scriptstyle{t\mapsto a_{i}}$

and the map $X^{0,\log}_{J}(\mathbb{C})\rightarrow
X^{an}\simeq(\mathbb{C}^{*})^{n-r}$ is the natural projection.

The natural map $X^{0,\log}_{J}\rightarrow\mathrm{Spec}(\mathbb{C})^{\log}$
induces a map $X_{J}^{0,\log}(\mathbb{C})\rightarrow S^{1}$, which locally
sends $((y_{1}\dots y_{r}),(a_{r+1}\dots a_{n}))$ to $\prod_{i}y_{i}$. Indeed,
$((y_{1}\dots y_{r}),(a_{r+1}\dots a_{n}))$ corresponds to the commutative
diagram

${\mathbb{N}^{r}}$${\mathbb{R}_{\geq 0}\times
S^{1}}$${\mathbb{\mathbb{C}}[x_{r+1}^{\pm 1}\dots x_{n}^{\pm
n}]}$${\mathbb{C}.}$$\scriptstyle{e_{i}\mapsto(0,y_{i})}$$\scriptstyle{e_{i}\mapsto
0}$$\scriptstyle{(x,y)\mapsto xy}$$\scriptstyle{t\mapsto a_{i}}$

and its image in $S^{1}$ it is obtained by precomposing it with the diagram,

${\mathbb{N}}$${\mathbb{N}^{r}}$${\mathbb{C}}$${\mathbb{C}[x_{r+1}^{\pm
1}\dots x_{n}^{\pm n}]}$$\scriptstyle{1\mapsto\sum
e_{i}}$$\scriptstyle{1\mapsto 0}$$\scriptstyle{e_{i}\mapsto 0}$

in which the bottom horizontal map is the canonical inclusion. Finally, one
obtains the diagram

${\mathbb{N}}$${\mathbb{R}_{\geq 0}\times
S^{1}}$${\mathbb{\mathbb{C}}}$${\mathbb{C},}$$\scriptstyle{e_{i}\mapsto(0,\prod_{i}y_{i})}$$\scriptstyle{(x,y)\mapsto
xy}$

which corresponds to the point $\prod y_{i}\in S^{1}$. In particular, the
fiber $X^{0,\log}_{J,1}(\mathbb{C})$ at $1$ of this morphism is the subset of
$X^{0,\log}_{J}(\mathbb{C})$, in which the product of the first $r$
coordinates is $1$.

####  3.2.2. The case of a semistable degeneration

Assume from now that $X\rightarrow C$ is a real strictly semistable
degeneration as in Section 3.1.2 and endow it with the logarithmic structure
as in Section 3.1.2. Choose an orientation $(U,\psi)$ around $0$, see Section
(2.3). Then, by Example 3.2.1.1(2), the orientation induces an isomorphism
$\pi^{-1}(U)\simeq[0,1)\times S^{1}$, where $\pi:C^{log,an}\rightarrow C^{an}$
is the natural map. Hence $C^{log,an}$ can be identified with the real
oriented blow-up of $C^{an}$ in $0$ (see e.g. [NO10, (1.2.3)]).

Then, we can analytify the commutative diagram

${X_{J}^{0,log}}$${X^{0}_{J}}$${X_{0}^{log}}$${X_{0}}$${0^{log}}$${0}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$

to get morphisms $X_{J}^{0,log}(\mathbb{C})\rightarrow
X_{0}^{log}(\mathbb{C})\rightarrow 0^{log,an}\simeq S^{1}$ with fibers at $1$
given by $X_{J,1}^{log}(\mathbb{C})\rightarrow X_{0,1}^{log}(\mathbb{C})$ and
a commutative cartesian diagram

${X_{J,1}^{0,log}(\mathbb{C})}$${X_{J}^{0,log}(\mathbb{C})}$${X^{0}_{J}(\mathbb{C})}$${X_{0,1}^{log}(\mathbb{C})}$${X_{0}^{log}(\mathbb{C})}$${X_{0}(\mathbb{C})}$${\\{1\\}}$${0^{\log}(\mathbb{C})=S^{1}.}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$

The main result that we are interested in and which motivates the use of
logarithmic geometry, is the following.

######  Theorem 3.2.2.1.

[NO10, Theorem 0.3] The topological space $X^{log}_{0,1}(\mathbb{C})$ is
homeomorphic to every smooth fiber of the morphism $X\rightarrow C$.

Hence, in order to study the Betti numbers of the complex general fiber of
$X\rightarrow C$, we can study the topology of $X^{log}_{0,1}(\mathbb{C})$ via
the morphism $X^{log}_{0,1}(\mathbb{C})\rightarrow X_{0}(\mathbb{C})$.

###  3.3. Stratifications

We now go further and we study $X_{0,1}^{log}(\mathbb{C})$, by stratifying it
in a way which is compatible with the logarithmic structure on
$X_{0}(\mathbb{C})$. Retain the notation of Section 3.2.2.

Let $\mathfrak{Z}:=\\{X^{0}_{\Delta}\\}_{\Delta\in P}$ be an algebraic
refinement of $\mathfrak{I}$. Let $I_{\Delta}$ be the subset
$I_{\Delta}\subseteq I$ such that $X^{0}_{\Delta}\subseteq X^{0}_{I_{\Delta}}$
and set $|\Delta|:=|I_{\Delta}|$. We endow $X^{0}_{\Delta}$ with the structure
of a log-variety $X^{0,log}_{\Delta}$ by restricting the log structure of
$X_{0}$ to $X^{0}_{\Delta}$. The most important properties of
$X^{0,log}_{\Delta}(\mathbb{C})$ and $X^{0,log}_{\Delta,1}(\mathbb{C})$, which
follows directly from the local description given in Example 3.2.1.1 are
summarized in the following remark.

######  Remark 3.3.1.

  1.   2. (1)

Example 3.2.1.1(7) shows that $X^{0,log}_{\Delta}(\mathbb{C})\rightarrow
X^{0}_{\Delta}(\mathbb{C})$ is a locally trivial fibration with fibers
$(S^{1})^{|\Delta|}$. In particular $X^{0,log}_{\Delta}(\mathbb{C})$ is a
smooth $C^{\infty}$-manifold of dimension $2\textrm{dim}(X_{0})+|\Delta|$.

  3. (2)

By Example 3.2.1.1(4), the morphism $X^{0}_{\Delta,1}(\mathbb{C})\rightarrow
X^{0}_{\Delta}(\mathbb{C})$ is a proper locally trivial fibration with fiber
$(S^{1})^{|\Delta|-1}$. In particular, $X^{0,log}_{\Delta}(\mathbb{C})$ is a
smooth $C^{\infty}$-manifold of dimension $2\textrm{dim}(X_{0})+|\Delta|-1$.

Since $\\{X^{0}_{\Delta}(\mathbb{C})\\}$ is a stratification of $X_{0}$ and
$\overline{X_{\Delta}^{0,\log}(\mathbb{C})}=X_{\Delta}^{\log}(\mathbb{C})\subseteq
X^{log}_{0}(\mathbb{C})$111This can be checked locally, hence we can assume
that $X_{0}^{\log}=Spec(\mathbb{C}[x_{1},\dots,x_{n}]/(x_{1}\dots x_{n}))$
endowed with the logarithmic structure of Example 3.1.1.2(5). Since in this
case $X_{0}^{\log}(\mathbb{C})$ is closed inside
$\mathbb{A}^{n,log}(\mathbb{C})\simeq(\mathbb{R}_{\geq 0}\times S^{1})^{n}$,
we can assume that $X_{0}^{log}$ is $\mathbb{A}^{n}=\prod\mathbb{A}^{1}$ with
the product log-structure of example 3.1.1.2(2). In this case the result
follows from a computation using that the map $(\mathbb{R}_{\geq 0}\times
S^{1})^{n}\simeq\mathbb{A}^{n,\log}(\mathbb{C})\rightarrow\mathbb{A}^{n}(\mathbb{C})=\mathbb{C}^{n}$
is the polar coordinates componentwise. , we see that
$\\{X^{0,log}_{\Delta}(\mathbb{C})\\}$ is a stratification of
$X^{log}_{0}(\mathbb{C})$. Since $X^{log}_{0,1}(\mathbb{C})\subseteq
X^{log}_{0}(\mathbb{C})$ is a closed subset, also
$\\{X^{0,log}_{\Delta,1}(\mathbb{C})\\}$ is a stratification of
$X^{log}_{0,1}(\mathbb{C})$.

###  3.4. Construction of the complex and statement of the main theorem

Retain the notation of Section 3.3 and let $A$ be a coefficient ring. There is
a natural spectral sequence (see e.g. [Pet16] and [Pet17, (3) pag. 2527])

(3.4.1)
$^{\mathbb{C}}E_{1}^{a,b}:=\bigoplus_{\textrm{dim}(X^{0}_{\Delta})=a}H_{c}^{a+b}(X^{0,log}_{\Delta,1}(\mathbb{C}),A)\Rightarrow
H_{c}^{a+b}(X_{0,1}^{log}(\mathbb{C}),A)\big{(}\simeq
H_{c}^{a+b}(X_{t}(\mathbb{C}),A)\big{)},$

where $t\in C(\mathbb{C})$ is any member different from $0$ and the last
isomorphism follows from Theorem 3.2.2.1.

Set

$C_{q,\mathfrak{Z},A}^{\bullet}:=\text{}^{\mathbb{C}}E_{1}^{\bullet,q}:$
$\underset{\textrm{dim}(X^{0}_{\Delta})=p-1}{\bigoplus}H_{c}^{p+q-1}(X^{0,log}_{\Delta,1}(\mathbb{C}),A)\rightarrow\underset{\textrm{dim}(X^{0}_{\Delta})=p}{\bigoplus}H_{c}^{p+q}(X^{0,log}_{\Delta,1}(\mathbb{C}),A)\rightarrow\underset{\textrm{dim}(X^{0}_{\Delta})=p+1}{\bigoplus}H_{c}^{p+q+1}(X^{0,log}_{\Delta,1}(\mathbb{C}),A).$

Now Theorem 1.3.1 can be stated more precisely as follows.

######  Theorem 3.4.2.

  1.   2. (1)

With the notation of Theorem 1.3.1, assume that

    1. (a)

$H^{i}(X^{0}_{\Delta}(\mathbb{R}),\mathbb{Z}/2\mathbb{Z})=0$ for all $i\geq 1$
and $X^{0}_{\Delta}\in\mathfrak{Z}$;

    2. (b)

$X^{0}_{\Delta}$ is a maximal variety, for all
$X^{0}_{\Delta}\in\mathfrak{Z}$.

Then, for every $t\in C(\mathbb{R})$ near $0$ one has:

$b_{p}(X_{t}(\mathbb{R})),\mathbb{Z}/2\mathbb{Z})\leq\sum_{0\leq q\leq
n}\textrm{dim}(H^{p}(C_{q,\mathfrak{Z},\mathbb{Z}/2\mathbb{Z}}^{\bullet}).$

  3. (2)

If in addition to (a) and (b), assume that

    1. (c)

the mixed Hodge structure on $H^{i}(X^{0}_{\Delta},\mathbb{Q})$ is pure of
type $(i,i)$ and $H^{i}(X^{0}_{\Delta},\mathbb{Z})$ is torsion free,

Then, for every $t\in C(\mathbb{R})$ near $0$ one has:

$(i)\quad\textrm{dim}(H^{q}(C_{p,\mathfrak{Z},\mathbb{Z}}^{\bullet}\otimes\mathbb{Q}))=h^{p,q}(X_{t})\quad\text{and}\quad(ii)\quad
C_{p,\mathfrak{Z},\mathbb{Z}}^{\bullet}\otimes\mathbb{Z}/2\mathbb{Z}\simeq
C_{p,\mathfrak{Z},\mathbb{Z}/2\mathbb{Z}}^{\bullet}.$

##  4\. Real log varieties and equivariant cohomology

In this section we recall some facts from the theory of real logarithmic
varieties. The theory appears explicitly in the literature in [Arg21, Sections
5-7] and in [AB21], and it is implicit in [Rau22]. We borrow part of the
presentation from [Arg21, Sections 5-7], where the author works in a slightly
more general setting and we complement Argüz results adding what is needed for
our purposes. In Section 4.1, we recall the basic definitions, examples and we
study the case of totally real semistable degenerations. In Section 4.2 we
study stratifications of these degenerations. In Section 4.3 we recall a few
fact around equivariant cohomology.

###  4.1. Real logarithmic structure

####  4.1.1. Definitions and examples

Assume now that $k=\mathbb{R}$. To give a log variety over $\mathbb{R}$ is
equivalent to give a log variety $(X,M)$ over $\mathbb{C}$ endowed with a pair
$(\sigma,\sigma_{M})$ where $\sigma:X(\mathbb{C})\rightarrow X(\mathbb{C})$ is
an anti-holomorphic involution and $\sigma_{M}:M\rightarrow M$ is an
involution of monoids making the diagram

${M}$${M}$${\mathcal{O}_{X}}$${\mathcal{O}_{X}.}$$\scriptstyle{\sigma_{M}}$$\scriptstyle{\sigma}$

commutative, where, by abuse of notation, we denote by $\sigma$ also the
involution on $\mathcal{O}_{X}$ induced by that on $X(\mathbb{C})$.

We endow $\mathbb{R}_{\geq 0}\times S^{1}$ with the involution $\tau$ given by
$\tau(x,r)=(x,\overline{r})$, so that the polar coordinates morphism
$\mathbb{R}_{\geq 0}\times S^{1}\rightarrow\mathbb{C}$ is equivariant once
$\mathbb{C}$ is endowed with the standard conjugation. Then $\tau$ and
$\sigma$ induce a natural involution on
$(X,M)(\mathbb{C}):=\mathrm{Hom}((\mathrm{Spec}(\mathbb{C}),\pi),(X,M))$ and
the real locus $(X,M)(\mathbb{R})\subseteq(X,M)(\mathbb{C})$ of $(X,M)$ is, by
definition, the fixed locus of this involution endowed with its subspace
topology.

If $(X,M)\rightarrow(Y,N)$ is a morphism of real log varieties, then the
induced morphism $(X,M)(\mathbb{C})\rightarrow(Y,N)(\mathbb{C})$ is
equivariant and hence it induces a morphism
$(X,M)(\mathbb{R})\rightarrow(Y,N)(\mathbb{R})$. In particular, there is a
natural morphism $(X,M)(\mathbb{R})\rightarrow X(\mathbb{R})$. We now give the
examples that are relevant for our paper.

######  Example 4.1.1.1.

  1.   2. (1)

If $X$ is endowed with the trivial log-structure, then
$(X,M)(\mathbb{R})\simeq X(\mathbb{R})$ is just the set of real points of $X$.

  3. (2)

Let $X=\mathrm{Spec}(\mathbb{R})$ endowed with the canonical log-structure.
Then the induced action on $(X(\mathbb{C}),\mathbb{N})$ is trivial on both
$X(\mathbb{C})=\\{\text{pt}\\}$ and $\mathbb{N}$. This induces on the
analytification $X^{\log}(\mathbb{C})\simeq S^{1}$ the standard conjugation,
so that $X^{\log}(\mathbb{R})=\\{+1,-1\\}$.

  4. (3)

Assume that $C=\mathrm{Spec}(\mathbb{R}[T])$, endowed with the log structure
from Example 3.1.1.2(3). Then, the involution on $(C(\mathbb{C}),\mathbb{N})$
is the standard complex conjugation on $C(\mathbb{C})\simeq\mathbb{C}$ and the
identity on $\mathbb{N}$. This induces on the analytification
$C^{\log}(\mathbb{C})\simeq\mathbb{R}_{\geq 0}\times S^{1}$ (see Example
3.2.1.1(3)) the identity of the first factor and the standard complex
conjugation on the second one, so that
$C^{log}(\mathbb{R})\simeq\mathbb{R}_{\geq 0}\times\\{\pm 1\\}$. The natural
map $C^{log}(\mathbb{R})\simeq\mathbb{R}_{\geq 0}\times\\{\pm 1\\}\rightarrow
C(\mathbb{R})\simeq\mathbb{R}$ sends $(x,1)$ to $x$ and $(x,-1)$ to $-x$.

  5. (4)

Assume that $X^{0}_{J}=\mathrm{Spec}(\mathbb{R}[x^{\pm 1}_{r+1}\dots x^{\pm
1}_{n}])$, endowed with the log structure
$\mathbb{N}^{r}\rightarrow\mathbb{R}[x^{\pm 1}_{r+1}\dots x^{\pm 1}_{n}]$ from
Example 3.1.1.2(6) sending all non zero elements to $0$. Then the involution
on $(X^{0}_{J}(\mathbb{C}),\mathbb{N}^{r})$ is the standard complex
conjugation on $X^{0}_{J}(\mathbb{C})\simeq(\mathbb{C}^{*})^{n-r}$ and the
trivial action on $\mathbb{N}^{r}$. Hence the induced action on
$X_{J}^{0,\log}(\mathbb{C})\simeq(S^{1})^{r}\times(\mathbb{C}^{*})^{n-r}$ (see
Example 3.2.1.1(3)) is the standard complex conjugation acting component-wise
and without any permutation of the factors, so that
$X_{J}^{0,\log}(\mathbb{R})\simeq\\{\pm 1\\}^{r}\times(\mathbb{R}^{*})^{n-r}$
and the map $X_{J}^{0,\log}(\mathbb{R})\simeq\\{\pm
1\\}^{r}\times(\mathbb{R}^{*})^{n-r}\rightarrow
X(\mathbb{R})\simeq(\mathbb{R}^{*})^{n-r}$ is the natural projection. The
natural map $X^{0,\log}_{J}(\mathbb{R})\simeq\\{\pm
1\\}^{r}\times(\mathbb{R}^{*})^{n-r}\rightarrow\mathrm{Spec}(\mathbb{R})^{\log}\simeq\\{\pm
1\\}$ sends $((y_{1}\dots y_{r}),(a_{r+1}\dots a_{r}))$ to $\prod_{i}y_{i}$,
as follows from the computation done in Example 3.2.1.1(4). In particular, the
fiber $X_{J,1}^{0,\log}(\mathbb{R})$ at $1$ of this morphism identifies with
the subset of $X^{0,\log}_{J}(\mathbb{R})\simeq\\{\pm
1\\}^{r}\times(\mathbb{R}^{*})^{n-r}$ in which the product of the first $r$
components is $1$. Hence $X^{0,\log}_{J,1}(\mathbb{R})\rightarrow
X^{0}_{J}(\mathbb{R})$ is a topological cover of degree $2^{|J|-1}$; see also
[Arg21, Proposition 8.5] and [Rau22, Theorem 1.3].

In all the examples of Example 4.1.1.1, the action of the involution on the
sheaf of monoids is trivial. Even if this is the only case we will need (since
the degenerations we are interested in are totally real), for completeness, we
give now an example in which the action is non-trivial, which comes from a
real semistable degeneration which in not totally real. The reader only
interested in Theorem 1.3.1 can skip the end of this section.

######  Example 4.1.1.2.

Let $X^{0}_{J}$ be $\mathrm{Spec}(\mathbb{C}[x_{r+1}^{\pm 1}\dots x_{n}^{\pm
1}])$ and endow it with the logarithmic structure associated to the morphism
$\mathbb{N}^{r}\rightarrow\mathrm{Spec}(\mathbb{C}[x_{r+1}^{\pm 1}\dots
x_{n}^{\pm 1}])$ sending all the non zero element to $0$. We consider the
standard conjugation action on $X^{0}_{J}(\mathbb{C})$ but now we consider
$\mathbb{N}^{r}$ endowed with the involution switching $e_{1}$ and $e_{2}$ and
acting trivially on the other $e_{i}$. This corresponds to the restriction to
some geometrically irreducible strata of the log structure coming from a
degeneration whose special fiber has the form
$\mathrm{Spec}(\mathbb{R}[x_{1}\dots x_{n}]/((x^{2}_{1}+x_{2}^{2})x_{3}\dots
x_{n})$, which is not totally real.

Then the induced involution on
$X^{0}_{J}(\mathbb{C})\simeq(S^{1})^{r}\times(\mathbb{C}^{*})^{n-r}$ sends
$((y_{1},y_{2},y_{3}\dots y_{r}),(a_{r+1}\dots a_{n}))$ to

$((\overline{y_{2}},\overline{y_{1}},\overline{y_{3}}\dots\overline{y_{r}}),(\overline{a_{r+1}}\dots\overline{a_{n}})),$

so that $X^{0,\log}_{J}(\mathbb{R})\subseteq X^{0,\log}_{J}(\mathbb{C})$
identifies with the subset made by the elements of the form

$((y_{1},\overline{y_{1}},\pm 1,\dots,\pm 1),(a_{r+1}\dots a_{n}))$

with $y_{1}\in S^{1}$ and $a_{i}\in\mathbb{R}^{*}$. Hence
$X^{0,\log}_{J}(\mathbb{R})$ identifies with $S^{1}\times\\{\pm
1\\}^{r-2}\times(\mathbb{R}^{*})^{n-r}$.

The natural map $S^{1}\times\\{\pm
1\\}^{r-2}\times(\mathbb{R}^{*})^{n-r}\simeq
X^{0,log}_{J}(\mathbb{R})\rightarrow
X^{0}_{J}(\mathbb{R})\simeq(\mathbb{R}^{*})^{n-r}$ is the natural projection,
which is no longer a topological cover of degree $2^{r}$ but a product of a
$2^{r-2}$ topological cover and an $S^{1}$-bundle over
$X^{0}_{J}(\mathbb{R})$. The fiber at $1$ of the other natural map
$S^{1}\times\\{\pm 1\\}^{r-2}\times(\mathbb{R}^{*})^{n-r}\simeq
X^{0,log}_{J}(\mathbb{R})\rightarrow\mathrm{Spec}(\mathbb{R})^{\log}\simeq\\{\pm
1\\}$ is given (since $y_{1}\overline{y_{1}}=1$) by the subset of
$X^{0,log}_{J}(\mathbb{R})$ made by elements
$((y_{1},\overline{y_{1}},y_{3}\dots y_{r}),(a_{r+1}\dots a_{n}))$ for which
$\prod_{i\geq 3}(y_{i})=1$, hence it is a product of an $S^{1}$-bundle and a
topological cover of degree $2^{r-3}$ over $X^{0}_{J}(\mathbb{R})$.

####  4.1.2. The case of a strictly semistable degeneration

Assume from now that $X\rightarrow C$ is a totally real strictly semistable
degeneration and endow it with the logarithmic structures as in Section 3.1.2.
Then we have a morphisms of real log-varieties $X^{log}\rightarrow C^{log}$
and $X_{0}^{log}\rightarrow 0^{log}$. Locally around $0$, the log structure
are isomorphic to one given in Example 4.1.1.1. Fix an orientation $(U,\phi)$
around $0$; see Section 2.3.

Recall the commutative diagram from Section 3.2.2

${X_{J,1}^{0,log}(\mathbb{C})}$${X_{J}^{0,log}(\mathbb{C})}$${X^{0}_{J}(\mathbb{C})}$${X_{0,1}^{log}(\mathbb{C})}$${X_{0}^{log}(\mathbb{C})}$${X_{0}(\mathbb{C})}$${\\{1\\}}$${0^{\log}(\mathbb{C})=S^{1}.}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$

The morphisms on the right part of the diagram are compatible with the
involutions acting on the various topological spaces appearing. Hence the
involutions acting on $X^{0,log}_{J}(\mathbb{C})$ and
$X^{log}_{0}(\mathbb{C})$ restrict to involutions on
$X^{0,log}_{J,1}(\mathbb{C})$ and $X_{0,1}^{log}(\mathbb{C})$ respectively,
making the whole diagram equivariant.

Taking fixed points we get a commutative diagram

${X_{J,1}^{0,log}(\mathbb{R})}$${X_{J}^{0,log}(\mathbb{R})}$${X^{0}_{J}(\mathbb{R})}$${X_{0,1}^{log}(\mathbb{R})}$${X_{0}^{log}(\mathbb{R})}$${X_{0}(\mathbb{R})}$${\\{1\\}}$${0^{\log}(\mathbb{R})=\\{\pm
1\\},}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$

where $X_{J,1}^{0,log}(\mathbb{R})$ and $X_{0,1}^{log}(\mathbb{R})$ are both
the fiber at $1$ of the morphisms $X_{J}^{0,log}(\mathbb{R})\rightarrow
X^{log}_{0}(\mathbb{R})\rightarrow\\{\pm 1\\}$ and the fixed points of the
involutions acting respectively on $X^{0,log}_{J,1}(\mathbb{C})$ and
$X_{1}^{0,log}(\mathbb{C})$. The main result that we are interested in and
which motivates the use of real logarithmic geometry, is the real analogue of
[NO10], which follows from the results in [Arg21] or in [Rau22].

######  Theorem 4.1.2.1 ([Arg21], [Rau22]).

The topological space $X^{log}_{0,1}(\mathbb{R})$ is homeomorphic to every
positive real smooth fiber near $0$ of the morphism $X\rightarrow C$.

###### Proof.

This is essentially the content of [Arg21, Proposition 7.4] or the one of
[Rau22, Theorem 1.1], which say 222We explain how to deduce this from the
results there.  • Even if [Arg21, Proposition 7.4] is stated in a very
particular case, the proof extends to nice (i.e. exact) proper and logarithmic
smooth morphism, since it (explicitly) mimics the proof of Theorem 4.1.2.1 in
[NO10]; • [Rau22, Theorem 1.1] is stated for morphisms around a disk. Since an
orientation around $0$ induces an equivariant isomorphism between a local
neighbourhood of $0$ with a disk and of a local neighbourhood of $0$ in
$C^{log}(\mathbb{R})$ with the real oriented blow-up, Rau’s results apply to
this situation. that $X^{log}(\mathbb{R})\rightarrow C^{log}(\mathbb{R})$ is a
locally trivial fibration. Therefore all smooth real fibers in the same
connected component of $C^{log}(\mathbb{R})$ are homeomorphic. Since the
orientation induces an equivariant isomorphism

$\pi^{-1}(U)\simeq[0,1)\times S^{1}\quad\text{so
that}\quad\pi_{\mathbb{R}}^{-1}(U(\mathbb{R}))\simeq[0,1)\times\\{\pm 1\\},$

and the fibers of $X^{log}(\mathbb{R})\rightarrow C^{log}(\mathbb{R})$ are
homeomorphic to the ones of $X(\mathbb{R})\rightarrow C(\mathbb{R})$ outside
$0$, the positive fibers of $X(\mathbb{R})\rightarrow C(\mathbb{R})$ near $0$
are all homeomorphic to $X^{log}_{0,1}(\mathbb{R})$. ∎

Hence, in order to study the Betti numbers of a positive smooth real general
fiber of $X(\mathbb{R})\rightarrow C(\mathbb{R})$, we can study the topology
of $X^{log}_{0,1}(\mathbb{R})$ via the morphism
$X^{log}_{0,1}(\mathbb{R})\rightarrow X_{0}(\mathbb{R})$.

###  4.2. Real stratifications

Let $\mathfrak{Z}:=\\{X^{0}_{\Delta}\\}_{\Delta\in P}$ be an algebraic
refinement of $\mathfrak{I}$ made by smooth real algebraic varieties and let
us retain the notation of Section 3.3. We endow $X^{0}_{\Delta}$ with the
structure of a real log-variety $X^{0,log}_{\Delta}$ by restricting the real
log structure of $X_{0}$ to $X^{0}_{\Delta}$. The most important properties of
$X^{0,log}_{\Delta}(\mathbb{R})$ and $X^{0,log}_{\Delta,1}(\mathbb{R})$, which
follows directly from the local description given in Example 3.2.1.1, are
summarized in the following remark.

######  Remark 4.2.1.

  1.   2. (1)

Example 3.2.1.1(4) shows that $X^{0,log}_{\Delta}(\mathbb{R})\rightarrow
X^{0}_{\Delta}(\mathbb{R})$ is a topological cover of degree $2^{|\Delta|}$.
In particular $X^{0,log}_{\Delta}(\mathbb{R})$ is a smooth
$C^{\infty}$-manifold of dimension $\textrm{dim}(X_{0})$.

  3. (2)

Example 3.2.1.1(4) shows that the morphism
$X^{0}_{\Delta,1}(\mathbb{R})\rightarrow X^{0}_{\Delta}(\mathbb{R})$ is a
topological cover of degree $2^{|\Delta|-1}$. In particular
$X^{0,log}_{\Delta,1}(\mathbb{R})$ is a smooth $C^{\infty}$-manifold of
dimension $\textrm{dim}(X_{0})$.

By Section 3.3, one has that $\\{X^{0,\log}_{\Delta,1}(\mathbb{C})\\}$ is a
stratification of $X^{log}_{0,1}(\mathbb{C})$. Since
$X^{log}_{0,1}(\mathbb{R})\subseteq X^{log}_{0,1}(\mathbb{C})$ is closed, also
$\\{X^{0,\log}_{\Delta,1}(\mathbb{R})\\}$ is a stratification of
$X^{log}_{0,1}(\mathbb{R})$.

We can now compute the Betti numbers of $X^{0,log}_{\Delta,1}(\mathbb{R})$ in
terms of the Betti numbers of $X^{0}_{\Delta}(\mathbb{R})$. This will be
important to transfer the assumptions $(a)$ and $(b)$ in Theorem 1.3.1 from
$X^{0}_{\Delta}(\mathbb{C})$ to $X^{0,log}_{\Delta,1}(\mathbb{C})$.

######  Lemma 4.2.2.

Assume that $H^{1}(X^{0}_{\Delta}(\mathbb{R}),\mathbb{Z}/2\mathbb{Z})=0$. Then
for every $i\geq 0$ one has

$b_{i}(X^{0,log}_{\Delta,1}(\mathbb{R}),A)=2^{|\Delta|-1}b_{i}(X^{0}_{\Delta}(\mathbb{R}),A).$

###### Proof.

By Remark 4.2.1, $X^{0}_{\Delta,1}(\mathbb{R})\rightarrow
X^{0}_{\Delta}(\mathbb{R})$ is a topological cover of degree $2^{|\Delta|-1}$.
Since $H^{1}(X^{0}_{\Delta}(\mathbb{R}),\mathbb{Z}/2\mathbb{Z})=0$, this cover
is the trivial one. Hence $X^{0}_{\Delta,1}(\mathbb{R})$ is the disjoint union
of $2^{|\Delta|-1}$ copies of $X^{0}_{\Delta}$, and the lemma follows. ∎

###  4.3. Equivariant cohomology

In this Section we recall a few generalities on equivariant cohomology as
presented in [BBF+60, Chapter IV]. This is an important tool in the proof of
Theorem 3.4.2 to compare complex and real Betti numbers. Since [BBF+60,
Chapter IV] deals with the case of cohomology and we need the compact
cohomology case, we give some details.

In this section we consider $G:=\mathbb{Z}/2\mathbb{Z}$ as an abelian group
and we let $X$ be a locally compact Hausdorff topological space on which $G$
acts. We write $X^{G}$ for the set of fixed points for the action of $G$ and
$X/G$ for the quotient of $X$. We assume that
$H_{c}^{i}(X,\mathbb{Z}/2\mathbb{Z})$ is finite dimensional for all $i\geq 0$
and that there exists some $n_{X}\in\mathbb{N}$ such that
$H_{c}^{i}(X,\mathbb{Z}/2\mathbb{Z})=0$ for all $i>n_{X}$.

####  4.3.1. Group cohomology

Let $M$ be a $\mathbb{Z}/2\mathbb{Z}$-vector space on which $G$ acts. Write
$H^{i}(G,M)$ for its $i^{th}$-cohomology group. If $M$ is finite, then
$H^{i}(G,M)$ is finite, there is an inequality
$\textrm{dim}(H^{i}(G,M))\leq\textrm{dim}(M)$ and if the action of $G$ on $M$
is trivial then $H^{i}(G,M)=M$ (see e.g. [BBF+60, IV, 2.1, Pag. 50]).

####  4.3.2. Classifying space

We fix an auxiliary integer $N\gg 0$, e.g. $N\geq n_{x}+1$. Let
$BG:=\mathbb{R}\mathbb{P}^{N}$ be the real projective space of dimension $N$,
which is the classifying space for $G$ and for dimension $N$ ([Bor54, Section
18]), in the sense that $\pi_{1}(BG)\simeq G$ and for every locally constant
sheaf $\mathcal{F}$ on $BG$ and every $x\in BG$, one has
$H^{i}(BG,\mathcal{F})=H^{i}(G,\mathcal{F}_{x})$ for all $i\leq N$. Let
$EG:=S^{N}$ be the unit sphere of dimension $N$, which is the universal cover
$EG\rightarrow BG$ of $BG$, endowed with the natural $G$-action.

####  4.3.3. G-equivariant cohomology with compact support

Define $X_{G}:=(X\times EG)/G$ where $G$ acts on $X\times EG$ diagonally.
Then, for every coefficient ring $A$, the equivariant cohomology of $X$ with
compact support and coefficient in $A$ is defined by
$H^{i}_{G,c}(X,A):=H_{c}^{i}(X_{G},A)$. It is important to keep in mind the
following morphisms:

${X/G}$${X_{G}}$${BG,}$$\scriptstyle{h}$$\scriptstyle{g}$

where $g:X_{G}\rightarrow X/G$ (resp. $h:X_{G}\rightarrow BG$) is induced by
the first (resp. the second) projection $X\times EG\rightarrow X\rightarrow
X/G$ (resp. $X\times EG\rightarrow EG\rightarrow BG$).

####  4.3.4. Smith-Thom spectral sequence

The morphism $h:X_{G}\rightarrow BG$ has fibers $X$, hence the Leray spectral
sequence with compact support for $h$ reads

${}_{X}E_{2}^{a,b}:=H_{c}^{a}(BG,H_{c}^{b}(X,A))\Rightarrow
H^{a+b}_{G,c}(X,A),$

where $H_{c}^{b}(X,A)$ is the constant local system associated to the group
$H_{c}^{b}(X,A)$. Since $BG$ is compact and the classifying space for $G$ and
for dimension $N$, one has
$H_{c}^{a}(BG,H_{c}^{b}(X,A))=H^{a}(BG,H_{c}^{b}(X,A))=H^{a}(G,H_{c}^{b}(X,A))$;
see Section 4.3.2. Therefore we get a spectral sequence

(4.3.4.1) $^{ST}_{X}E_{2}^{a,b}:=H^{a}(G,H_{c}^{b}(X,A))\Rightarrow
H^{a+b}_{G,c}(X,A),$

which we call the the Smith-Thom spectral sequence (with compact support).

####  4.3.5. Maximality

Recall that, by [BBF+60, Application 3.7(b), Pag 54], for $n_{X}<m<N$, the
inclusion $(X^{G})_{G}=X^{G}\times BG\subseteq X_{G}$ induces a canonical
isomorphism $H^{m}_{G,c}(X,\mathbb{Z}/2\mathbb{Z})\simeq H^{m}_{c}(X^{G}\times
BG,\mathbb{Z}/2\mathbb{Z})\simeq H_{c}^{*}(X^{G},\mathbb{Z}/2\mathbb{Z})$.
Since
$\textrm{dim}(H^{a}(G,H_{c}^{b}(X,\mathbb{Z}/2\mathbb{Z}))\leq\textrm{dim}(H_{c}^{b}(X,\mathbb{Z}/2\mathbb{Z}))$,
the spectral sequence (4.3.4.1) gives an inequality

(4.3.5.1) $b_{c,*}(X^{G})\leq b_{c,*}(X),$

where $b_{c,*}$ denotes the total Betti number with compact support. We can
now recall the definition of maximality in this context.

######  Definition 4.3.5.2.

We say that $X$ is maximal (in the sense of Smith-Thom) if
$b_{c,*}(X^{G})=b_{c,*}(X)$.

The Smith-Thom spectral sequence (4.3.4.1), together with the inequality
$|H^{a}(G,H_{c}^{b}(X,\mathbb{Z}/2\mathbb{Z}))|\leq|H_{c}^{b}(X,\mathbb{Z}/2\mathbb{Z})|$,
implies that $X$ is maximal if and only if ${}^{ST}_{X}E_{2}^{a,b}$
degenerates at $E_{2}$ and the action of $G$ on
$H_{c}^{b}(X,\mathbb{Z}/2\mathbb{Z})$ is trivial for all $b$; see e.g.
[BBF+60, 4.1, Pag. 55].

####  4.3.6. Filtrations and maximality

Assume now that $X$ is maximal, so that the spectral sequence degenerates at
$E_{2}$ and the action of $G$ on $H_{c}^{b}(X,\mathbb{Z}/2\mathbb{Z})$ is
trivial for all $b$.

Then, for every $n_{X}<m<N$ the spectral sequence (4.3.4.1) induces a
canonical decreasing filtration

$0=\text{}_{m}F_{m+1}\subseteq\text{}_{m}F_{m}\subseteq\dots\subseteq\text{}_{m}F_{i+1}\subseteq\text{}_{m}F_{i}\subseteq\dots\subseteq\text{}_{m}F_{0}=H_{c}^{*}(X^{G},\mathbb{Z}/2\mathbb{Z})(\simeq
H_{G,c}^{m}(X,\mathbb{Z}/2\mathbb{Z}))$

such that

${}_{m}F_{i}/_{m}F_{i+1}\simeq\text{}^{ST}_{X}E_{2}^{i,m-i}=H^{i}(G,H_{c}^{m-i}(X,\mathbb{Z}/2\mathbb{Z}))=H_{c}^{m-i}(X,\mathbb{Z}/2\mathbb{Z}),$

where the last equality follows from the fact that the action of $G$ on
$H_{c}^{m-i}(X,\mathbb{Z}/2\mathbb{Z})$ is trivial; see Section 4.3.1.

##  5\. Proof of Theorem 3.4.2(1)

Retain the notation and assumptions of Theorem 3.4.2(1). Up to a choice of an
orientation $(U,\psi)$ around $0\in C(\mathbb{R})$, we can assume that
$X_{t}$, for some $t$ near $0$, is positive; see Section 2.3. In order to
simplify the notation, in this section, for every topological space $Z$, we
write

$H^{i}(Z):=H^{i}(Z,\mathbb{Z}/2\mathbb{Z});\quad
H^{i}_{c}(Z):=H_{c}^{i}(Z,\mathbb{Z}/2\mathbb{Z});\quad
b_{i,c}:=\textrm{dim}(H^{i}_{c}(Z));\quad b_{c,*}:=\sum_{i}b_{i,c}.$

As explained in Section 1.4 the strategy is the following:

  1. (1)

By using the work done in the previous sections, we construct a topological
space $X_{0,1}^{log}(\mathbb{C})$ endowed with an involution and with a
morphism $X_{0,1}^{log}(\mathbb{C})\rightarrow X_{0}(\mathbb{C})$ compatible
with the involutions, such that $X_{0,1}^{log}(\mathbb{C})$ is homeomorphic to
the general smooth fiber of the morphism $X(\mathbb{C})\rightarrow
C(\mathbb{C})$ and the set of fixed points $X_{0,1}^{log}(\mathbb{R})$ is
homeomorphic to a positive real fiber near $0$;

  2. (2)

By using the stratification $\\{X^{0}_{\Delta}\\}$ and the geometry of the
morphism $X_{0,1}^{log}(\mathbb{C})\rightarrow X_{0}(\mathbb{C})$, we stratify
$X_{0,1}^{log}(\mathbb{C})$ and $X_{0,1}^{log}(\mathbb{R})$ in a compatible
way and we transfer the assumptions (a)-(b) of Theorem 3.4.2 to the these
stratifications;

  3. (3)

Using the assumption $(a)$ of Theorem 3.4.2 and the spectral sequence for a
stratified space, we construct a chain complex
${}^{\mathbb{R}}E_{1}^{\bullet,0}$ whose cohomology computes the Betti numbers
of $X_{0,1}^{log}(\mathbb{R})$;

  4. (4)

Using the assumption $(b)$ of Theorem 3.4.2 and the spectral sequence for
equivariant cohomology (combined with the assumption $(a)$), we construct a
filtration $F^{\bullet}_{i}\subseteq$ ${}^{\mathbb{R}}E_{1}^{\bullet,0}$ such
that $F^{\bullet}_{i}/F^{\bullet}_{i+1}$ is isomorphic to a complex depending
only on the complex geometry of $X\rightarrow C$;

  5. (5)

We conclude the proof using the spectral sequence for a filtered complex.

###  5.1. Construction of $X_{0,1}^{log}(\mathbb{C})$ and
$X_{0,1}^{log}(\mathbb{R})$

We endow $X\rightarrow C$ with the real logarithmic structure defined in
Section 4.1.2 (see also Sections 3.1.2 and 3.2.2) and recall the commutative
diagram

${X_{0,1}^{log}(\mathbb{C})}$${X_{0}^{log}(\mathbb{C})}$${X_{0}(\mathbb{C})}$${\\{1\\}}$${0^{\log}(\mathbb{C})=S^{1},}$$\scriptstyle{\alpha}$$\scriptstyle{\pi}$

compatible with the involutions and its real counterpart

${X_{0,1}^{log}(\mathbb{R})}$${X_{0}^{log}(\mathbb{R})}$${X_{0}(\mathbb{R})}$${\\{1\\}}$${0^{\log}(\mathbb{R})=\\{\pm
1\\}.}$$\scriptstyle{\alpha_{\mathbb{R}}}$$\scriptstyle{\pi_{\mathbb{R}}}$

By Theorems 3.2.2.1 and 4.1.2.1, the general smooth fiber of
$X(\mathbb{C})\rightarrow C(\mathbb{C})$ is homeomorphic to
$X_{0,1}^{log}(\mathbb{C})$ and the general positive smooth fiber near $0$ of
$X(\mathbb{R})\rightarrow C(\mathbb{R})$ is homeomorphic to
$X_{0,1}^{log}(\mathbb{R})$. Hence, to prove Theorem 3.4.2(1), we can replace
$X_{t}(\mathbb{R})$ with $X_{0,1}^{log}(\mathbb{R})$.

###  5.2. Stratifications

As in Sections 3.3 and 4.2, the collections

$\\{X^{0,log}_{\Delta,1}(\mathbb{C}):=\alpha^{-1}(X^{0}_{\Delta}(\mathbb{C}))\\}\quad\text{and}\quad\\{X^{0,log}_{\Delta,1}(\mathbb{R}):=\alpha_{\mathbb{R}}^{-1}(X^{0}_{\Delta}(\mathbb{R}))\\}$

give equivariant stratifications of $X^{log}_{0,1}(\mathbb{C})$ and
$X^{log}_{0,1}(\mathbb{R})$ respectively. Recall that, by Remarks 3.3.1 and
4.2.1, one has that

  1. (1)

$X^{log}_{0,1}(\mathbb{C})$ is a smooth $C^{\infty}$-manifold of dimension
$2\textrm{dim}(X^{0}_{\Delta})+|\Delta|-1$;

  2. (2)

$X^{log}_{0,1}(\mathbb{R})$ is a smooth $C^{\infty}$-manifolds of dimension
$\textrm{dim}(X^{0}_{\Delta})$.

The following is the key lemma that allows us to transfer the assumptions (a)
and (b) of Theorem 3.4.2 from the stratifications of $X_{0}$ to the ones of
$X^{\log}_{0,1}(\mathbb{C})$ and $X^{\log}_{0,1}(\mathbb{R})$.

######  Lemma 5.2.1.

  1.   2. (a)

$H^{i}(X^{0,\log}_{\Delta,1}(\mathbb{R}))=0$ for $i\geq 1$;

  3. (b)

$X^{0,\log}_{\Delta,1}(\mathbb{C})$ is a maximal variety.

###### Proof.

By Remark 4.2.1, one has that $X^{0,\log}_{\Delta,1}(\mathbb{R})\rightarrow
X^{0}_{\Delta}(\mathbb{R})$ is a topological cover of degree $2^{|\Delta|-1}$.
Since $H^{1}(X^{0}_{\Delta}(\mathbb{R}),\mathbb{Z}/2\mathbb{Z})=0$ by
assumption, this cover is the trivial one. Hence
$X^{0}_{\Delta,1}(\mathbb{R})$ is the disjoint union of $2^{|\Delta|-1}$
copies of $X^{0}_{\Delta}(\mathbb{R})$, so that

(5.2.2) $b_{i}(X^{0,\log}_{\Delta,1}(\mathbb{C}))=0\quad\text{for $i\geq
1$}\quad\text{and}\quad
b_{*}(X^{0,\log}_{\Delta,1}(\mathbb{C}))=2^{|\Delta|-1}b_{*}(X^{0}_{\Delta}(\mathbb{R})),$

and, in particular, we get (a).

To prove (b), we first observe that, by the Smith-Thom inequality (1.1.1), one
has

$b_{*}(X^{0,log}_{\Delta,1}(\mathbb{R}))\leq
b_{*}(X^{0,log}_{\Delta,1}(\mathbb{C})),$

hence it is enough to prove that

$b_{*}(X^{0,log}_{\Delta,1}(\mathbb{C}))\leq
b_{*}(X^{0,log}_{\Delta,1}(\mathbb{R})).$

First of all, one has the following relations

$b_{*}(X^{0,log}_{\Delta,1}(\mathbb{R}))=2^{|\Delta|-1}b_{*}(X^{0}_{\Delta}(\mathbb{R}))=2^{|\Delta|-1}b_{*}(X^{0}_{\Delta}(\mathbb{C})),$

where the first equality follows from (5.2.2) and the second one from the
assumption (b) of Theorem 3.4.2 on $X^{0}_{\Delta}(\mathbb{C}).$ Hence, in
order to conclude, it is enough show that

$b_{*}(X^{0,log}_{\Delta,1}(\mathbb{C}))\leq
2^{|\Delta|-1}b_{*}(X^{0}_{\Delta}(\mathbb{C})).$

By Example 3.2.1.1(7) the morphism
$\alpha:X^{0,\log}_{\Delta,1}(\mathbb{C})\rightarrow
X^{0}_{\Delta}(\mathbb{C})$ is a locally trivial fibration with fiber
$(S^{1})^{|\Delta|-1}$. The Leray spectral sequence associated to such
morphism $\alpha$

$E_{2}^{a,b}:=H^{a}(X^{0}_{\Delta}(\mathbb{C}),R^{b}\alpha_{*}\mathbb{Z}/2\mathbb{Z})\Rightarrow
H^{a+b}(X^{0,\log}_{\Delta,1}(\mathbb{C}))$

shows that

$b_{i}(X^{0,\log}_{\Delta,1}(\mathbb{C}))\leq\sum_{a}\textrm{dim}(H^{a}(X^{0}_{\Delta}(\mathbb{C}),R^{i-a}\alpha_{*}\mathbb{Z}/2\mathbb{Z})).$

By Lemma 5.2.3, proved later, the locally constant sheaf
$R^{i}\alpha_{*}\mathbb{Z}/2\mathbb{Z}$ is constant so that

$R^{i}\alpha_{*}\mathbb{Z}/2\mathbb{Z}\simeq(\mathbb{Z}/2\mathbb{Z})^{\textrm{dim}(\displaystyle{H^{i}((S^{1})^{|\Delta|-1})})}.$

Hence

$\dim(H^{a}(X^{0}_{\Delta}(\mathbb{C}),R^{i-a}\alpha_{*}\mathbb{Z}/2\mathbb{Z}))=\dim(H^{a}(X^{0}_{\Delta}(\mathbb{C}),\mathbb{Z}/2\mathbb{Z}^{\dim(\displaystyle{H^{i-a}((S^{1})^{|\Delta|-1})})}))=$
$=b_{a}(X^{0}_{\Delta}(\mathbb{C}))\dim(H^{i-a}((S^{1})^{|\Delta|-1}))=b_{a}(X^{0}_{\Delta}(\mathbb{C}))\binom{|\Delta|-1}{i-a}.$

This implies that

$\sum_{i}b_{i}(X^{0,\log}_{\Delta,1}(\mathbb{C}))\leq\sum_{i}\sum_{a}b_{a}(X^{0}_{\Delta}(\mathbb{C}))\binom{|\Delta|-1}{i-a}=\sum_{a}\binom{|\Delta|-1}{a}\sum_{i}b_{i-a}(X^{0}_{\Delta}(\mathbb{C}))$
$\leq\sum_{a}\binom{|\Delta|-1}{a}\sum_{i}b_{i}(X^{0}_{\Delta}(\mathbb{C}))=2^{|\Delta|-1}\sum_{a}b_{a}(X^{0}_{\Delta}(\mathbb{C})).$

∎

######  Lemma 5.2.3.

$R^{i}\alpha_{*}\mathbb{Z}/2\mathbb{Z}$ is a constant sheaf.

###### Proof.

Let the notation be as in the following canonical commutative diagram

${X^{0,\log}_{\Delta,1}(\mathbb{C})}$${X^{0,\log}_{\Delta}(\mathbb{C})}$${X_{0}^{\log}(\mathbb{C})}$${X^{0}_{\Delta}(\mathbb{C})}$${X_{0}(\mathbb{C}).}$$\scriptstyle{j}$$\scriptstyle{\alpha}$$\scriptstyle{h}$$\scriptstyle{\pi}$

We first show that $R^{i}\pi_{*}\mathbb{Z}/2\mathbb{Z}$ is constant. By
[KN99a, Lemma 1.5], one has that
$R^{i}\pi_{*}\mathbb{Z}/2\mathbb{Z}\simeq(\bigwedge^{i}\overline{M}_{X^{0,\log}_{\Delta}}^{gp})\otimes\mathbb{Z}/2\mathbb{Z}$,
so that it is enough to show that $\overline{M}_{X^{0,\log}_{\Delta}}^{gp}$ is
constant. By [Nak00, Lemma 1.8.1] one has

$\overline{M}_{X_{0}^{\log}}^{gp}\simeq\bigoplus_{i\in I}(X_{i}\rightarrow
X)_{*}\mathbb{Z}$

so that

$\overline{M}_{X^{0,\log}_{\Delta}}^{gp}=h^{*}\overline{M}_{X_{0}^{\log}}^{gp}=\bigoplus_{i\in
I}h^{*}(X_{i}\rightarrow X)_{*}\mathbb{Z}=\bigoplus_{i\in
I\text{}|\text{}X^{0}_{\Delta}\subseteq X_{i}}\mathbb{Z}.$

We now deduce that this implies that $R^{i}\alpha_{*}\mathbb{Z}/2\mathbb{Z}$
is constant. Since $j$ is a closed immersion
$R^{i}\alpha_{*}\mathbb{Z}/2\mathbb{Z}=R^{i}\pi_{*}(j_{*}\mathbb{Z}/2\mathbb{Z})$,
therefore it is enough to show that
$R^{i}\pi_{*}\mathbb{Z}/2\mathbb{Z}\rightarrow
R^{i}\pi_{*}(j_{*}\mathbb{Z}/2\mathbb{Z})$ is surjective. This can be checked
locally on $X^{0}_{\Delta}(\mathbb{C})$. By Example 3.2.1.1(4), we can assume
that $X^{0,log}_{\Delta}(\mathbb{C})\simeq(S^{1})^{|\Delta|}\times
X^{0}_{\Delta}(\mathbb{C})$ and $X^{0,log}_{\Delta,1}(\mathbb{C})$ is the
subset defined by the product of the first $|\Delta|$ components equal to $1$.
In this case the result follows from the fact that the inclusion
$X^{0,log}_{\Delta,1}(\mathbb{C})\subseteq X^{0,log}_{\Delta}(\mathbb{C})$ has
a (non-canonical) retraction, depending on the choice of one of the first
$|\Delta|$ components. ∎

###  5.3. Computations of the real Betti numbers

Let us consider the natural spectral sequence (see e.g. [Pet16] and [Pet17,
(3) pag. 2527])

(5.3.1)
$^{\mathbb{R}}E_{1}^{a,b}:=\bigoplus_{\textrm{dim}(X^{0}_{\Delta})=a}H_{c}^{a+b}(X^{0,log}_{\Delta,1}(\mathbb{R}))\Rightarrow
H_{c}^{a+b}(X_{0,1}^{log}(\mathbb{R})).$

Since $X^{0,log}_{\Delta,1}(\mathbb{R})$ is a smooth $C^{\infty}$-manifolds of
dimension $\textrm{dim}(X^{0}_{\Delta})$, we can apply Poincaré duality to
deduce that

$\bigoplus_{\textrm{dim}(X^{0}_{\Delta})=a}H_{c}^{a+b}(X^{0,log}_{\Delta,1}(\mathbb{R}))\simeq\bigoplus_{\textrm{dim}(X^{0}_{\Delta})=a}H^{-b}(X^{0,log}_{\Delta,1}(\mathbb{R})).$

By Lemma 5.2.1(a), we get that ${}^{\mathbb{R}}E_{1}^{a,b}=0$ if $b\neq 0$.
Hence the first page of the spectral sequence (5.3.1) reduces to the line
${}^{\mathbb{R}}E_{1}^{\bullet,0}$:

(5.3.2)
$\small{\dots\rightarrow\bigoplus_{\textrm{dim}(X^{0}_{\Delta})=a-1}H_{c}^{a-1}(X^{0,log}_{\Delta,1}(\mathbb{R}))\xrightarrow{d_{a-1}}\bigoplus_{\textrm{dim}(X^{0}_{\Delta})=a}H_{c}^{a}(X^{0,log}_{\Delta,1}(\mathbb{R}))\xrightarrow{d_{a}}\bigoplus_{\textrm{dim}(X^{0}_{\Delta})=a+1}H_{c}^{a+1}(X^{0,log}_{\Delta,1}(\mathbb{R}))\rightarrow\dots}$

so that from (5.3.1) and Theorem 4.1.2.1, one gets

(5.3.3)
$H_{c}^{a}(X_{t}(\mathbb{R}))=H^{a}(\text{}^{\mathbb{R}}E_{1}^{\bullet,0}),$

for $t$ positive and near $0$. Hence, in order to conclude the proof of
Theorem 3.4.2,(1) it is enough to bound the dimension of
$H^{a}(\text{}^{\mathbb{R}}E_{1}^{\bullet,0})$.

###  5.4. Construction of the filtration

Since $X^{0,log}_{\Delta,1}(\mathbb{C})$ and
$X^{0,log}_{\Delta,1}(\mathbb{R})$ are smooth $C^{\infty}$-varieties, by
Poincaré duality and Lemma 5.2.1(b), one has that
$X^{0,log}_{\Delta,1}(\mathbb{C})$ is maximal. The dimension of
$X^{0,log}_{\Delta,1}(\mathbb{R})$ is $\dim(X^{0}_{\Delta})$, therefore we
have

$H_{c}^{\textrm{dim}(\Delta)}(X^{0,log}_{\Delta,1}(\mathbb{R})))=H_{c}^{*}(X^{0,log}_{\Delta,1}(\mathbb{R})).$

In order to get a filtration of
$H_{c}^{\textrm{dim}(X^{0}_{\Delta})}(X^{0,log}_{\Delta,1}(\mathbb{R}))$, we
use the tools of Section 4.3.6. Once we set $m=2\dim(X_{0})+1$ and choose
$N\gg m$, one gets (using the construction of Section 4.3.6) a decreasing
filtration $\text{}_{m}F_{\Delta,\bullet}$

$\dots\subseteq\text{}_{m}F_{\Delta,i+1}\subseteq\text{}_{m}F_{\Delta,i}\subseteq\dots\subseteq\text{}_{m}F_{\Delta,0}=H^{\dim(X^{0}_{\Delta})}_{c}(X^{0,log}_{\Delta,1}(\mathbb{R}))\big{(}\simeq
H_{G,c}^{m}(X^{0,log}_{\Delta,1}(\mathbb{C})\big{)}$

with graded quotients

$\mathrm{Gr}_{{}_{m}F_{\Delta,\bullet}}^{i}:=\text{}_{m}F_{\Delta,i}/_{m}F_{\Delta,i+1}\simeq
H_{c}^{m-i}(X^{0,log}_{\Delta,1}(\mathbb{C}))$

Let $X^{0}_{\Delta}$ and $X^{0}_{\Delta^{\prime}}$ be two strata such that
$\textrm{dim}(X^{0}_{\Delta})=a-1=\textrm{dim}(X^{0}_{\Delta^{\prime}})-1$ and
$X^{0}_{\Delta}\subseteq X_{\Delta^{\prime}}$ so that there is a commutative
diagram

${H_{c}^{m}(X^{0,\log}_{\Delta,1}(\mathbb{C})_{G})}$${H_{c}^{m+1}(X^{0,\log}_{\Delta^{\prime},1}(\mathbb{C})_{G})}$${H_{c}^{m}(X^{0,\log}_{\Delta,1}(\mathbb{R})\times
BG)}$${H_{c}^{m+1}(X^{0,\log}_{\Delta^{\prime},1}(\mathbb{R})\times
BG)}$${\underset{k}{\bigoplus}H_{c}^{m-k}(X^{0,\log}_{\Delta,1}(\mathbb{R}))\otimes
H_{c}^{k}(BG)}$${\underset{k}{\bigoplus}H_{c}^{m+1-k}(X^{0,\log}_{\Delta^{\prime},1}(\mathbb{R}))\otimes
H_{c}^{k}(BG)}$${H_{c}^{a-1}(X^{0,\log}_{\Delta,1}(\mathbb{R}))\otimes
H_{c}^{m-a+1}(BG)}$${H_{c}^{a}(X^{0,\log}_{\Delta^{\prime},1}(\mathbb{R}))\otimes
H_{c}^{m-a+1}(BG)}$${H_{c}^{a-1}(X^{0,\log}_{\Delta,1}(\mathbb{R}))}$${H_{c}^{a}(X^{0,\log}_{\Delta^{\prime},1}(\mathbb{R})),}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where, for $\\#\in\\{\Delta,\Delta^{\prime}\\}$,

  * •

the horizontal maps are induced from the stratifications
$\\{X^{0,\log}_{\Delta,1}(\mathbb{C})_{G}\\}$,
$\\{X^{0,\log}_{\Delta,1}(\mathbb{R})\times BG\\}$ and
$\\{X^{0,\log}_{\Delta,1}(\mathbb{R})\\}$ of $X^{\log}_{0,1}(\mathbb{C})_{G}$,
$X^{\log}_{0,1}(\mathbb{R})\times BG$ and $X^{\log}_{0,1}(\mathbb{R})$
respectively;

  * •

the upper vertical maps are induced by the inclusions
$X^{0,\log}_{1,\\#}(\mathbb{R})\times BG\subseteq
X^{0,\log}_{1,\\#}(\mathbb{C})_{G}$ and they are isomorphisms by [BBF+60, IV,
Pag. 55, Application 3.7(b)] (see Section 4.3.5);

  * •

the remaining top-to-bottom vertical maps are induced respectively by the
Kunneth formula, the fact that $H_{c}^{i}(X^{0,\log}_{1,\\#}(\mathbb{R}))=0$
for $i\neq\textrm{dim}(X^{0}_{\Delta})$ and
$H_{c}^{m-a+1}(BG)\simeq\mathbb{Z}/2\mathbb{Z}$.

Since the morphisms
$H_{c}^{i}(X^{0,\log}_{\Delta,1}(\mathbb{C})_{G})\rightarrow
H_{c}^{i+1}(X^{0,\log}_{\Delta^{\prime},1}(\mathbb{C})_{G})$ and
$H_{c}^{i}(X^{0,\log}_{\Delta,1}(\mathbb{C}))\rightarrow
H_{c}^{i+1}(X^{0,\log}_{\Delta^{\prime},1}(\mathbb{C}))$ give rise to a
morphism of spectral sequences

${{}^{ST}_{X^{0,\log}_{\Delta,1}(\mathbb{C})}E_{2}^{a,b}:=H^{a}(G,H_{c}^{b}(X^{0,\log}_{\Delta,1}(\mathbb{C})))}$${H_{c}^{a+b}(X^{0,\log}_{\Delta,1}(\mathbb{C})_{G})}$${{}^{ST}_{X^{0,\log}_{\Delta^{\prime},1}(\mathbb{C})}E_{2}^{a,b+1}:=H^{a}(G,H_{c}^{b+1}(X^{0,\log}_{\Delta^{\prime},1}(\mathbb{C})))}$${H_{c}^{a+b+1}(X^{0,\log}_{\Delta^{\prime},1}(\mathbb{C})_{G}),}$

the morphism $H^{a-1}_{c}(X^{0,log}_{\Delta,1}(\mathbb{R}))\rightarrow
H^{a}_{c}(X^{0,log}_{\Delta^{\prime},1}(\mathbb{R}))$ sends
${}_{m}F_{\Delta,i}$ to ${}_{m+1}F_{\Delta^{\prime},i}$.

Therefore, if we set

${}_{m}F^{p}_{i}:=\bigoplus_{\textrm{dim}(X^{0}_{\Delta})=p}\text{
}_{m+p}F_{\Delta,i}$

we get a decreasing filtration $F^{\bullet}_{\bullet}$ of
${}^{\mathbb{R}}E_{1}^{\bullet,0}$ (Section 5.3)

${{}_{m}F^{a-1}_{i+1}}$${{}_{m}F^{a}_{i+1}}$${{}_{m}F^{a+1}_{i+1}}$${{}_{m}F^{a-1}_{i}}$${{}_{m}F^{a}_{i}}$${{}_{m}F^{a+1}_{i}}$${\dots}$${\dots}$${\dots}$${\dots\underset{\textrm{dim}(X^{0}_{\Delta})=a-1}{\bigoplus}H_{c}^{a-1}(X^{0,log}_{\Delta,1}(\mathbb{R}))}$${\underset{\textrm{dim}(X^{0}_{\Delta})=a}{\bigoplus}H_{c}^{a}(X^{0,log}_{\Delta,1}(\mathbb{R}))}$${\underset{\textrm{dim}(X^{0}_{\Delta})=a+1}{\bigoplus}H_{c}^{a+1}(X^{0,log}_{\Delta,1}(\mathbb{R}))\dots}$

such that the graded quotient

$\mathrm{Gr}_{F_{\bullet}}^{i,\bullet}:=\text{ }_{m}F^{\bullet}_{i}/\text{
}_{m}F^{\bullet}_{i+1}$

is the complex ${}^{\mathbb{C}}E_{1}^{\bullet,m-i}$ of Section 3.4

${{}_{m}F^{a-1}_{i}/_{m}F_{i+1}^{a-1}}$${{}_{m}F^{a}_{i}/_{m}F_{i+1}^{a}}$${{}_{m}F^{a+1}_{i}/_{m}F_{i+1}^{a+1}}$${\underset{\textrm{dim}(X^{0}_{\Delta})=a-1}{\bigoplus}H_{c}^{m+a-i-1}(X^{0,log}_{\Delta,1}(\mathbb{C}))}$${\underset{\textrm{dim}(X^{0}_{\Delta})=a}{\bigoplus}H_{c}^{m+a-i}(X^{0,log}_{\Delta,1}(\mathbb{C}))}$${\underset{\textrm{dim}(X^{0}_{\Delta})=a+1}{\bigoplus}H_{c}^{m+a-i+1}(X^{0,log}_{\Delta,1}(\mathbb{C})).}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

###  5.5. End of the proof

Recall the spectral sequence for a filtered complex

${}^{F}E_{1}^{r,s}:=H^{r+s}(\mathrm{Gr}_{F_{\bullet}}^{r,\bullet})\simeq
H^{r+s}(\text{}^{\mathbb{C}}E_{1}^{\bullet,m-r})\Rightarrow
H^{r+s}(^{\mathbb{R}}E_{1}^{\bullet,0}).$

Since

$H^{r+s}(\text{}^{\mathbb{R}}E_{1}^{\bullet,0})\simeq
H^{r+s}(X_{0,1}^{\log}(\mathbb{R}))\simeq H^{r+s}(X_{t}(\mathbb{R})),$

where the first isomorphism follows from (5.3.3) and second from Theorem
4.1.2.1 (and the compactness of $X_{t}(\mathbb{R})$), we get that

$\textrm{dim}(H^{p}(X_{t}(\mathbb{R}),\mathbb{Z}/2\mathbb{Z}))=\sum_{q}\textrm{dim}(\text{}^{F}E_{\infty}^{q,p-q})\leq\sum_{q}\textrm{dim}(\text{}^{F}E_{1}^{q,p-q})=\sum_{q}\textrm{dim}(H^{p}(\text{}^{\mathbb{C}}E_{1}^{\bullet,m-q}))=\sum_{q}\textrm{dim}(H^{p}(\text{}^{\mathbb{C}}E_{1}^{\bullet,q})).$

##  6\. Proof of Theorem 3.4.2(2)

###  6.1. Notation

Retain the notation and the assumption of Theorem 3.4.2(2). We set

$X^{\log}_{0,1}(\mathbb{C}):=X^{\log}_{0,1}\quad,\quad
X^{0,\log}_{\Delta,1}(\mathbb{C}):=\widetilde{X}^{0}_{\Delta}$

and for every topological space $Z$ we write $H^{i}(Z):=H^{i}(Z,\mathbb{Q})$.
Recall that there are morphisms

(6.1.1) $\pi:X^{\log}_{0,1}\rightarrow
X_{0}\quad\text{and}\quad\pi_{\Delta}:\widetilde{X}^{0}_{\Delta}\rightarrow
X^{0}_{\Delta}$

and stratifications

(6.1.2) $\coprod
X^{0}_{\Delta}=X_{0}\quad\text{and}\quad\coprod\widetilde{X}^{0}_{\Delta}=X^{log}_{0,1}.$

For a $\mathbb{Q}$-mixed Hodge structure $V$, we write $\mathrm{Gr}^{W}_{i}$
for its weight $i$-graded piece.

Fix $q\in\mathbb{N}$.

###  6.2. Strategy

We detail the strategy to prove Theorem 3.4.2(2).

  1. (1)

In order to prove point $(ii)$, what one needs to do is to transfer the
assumption on the torsion freeness on the cohomology of $X^{0}_{\Delta}$ to
the one of $\widetilde{X}^{0}_{\Delta}$. This reduces to study the Leray
spectral sequence with integer coefficient for the morphism
$\pi_{\Delta}:\widetilde{X}^{0}_{\Delta}\rightarrow X^{0}_{\Delta}$ and to
show that it degenerates in $E_{2}$ (Lemma 6.4.1.1).

  2. (2)

We then deal with point $(i)$. This involves a careful analysis of different
spectral sequences coming from geometry. Following [IKMZ19], we first show
that the $(p,q)$-hodge number of the general fiber can be computed as the
weight $2q$-part of the limiting Hodge structure on $H^{p+q}(X^{\log}_{0,1})$
(Lemma 6.4.2.3);

  3. (3)

The Leray spectral sequence ${}^{L}E_{2}$ for
$X^{\log}_{0,1}\displaystyle{\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}}X_{0}$
is a spectral sequence of mixed Hodge structure which degenerates in
${}^{L}E_{3}$, hence $\mathrm{Gr}^{W}_{2q}H^{p+q}$ can be computed via its
second page (Section 6.3.3);

  4. (4)

We then filter the complex $\text{}^{\mathbb{C}}E_{1}^{\bullet,q}$ by using
the filtration induced on every term by the Leray spectral sequence for
$\widetilde{X}^{0}_{\Delta}\displaystyle{\stackrel{{\scriptstyle\pi_{\Delta}}}{{\longrightarrow}}}X^{0}_{\Delta}$
and we consider the spectral sequence ${}^{F}E_{1}$ for the filtered complex
(Section 6.5.1);

  5. (5)

Exploiting the assumptions on weights, we show that the first page of
${}^{F}E_{1}$ is isomorphic, up to change the indexing, to the second page of
${}^{L}E_{2}$. Hence the cohomology of the lines of ${}^{F}E_{1}$ computes the
weights of the limiting Hodge structure (Section 6.5.2);

  6. (6)

This arguments bounds the cohomology of
$\text{}^{\mathbb{C}}E_{1}^{\bullet,q}$ with the Hodge numbers of the generic
fiber and to get an actual equality, one has to do a few computations
recalling how $\text{}^{\mathbb{C}}E_{1}^{\bullet,q}$ was constructed (Section
6.4).

######  Remark 6.2.1.

Another natural strategy to prove Theorem 3.4.2(ii) is to try to reduce to
finite fields, by the specialization arguments from [Nak00], and to use the
theory of Frobenius weights there. While this might be possible, one issue is
that the existence of the spectral sequence with étale cohomology with compact
support for logarithmic schemes has not been proved. While this spectral
sequence seems likely to exist and it seems to be useful to study it, doing it
would take us far away from the techniques used in the rest of this paper.
Hence we preferred to remain in the complex analytic setting, at the cost of
doing a somehow more involved argument. We hope to come back to the étale case
problem in the near future.

Before starting with the actual proof of Theorem 3.4.2(ii) in Sections 6.4 and
6.5, we collect some preliminaries on the nearby cycles functors in Section
6.3.

###  6.3. Nearby cycles and spectral sequences

####  6.3.1. Computation of nearby cycles over open strata

For the lack of a reference, we collect a few basic results on the nearby
cycle functors $R^{i}\pi_{\Delta,*}A$ for an abelian group $A$. This is
essentially a more refined version of Lemma 5.2.3. The analogue description in
the étale case has been done in [Nak00, Appendix A.1].

First, observe that $\pi_{\Delta,*}A\simeq A$, so that we want to compute
$R^{i}\pi_{\Delta,*}A$ for $i\geq 1$. Recall that
$\pi_{\Delta}:\widetilde{X}_{\Delta,1}\rightarrow X^{0}_{\Delta}$ factorises
as

$\widetilde{X}^{0}_{\Delta}\xrightarrow{\alpha_{\Delta}}X^{0,log}_{\Delta}\xrightarrow{\beta_{\Delta}}X^{0}_{\Delta}.$

Hence, since $\alpha_{\Delta}$ is a closed immersion,

$R^{1}\pi_{\Delta*}A\simeq R^{1}\beta_{\Delta,*}\alpha_{\Delta,*}A.$

Moreover, since $\pi_{\Delta}:\widetilde{X}^{0}_{\Delta}\rightarrow
X^{0}_{\Delta}$ is a locally trivial $(S^{1})^{|\Delta|-1}$-bundle, the
natural cup product map

$\wedge^{i}R^{1}\pi_{\Delta,*}A\xrightarrow{\simeq}R^{i}\pi_{\Delta,*}A$

is an isomorphism.

Similar reasonings show that

(6.3.1.1)
$\wedge^{i}R^{1}\beta_{\Delta,*}A\xrightarrow{\simeq}R^{i}\beta_{\Delta,*}A$

is an isomorphism and, as in Lemma 5.2.3, by [KN99a, Lemma 1.5] and [Nak00],
there is a canonical isomorphism

(6.3.1.2) $R^{1}\beta_{\Delta,*}A\simeq\bigoplus_{X^{0}_{\Delta}\subseteq
X_{i}}A(-1).$

Working locally as in Lemma 5.2.3, one sees that the natural morphism

$R^{1}\beta_{\Delta,*}A\rightarrow R^{1}\pi_{\Delta,*}A\simeq
R^{1}\beta_{\Delta,*}\alpha_{\Delta,*}A$

is surjective and fits into an exact sequence

$0\rightarrow
A(-1)\xrightarrow{diag}R^{1}\beta_{\Delta,*}A\simeq\bigoplus_{X^{0}_{\Delta}\subseteq
X_{i}}A(-1)\rightarrow R^{1}\pi_{\Delta,*}A\rightarrow 0$

where $diag$ is the diagonal map. Observe that the choice of an $i$ such that
$X^{0}_{\Delta}\subseteq X_{i}$, induces a (non-canonical) splitting of the
exact sequences. Putting all together, we get a natural exact sequence

$0\rightarrow A(-1)\otimes R^{i-1}\pi_{\Delta,*}A\rightarrow
R^{i}\beta_{\Delta,*}A\rightarrow R^{i}\pi_{\Delta,*}A\rightarrow 0$

which splits (non-canonically). In particular there is a canonical short exact
sequence

(6.3.1.3) $0\rightarrow H_{c}^{j}(X^{0}_{\Delta},A(-1)\otimes
R^{i-1}\pi_{\Delta,*}A)\rightarrow
H_{c}^{j}(X^{0}_{\Delta},R^{i}\beta_{\Delta,*}A)\rightarrow
H_{c}^{j}(X^{0}_{\Delta},R^{i}\pi_{\Delta,*}A)\rightarrow 0,$

which splits (non canonically).

####  6.3.2. Weights and Hodge structures on open strata

Now by (6.3.1.2) and (6.3.1.1)

$H_{c}^{j}(X^{0}_{\Delta},R^{i}\beta_{\Delta,*}A)\simeq
H_{c}^{i}(X^{0}_{\Delta},\wedge^{i}R^{1}\beta_{\Delta,*}A)\simeq
H_{c}^{i}(X^{0}_{\Delta},\wedge^{i}(\bigoplus_{X^{0}_{\Delta}\subseteq
X_{i}}A(-1)))\simeq H_{c}^{j}(X^{0}_{\Delta},A(-i))^{\binom{|\Delta|-1}{i}}.$

Combining this with (6.3.1.3) and Poincaré Duality we get the following.

######  Lemma 6.3.2.1.

If $X^{0}_{\Delta}$ has dimension $d$, then
$H_{c}^{j}(X^{0}_{\Delta},R^{i}\pi_{*}\mathbb{Q})$ carries a natural Hodge
structure pure of weight $2j-2d+2i$

We now apply Lemma 6.3.2.1 to the Leray spectral sequences (with compact
support)

(6.3.2.2)
$\quad_{\Delta}^{L}E^{a,b}_{2}:=H_{c}^{a}(X^{0}_{\Delta},R^{b}\pi_{\Delta,*}\mathbb{Q})\Rightarrow
H_{c}^{a+b}(\widetilde{X}^{0}_{\Delta}).$

associated to the morphisms
$\pi_{\Delta}:\widetilde{X}^{0}_{\Delta}\rightarrow X^{0}_{\Delta}$.

######  Lemma 6.3.2.3.

${}_{\Delta}^{L}E^{a,b}_{2}$ degenerates in $E_{2}$.

###### Proof.

By Lemma 6.3.2.1, the term ${}_{\Delta}^{L}E^{a,b}_{2}$ is pure of weight
$2a-2\textrm{dim}(X_{\Delta}^{0})+2b$. Since there are no morphism between
mixed Hodge structures of different weights and the morphisms are compatible
with the Hodge structures, the maps
${}_{\Delta}^{L}E^{a,b}_{r}\rightarrow_{\Delta}^{L}E^{a+r,b-r+1}_{r}$ are
trivial for all $a,b,r\in\mathbb{Z}$. ∎

####  6.3.3. Vanishing cycles spectral sequence

Consider now the Leray spectral sequence for the morphism
$\pi:X_{0,1}^{log}\rightarrow X_{0}$

(6.3.3.1) $^{L}E^{a,b}_{2}:=H^{a}(X_{0},R^{b}\pi_{*}\mathbb{Q})\Rightarrow
H^{a+b}(X^{\log}_{0,1}).$

By [Ill02, Corollary 8.4], the sheaf $R^{b}\pi_{*}\mathbb{Q}$ identifies with
the classical sheaf of vanishing cycles $R^{b}\Psi\mathbb{Q}$. Hence, by
[DS04, Section 1.4], it is a spectral sequence of mixed Hodge structures,
called also the the vanishing cycles spectral sequence. By [Sch73] (see also
[SZ90] and the discussion in [Ill02, Section 8.8, before (8.8.6)]), it
degenerates in $E_{3}$.

Since taking graded pieces for the weight filtration is an exact operation, we
can consider the weight $2q$-graded piece of the spectral sequence (6.3.3.1)

(6.3.3.2) $\mathrm{Gr}^{W}_{2q}\text{
}^{L}E^{a,b}_{2}:=\mathrm{Gr}^{W}_{2q}H^{a}(X_{0},R^{b}\pi_{*}\mathbb{Q})\Rightarrow\mathrm{Gr}^{W}_{2q}H^{a+b}(X^{\log}_{0,1})$

which still degenerates in $E_{3}$, so that

(6.3.3.3)
$\mathrm{Gr}^{W}_{2q}H^{p+q}(X^{\log}_{0,1})=\sum_{i}\mathrm{Gr}^{W}_{2q}\text{
}^{L}E^{i,p+q-i}_{3}.$

###  6.4. Preliminary reductions

####  6.4.1. Using torsion freeness

We first prove point $(ii)$ in Theorem 3.4.2(2). This is a consequence of the
following corollary of Lemma 6.3.2.3, which transfers part of the
$(c)$-assumptions from $X^{0}_{\Delta}$ to $\widetilde{X}^{0}_{\Delta}$.

######  Corollary 6.4.1.1.

$H_{c}^{n}(\widetilde{X}^{0}_{\Delta},\mathbb{Z})$ is torsion free.

###### Proof.

Let us consider the rational and the integer Leray spectral sequences
${}_{\mathbb{Q}}^{L}E_{2}$ and ${}_{\mathbb{Z}}^{L}E_{2}$ associated to the
morphism $\pi_{\Delta}:\widetilde{X}^{0}_{\Delta}\rightarrow X^{0}_{\Delta}$.
Thanks to Lemma 6.3.2.3, we know that ${}_{\mathbb{Q}}^{L}E_{2}$ degenerates
in $E_{2}$. On the other hand, by hypothesis of Theorem 3.4.2 and the split
exact sequence (6.3.1.3), the abelian group ${}_{\mathbb{Z}}^{L}E_{2}^{i,n-i}$
has no torsion for all $i$. Therefore the differential
${}_{\mathbb{Z}}^{L}E_{r}^{a,b}\rightarrow_{\mathbb{Z}}^{L}E_{r}^{a+r,b-r+1}$
has to be trivial for all $a,b,r\in\mathbb{N}$, so that
${}^{L}_{\mathbb{Z}}E_{2}$ degenerates in $E_{2}$. Hence, there exists a
decreasing filtration $F^{s}\subseteq F^{s-1}\subseteq\dots\subseteq
H_{c}^{n}(\widetilde{X}^{0}_{\Delta},\mathbb{Z})$ such that
$F_{i}/F_{i+1}\simeq\text{ }_{\mathbb{Z}}^{L}E_{2}^{i,n-i}$ has no torsion.
This implies that $H_{c}^{n}(\widetilde{X}^{0}_{\Delta},\mathbb{Z})$ has no
torsion and the proof of the lemma is concluded. ∎

Thanks to Corollary 6.4.1.1
${}^{\mathbb{C}}_{\mathbb{Z}}E^{\bullet,q}_{1}\otimes\mathbb{Z}/2\mathbb{Z}\simeq\text{
}_{\mathbb{Z}/2\mathbb{Z}}^{\quad\mathbb{C}}E^{\bullet,q}_{1}$ and point
$(ii)$ of Theorem 3.4.2(2) is proved.

####  6.4.2. Reduction to a weight computation

We now start the proof of (i), by reducing it to a proof of the inequality
(6.4.2.2) in the following Lemma.

######  Lemma 6.4.2.1.

Assume that for all $p,q\in\mathbb{N}$ the inequality

(6.4.2.2) $\textrm{dim}(H^{p}(\text{
}^{\mathbb{C}}E^{\bullet,q}_{1}))\leq\mathrm{Gr}^{W}_{2q}H^{p+q}(X_{0,1}^{log}).$

holds. Then one has $\mathrm{Gr}^{W}_{2q}H^{p+q}(X_{0,1}^{log})=h^{p,q}$.

In order to prove Lemma 6.4.2.1, we start relating the Hodge numbers of the
general fiber to the weight filtration in the cohomology of $X^{\log}_{0,1}$.

######  Lemma 6.4.2.3.

One has the equality
$h^{p,q}(X_{t})=\dim(\mathrm{Gr}^{W}_{2q}H^{p+q}(X^{\log}_{0,1}))$ and the
mixed Hodge structure on $H^{p+q}(X^{\log}_{0,1})$ has only even weights.

###### Proof.

This is essentially proved in [IKMZ19, Middle of Page 31]. We briefly recall
the argument. For every subset $J\subseteq I$ consider the spectral sequence

${}^{\mathbb{C}}E^{a,b}_{1}:=\bigoplus_{\displaystyle{\begin{subarray}{c}X^{0}_{\Delta}\subseteq
X_{J},\\\
\textrm{dim}(X^{0}_{\Delta})=a\end{subarray}}}H_{c}^{a+b}(X^{0}_{\Delta})\Rightarrow
H^{a+b}(X_{J}).$

By assumption and Poincaré duality, ${}^{\mathbb{C}}E^{a,b}_{1}$ is pure of
type $(b,b)$, hence $H^{n}(X_{J})=0$ for $n$ odd and
$H^{2n}(X_{J})\simeq\mathbb{Q}(-n)^{n_{J}}$ for some $n_{J}\in\mathbb{Z}$.
Hence, the weight spectral sequence

$E_{2}^{i,j}:=\bigoplus_{a\geq\max{\\{0,i\\}}}\bigoplus_{|J|=2a-i}H^{2i+j-2a}(X_{J})(i-a)\Rightarrow
H^{i+j}(X^{\log}_{0,1})$

shows that $W^{2i}=F_{p+q-i}$, where $W^{j}$ is the increasing weight
filtration on $H^{p+q}(X^{\log}_{0,1})$ and $F_{j}$ the decreasing Hodge
filtration. Hence

$h^{p,q}(X_{t})=h^{q,p}(X_{t})=\textrm{dim}(F_{p}/F_{p+1})=\dim(W^{2q}/W^{2q-2})=\textrm{dim}(\mathrm{Gr}^{W}_{2q}H^{p+q}(X^{\log}_{0,1})).$

∎

###### Proof of Lemma 6.4.2.1.

Lemma 6.4.2.3 implies that

$\dim(H^{p+q}(X_{0,1}^{log}))=\sum_{i}\mathrm{Gr}^{W}_{2i}H^{p+q}(X_{0,1}^{log}).$

Therefore, by the spectral sequence (3.4.1), one gets the inequality

$\sum_{i}\textrm{dim}(H^{i}(\text{
}^{\mathbb{C}}E^{\bullet,p+q-i}_{1}))\geq\sum_{i}\mathrm{Gr}^{W}_{2i}H^{p+q}(X_{0,1}^{log}).$

The assumption (6.4.2.2) implies that

$\textrm{dim}(H^{p}(\text{
}^{\mathbb{C}}E^{\bullet,q}_{1}))=\mathrm{Gr}^{W}_{2q}H^{p+q}(X_{0,1}^{log}).$

Hence, again by Lemma 6.4.2.3, one has
$\mathrm{Gr}^{W}_{2q}H^{p+q}(X_{0,1}^{log})=h^{p,q}(X_{t})$. ∎

###  6.5. End of the proof

By Lemma 6.4.2.1, in order to end the proof of Theorem 3.4.2, it remains to
prove the inequality (6.4.2.2).

####  6.5.1. Filtering the complex

Recall the spectral sequences

${}^{r}_{q}E^{a,b}_{1}:=\bigoplus_{\textrm{dim}(X^{0}_{\Delta})=a}H_{c}^{a+b}(X^{0}_{\Delta},R^{q-r}\pi_{\Delta,*}\mathbb{Q})\Rightarrow
H^{a+b}(X_{0},R^{q-r}\pi_{*}\mathbb{Q})$

associated to the stratifications (6.1.2). By Lemma 6.3.2.3, there exists a
decreasing filtration ${}_{n}F_{i,\Delta}$ of
$H_{c}^{n}(\widetilde{X}_{\Delta}^{0})$ such that
${}_{n}F_{i,\Delta}/_{n}F_{i+1,\Delta}\simeq
H_{c}^{i}(X_{\Delta}^{0},R^{n-i}\pi_{\Delta,*}\mathbb{Q})$.

If $\textrm{dim}(X^{0}_{\Delta})$ and $\textrm{dim}(X^{0}_{\Delta^{\prime}})$
are respectively $m-1$ and $m$, and $X^{0}_{\Delta}\subseteq
X_{\Delta^{\prime}}$, there is a morphism of spectral sequences

${{}^{L}_{\Delta}E_{2}^{a,b}:=H_{c}^{a}(X^{0}_{\Delta},R^{b}\pi_{\Delta,*}\mathbb{Q})}$${H_{c}^{a+b}(\widetilde{X}^{0}_{\Delta})}$${{}^{L}_{\Delta^{\prime}}E_{2}^{a+1,b}:=H_{c}^{a+1}(X^{0}_{\Delta^{\prime}},R^{b}\pi_{\Delta^{\prime},*}\mathbb{Q})}$${H_{c}^{a+b+1}(\widetilde{X}^{0}_{\Delta^{\prime}}),}$

so that the morphism $H^{n-1}_{c}(\widetilde{X}^{0}_{\Delta})\rightarrow
H^{n}_{c}(\widetilde{X}^{0}_{\Delta^{\prime}})$ sends ${}_{n}F_{\Delta,i}$ to
${}_{n+1}F_{\Delta^{\prime},i+1}$. Hence, setting

${}^{q}F^{p}_{i}:=\bigoplus_{\dim(\widetilde{X}^{0}_{\Delta})=p}\text{
}_{p+q}F_{\Delta^{\prime},i+p}\subseteq\bigoplus_{\dim(\widetilde{X}^{0}_{\Delta})=p}H_{c}^{p+q}(\widetilde{X}^{0}_{\Delta})=\text{
}^{\mathbb{C}}E_{1}^{p,q},$

we get a decreasing filtration

${}^{q}F^{\bullet}_{\bullet}:=\quad^{q}F^{\bullet}_{i+1}\subseteq^{q}F^{\bullet}_{i}\subseteq\dots\subseteq\text{
}^{q}F^{\bullet}_{0}=\text{ }^{\mathbb{C}}E^{\bullet,q}_{1}$

such that

$\mathrm{Gr}_{\text{ }^{q}F_{\bullet}}^{i,\bullet}:=\text{
}^{q}F^{\bullet}_{i}/^{q}F^{\bullet}_{i+1}\simeq\text{}^{i}_{q}E_{1}^{\bullet,i}.$

Hence the spectral sequence for a filtered complex reads

(6.5.1.1) $\text{}_{q}^{F}E_{1}^{a,b}=H^{a+b}(\mathrm{Gr}_{\text{
}^{q}F_{\bullet}}^{a,\bullet})=H^{a+b}(\text{}^{a}_{q}E_{1}^{\bullet,a})=\text{
}^{a}_{q}E^{a+b,a}_{2}\Rightarrow
H^{a+b}(\text{}^{\mathbb{C}}E^{\bullet,q}_{1}).$

####  6.5.2. Computation of the spectral sequence of the filtered complex

By (6.5.1.1)

(6.5.2.1)
$\dim(H^{p}(\text{}^{\mathbb{C}}E^{\bullet,q}))\leq\sum_{i}\dim(\text{
}_{q}^{F}E_{2}^{i,p-i}).$

We now fully employ the assumption on the weights by using Lemma 6.3.2.1.

######  Lemma 6.5.2.2.

For every $r\in\mathbb{N}$ the spectral sequence

${}^{r}_{q}E^{a,b}_{1}:=\bigoplus_{\textrm{dim}(X^{0}_{\Delta})=a}H_{c}^{a+b}(X^{0}_{\Delta},R^{q-r}\pi_{\Delta,*}\mathbb{Q})\Rightarrow
H^{a+b}(X_{0},R^{q-r}\pi_{*}\mathbb{Q}),$

degenerates in $E_{2}$ and it induces a natural isomorphism

${}^{r}_{q}E_{2}^{a,b}\simeq\mathrm{Gr}^{W}_{2b+2q-2r}H^{a+b}(X_{0},R^{q-r}\pi_{*}\mathbb{Q}).$

###### Proof.

Since $H_{c}^{a+b}(X^{0}_{\Delta},R^{q-r}\pi_{\Delta,*}\mathbb{Q})$ is pure of
weight $2b+2q-2r$ by Lemma 6.3.2.1 and there are no morphisms between Hodge
structures of different weights, the spectral sequence degenerates in $E_{2}$
and it induces the displayed isomorphism. ∎

By (6.5.1.1) one has

(6.5.2.3) $\text{}^{F}_{q}E_{1}^{a,b}=\text{}^{a}_{q}E^{a+b,a}_{2},$

which by Corollary 6.5.2.2, is canonically isomorphic to

$\text{}^{a}_{q}E^{a+b,a}_{2}\simeq\mathrm{Gr}^{W}_{2q}H^{2a+b}(X_{0},R^{q-a}\pi_{*}\mathbb{Q}),$

which by definition (see (6.3.3.2)) is $\mathrm{Gr}^{W}_{2q}\text{
}^{L}E^{2a+b,q-a}_{2}$. Hence the bijection
$\varphi:\mathbb{Z}^{2}\rightarrow\mathbb{Z}^{2}$ sending $(x,y)$ to
$(2x+y,q-x)$ (with inverse $(x,y)\mapsto(q-y,2y+x-2q)$), induces a natural
isomorphism

(6.5.2.4) $^{F}_{q}E^{a,b}_{1}\simeq\mathrm{Gr}^{W}_{2q}\text{
}^{L}E^{2a+b,q-a}_{2}.$

So one gets

$H^{p}(\text{ }^{\mathbb{C}}E_{1}^{\bullet,q})\leq\sum_{i}\dim(\text{
}_{q}^{F}E_{2}^{i,p-i})=\sum_{i}\textrm{dim}(\mathrm{Gr}^{W}_{2q}\text{
}^{L}E^{p+i,q-i}_{3})=\sum_{i}\textrm{dim}(\mathrm{Gr}^{W}_{2q}\text{
}^{L}E^{p+i,q-i}_{\infty})=\mathrm{Gr}^{W}_{2q}H^{p+q}(X^{log}_{0,1}),$

where, to summarize, the first inequality comes from (6.5.1.1), the second
equality from (6.5.2.4), the third one from the degeneration of
$\mathrm{Gr}^{W}_{2q}\text{ }^{L}E$ in $E_{3}$ (Section 6.3.3) and the last
one from (6.3.3.1). Hence we proved the inequality (6.4.2.2) and the proof of
Theorem 3.4.2(2) is concluded.

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Emiliano Ambrosi, Université de Strasbourg

eambrosi’at’unistra.fr:  Institut de Recherche Mathématique Avancée (IRMA)
Matilde Manzaroli, Universität Tübingen

matilde.manzaroli’at’uni-tuebingen.de: Mathematisches Institut