# Hilali conjecture and complex algebraic varieties

Shoji Yokura  Graduate School of Science and Engineering, Kagoshima
University, 21-35 Korimoto 1-chome, Kagoshima 890-0065, Japan
<EMAIL_ADDRESS>
###### Abstract.

A simply connected topological space is called _rationally elliptic_ if the
rank of its total homotopy group and its total (co)homology group are both
finite. A well-known Hilali conjecture claims that for a rationally elliptic
space its homotopy rank _does not exceed_ its (co)homology rank. In this
paper, after recalling some well-known fundamental properties of a rationally
elliptic space and giving some important examples of rationally elliptic
spaces and rationally elliptic singular complex algebraic varieties for which
the Hilali conjecture holds, we give some revised formulas and some
conjectures. We also discuss some topics such as mixd Hodge polynomials
defined via mixed Hodge structures on cohomology group and the dual of the
homotopy group, related to the “Hilali conjecture _modulo product_ ”, which is
an inequality between the usual homological Poincaré polynomial and the
homotopical Poincaré polynomial.

##  1\. Introduction

Homotopy group and homology group are clearly two very fundamental and
important invariants in geometry and topology; they are related to each other,
just like “two wheels of a car” or “two sides of a coin”. The Hilali
conjecture111We note that there are also _relative Hilali conjectures_ for a
map instead of a space (see [12, 45, 46, 50, 51]). is a very simple inequality
concerning the dimensions (called “homotopy” and “homology” dimensions) of
these two invariants:

(1.1)
$\operatorname{dim}\left(\pi_{*}(X)\otimes{\mathbb{Q}}\right)\leq\operatorname{dim}H_{*}(X;{\mathbb{Q}})\quad(<\infty)$

Usually people may think (as the author thought before) that for most spaces
(excluding strange or very wild spaces) these dimensions are both finite, but
it turns out that it is not the case, as remarked in [17, §32 Elliptic
spaces]. Namely, in a sense, topological spaces whose homotopy and homology
dimensions are both finite belongs to a special class of spaces, called
_rationally elliptic_ spaces. If one of the dimensions is not finite, such a
space is called _rationally hyperbolic_. The Hilali conjecture claims the
above inequality (1.1) for _any rationally elliptic space_ $X$ belonging to
this “special” class, namely, in words

homotopy dimension never exceeds homology dimension.

Complex algebraic varieties are certainly “special” spaces in the category of
topological spaces. In this paper we consider the Hilali conjecture for
complex algebraic varieties. Unfortunately, even complex algebraic varieties
are not “special enough” to discuss the Hilali conjecture, because a complex
algebraic variety $X$ has to be _rationally elliptic_ in order to see whether
the Hilali conjecture holds or not. It turns out that, as proved by Stephen
Halperin for the first time in 1977 and later by John B. Friedlander and S.
Halperin in 1979, rationally elliptic spaces satisfy _“very stringent
restrictions or properties_ (see [17, §32 Elliptic spaces]). This fact might
be good since one could restrict oneself to very special spaces, but bad since
it would be hard to control such stringent properties.

The purposes of the present paper are as follows:

  * •

To propagandize or advertise the Hilali conjecture,

  * •

To inform that for “most spaces”, or “generically”, either one of the homology
dimension and the homotopy dimension is infinite, thus these two dimensions
are both finite only for a very special class of spaces and such a space is
called _rationally elliptic_ and the Hilali conjecture is for such a space.

  * •

To show that spheres and complex projective spaces are fundamental and
important spaces for discussing the Hilali conjecture and the notion of
rational ellipticity.

  * •

To give revised versions of some known formulas and theorems.

  * •

To give some reasonable questions and conjectures in order to deal with the
Hilali conjecture furthermore.

The paper is organized as follows. In §2 we recall the notion of rational
ellipticity, in particular its importance in connection with well-known
conjectures such as Bott conjecture, Hopf conjecture and Gromov conjecture in
Riemannian geometry. In §3 we recall the Hilali conjecture and see that
spheres and complex projective spaces (which are very fundamental and
important spaces, building other interesting spaces and complex algebraic
varieties) and Cartesian products of them are rationally elliptic and also
satisfy the Hilali conjecture. Related to these examples we discuss singular
complex algebraic varieties homeomorphic and/or homotopic to the complex
projective spaces. In §4 we discuss stringent properties of a rationally
elliptic space, which were discovered by J. B. Friedlander and S. Halperin, in
particular as typical models we consider spheres, complex projective spaces
and Cartesian products of them. We also give some revised or strengthened
versions of some known formulas and theorems. In §§5-7 we recall some known
results about rationally elliptic toric and Kähler manifolds and based on
these results and discussion in this paper we give a conjecture claiming that
_the Hilali conjecture would hold for any complex algebraic variety, singular
or non-singular, provided that it is rationally elliptic._ In §§8-10 we
discuss what is called “Hilali conjecture _modulo product_ ” concerning the
usual (homological) Poincaré polynomial and the homotopical Poincaré
polynomial and also the mixed Hodge polynomial and the homotopical mixed Hodge
polynomial (based on the existence of mixed Hodge structure), which specialize
to the corresponding Poincaré polynomials. At the very end we give a
conjecture (due to Anatoly Libgober) claiming that _if a quasi-projective
variety is rationally elliptic, then the mixed Hodge structure is of
Hodge–Tate type_.

Finally we remark that it is very standard or usual to discuss rationally
elliptic spaces and the Hilali conjecture, in rational homotopy theory,
appealing to Sullivan’s model theory, but in this paper we do not do so for
this kind of presentation.

##  2\. Rationally elliptic space

We let

$\pi_{*}(X)\otimes{\mathbb{Q}}:=\bigoplus_{k\geq
1}\pi_{k}(X)\otimes{\mathbb{Q}}\quad\text{and}\quad
H_{*}(X;{\mathbb{Q}}):=\bigoplus_{k\geq 0}H_{k}(X;{\mathbb{Q}}).$

A simply connected topological space $X$ is called _rationally elliptic_
222This terminology has nothing to do with an _elliptic_ curve in Algebraic
Geometry. In [25] this name was not used. In [20, §1 Introduction] it was
called a _space of type F_ and in [22] this name seems to start being used. if

$\dim\left(\pi_{*}(X)\otimes{\mathbb{Q}}\right)=\sum_{k\geq
2}\dim\left(\pi_{k}(X)\otimes{\mathbb{Q}}\right)<\infty,\,\,\dim
H_{*}(X;{\mathbb{Q}})=\sum_{k\geq 0}\dim H_{k}(X;{\mathbb{Q}})<\infty.$

Here we note that these dimensions are respectively the ranks of the total
homotopy group $\pi_{*}(X)=\bigoplus_{k\geq 2}\pi_{k}(X)$ and the total
homology group $H_{*}(X;\mathbb{Z})=\bigoplus_{k\geq 0}H_{k}(X;\mathbb{Z})$.
From now on, $\dim\left(\pi_{*}(X)\otimes{\mathbb{Q}}\right)$ and $\dim
H_{*}(X;{\mathbb{Q}})$ shall be simply called “homotopy” and “homology”
dimension, respectively. The cohomology $H^{*}(X;{\mathbb{Q}})$ can be also
used instead of the homology group, but by the universal coefficient theorem
we have $\dim H^{*}(X;{\mathbb{Q}})=\dim H_{*}(X;{\mathbb{Q}})$.

Importance of this _rational ellipticity_ , in particular, comes from the
following well-known conjecture attributed to Raoul Bott (cf. [3]) in
Riemannian Geometry:

######  Conjecture 2.1 (Bott conjecture333This is supposed to be one of the
central conjectures in Riemannian geometry).

A compact simply connected Riemannian manifold with a non-negative sectional
curvature is rationally elliptic.

As to this conjecture, we cite the following remark from Grove–Halperin’s
paper [22, Introduction, p.380]:

_“This conjecture has been attributed to Bott. Our interest in it was first
stimulated by D. Toledo. An assertion equivalent to the conjecture is that the
integers $\rho_{p}=\sum_{q\leq p}\dim H_{q}(\Omega M;\mathbb{Q})$ grow only
sub-exponentially in $p$ (i.e., $\forall C>1,\forall k\in\mathbb{N},\exists
p>k:\rho_{p}<C^{p}$), in particular the principal result of [3] is a much
weaker version of this conjecture.”_

Hence Bott conjecture is sometimes called “Bott–Grove–Halperin conjecture”.
Later we will see that if $M$ is rationally elliptic and $\dim M=n$, then we
have

  1. (1)

$\chi(M)\geq 0$.

  2. (2)

$\dim H_{*}(M;\mathbb{Q})\leq 2^{n}$

Thus, _a positive answer to Bott conjecture_ would imply the following
Hopf–Chern conjecture and Gromov conjecture (in the case when
$\mathbb{F}={\mathbb{Q}}$), which are still open:

######  Conjecture 2.2 (Hopf conjecture [32](also see [4]), 1953 or Chern
conjecture [10], 1966).

A compact simply connected even-dimensional Riemannian manifold $M$ with non-
negative sectional curvature has non-negative Euler characteristic,
$\chi(M)\geq 0$.

######  Conjecture 2.3 (Gromov conjecture [21],1981).

A simply connected complete manifold $M$ (of dimension $n$) with a non-
negative sectional curvature satisfies $\dim H_{*}(M;\mathbb{F})\leq 2^{n}$
for _any field $\mathbb{F}$_.

######  Remark 2.4.

To be more precise about Gromov conjecture, Mikhael L. Gromov has proved that
there is a constant $C(n)$ such that $\dim H_{*}(M;\mathbb{F})\leq C(n)$ and
then conjectured that the upper bound of $C(n)$ is $2^{n}$, i.e., $C(n)\leq
2^{n}$.

As to Bott’ conjecture, Xiaoyang Chen [15] has recently proved

######  Theorem 2.5.

A simply connected Riemannian manifold with entire Grauert tube is rationally
elliptic.

László Lempert and Róbert Szöke [34] has proved that “entire Grauert tube”
implies “non-negative sectional curvature”, thus we can say that X. Chen has
affirmatively solved Bott–Grove–Halperin conjecture _under the stronger
assumption_ of “entire Grauert tube”.

##  3\. Hilali conjecture

Mohamed Rachid Hilali made the following conjecture in his thesis [29] (also
see a recent survey paper by Yves Félix and S. Halperin [20, §10]):

######  Conjecture 3.1 (Hilali Conjecture, 1990).

If $X$ is a rationally elliptic space, then we have

$\dim(\pi_{*}(X)\otimes{\mathbb{Q}})\leq\dim H_{*}(X;{\mathbb{Q}}).$

The conjecture is very simple in the sense that it is just an inequality of
the dimensions of homotopy and homology (which are fundamental, important,
well-known and well-studied invariants in geometry and topology, even in
mathematical physics), thus it seems to be quite tractable and to be solvable
easily, but no one has found a counterexample yet, thus it is still open in
even more than 30 years after it was conjectured.

###  3.1. Fundamental examples for which the Hilali conjecture holds.

######  Example 3.2.

The following results follow from the well-known _Serre Finiteness Theorem_
[41]:

$\pi_{i}(S^{2k})\otimes{\mathbb{Q}}=\begin{cases}{\mathbb{Q}}&\,i=2k,4k-1,\\\
\,0&\,i\not=2k,4k-1,\end{cases}\qquad\pi_{i}(S^{2k+1})\otimes{\mathbb{Q}}=\begin{cases}{\mathbb{Q}}&\,i=2k+1,\\\
\,0&\,i\not=2k+1.\end{cases}$

Hence we see that

$2=\dim(\pi_{*}(S^{2k})\otimes{\mathbb{Q}})=\dim
H_{*}(S^{2k};{\mathbb{Q}})=1+\dim H_{2k}(S^{2k};{\mathbb{Q}})=2.$
$1=\dim(\pi_{*}(S^{2k+1})\otimes{\mathbb{Q}})<\dim
H_{*}(S^{2k+1};{\mathbb{Q}})=1+\dim H_{2k+1}(S^{2k+1};{\mathbb{Q}})=2.$

######  Example 3.3.

$\pi_{k}(\mathbb{CP}^{n})\otimes\mathbb{Q}=\begin{cases}\mathbb{Q}\quad\text{for
$k=2,2n+1$,}\\\ 0\quad\text{for $k\not=2,2n+1$,}\\\ \end{cases}$

This follows from the long exact sequence of a fibration $S^{1}\hookrightarrow
S^{2n+1}\to\mathbb{CP}^{n}$:

$\cdots\to\pi_{k}(S^{1})\to\pi_{k}(S^{2n+1})\to\pi_{k}(\mathbb{CP}^{n})\to\pi_{k-1}(S^{1})\to\pi_{k-1}(S^{2n+1})\to\cdots$

The following is known:

$H_{k}(\mathbb{CP}^{n};\mathbb{Q})=\begin{cases}\mathbb{Q}\quad\text{for $k=0$
and $2\leq k\leq 2n$ for even $k$,}\\\ 0\quad\text{otherwise.}\\\ \end{cases}$

Hence we have

$2=\dim(\pi_{*}(\mathbb{CP}^{n})\otimes{\mathbb{Q}})\leq\dim
H_{*}(\mathbb{CP}^{n};{\mathbb{Q}})=1+n\,\,(n\geq 1).$

Before giving more examples, let us observe the following:

######  Proposition 3.4.

If $X_{1},\cdots,X_{m}$ are rationally elliptic and satisfy the Hilali
conjecture, then $X_{1}\times\cdots\times X_{m}$ is also rationally elliptic
and satisfies the Hilali conjecture.

###### Proof.

Rational ellipticity of $X_{1}\times\cdots\times X_{m}$ follows from homotopy
being “additive” and homology being “multiplicative”:

$\dim(\pi_{*}(X_{1}\times\cdots\times
X_{m})\otimes{\mathbb{Q}})=\dim(\pi_{*}(X_{1})\otimes{\mathbb{Q}})+\cdots+\dim(\pi_{*}(X_{m})\otimes{\mathbb{Q}})<\infty.$
$\dim H_{*}(X_{1}\times\cdots\times X_{m};{\mathbb{Q}})=\dim
H_{*}(X_{1};{\mathbb{Q}})\times\cdots\times\dim
H_{*}(X_{m};{\mathbb{Q}})<\infty.$

$X_{1}\times\cdots\times X_{m}$ satisfying the Hilalic conjecture, i.e.,

$\dim(\pi_{*}(X_{1}\times\cdots\times X_{m})\otimes{\mathbb{Q}})\leq\dim
H_{*}(X_{1}\times\cdots\times X_{m};{\mathbb{Q}}),$

which follows from Lemma 3.5 (the proof of which is by induction, left for the
reader) below. ∎

######  Lemma 3.5.

Let $a_{i},b_{i}$ ($1\leq i\leq m$) be real numbers such that $a_{i}\leq
b_{i}$ and $b_{i}\geq 1$. If either $2\leq b_{i}$ or $0=a_{i}<b_{i}=1$ for
each $i$, then $a_{1}+\cdots+a_{m}\leq b_{1}\times\cdots\times b_{m}$.

$``0=a_{i}<b_{i}=1"$ comes from the following:

$b_{i}=\dim H_{*}(X_{i};{\mathbb{Q}})=1\Longrightarrow
a_{i}=\dim(\pi_{*}(X_{i})\otimes{\mathbb{Q}})=0$

which follows from the following Serre Theorem (usually called Whitehead–Serre
Theorem):

######  Theorem 3.6.

If $f:X\to Y$ is a continuous map of simply connected spaces, the following
are equivalent

  1. (1)

$\pi_{*}(f)\otimes{\mathbb{Q}}:\pi_{*}(X)\otimes{\mathbb{Q}}\cong\pi_{*}(Y)\otimes{\mathbb{Q}}$
is an isomorphism.

  2. (2)

$H_{*}(f;{\mathbb{Q}}):H_{*}(X;{\mathbb{Q}})\cong H_{*}(Y;{\mathbb{Q}})$ is an
isomorphism.

Indeed, $\dim H_{*}(X;{\mathbb{Q}})=1$, i.e.,
$H_{0}(X;{\mathbb{Q}})={\mathbb{Q}}$ and $H_{i}(X;{\mathbb{Q}})=0(i\geq 1)$,
implies that
$H_{*}(a_{X};{\mathbb{Q}}):H_{*}(X;{\mathbb{Q}})={\mathbb{Q}}\cong
H_{*}(pt;{\mathbb{Q}})={\mathbb{Q}}$ for a constant map $a_{X}:X\to pt$. Then
the above Serre Theorem implies the homotopy isomorphism
$\pi_{*}(a_{X})\otimes{\mathbb{Q}}:\pi_{*}(X)\otimes{\mathbb{Q}}\cong\pi_{*}(pt)\otimes{\mathbb{Q}}=0$,
i.e., $\dim(\pi_{*}(X_{i})\otimes{\mathbb{Q}})=0$.

######  Example 3.7.

It follows from Proposition 3.4 above that

$S^{n_{1}}\times\cdots\times S^{n_{k}}$,
$\mathbb{CP}^{m_{1}}\times\cdots\times\mathbb{CP}^{m_{s}}$ and
$S^{n_{1}}\times\cdots\times
S^{n_{k}}\times\mathbb{CP}^{m_{1}}\times\cdots\times\mathbb{CP}^{m_{s}}$

are rationally elliptic and satisfies the Hilali conjecture.

######  Example 3.8.

A simply connected compact Lie group $G$ is rationally elliptic and satisfies
the Hilali conjecture. Because it is well-known (by Heinz Hopf) that $G$ is
rationally homotopy equivalent to the product of odd-dimensional spheres,
i.e., there is a map $f:G\to S^{2k_{1}+1}\times\cdots\times S^{2k_{n}+1}$ such
that
$\pi_{*}(f)\otimes{\mathbb{Q}}:\pi_{*}(G)\otimes{\mathbb{Q}}\cong\pi_{*}(S^{2k_{1}+1}\times\cdots\times
S^{2k_{n}+1})\otimes{\mathbb{Q}}.$ Hence, by Theorem 3.6 (Serre theorem),
$H_{*}(f;{\mathbb{Q}}):H_{*}(G;{\mathbb{Q}})\cong
H_{*}(S^{2k_{1}+1}\times\cdots\times S^{2k_{n}+1};{\mathbb{Q}}).$

We note that $\mathbb{CP}^{m_{1}}\times\cdots\times\mathbb{CP}^{m_{s}}$ is a
rationally elliptic _Kähler manifold_ and also a rationally elliptic _smooth
toric variety_. We will come back to rationally elliptic Kähler manifolds and
smooth toric varieties later again.

###  3.2. Rationally elliptic singular varieties for which the Hilali
conjecture holds.

$\mathbb{CP}^{m_{1}}\times\cdots\times\mathbb{CP}^{m_{s}}$ is a non-singular
complex algebraic variety. How about a singular complex algebraic variety?

######  Example 3.9.

A cuspidal curve is a singular curve whose singularities are cusps. A cuspidal
curve is homeomorphic to $\mathbb{CP}^{1}$. Hence, it is a rationally elliptic
and satisfies the Hilali conjecture.

######  Remark 3.10.

A classification of cuspidal curves seems to be still open. Mariusz Koras and
Karol Palka [33] have recently proved that a cuspidal curve can have _at most
4 cusps._

A cuspidal curve is a singular curve which is _homeomorphic to
$\mathbb{CP}^{1}$_.

######  Question 3.11.

How about a singular complex algebraic variety444Note that this variety is not
_a fake projective space_ , which is a non-singular complex algebraic variety
which has the same Betti numbers as a complex projective space, but not
isomorphic to it. which is homeomorphic to $\mathbb{CP}^{n}\,\,(n\geq 2)$ or
homotopic to $\mathbb{CP}^{n}\,\,(n\geq 1)$?

As to $\mathbb{CP}^{2}$, Lawrence Brenton [6]555The author asked Alex Dimca
for some work concerning Quesion 3.11 and then he immediately answered with
his related works [2, 11]. In the reference of [11] is cited Brenton’s paper
[6]. The author would like to thank A. Dimca for sending these two papers [2,
11]. has constructed singular surfaces which are homotopic to
$\mathbb{CP}^{2}$. As far as the author knows, there seems to be no such a
paper available for $\mathbb{CP}^{n}$ with $n\geq 3$.

######  Example 3.12.

Let $C_{1},\cdots,C_{s}$ are cuspidal curves and $B_{1},\cdots,B_{t}$ be
Brenton’s singular surfaces. Then $C_{1}\times\cdots\times C_{s}$ and
$B_{1}\times\cdots\times B_{t}$ are respectively $s$-dimensional and
$2t$-dimensional singular varieties which are rationally elliptic and also
satisfies the Hilali conjecture.
$\mathbb{CP}^{m_{1}}\times\cdots\times\mathbb{CP}^{m_{r}}\times
C_{1}\times\cdots\times C_{s}\times B_{1}\times\cdots\times B_{t}$ is a
$2(m_{1}+\cdots+m_{r})+s+2t$-dimensional rationally elliptic singular variety
and also satisfies the Hilali conjecture.

These examples are trivial ones. Here is a naive question:

######  Question 3.13.

Is there a characterization of a rationally elliptic singular complex
algebraic variety which satisfies the Hilali conjecture?

As to the above Question 3.13, we would like to pose the following very naive
or simple-minded question:

######  Question 3.14.

Is it correct that a rationally elliptic complex algebraic variety is always
homeomorphic or homotopic to the product of complex projective spaces
$\mathbb{CP}^{m_{1}}\times\cdots\times\mathbb{CP}^{m_{s}}$?

We want to or should close this section with the following famous theorem. If
we consider a _nonsingular_ surface which is homotopic to $\mathbb{CP}^{2}$,
we have the following famous theorem due to Shing-Tung Yau [47], which is an
answer to Severi’s old problem666Francesco Severi posed the question of
whether there was a complex surface homeomorphic to $\mathbb{CP}^{2}$ but not
biholomorphic to it. So, S.-T. Yau gave a negative answer even in a weaker
version of “homeomorphic” being replaced by “homotopic”. [42] :

######  Theorem 3.15 (S.-T. Yau).

Any complex surface which is homotopic to $\mathbb{CP}^{2}$ is biholomorphic
to $\mathbb{CP}^{2}$.

######  Remark 3.16.

It is quite natural to ask if a similar statement holds for $\mathbb{CP}^{n}$
with $n\geq 3$. As far as the author knows by literature search, there seems
to be no result available for this naive question.

##  4\. Fundamental properties of rationally elliptic spaces

In this section we recall some fundamental properties of rationally elliptic
spaces for later use of them.

###  4.1. S. Halperin’s Theorems

$\chi^{\pi}(X):=\sum_{k}(-1)^{k}\dim(\pi_{k}(X)\otimes{\mathbb{Q}})=\dim(\pi_{\operatorname{even}}(X)\otimes{\mathbb{Q}})-\dim(\pi_{\operatorname{odd}}(X)\otimes{\mathbb{Q}})$

is called _homotopy Euler–Poincaré characteristic_ , which is surely a
homotopy version of Euler–Poincaré characteristic:

$\chi(X)=\sum_{k}(-1)^{k}\dim(H_{k}(X;{\mathbb{Q}})=\dim
H_{\operatorname{even}}(X;{\mathbb{Q}})-\dim
H_{\operatorname{odd}}(X;{\mathbb{Q}}).$

First let us look at those of $S^{n}$ and $\mathbb{CP}^{n}$ (see Examples 3.2
and 3.3 above).

$\chi^{\pi}(S^{2k})=(-1)^{2k}+(-1)^{4k-1}=0$ and
$\chi(S^{2k})=1+(-1)^{2k}=2>0$ $\chi^{\pi}(S^{2k+1})=(-1)^{2k+1}=-1<0$ and
$\chi(S^{2k+1})=1+(-1)^{2k+1}=0$.
$\chi^{\pi}(\mathbb{CP}^{n})=(-1)^{2}+(-1)^{2n+1}=0$ and
$\chi(\mathbb{CP}^{n})=1+\sum_{i=1}^{n}(-1)^{2i}=1+n>0.$

Hence we have

$\chi^{\pi}(S^{n_{1}}\times\cdots\times
S^{n_{j}}\times\mathbb{CP}^{m_{1}}\times\cdots\times\mathbb{CP}^{m_{s}})\leq
0$ and $\hskip 170.71652pt\chi(S^{n_{1}}\times\cdots\times
S^{n_{j}}\times\mathbb{CP}^{m_{1}}\times\cdots\times\mathbb{CP}^{m_{s}})\geq
0$ $\chi^{\pi}(S^{2n_{1}}\times\cdots\times
S^{2n_{j}}\times\mathbb{CP}^{m_{1}}\times\cdots\times\mathbb{CP}^{m_{s}})=0$
and  $\hskip 170.71652pt\chi(S^{2n_{1}}\times\cdots\times
S^{2n_{j}}\times\mathbb{CP}^{m_{1}}\times\cdots\times\mathbb{CP}^{m_{s}})>0.$

As observed above, $S^{n_{1}}\times\cdots\times
S^{n_{j}}\times\mathbb{CP}^{m_{1}}\times\cdots\times\mathbb{CP}^{m_{s}}$ is
rationally elliptic. It turns out that this property “$\chi^{\pi}(X)\leq 0$
and $\chi(X)\geq 0$” always holds for any rationally elliptic space $X$, as
proved by S. Halperin [25]:

######  Theorem 4.1.

[25, Theorem 1, p.174] Let $X$ be a rationally elliptic space. Then
$\chi^{\pi}(X)\leq 0$ and $\chi(X)\geq 0$. Moreover, the following are
equivalent:

  1. (1)

$\chi^{\pi}(X)=0$,

  2. (2)

$\chi(X)>0,$

  3. (3)

$H_{odd}(X)\otimes{\mathbb{Q}}=0$.

######  Remark 4.2.

Thus an affirmative solution of Bott conjecture implies Hopf conjecture.

######  Remark 4.3.

The equivalence of (1), (2) and (3) above was posed as a question in Dennis
Sullivan’s famous paper [44]. In [28] Kathryn Hess wrote that Halperin’s paper
was _“the first major paper on the structure and properties of Sullivan models
after the work of Sullivan himself”_.

######  Remark 4.4.

According to [25], a special case of Theorem 4.1 was already solved by Henri
Cartan [7] and the proof of Theorem 4.1 is reduced to this special case.

Next let us look at

$\operatorname{dim}(X)$, Poincaré polynomial $P_{X}(t)=\sum_{k}\dim
H_{k}(X;\mathbb{Q})\,t^{k}$ and $\chi(X)$

of the above examples (cf. Examples 3.2 and 3.3). First we recall the rational
homotopy groups of $S^{2k}$ and $\mathbb{CP}^{n}$ in the following forms:

$\pi_{i}(S^{2k})\otimes{\mathbb{Q}}=\begin{cases}{\mathbb{Q}}&\,i=2k,2(2k)-1\\\
\,0&\,i\not=2k,2(2k)-1\end{cases},\,\,\pi_{k}(\mathbb{CP}^{n})\otimes\mathbb{Q}=\displaystyle\begin{cases}\mathbb{Q}\quad\text{for
$k=2,2n+1=2(n+1)-1$,}\\\ 0\quad\text{for $k\not=2,2n+1=2(n+1)-1$,}\\\
\end{cases}$

  1. (1)

$\operatorname{dim}(S^{2k})=2k=2(2k)-1-(2k-1),P_{S^{2k}}(t)=1+t^{2k}=\frac{1-t^{2(2k)}}{1-t^{2k}},\chi(S^{2k})=2=\frac{2k}{k},$

  2. (2)

$\operatorname{dim}(\mathbb{CP}^{n})=2n=2(n+1)-1-(2\cdot
1-1),P_{\mathbb{CP}^{n}}(t)=1+\sum_{i=1}^{n}t^{2i}=\frac{1-t^{2(n+1)}}{1-t^{2\cdot
1}},$ $\chi(\mathbb{CP}^{n})=n+1=\frac{n+1}{1}.$

  3. (3)

$\displaystyle\operatorname{dim}$
$\displaystyle\left(\prod_{i=1}^{k}S^{2n_{i}}\times\prod_{j=1}^{s}\mathbb{CP}^{m_{j}}\right)=\sum_{i=1}^{k}2n_{i}+\sum_{j=1}^{s}2m_{j}$
$\displaystyle=\sum_{i=1}^{k}\\{2(2n_{i})-1\\}+\sum_{j=1}^{s}\\{2(m_{j}+1)-1\\}-\left(\sum_{i=1}^{k}\\{2\cdot
n_{i}-1\\}+\sum_{j=1}^{s}\\{2\cdot 1-1\\}\right).$ $\displaystyle
P_{\prod_{i=1}^{k}S^{2n_{i}}\times\prod_{j=1}^{s}\mathbb{CP}^{m_{j}}}(t)=\prod_{i=1}^{k}P_{S^{2n_{i}}}(t)\prod_{j=1}^{s}P_{\mathbb{CP}^{m_{j}}}(t)$
$\displaystyle\qquad\qquad=\prod_{i=1}^{k}\frac{1-t^{2(2n_{i})}}{1-t^{2n_{i}}}\prod_{j=1}^{s}\frac{1-t^{2(m_{j}+1)}}{1-t^{2}}=\frac{\prod_{i=1}^{k}(1-t^{2(2n_{i})})\prod_{j=1}^{s}(1-t^{2(m_{j}+1)})}{\prod_{i=1}^{k}(1-t^{2n_{i}})\cdot(1-t^{2\cdot
1})^{s}},$
$\chi\left(\prod_{i=1}^{k}S^{2n_{i}}\times\prod_{j=1}^{s}\mathbb{CP}^{m_{j}}\right)=2^{k}\prod_{j=1}^{s}(m_{j}+1)=\frac{\prod_{i=1}^{k}2n_{i}\prod_{j=1}^{s}(m_{j}+1)}{\prod_{i=1}^{k}n_{i}\prod_{j=1}^{s}1}.$

In fact, these things hold for any rationally elliptic space, which was proved
also by S. Halperin [25], as shown below. Let $X$ be rationally elliptic.

Let $y_{1},\cdots,y_{q}$ be a basis of
$\pi_{\operatorname{odd}}(X)\otimes{\mathbb{Q}}$
($\dim(\pi_{\operatorname{odd}}(X)\otimes{\mathbb{Q}})=q$) and
$x_{1},\cdots,x_{r}$ be a basis of
$\pi_{\operatorname{even}}(X)\otimes{\mathbb{Q}}$
($\dim(\pi_{\operatorname{even}}(X)\otimes{\mathbb{Q}})=r$). If
$y_{j}\in\pi_{2b_{j}-1}(X)\otimes{\mathbb{Q}}$ and
$x_{i}\in\pi_{2a_{i}}(X)\otimes{\mathbb{Q}}$, $2b_{j}-1$ and $2a_{i}$ are
called _degrees_ of $y_{j}$ and $x_{j}$. $(b_{1},\cdots,b_{q})$ and
$(a_{1},\cdots,a_{r})$ are respectively called _$b$ -exponents_ and _$a$
-exponetns_ of $X$ (note that they are called “odd” exponents and “even”
exponents in Félix–Halperin–Thomas’s book [17, §32 Elliptic spaces,
Definition, p.441]). E.g., for
$X=\prod_{i=1}^{k}S^{2n_{i}}\times\prod_{j=1}^{s}\mathbb{CP}^{m_{j}}$, we have
$\pi_{2\cdot 2n_{1}-1}(X)\otimes\mathbb{Q}=\cdots=\pi_{2\cdot
2n_{k}-1}(X)\otimes\mathbb{Q}=\pi_{2(m_{1}+1)-1}(X)\otimes\mathbb{Q}=\cdots=\pi_{2(m_{s}+1)-1}(X)\otimes\mathbb{Q}=\mathbb{Q}$
and
$\pi_{2n_{1}}(X)\otimes\mathbb{Q}=\cdots=\pi_{2n_{k}}(X)\otimes\mathbb{Q}=\mathbb{Q},\quad\pi_{2\cdot
1}(X)\otimes\mathbb{Q}=\cdots=\underbrace{\mathbb{Q}\oplus\cdots\mathbb{Q}}_{s}$.
$(2n_{1},\cdots,2n_{k},m_{1}+1,\cdots,m_{s}+1)$ are $b$-exponents and
$(n_{1},\cdots,n_{k},\underbrace{1,\cdots,1}_{s})$ are $a$-exponents of $X$.

The largest integer $n_{X}$ such that $H_{n_{X}}(X;{\mathbb{Q}})\not=0$ is
called _formal dimension_ of $X$.

######  Theorem 4.5.

[25, Theorem 3’, p.188, Corollary 2, p.198] Let the symbols be as above.

  1. (1)

$n_{X}=\sum_{j=1}^{q}(2b_{j}-1)-\sum_{i=1}^{r}(2a_{i}-1)$

  2. (2)

Betti numbers $\beta_{i}=\dim H_{i}(X;\mathbb{Q})$ satisfy Poincaré duality,
i.e., $\beta_{i}=\beta_{n_{X}-i}$.

  3. (3)

In the case when $\chi^{\pi}(X)=q-r=0$ (thus $\chi(X)>0$), i.e., $q=r$,
Poincaré polynomial $P_{X}(t)=\sum_{k}\dim H_{k}(X;\mathbb{Q})\,t^{k}$ of $X$
is expressed by $b$-exponents and $a$-exponents:

$P_{X}(t)=\frac{\prod_{i=1}^{q}(1-t^{2b_{i}})}{\prod_{i=1}^{q}(1-t^{2a_{i}})}.$

In particular,

(4.6)
$\chi(X)=P_{X}(-1)=P_{X}(1)=\dim(H_{*}(X)\otimes\mathbb{Q})=\frac{\prod_{i=1}^{q}b_{i}}{\prod_{i=1}^{q}a_{i}}.$

######  Remark 4.7.

Here we just remark that (4.6) is due to the following modification:

$\displaystyle
P_{X}(t)=\frac{\prod_{i=1}^{q}(1-t^{2b_{i}})}{\prod_{i=1}^{q}(1-t^{2a_{i}})}$
$\displaystyle=\frac{\prod_{i=1}^{q}(1-(t^{2})^{b_{i}})}{\prod_{i=1}^{q}(1-(t^{2})^{a_{i}})}$
$\displaystyle=\frac{(1-t^{2})^{q}\prod_{i=1}^{q}(1+t^{2}+\cdots+(t^{2})^{b_{i}-1})}{(1-t^{2})^{q}\prod_{i=1}^{q}(1+t^{2}+\cdots+(t^{2})^{a_{i}-1})}$
$\displaystyle=\frac{\prod_{i=1}^{q}(1+t^{2}+\cdots+(t^{2})^{b_{i}-1})}{\prod_{i=1}^{q}(1+t^{2}+\cdots+(t^{2})^{a_{i}-1})}.$

###  4.2. J. B. Friedlander–S. Halperin’s Theorems

In this section we recall some fundamental results of Friedlander–Halperin
[20].

######  Definition 4.8.

[20, Introduction, _Definition_ , p.117-118] Let $B=(b_{1},\cdots,b_{q})$ and
$A=(a_{1},\cdots,a_{r})$ be sequences of positive integers.

  1. (1)

We say $(B;A)$ satisfies _strong arithmetic condition_ (abbr. S.A.C.) if for
every subsequence $A^{*}$ of $A$ of length $s$ ($1\leq s\leq r$) there exists
at least $s$ elements $b_{j}$’s of $B$ such that

$b_{j}=\sum_{a_{i}\in A^{*}}\gamma_{ij}a_{i}$

where $\gamma_{ij}$ is _a non-negative integer_ such that $\sum_{a_{i}\in
A^{*}}\gamma_{ij}\geq 2.$

  2. (2)

If $\sum_{a_{i}\in A^{*}}\gamma_{ij}\geq 2$ is not required, then we say that
$(B;A)$ satisfies _arithmetic condition_ (abbr. A.C.).

Thus, in both cases, it is _necessary_ that $r\leq q$ (by considering $s=r$).

######  Example 4.9.

  1. (1)

$B=(3,4,6)$, $A=(2,3,4)$. $(B;A)$ satisfies A.C., but not S.A.C..

  2. (2)

$B=(4,6,8)$, $A=(2,3,4)$. $(B;A)$ satisfies S.A.C. (hence A.C.).

  3. (3)

$B=(3,4,5,5,8)$, $A=(2,3,4)$. $(B;A)$ satisfies A.C., but not S.A.C..

  4. (4)

$B=(3,4,5,6,8)$, $A=(2,3,4)$. $(B;A)$ satisfies S.A.C. (hence A.C.).

  5. (5)

$B=(3,5,7)$, $A=(2,4)$. $(B;A)$ does not satisfy A.C. (hence not S.A.C.).

######  Theorem 4.10.

[20, Theorem 1, p.118] Let $B=(b_{1},\cdots,b_{q})$ and
$A=(a_{1},\cdots,a_{r})$ be finite sequences of positive integers. The
following are equivalent:

  1. (1)

$(B;A)$ satisfies S.A.C.

  2. (2)

$B$ and $A$ are $b$-exponents and $a$-exponents of a rationally elliptic space
$X$.

The above Theorem 4.10 is equivalent to Theorem 4.12 below:

Let $R=\mathfrak{K}[u_{1},\cdots,u_{r}]$ the ring of polynomials in $r$
variables $u_{i}$ of degree $a_{i}$ over an infinite field $\mathfrak{K}$. Let

$\Phi_{i}:=\\{\sigma_{ij}\\}_{j=1,\cdots,\ell_{i}},i=1,\cdots,q$

be families of non-linear monomials $\sigma_{ij}$ of degree $b_{i}$ in the
variables $u_{1},\cdots,u_{r}$. Then $(B,A)$ satisfies S.A.C. if and only if
the families $\Phi_{1},\cdots,\Phi_{q}$ satisfies the following P.C.:

######  Definition 4.11.

[20, _Definition_ , p.119] The families $\Phi_{1},\cdots,\Phi_{q}$ satisfy
P.C. (polynomial condition) if and only if for each $s$ and for each set of
$s$ variables $u_{i_{1}},\cdots,u_{i_{s}}$ there are at least $s$ families
$\Phi_{m_{1}},\cdots,\Phi_{m_{s}}$ containing a non-linear monomial in
$\mathfrak{K}[u_{i_{1}},\cdots,u_{i_{s}}]$.

######  Theorem 4.12.

[20, Theorem 3, p.119] Assume that $\Phi_{1},\cdots,\Phi_{q}$ are sets of
monomials as above. $\Phi_{1},\cdots,\Phi_{q}$ satisfy P.C. if and only if
there are polynomials $f_{1},\cdots,f_{q}$ of the form

$f_{i}=\sum_{j=1}^{\ell_{i}}c_{ij}\sigma_{ij},\quad
c_{ij}\in\mathfrak{K},\sigma_{ij}\in\Phi_{i}\,\,(1\leq j\leq\ell_{i})$

such that
$\operatorname{dim}_{\mathfrak{K}}\left(\displaystyle\frac{\mathfrak{K}[u_{1},\cdots,u_{r}]}{(f_{1},\cdots,f_{q})}\right)<\infty.$

The major part of Friedlander–Halperin’s paper [20] is devoted to the proof of
this “algebro-geometric” Theorem 4.12.

######  Corollary 4.13.

[20, 2.5 Lemma, p.121] If $B=(b_{1},b_{2},\cdots,b_{q})$; $b_{1}\geq
b_{2}\geq\cdots\geq b_{q}$, and $A=(a_{1},a_{2},\cdots,a_{r})$; $a_{1}\geq
a_{2}\geq\cdots\geq a_{r}$. If $(B;A)$ satisfies S.A.C, then $b_{i}\geq
2a_{i}\quad(1\leq i\leq r).$

######  Corollary 4.14.

[20, 1.3 Corollary, p.118, 2.6 Proposition, p.121]

  1. (1)

$n_{X}\geq q+r=\dim(\pi_{*}(X)\otimes{\mathbb{Q}})$.

  2. (2)

$n_{X}\geq\sum_{j=1}^{q}b_{j}.$

  3. (3)

$2n_{X}-1\geq\sum_{j=1}^{q}(2b_{j}-1).$ (Equality holds for $X=S^{2n}$;
$2(2n)-1=4n-1.$)

  4. (4)

$n_{X}\geq\sum_{i=1}^{r}2a_{i}.$ (Equality holds for $X=S^{2n}$;
$n_{X}=2n=2a$)

###### Proof.

All these inequalities will be used later. So, for the sake of convenience of
the reader, the proof is given below.

$\displaystyle\quad(1)\quad
n_{X}=\sum_{j=1}^{q}(2b_{j}-1)-\sum_{i=1}^{r}(2a_{i}-1){}$
$\displaystyle=\sum_{j=1}^{q}(b_{j}-1)+\sum_{j=1}^{q}b_{j}-\sum_{i=1}^{r}2a_{i}+r$
$\displaystyle\geq\sum_{j=1}^{q}(b_{j}-1)+\sum_{j=1}^{r}b_{j}-\sum_{i=1}^{r}2a_{i}+r\quad\text{(since
$q\geq r$)}$ $\displaystyle\geq
q+\sum_{i=1}^{r}(b_{i}-2a_{i})+r\quad\text{(since $b_{j}\geq 2$)}$
$\displaystyle\geq q+r\quad\text{(since $b_{i}\geq 2a_{i}$)}$
$\displaystyle\quad(2)\quad
n_{X}=\sum_{j=1}^{q}(2b_{j}-1)-\sum_{i=1}^{r}(2a_{i}-1)$
$\displaystyle=\sum_{j=1}^{q}b_{j}+\sum_{j=1}^{q}(b_{j}-1)-\sum_{i=1}^{r}(2a_{i}-1)$
$\displaystyle\geq\sum_{j=1}^{q}b_{j}+\sum_{j=1}^{r}(b_{j}-1)-\sum_{i=1}^{r}(2a_{i}-1)\quad\text{(since
$q\geq r$)}$
$\displaystyle=\sum_{j=1}^{q}b_{j}+\sum_{i=1}^{r}(b_{i}-2a_{i})\geq\sum_{j=1}^{q}b_{j}\quad\text{(since
$b_{i}\geq 2a_{i}$)}$

(3) $2n_{X}\geq\sum_{j=1}^{q}2b_{j}\Longrightarrow
2n_{X}-q\geq\sum_{j=1}^{q}(2b_{j}-1)\Longrightarrow
2n_{X}-1\geq\sum_{j=1}^{q}(2b_{j}-1)$.

(4) $n_{X}\geq\sum_{j=1}^{q}b_{j}\Longrightarrow
n_{X}\geq\sum_{j=1}^{r}b_{j}\Longrightarrow n_{X}\geq\sum_{i=1}^{r}2a_{i}$. ∎

######  Remark 4.15.

In fact, as one can see, the inequalities in (1), (3) and (4) can be made
sharper ones as follows:

  1. (1)

$n_{X}\geq 3q-r\geq 2q\geq\dim(\pi_{*}(X)\otimes{\mathbb{Q}})$. It suffices to
show $n_{X}\geq 3q-r$, whose proof is a slight revision of the above proof.
Indeed,

$\displaystyle n_{X}=\sum_{j=1}^{q}(2b_{j}-1)-\sum_{i=1}^{r}(2a_{i}-1)$
$\displaystyle=\sum_{j=1}^{q}(b_{j}-1)+\sum_{j=1}^{q}b_{j}-\sum_{i=1}^{r}2a_{i}+r$
$\displaystyle\geq\sum_{j=1}^{q}(b_{j}-1)+\sum_{j=r+1}^{q}b_{j}+\sum_{j=1}^{r}b_{j}-\sum_{i=1}^{r}2a_{i}+r$
$\displaystyle\geq q+2(q-r)+\sum_{i=1}^{r}(b_{i}-2a_{i})+r\quad\text{(since
$b_{j}\geq 2$)}$ $\displaystyle\geq q+2(q-r)+r=3q-r.\quad\text{(since
$b_{i}\geq 2a_{i}$)}$

  2. (3)

$2n_{X}-q\geq\sum_{j=1}^{q}(2b_{j}-1)$ is sharper than
$2n_{X}-1\geq\sum_{j=1}^{q}(2b_{j}-1)$.

  3. (4)

$\sum_{j=1}^{q}b_{j}=\sum_{j=1}^{r}b_{j}+\sum_{j=r+1}^{q}b_{j}\geq\sum_{i=1}^{r}2a_{i}+2(q-r)=\sum_{i=1}^{r}2a_{i}-2\chi^{\pi}(X).$
Hence $n_{X}\geq\sum_{i=1}^{r}2a_{i}-2\chi^{\pi}(X)$ is sharper than
$n_{X}\geq\sum_{i=1}^{r}2a_{i}$. Note that $\chi^{\pi}(X)\leq 0$.

######  Proposition 4.16.

(cf. [14, Proposition 2.2]) If $X$ is rationally elliptic and $\chi(X)>0$
(such a space is called positively elliptic), then the Hilali conjecture
holds.

###### Proof.

Here we give a much easier proof different from that of [14, Proposition 2.2],
_as a corollary_ of Friedlander–Halperin’s results. Since $\chi(X)>0$,
$\chi^{\pi}(X)=\dim(\pi_{\operatorname{even}}(X)\otimes{\mathbb{Q}})-\dim(\pi_{\operatorname{odd}}(X)\otimes{\mathbb{Q}})=r-q=0$,
i.e., $r=q$. Hence

$\dim(H_{*}(X)\otimes\mathbb{Q})=\frac{\prod_{i=1}^{q}b_{i}}{\prod_{i=1}^{q}a_{i}}{}\geq\frac{\prod_{i=1}^{q}2a_{i}}{\prod_{i=1}^{q}a_{i}}{}=\prod_{i=1}^{q}2=2^{q}.$

$\dim(\pi_{*}(X)\otimes{\mathbb{Q}})=\dim(\pi_{\operatorname{even}}(X)\otimes{\mathbb{Q}})+\dim(\pi_{\operatorname{odd}}(X)\otimes{\mathbb{Q}})=2q\leq
2^{q}\leq\dim(H_{*}(X)\otimes\mathbb{Q}).$ ∎

In the case of $q>r$, an explicit description of the Poincaré polynomial
$P_{X}(t)$ is still open (as far as the author knows). However we do have the
following bound for $P_{X}(t)$:

######  Theorem 4.17.

[20, 2.8 Proposition, p.121, _Proof of Proposition 2.8_ , pp.131-132]

(4.18)
$Q_{X}(t):=\frac{\prod_{i=1}^{q}(1-t^{2b_{i}})}{(1-t)^{q-r}\prod_{j=1}^{r}(1-t^{2a_{j}})}=1+\cdots+c_{m}t^{m}+\cdots+c_{n_{X}}t^{n_{X}}$

is such that each $c_{m}$ is non-negative and
$\operatorname{dim}H_{m}(X;\mathbb{Q})\leq c_{m}$ for each $m$. So,

$P_{X}(t)\leq Q_{X}(t)$, in particular,
$\operatorname{dim}H_{*}(X;\mathbb{Q})\leq Q_{X}(1)$.

The difficult part is $\operatorname{dim}H_{m}(X;\mathbb{Q})\leq c_{m}$, for
which they use _transcendence degree or Krull dimension and Macaulay’s
theorem_ in commutative ring theory.

######  Theorem 4.19.

[20, 2.9 Corollary, p.121]

$\dim H_{*}(X;{\mathbb{Q}})\leq
2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}\leq(2n_{X})^{n_{X}}$

###### Proof.

We write down a proof for the sake of convenience of the reader.

$\displaystyle
Q_{X}(t)=\frac{\prod_{i=1}^{q}(1-t^{2b_{i}})}{(1-t)^{q-r}\prod_{i=1}^{r}(1-t^{2a_{i}})}$
$\displaystyle=\frac{(1-t^{2})^{q}\prod_{j=1}^{q}(1+t^{2}+\cdots+(t^{2})^{b_{j}-1})}{(1-t)^{q-r}(1-t^{2})^{r}\prod_{i=1}^{r}(1+t^{2}+\cdots+(t^{2})^{a_{i}-1})}$
$\displaystyle=\frac{(1-t^{2})^{q-r}\prod_{j=1}^{q}(1+t^{2}+\cdots+(t^{2})^{b_{j}-1})}{(1-t)^{q-r}\prod_{i=1}^{r}(1+t^{2}+\cdots+(t^{2})^{a_{i}-1})}$
$\displaystyle=\frac{(1+t)^{q-r}\prod_{j=1}^{q}(1+t^{2}+\cdots+(t^{2})^{b_{j}-1})}{\prod_{i=1}^{r}(1+t^{2}+\cdots+(t^{2})^{a_{i}-1})},$

which implies that
$Q_{X}(1)=2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}.$ Or we do
the following modification:

$\displaystyle
Q_{X}(t)=\frac{\prod_{i=1}^{q}(1-t^{2b_{i}})}{(1-t)^{q-r}\prod_{i=1}^{r}(1-t^{2a_{i}})}$
$\displaystyle=\frac{(1-t)^{q}\prod_{j=1}^{q}(1+t+\cdots+t^{2b_{j}-1})}{(1-t)^{q-r}(1-t)^{r}\prod_{i=1}^{r}(1+t+\cdots+t^{2a_{i}-1})}$
$\displaystyle=\frac{\prod_{j=1}^{q}(1+t+\cdots+t^{2b_{j}-1})}{\prod_{i=1}^{r}(1+t+\cdots+t^{2a_{i}-1})}$

which also implies
$Q_{X}(1)=\frac{\prod_{j=1}^{q}2b_{j}}{\prod_{i=1}^{r}2a_{i}}=2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}.$
Hence

$\dim H_{*}(X;{\mathbb{Q}})=P_{X}(1)\leq Q_{X}(1)\leq
2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}.$

As to the second inequality
$2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}\leq(2n_{X})^{n_{X}}$,
their proof is not written in [20], but it must be as follows:

$2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}\leq
2^{n_{X}}\prod_{j=1}^{q}n_{X}\leq 2^{n_{X}}(n_{X})^{n_{X}}=(2n_{X})^{n_{X}}.$

Because $n_{X}\geq q+r$ and $n_{X}\geq\sum_{j=1}^{q}b_{j}$. ∎

In fact, a sharper and better one holds:

(4.20) $2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}\leq
2^{n_{X}},$

thus we get

(4.21) $\dim H_{*}(X;{\mathbb{Q}})\leq 2^{n_{X}}$

So, Gromov conjecture holds for $\mathbb{F}=\mathbb{Q}$.

###### Proof of (4.20) (by Halperin [26]).

$\dim H_{*}(X;{\mathbb{Q}})\leq
2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}={}\frac{\prod_{j=1}^{q}2b_{j}}{\prod_{i=1}^{r}2a_{i}}{}<\prod_{j=1}^{q}2b_{j}\leq{}\prod_{j=1}^{q}2^{b_{j}}={}2^{\sum_{j=1}^{q}b_{j}}\leq
2^{n_{X}}.$

∎

######  Remark 4.22.

The above inequality (4.21) is in [18, Theorem 2.75, p.85], whose references
are Halperin’s paper [25] and Félix–Halperin–Thomas’s book [17], but such a
formula is not written anywhere in these two references. So, we understand
that $(2n_{X})^{n_{X}}$ in the above inequality
$2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}\leq(2n_{X})^{n_{X}}$
was _not a misprint_ of $2^{n_{X}}$.

######  Remark 4.23.

It might be interesting to obtain a better or sharper inequality for the
inequalities $\dim H_{*}(X;{\mathbb{Q}})\leq
2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}\leq 2^{n_{X}}$.

  1. (1)

For example, a bit sharper one is the following. If we use the descending
order for $b$-exponents and $a$-exponents as in Corollary 4.13, $b_{1}$ is the
maximum of the $b$-exponents and $a_{r}$ is the minimum of the $a$-exponents.
Hence we have the following inequalities

(4.24) $\dim H_{*}(X;{\mathbb{Q}})\leq
2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}\leq
2^{q-r}\frac{(b_{1})^{q}}{(a_{r})^{r}}=\frac{(2b_{1})^{q}}{(2a_{r})^{r}}\leq\frac{(2b_{1})^{q}}{2^{r}}=2^{q-r}(b_{1})^{q}.$

E.g., if $X=\mathbb{CP}^{n}$, then we have $\dim
H_{*}(\mathbb{CP}^{n};{\mathbb{Q}})\leq 2^{q-r}(b_{1})^{q}=n+1.$

  2. (2)

If $a_{r}\geq 2$, then we get a bit better one:

(4.25) $\dim H_{*}(X;{\mathbb{Q}})\leq
2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}\leq\frac{(2b_{1})^{q}}{(2a_{r})^{r}}=2^{q-r}\frac{(b_{1})^{q}}{(a_{r})^{r}}$

E.g., if $X=\mathbb{S}^{2n}$, then we have $\dim
H_{*}(S^{2n};{\mathbb{Q}})\leq\displaystyle
2^{q-r}\frac{(b_{1})^{q}}{(a_{r})^{r}}=\frac{2n}{n}=2.$

  3. (3)

If we use the following formula for geometric and arithmetic means

$\left(\prod_{i=1}^{n}x_{i}\right)^{\frac{1}{n}}\leq\frac{1}{n}\sum_{i=1}^{n}x_{i}.$

we can get the following inequality:

(4.26) $\dim H_{*}(X;{\mathbb{Q}})\leq
2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}\leq\left(\frac{2n_{X}}{q}\right)^{q}.$

Indeed,

$2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}\leq
2^{q-r}\left(\frac{1}{q}\sum_{j=1}^{q}b_{j}\right)^{q}\leq\left(\frac{2}{q}\right)^{q}\left(\sum_{j=1}^{q}b_{j}\right)^{q}\leq\left(\frac{2n_{X}}{q}\right)^{q}.$

E.g., $\dim H_{*}(\mathbb{CP}^{n};{\mathbb{Q}})\leq 2(2n)=4n$ and $\dim
H_{*}(S^{2n};{\mathbb{Q}})\leq 2(2n)=4n$.

  4. (4)

In fact, the proof of (4.20) can be modified to get a better inequality:

$\dim H_{*}(X;{\mathbb{Q}})\leq
2^{q-r}\frac{\prod_{j=1}^{q}b_{j}}{\prod_{i=1}^{r}a_{i}}=\frac{\prod_{j=1}^{q}2b_{j}}{\prod_{i=1}^{r}2a_{i}}\leq\frac{\prod_{j=1}^{q}2b_{j}}{2^{r}}\leq\frac{\prod_{j=1}^{q}2^{b_{j}}}{2^{r}}=\frac{2^{\sum_{j=1}^{q}b_{j}}}{2^{r}}\leq
2^{n_{X}-r}.$

######  Remark 4.27.

In [39, Theorem 1] Andrey V. Pavlov showed the following inequalities:

(4.28) $\dim H_{k}(X;{\mathbb{Q}})\leq\binom{n_{X}}{m},$ (4.29) $\dim
H_{k}(X;{\mathbb{Q}})\leq\sum_{k+2\ell=m}\binom{q-r}{k}\binom{p}{\ell}.$

(4.28) can be shown by $c_{m}\leq\binom{n_{X}}{m}$, where $c_{m}$ is the
coefficient of $t^{m}$ of the polynomial $Q_{X}(t)$ (see 4.18). A crucial key
of Pavlov’s theorem is that the roots of the polynomial $Q_{X}(t)$ are roots
of the unity. Clearly (4.28) also implies $\dim H_{*}(X;{\mathbb{Q}})\leq
2^{n_{X}}$. In fact, (4.29) implies the following inequality [39, Corollary ]:

$\dim H_{m}(X;{\mathbb{Q}})\leq\frac{1}{2}\binom{n_{X}}{m}(m\not=0,n_{X}),$

which implies $\dim H_{*}(X;{\mathbb{Q}})\leq 2^{n_{X}-1}+1$.

######  Remark 4.30.

If one finds a compact Riemannian manifold $M$ of dimension $n$ with non-
negative sectional curvature or entire Grauert tube such that $2^{n-1}+1<\dim
H_{*}(X;{\mathbb{Q}})\leq 2^{n}$, then this manifold would be a counterexample
to Bott conjecture, although it still satisfies Gromov conjecture.

Before closing this section, we want to revise Theorem 4.17 a bit. In this
theorem we assume that $q>r$ and $r\geq 1$. For example, as in the case when
$X=S^{2n+1}$ is an odd sphere, it can happen that $r=0$. In fact, for any
rationally elliptic space $X$, Theorem 4.17 still holds in the following
sense:

######  Theorem 4.31.

For any rationally elliptic space $X$ with $b$-exponents
$(b_{1},\cdots,b_{q})$ and $a$-exponents $(a_{1},\cdots,a_{r})$, we have

$P_{X}(t)\leq\frac{\prod_{i=1}^{q}(1-t^{2b_{i}})}{(1-t)^{q-r}\prod_{j=1}^{r}(1-t^{2a_{j}})}.$

Here, if $r=0$, then it is understood that $\prod_{j=1}^{r}(1-t^{2a_{j}})=1$.

###### Proof.

First we should note that $q\geq r$ since $X$ is rationally elliptic, thus
$\chi^{\pi}(X)=r-q\leq 0$.

  1. (1)

In the case when $q=r$, the _equality_ holds as shown in Theorem 4.5 above.

  2. (2)

In the case when $q>r$ and $r\geq 1$, it is nothing but Theorem 4.17.

  3. (3)

It remains to see the case when $r=0$. We consider the product space $X\times
S^{2}$. Then, since we have $S_{2}(S^{2})\otimes{\mathbb{Q}}={\mathbb{Q}}$ and
$S_{3}(S^{2})\otimes{\mathbb{Q}}={\mathbb{Q}}$ with $3=2\cdot 2-1$, we have
that $b$-exponents and $a$-exponents of $X\times S^{2}$ are respectively
$\\{b_{1},\cdots,b_{q},2\\}$ and $\\{1\\}$. Therefore it follows from Theorem
4.17 that we have

(4.32) $P_{X\times
S^{2}}(t)\leq\frac{\left\\{\prod_{i=1}^{q}(1-t^{2b_{i}})\right\\}(1-t^{2\cdot
2})}{(1-t)^{(q+1)-1}(1-t^{2\cdot
1})}=\frac{\left\\{\prod_{i=1}^{q}(1-t^{2b_{i}})\right\\}(1+t^{2})}{(1-t)^{q}}.$

Since $P_{X\times S^{2}}(t)=P_{X}(t)\times
P_{S^{2}}(t)=P_{X}(t)\times(1+t^{2})$, cancelling out the term $(1+t^{2})$ of
the both sides of (4.32), the above inequality becomes

(4.33) $P_{X}(t)\leq\frac{\prod_{i=1}^{q}(1-t^{2b_{i}})}{(1-t)^{q}}.$

∎

######  Example 4.34.

For example, consider $X=S^{2b_{1}-1}\times\cdots\times S^{2b_{q}-1}$. Then we
have

$P_{X}(t)=\prod_{i=1}^{q}(1+t^{2b_{i}-1})<\frac{\prod_{i=1}^{q}(1-t^{2b_{i}})}{(1-t)^{q}}=\prod_{i=1}^{q}(1+t+t^{2}+\cdots+t^{2b_{i}-1}).$

The equality does not hold since $b_{i}\geq 2$.

##  5\. Rationally elliptic smooth toric varieties

Indranil Biswas, Vicente Muñoz and Aniceto Murillo [5] proved

######  Theorem 5.1 (Biswas–Muñoz–Murillo).

If a compact smooth troic variety $X$ is rationally elliptic, then its
Poincaré polynomial $P_{X}(t)$ is equal to that of a product of complex
projective spaces. I.e., if $dimX=n$, then
$P_{X}(t)=P_{\mathbb{CP}^{m_{1}}}(t)\times\cdots\times
P_{\mathbb{CP}^{m_{k}}}(t)$ with $n=m_{1}+\cdots m_{k}$.

Thus, since $X$ is rationally elliptic and $\chi(X)>0$, as a corollary we get

######  Corollary 5.2.

The Hilali conjecture holds for a rationally elliptic smooth toric variety.

In fact, the above theorem also holds for homotopical Poincaré polynomial
$P^{\pi}_{X}(t)$, although we do not need it for the above corollary:

######  Theorem 5.3 (A. Libgober and S. Yokura [35]).

If a compact smooth toric variety $X$ is rationally elliptic, then its
homotopical Poincaré polynomial $P^{\pi}_{X}(t)$ is equal to that of a product
of complex projective spaces in the above theorem. I.e., if $dimX=n$, then
$P^{\pi}_{X}(t)=P^{\pi}_{\mathbb{CP}^{m_{1}}}(t)+\cdots+P^{\pi}_{\mathbb{CP}^{m_{k}}}(t)$
with $n=m_{1}+\cdots m_{k}$.

##  6\. Rationally elliptic Kähler manifolds

Jaume Amorós and I. Biswas [1, Theorem 1.1] proved

######  Theorem 6.1 (Amorós–Biswas).

A simply connected compact complex Kähler surface is rationally elliptic if
and only if it belongs to the following list:

(i) $\mathbb{CP}^{2}$,

(ii) Hirzebruch surfaces ( a ruled surface over $\mathbb{CP}^{1}$)
$\mathbb{S}_{h}=\mathbb{P}_{\mathbb{CP}^{1}}(\mathcal{O}\oplus\mathcal{O}(h))$
for $h\geq 0$,

(iii) a simply connected general type surface $X$ with $q(X)=p_{g}(X)=0$,
$K_{X}^{2}=8$ and $c_{2}(X)=4$.

######  Remark 6.2.

[1, Remark 3.1, p.1173] $X$ in (iii) is called _fake quadric_. Friedrich
Hirzebruch asked whether fake quadrics exist. This question remains open. By
Freedman’s theorem any fake quadric, _if it exists_ , is homeomorphic to
quadric $\mathbb{S}_{0}=\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ or Hirzebruch
surface
$\mathbb{S}_{1}=\mathbb{P}_{\mathbb{CP}^{1}}(\mathcal{O}\oplus\mathcal{O}(1))$
(the blow-up of $\mathbb{CP}^{2}$ at a point).

######  Remark 6.3.

  1. (1)

Blow-up of $\mathbb{CP}^{2}$ at two points is Kähler, since blow-up of a
Kähler manifold at a point is still Kähler, but _not_ rationally elliptic,
because it is not in the above list. So, _blow-up sometimes destroys rational
ellipticity_.

  2. (2)

F. Hirzebruch (Math. Ann., 1951) showed that

$\mathbb{S}_{h}$ is diffeomorphic to $\mathbb{S}_{k}$ $\Longleftrightarrow
h\equiv k(mod\,\,2)$.

Thus a simply connected rationally elliptic compact Kähler surface is
homeomorphic to $\mathbb{CP}^{2}$ or
$\mathbb{S}_{0}=\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ or
$\mathbb{S}_{1}=\mathbb{CP}^{2}\\#\overline{\mathbb{CP}^{2}}$ (the blow-up of
$\mathbb{CP}^{2}$ at a point).

  3. (3)

$\mathbb{S}_{h}$ is a $\mathbb{CP}^{1}=S^{2}$-bundle over
$\mathbb{CP}^{1}=S^{2}$,
$\pi_{*}(\mathbb{S}_{h})=\pi_{*}(S^{2})\oplus\pi_{*}(S^{2})$ (by the long
exact sequence and the existence of a section ($\infty$-section)).
$H^{*}(\mathbb{S}_{h})\cong H^{*}(\mathbb{CP}^{1})\otimes
H^{*}(\mathbb{CP}^{1})$ since $\pi_{1}(\mathbb{CP}^{1})=0$. So,
$\dim(\pi_{*}(\mathbb{S}_{h})\otimes{\mathbb{Q}})=\dim
H_{*}(\mathbb{S}_{h};{\mathbb{Q}})=4$. So, $\mathbb{S}_{h}$ satisfies Hilali
conjecture.

$\mathbb{CP}^{2}$ is rationally elliptic and the blown-up
$Blow_{1}(\mathbb{CP}^{2})$ of $\mathbb{CP}^{2}$ at a point is still
rationally elliptic, but the blown-up $Blow_{2}(\mathbb{CP}^{2})$ of
$\mathbb{CP}^{2}$ at two points is not rationally elliptic any more. Thus, it
would be reasonable to pose the following question:

######  Question 6.4.

  1. (1)

Given a rationally elliptic complex manifold $M$, can one characterize
properties of $M$ so that the blown-up of $M$ at a point is still rationally
elliptic?

  2. (2)

In general, given a rationally elliptic complex manifold $M$, can one
characterize properties of $M$ so that the blown-up of $M$ at $k$ points is
still rationally elliptic?

######  Theorem 6.5.

[1, Theorem 1.3] If $X$ is a simply connected compact Kähler threefold which
is rationally elliptic, then the Hodge numbers satisfy that $h^{p,q}=0$ for
$p\not=q$ and $h^{p,p}$ is one of the following:

  1. (1)

$h^{0,0}=h^{1,1}=h^{2,2}=h^{3,3}=1$

  2. (2)

$h^{0,0}=h^{3,3}=1$, $h^{1,1}=h^{2,2}=2.$

  3. (3)

$h^{0,0}=h^{3,3}=1$, $h^{1,1}=h^{2,2}=3.$

######  Remark 6.6.

The Poincaré polynomials of the above Kähler threefold are respectively

  1. (1)

$1+t^{2}+t^{4}+t^{6}$,

  2. (2)

$1+2t^{2}+2t^{4}+t^{6}=(1+t^{2})(1+t^{2}+t^{4})$,

  3. (3)

$1+3t^{2}+3t^{4}+t^{6}=(1+t^{2})^{3}$.

which are respectively the same as the Poincaré polynomial of

  1. (1)

$\mathbb{CP}^{3}$,

  2. (2)

$\mathbb{CP}^{1}\times\mathbb{CP}^{2}$,

  3. (3)

$\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\mathbb{CP}^{1}$.

Yang Su and Jianqiang Yang [43] roved

######  Theorem 6.7.

If $X$ is a simply connected compact Kähler fourfold which is rationally
elliptic, then the odd Betti numbers of $X$ are all zero (as above) and the
Hodge numbers of $X$ is one of the following: $h^{p,q}=0$ if $p\not=q$ and
$h^{p,p}$ is as follows:

  1. (1)

$h^{0,0}=h^{1,1}=h^{2,2}=h^{3,3}=h^{4,4}=1$

  2. (2)

$h^{0,0}=h^{1,1}=h^{3,3}=h^{4,4}=1$, $h^{2,2}=2.$

  3. (3)

$h^{0,0}=h^{4,4}=1$, $h^{1,1}=h^{2,2}=h^{3,3}=2.$

  4. (4)

$h^{0,0}=h^{4,4}=1$, $h^{1,1}=h^{3,3}=2$, $h^{2,2}=3.$

  5. (5)

$h^{0,0}=h^{4,4}=1$, $h^{1,1}=h^{3,3}=3$, $h^{2,2}=4.$

  6. (6)

$h^{0,0}=h^{4,4}=1$, $h^{1,1}=h^{3,3}=4$, $h^{2,2}=6.$

######  Remark 6.8.

The Poincaré polynomials of the above Kähler threefold are respectively

  1. (1)

$1+t^{2}+t^{4}+t^{6}+t^{8}$,

  2. (2)

$1+t^{2}+2t^{4}+t^{6}+t^{8}=(1+t^{4})(1+t^{2}+t^{4})$,

  3. (3)

$1+2t^{2}+2t^{4}+2t^{6}+t^{8}=(1+t^{2})(1+t^{2}+t^{4}+t^{6})$,

  4. (4)

$1+2t^{2}+3t^{4}+2t^{6}+t^{8}=(1+t^{2}+t^{4})^{2}$,

  5. (5)

$1+3t^{2}+4t^{4}+3t^{6}+t^{8}=(1+t^{2})^{2}(1+t^{2}+t^{4})$,

  6. (6)

$1+4t^{2}+6t^{4}+4t^{6}+t^{8}=(1+t^{2})^{4}$,

which are respectively the same as the Poincaré polynomial of

  1. (1)

$\mathbb{CP}^{4}$,

  2. (2)

$S^{4}\times\mathbb{CP}^{2}$ (note that $S^{4}$ is not Kähler),

  3. (3)

$\mathbb{CP}^{1}\times\mathbb{CP}^{3}$,

  4. (4)

$\mathbb{CP}^{2}\times\mathbb{CP}^{2}$,

  5. (5)

$\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\mathbb{CP}^{2}$,

  6. (6)

$\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\mathbb{CP}^{1}$.

So, whatever the Hodge number $h^{p,p}$ is, the Hilali conjecture holds for a
_rationally elliptic_ Kähler manifold $X$ of dimension $\leqq 4$, since
$\chi(X)>0$. How about higher dimension bigger than $4$? As far as the author
knows, there is no work available.

However, speaking of Kähler manifolds, the following famous theorem for
_formality_ (see [13] and [17]) should be mentioned: Pierre Deligne, Phillip
Griffiths, John Morgan and D. Sullivan [13] proved the following

######  Theorem 6.9 (Deligne–Griffiths–Morgan–Sullivan).

Any compact Kähler manifold is a formal space.

In fact, M. R. Hilali and My Ismail Mamouni [30, Theorem 2] proved the
following

######  Theorem 6.10 (Hilali–Mamouni).

The Hilali conjecture holds for any rationally elliptic formal space.

Thus as a corollary, although we do knot know about Hodge numbers $h^{p,q}$
explicitly, we can have

######  Corollary 6.11.

The Hilali conjecture holds for any rationally elliptic Kähler manifold of any
dimension.

Speaking of formal space, here are some examples of _formal singular
varieties_ , for which see D. Chataur–J. Cirici’s paper [9, §4]

  1. (1)

Complete intersections with _isolated singularities_.

  2. (2)

Projective varieties with _isolated singularities_ , satisfying that there
exists a resolution of singularities such that _the exceptional divisor is
smooth_.

  3. (3)

Projective varieties whose singularities are only isolated _ordinary multiple
points_.

  4. (4)

_Projective cones_ over smooth projective varieties.

Here is the following question:

######  Question 6.12.

Among the above list (1), (2), (3) and (4) of formal singular varieties, which
are rationally elliptic?

Answering this question seems (to at least the author) to be not easy, as the
following simple examples show:

######  Example 6.13.

The projective cone over $\mathbb{CP}^{1}$ is $\mathbb{CP}^{2}$ and in
general, the projective cone over $\mathbb{CP}^{n}$ is $\mathbb{CP}^{n+1}$.

######  Example 6.14.

Let us denote the projective cone over $X$ by $pc(X)$. The projective cone
$pc\left(\mathbb{CP}^{1}\times\mathbb{CP}^{1}\right)$ of the quadric
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ embedded into $\mathbb{CP}^{3}$, is not
rationally elliptic, although $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ is
rationally elliptic. This is due to the following. The Poincaré polynomial

$P_{pc(\mathbb{CP}^{1}\times\mathbb{CP}^{1})}(t)=1+t^{2}(1+t^{2})^{2}=1+t^{2}+2t^{4}+t^{6},$

which is by the following decomposition in the Grothendieck ring:

$[pc\left(\mathbb{CP}^{1}\times\mathbb{CP}^{1}\right)]=[c]+[\mathbb{C}][\mathbb{CP}^{1}\times\mathbb{CP}^{1}]$

Here $c$ is the cone point and $[\mathbb{C}]=[\mathbb{CP}^{1}-c]$. So,
$1=\beta_{2}\not=\beta_{4}=\beta_{6-2}=2$, i.e., the Poincaré duality does not
hold. I.e.,
$\operatorname{dim}(\pi_{*}(pc(\mathbb{CP}^{1}\times\mathbb{CP}^{1})\otimes\mathbb{Q})=\infty$.
Therefore _the simple operation of taking the projective cone_ also sometimes
destroy rational ellipticity. Here we note that this projective cone over the
quadric is also considered in [8, Example 2.2.4] as an example of a singular
variety whose cohomology does not satisfy the Poincaré duality, but whose
_intersection homology_ restores the Poincaré duality;
$H^{2}(pc\left(\mathbb{CP}^{1}\times\mathbb{CP}^{1}\right);{\mathbb{Q}})={\mathbb{Q}}$,
but
$IH^{2}(pc\left(\mathbb{CP}^{1}\times\mathbb{CP}^{1}\right);{\mathbb{Q}})={\mathbb{Q}}\oplus{\mathbb{Q}}$.

##  7\. Formal dimension and Hilali conjecture

At the moment we do not know _a characterization of rationally elliptic
algebraic varieties, smooth or singular_. However, _as long as its complex
dimension $n$ is less than or equal to $10$, i.e., $n_{X}=2n\leq 20$_, the
Hilali conjecture always holds for such a variety, due to the following
results:

######  Theorem 7.1 (M.R. Hilali and M.I. Mamouni [31]).

If $n_{X}\leq 10$, the Hilali conjecture holds.

This was extended up to $n_{X}=16$:

######  Theorem 7.2 (O. Nakamura and T.Yamaguchi [38]).

If $n_{X}\leq 16$, the Hilali conjecture holds.

Furthermore this was extended up to $n_{X}=20$:

######  Theorem 7.3 (S. Cattalani and A. Milivojevic [14]).

If $n_{X}\leq 20$, the Hilali conjecture holds.

As far as the author knows, we do not know whether the formal dimension
$n_{X}$ has been extended _bigger than 20_.

Before going to the next section, based on what we observed so far, we want to
make the following naive conjecture:

######  Conjecture 7.4.

If $X$ is a rationally elliptic complex algebraic variety, singular or non-
singular, then its Poincaré polynomial $P_{X}(t)$ is the same as that of the
product of even-dimensional spheres and complex projective spaces:

$P_{X}(t)=\prod_{i=1}^{k}P_{S^{2n_{i}}}(t)\times\prod_{j=1}^{s}P_{\mathbb{CP}^{m_{j}}}(t).$

######  Remark 7.5.

If Conjecture 7.4 is correct, then, since $\chi(X)>0$, as a corollary we would
get that _the Hilali conjecture always holds for any rationally elliptic
complex algebraic variety, singular or non-singular._

##  8\. Hilali Conjecture modulo “products”

$\dim\pi_{*}(X)\otimes{\mathbb{Q}}$, $\dim H_{*}(X;{\mathbb{Q}})$,
$\chi^{\pi}(X)$, $\chi(X)$ are special values of Poincaré polynomials:

$P_{X}^{\pi}(t):=\sum_{k}\dim\pi_{k}(X;{\mathbb{Q}})t^{k},\quad
P_{X}(t):=\sum_{k}\dim H_{k}(X;{\mathbb{Q}})t^{k}.$
$\dim(\pi_{*}(X)\otimes{\mathbb{Q}})=P_{X}^{\pi}(1),\quad\dim
H_{*}(X;{\mathbb{Q}})=P_{X}(1)$
$\chi^{\pi}(X)=P_{X}^{\pi}(-1),\quad\chi(X)=P_{X}(-1).$

For the three “distinguished values” for $t$, we have the following
inequalities, although the third one is conjectural:

  1. (1)

$t=-1$: $P_{X}^{\pi}(-1)<P_{X}(-1)$ (By Halperin’s theorem).

  2. (2)

$t=0$: $0=P_{X}^{\pi}(0)<P_{X}(0)=1.$

  3. (3)

$t=1$: $P_{X}^{\pi}(1)\leq P_{X}(1)$ (Hilali conjecture).

How about the other values $t$ for comparing $P_{X}^{\pi}(t)$ and $P_{X}(t)$?
Of course $P_{X}^{\pi}(t)\leq P_{X}(t)$ does not necessarily hold for some
values $t$, as shown below.

######  Example 8.1.

$P_{S^{2}}^{\pi}(t)=t^{2}+t^{3}$ and $P_{S^{2}}(t)=1+t^{2}$.
$P_{S^{2}}^{\pi}(1)=P_{S^{2}}(1)=2.$

$P_{S^{2}}^{\pi}(2)=2^{2}+2^{3}=12>P_{S^{2}}(2)=1+2^{2}=5.$

“homotopy is additive” and “homology is multiplicative”, as we used above!
Namely,

$P_{X\times Y}^{\pi}(t)=P_{X}^{\pi}(t)+P_{Y}^{\pi}(t),\quad P_{X\times
Y}(t)=P_{X}(t)\times P_{Y}(t).$

In particular we have
$P_{X^{n}}^{\pi}(1)=nP_{X}^{\pi}(1),P_{X^{n}}(1)=(P_{X}(1))^{n}.$ Here $X^{n}$
is the Cartesian product $X^{n}=\underbrace{X\times\cdots\times X}_{n}$.

In [48] the author showed the following theorem:

######  Theorem 8.2 (Hilali Conjecture modulo “products”).

For any rationally elliptic space $X$, there exists an integer $n_{0}(X)$ such
that for any integer $n\geq n_{0}(X)$

$P_{X^{n}}^{\pi}(1)<P_{X^{n}}(1).$

This theorem was motivated by an elementary fact in Calculus:

######  Theorem 8.3.

If $|r|<1$, then $\displaystyle\lim_{n\to\infty}nr^{n}=0$.

If $P_{X}(1)=1$, then $P^{\pi}_{X}(1)=0$ (by Theorem 3.6 (Serre Theorem) ),
hence for any integer $n\geqq 1$ we have

$0=P^{\pi}_{X^{n}}(1)<P_{X^{n}}(1)=1.$

So let $P_{X}(1)\geqq 2$, thus $\frac{1}{P_{X}(1)}<1$. Hence we have

$\lim_{n\to\infty}n\Bigl{(}\frac{1}{P_{X}(1)}\Bigr{)}^{n}=0.$

Therefore, whatever the value $P^{\pi}_{X}(1)$ is, we have

$\displaystyle\lim_{n\to\infty}nP^{\pi}_{X}(1)\Bigl{(}\frac{1}{P_{X}(1)}\Bigr{)}^{n}=\displaystyle\lim_{n\to\infty}\frac{nP^{\pi}_{X}(1)}{(P_{X}(1))^{n}}=0.$

Hence there exists some integer $n_{0}$ such that for all $n\geqq n_{0}$ we
have
$\frac{P^{\pi}_{X^{n}}(1)}{P_{X^{n}}(1)}=\frac{nP^{\pi}_{X}(1)}{(P_{X}(1))^{n}}<1.$
Therefore there exists some integer $n_{0}$ such that for all $n\geqq n_{0}$

$P^{\pi}_{X^{n}}(1)<P_{X^{n}}(1).$

##  9\. “Hilali conjecture modulo products” for Poincaré polynomials

So, in the same way, for any positive real number $r$, there exists an integer
$n_{r}(X)$ such that for all $n\geq n_{r}(X)$

$P^{\pi}_{X^{n}}(r)<P_{X^{n}}(r).$

Surely the integer $n_{r}(X)$ depends on the choice of the number $r$. In
fact, in [35, Theorem 1.2] we showed the existence of some integer which does
not depend on the choice of the chosen number $r$, as follows:

######  Theorem 9.1.

Let $\epsilon$ be a positive real number. Then there exists a positive integer
$n(\epsilon)$ such that for any $n\geq n(\epsilon)$

(9.2) $P_{X^{n}}^{\pi}(t)<P_{X^{n}}(t),\forall t\in[\epsilon,\infty).$

######  Remark 9.3.

In the above inequality (9.2) $[\epsilon,\infty)$ cannot be replace by
$[0,\infty)$ and it is crucial that $\epsilon$ is _positive_.

######  Definition 9.4 (A. Libgober and S. Yokura [35]).

Let $\mathfrak{pp}(X;\epsilon)$ be the _smallest one_ of the integer
$n(\epsilon)$ such that for any $n\geq n(\epsilon)$

$P_{X^{n}}^{\pi}(t)<P_{X^{n}}(t),\forall t\in[\epsilon,\infty).$

It is called the _stabilization threshold_.

In particular, for $\epsilon=1$, for example we have:

  1. (1)

$\mathfrak{pp}(S^{2n+1};1)=1$,

  2. (2)

$\mathfrak{pp}(S^{2n};1)=3$,

  3. (3)

$\mathfrak{pp}(\mathbb{C}P^{1},1)=3$ and $\mathfrak{pp}(\mathbb{C}P^{n},1)=2$
if $n\geq 2$.

So, if $X=S^{n},\mathbb{CP}^{n}$, $\mathfrak{pp}(X;1)\leq 3.$ We asked
ourselves “$\mathfrak{pp}(X;1)\leq 3$ for any $X$?” Many trials did not work.
However, using Hilali conjecture, we [35, Theorem 3.1] could show:

######  Theorem 9.5.

If $X$ is a rationally elliptic space X satisfying the Hilali conjecture, then

(9.6) $\mathfrak{pp}(X;1)\leq 3.$

Without using the Hilali conjecture, we [35, Proposition 3.7] have the
following theorem _via the formal dimension_ :

######  Theorem 9.7.

If $X$ is a rationally elliptic space X of formal dimension $n\geq 3$, then

$\mathfrak{pp}(X;1)\leq n.$

Note that we need $n\geq 3$, because the formal dimension of
$\mathbb{CP}^{1}=2$, but $\mathfrak{pp}(\mathbb{C}P^{1},1)=3$.

######  Corollary 9.8.

For any rationally elliptic complex algebraic variety of complex dimension
$n$, $\mathfrak{pp}(X;1)\leq 2n.$

######  Remark 9.9.

If we could find an example such that $\mathfrak{pp}(X;1)=4$ or
$\mathfrak{pp}(X;1)>4$, then it would be a counterexample to the Hilali
conjecture.

######  Question 9.10.

Is there any “reason” for why we have $3$ in the above inequality (9.6)?

##  10\. “Hilali conjecture modulo products” for mixed Hodge polynomials

Using mixed Hodge structures on the cohomology group and the dual of homotopy
group, one can define the following (co)homological) mixed Hodge polynomial
$MH_{X}(t,u,v)$ and the homotopical mixed Hodge polynomial
$MH_{X}^{\pi}(t,u,v)$ of a complex algebraic variety $X$:

$MH_{X}(t,u,v):=\sum_{k,p,q}\dim\Bigl{(}Gr_{F^{\bullet}}^{p}Gr^{W_{\bullet}}_{p+q}H^{k}(X;\mathbb{C})\Bigr{)}t^{k}u^{p}v^{q},$
$MH^{\pi}_{X}(t,u,v):=\sum_{k,p,q}\dim\Bigl{(}Gr_{\tilde{F}^{\bullet}}^{p}Gr^{\tilde{W}_{\bullet}}_{p+q}((\pi_{k}(X)\otimes\mathbb{C})^{\vee})\Bigr{)}t^{k}u^{p}v^{q}.$

Here $(W_{\bullet},F^{\bullet})$ is the mixed Hodge structure of the
cohomology group and $(\widetilde{W}_{\bullet},\widetilde{F}^{\bullet})$ is
the mixed Hodge structure of the dual of homotopy groups. There exists a mixed
Hodge structure on the homotopy groups of complex algebraic varieties as well
(see [37], [23, 24]).

Here we note that the Poincaré polynomials $P_{X}(t)$ and $P^{\pi}_{X}(t)$ are
the specializations of these mixed Hodge
polynomials:$P_{X}(t)=MH_{X}(t,1,1),P^{\pi}_{X}(t)=MH^{\pi}_{X}(t,1,1).$

Then we [35, Theorem 1.10] showed the following:

######  Theorem 10.1.

Let $\epsilon$ be a positive real number. Then there exists a positive integer
$n(\epsilon)$ such that for any $n\geq n(\epsilon)$

$MH_{X^{n}}^{\pi}(t,u,v)<MH_{X^{n}}(t,u,v),\forall(t,u,v)\in[\epsilon,r]\times[\epsilon,r]\times[\epsilon,r].$

######  Remark 10.2.

At the moment we do no know whether
$[\epsilon,r]\times[\epsilon,r]\times[\epsilon,r]$ can be replaced by
$[\epsilon,\infty)\times[\epsilon,\infty)\times[\epsilon,\infty)$.

In fact, we [35, Theorem 4.2] showed the following theorem, which is a
stronger version of Theorem 5.1 (Biswas–Muñoz–Murillo’s theorem [5]):

######  Theorem 10.3.

The homotopical and cohomological mixed Hodge polynomials of a rationally
elliptic toric manifold $X$ of complex dimension $n$ is equal to those of a
product of complex projective spaces, i.e.,

$MH_{X}(t,u,v)=\prod_{i=1}^{k}MH_{\mathbb{CP}^{n_{i}}}(t,u,v)=\prod_{i=1}^{k}\left(1+t^{2}uv+\cdots+t^{2i}(uv)^{i}+\cdots+t^{2n_{i}}(uv)^{n_{i}}\right).$
$MH^{\pi}_{X}(t,u,v)=\sum_{i=1}^{k}MH^{\pi}_{\mathbb{CP}^{n_{i}}}(t,u,v)=\sum_{i=1}^{k}\left(t^{2}uv+t^{2n_{i}+1}(uv)^{n_{i}+1}\right).$

######  Conjecture 10.4 (by A. Libgober).

If a quasi-projective variety is rationally elliptic, then its mixed Hodge
structure is of Hodge–Tate type (or called ballanced [19]), i.e., the non-zero
mixed Hodge numbers are $h^{k,p,q}$ with $p=q$.

If Conjecture 10.4 is correct, then for any rationally elliptic quasi-
projective variety $X$ the mixed Hodge polynomial $MH_{X}(t,u,v)$ is a
polynomial of $t$ and $uv$, as in Theorem 10.3.

Acknowledgements: This is an extended version of the author’s talk at Mini
workshop “Various aspects of singularities” held at the University of Tokyo,
March 30–31, 2023. The author would like to thank the organizers, Toru Ohmoto,
Hiraku Kawanoue, Kei-ichi Watanabe and Shihoko Ishii, for the invitation to
the workshop and in particular Prof. S. Ishii for support for travel expenses.
The author would also like to thank Anatoly Libgober and Alex Dimca for useful
comments and suggestions. The author is supported by JSPS KAKENHI Grant
Numbers JP19K03468 and 23K03117.

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