# Existence of real algebraic hypersurfaces with many prescribed components

Michele Ancona

###### Abstract.

Given a real algebraic variety $X$ of dimension $n$, a very ample divisor $D$
on $X$ and a smooth closed hypersurface $\Sigma$ of $\mathbf{R}^{n}$, we
construct real algebraic hypersurfaces in the linear system
$\absolutevalue{mD}$ whose real locus contains many connected components
diffeomorphic to $\Sigma$. As a consequence, we show the existence of real
algebraic hypersurfaces in the linear system $\absolutevalue{mD}$ whose Betti
numbers grow by the maximal order, as $m$ goes to infinity. As another
application, we recover a result by D. Gayet on the existence of many disjoint
lagrangians with prescribed topology in any smooth complex hypersurface of
$\mathbf{C}\mathbf{P}^{n}$. The results in the paper are proved more generally
for complete intersections. The proof of our main result uses probabilistic
tools.

Institut de Recherche Mathématique Avancée, Université de Strasbourg.
_E-mail address_<EMAIL_ADDRESS>
##  1\. Introduction

This paper deals with the topology of real algebraic varieties. A real
algebraic variety $X$ is an algebraic variety defined over $\mathbf{R}$. We
denote its real and complex locus respectively by $X(\mathbf{R})$ and
$X(\mathbf{C})$ and its dimension by $n$. We assume that $X$ is smooth and
projective and that its real locus is non empty.

###  1.1. Topology of real algebraic hypersurfaces

A major restriction for the topology of real algebraic varieties is given by
Smith-Thom inequality [18], which says that the sum of the
$\mathbf{Z}/2$-Betti numbers of the real locus of a real algebraic variety is
smaller or equal than the sum of the $\mathbf{Z}/2$-Betti numbers of its
complex locus, that is

$b\left(X(\mathbf{R})\right):=\displaystyle\sum_{i=0}^{n}b_{i}\left(X(\mathbf{R}),\mathbf{Z}/2\right)\leq\sum_{i=0}^{2n}b_{i}\left(X(\mathbf{R}),\mathbf{Z}/2\right)=:b\left(X(\mathbf{C})\right).$

In the case of equality, the variety is called a _maximal variety_. For real
algebraic curves, Smith-Thom inequality is known as Harnack-Klein inequality
[11, 14] and can be written $b_{0}\big{(}C(\mathbf{R})\big{)}\leq g+1$, where
$g$ is the genus of $C(\mathbf{C})$.

A fundamental question in real algebraic geometry asks whether a given real
algebraic variety contains maximal hypersurfaces in some given linear system.
For example, for any $m\in\mathbf{N}$, there exists a degree $m$ maximal real
algebraic curves in $\mathbf{P}^{2}$, see [11]. However, in general, the
answer to this question is negative. For example, given a non-maximal real
algebraic curve $C$, the surface $C\times C$ does not contain any maximal
curves. Nevertheless, such surface contains _asymptotically maximal
hypersurfaces_. To give the definition of asymptotically maximal
hypersurfaces, let us fix a very ample divisor $D$ of $X$ and consider the
linear system $\absolutevalue{mD},m>0$, consisting of real divisors linearly
equivalent to $mD$. By Bertini theorem, a generic element $Z$ of
$\absolutevalue{mD}$ is smooth and, by Ehresmann theorem, the topology of the
complex locus $Z(\mathbf{C})$ of $Z$ only depends on $D$ and on $m$. One can
then compute the asymptotics
$b\left(Z(\mathbf{C})\right)=D^{n}m^{n}+O(m^{n-1})$, as $m\rightarrow\infty$,
where $D^{n}$ denotes the top self-intersection number of $D$ (which is a
positive integer) and $n$ is the dimension of $X$. By Smith-Thom inequality,
we then have

$b\left(Z(\mathbf{R})\right)\leq D^{n}m^{n}+O(m^{n-1})$

and a sequence $Z_{m}\in\absolutevalue{mD}$ is called asymptotically maximal
if

$\displaystyle\lim_{m\rightarrow\infty}\frac{b\left(Z_{m}(\mathbf{R})\right)}{D^{n}m^{n}}\rightarrow
1.$

Asymptotically maximal hypersurfaces exists in projective spaces [13], in
toric varieties [3] and in surfaces [7, Theorem 5]. However, one does not know
if any real algebraic variety contains asymptotically maximal hypersurfaces
and it is not clear (at least for the author) if one should expect this
property to be true for any $X$.

One of the consequence of the main result of this paper is that _any_ real
algebraic variety always contains real algebraic hypersurfaces whose Betti
numbers grow as the maximal possible order. More precisely we obtain the
following result.

######  Theorem 1.1.

Let $X$ be a real algebraic variety of dimension $n$ and $D$ be a real very
ample divisor of $X$. Then there exists $c>0$ and $m_{0}\in\mathbf{N}$ such
that, for any $m\geq m_{0}$, there exists $Z\in\absolutevalue{mD}$ with
$b_{i}\left(Z(\mathbf{R})\right)\geq cm^{n}$ for any $i\in\\{0,\dots,n-1\\}$.

###  1.2. Existence of hypersurfaces with prescribed components

We actually prove a more precise result than Theorem 1.1. Indeed, for any
smooth closed (not necessarly connected) hypersurface $\Sigma$ of
$\mathbf{R}^{n}$, we can produce real algebraic hypersurfaces
$Z_{m}\in\absolutevalue{mD}$ such that $Z_{m}(\mathbf{R})$ contains at least
$cm^{n}$ connected component diffeomorphic to $\Sigma$. This is the content of
Theorem 1.3, which is the main result of our paper. In order to state it, we
need the following definition.

######  Definition 1.2.

Let $\Sigma$ be a closed smooth manifold, not necessarly connected. For any
smooth manifold $S$, we denote by $\mathcal{N}_{\Sigma}(S)$ the number of
connected components of $S$ that are diffeomorphic to $\Sigma$.

######  Theorem 1.3.

Let $X$ be a real algebraic variety of dimension $n$ and $D$ be a real very
ample divisor of $X$. For any smooth closed hypersurface $\Sigma$ of
$\mathbf{R}^{n}$, there exists $c>0$ and $m_{0}\in\mathbf{N}$ such that, for
any $m\geq m_{0}$, there exists $Z\in\absolutevalue{mD}$ with
$\mathcal{N}_{\Sigma}\left(Z(\mathbf{R})\right)\geq cm^{n}$.

Theorem 1.1 then follows from Theorem 1.3 by taking as $\Sigma$ any
hypersurface with $b_{i}(\Sigma)\geq 1$ for any $i\in\\{0,\dots,n-1\\}$.

The proof of Theorem 1.3 is probabilistic and it is given in Section 2. The
idea is to construct a probability measure on $\absolutevalue{mD}$ for which
the expected value
$\mathbf{E}\left(\mathcal{N}_{\Sigma}(Z(\mathbf{R}))\right)$ of
$\mathcal{N}_{\Sigma}(Z(\mathbf{R}))$, for $Z\in\absolutevalue{mD}$, is at
least $cm^{n}$, for $m$ large enough, where $c$ is a positive constant
independent of $m$. This is the content of Theorem 2.3. Remark that in Section
1.4 we will state a generalization of Theorem 1.3 for complete intersections.

###  1.3. Existence of many lagrangians in complex hypersurfaces

We now give an application of Theorem 1.3 in symplectic geometry. Recall that
the complex projective space $\mathbf{C}\mathbf{P}^{n}$ is equipped with a
natural symplectic form $\omega_{FS}$, called the Fubini-Study symplectic
form. Any smooth complex hypersurface of $\mathbf{C}\mathbf{P}^{n}$ inherits,
by restriction, a symplectic form. Recall also that a lagrangian of a
symplectic manifold $M$ is a smooth closed submanifold of dimension half the
dimension of $M$ and for which the restriction of the symplectic form is zero
everywhere. For example, $\mathbf{R}\mathbf{P}^{n}$ is a lagrangian of
$\mathbf{C}\mathbf{P}^{n}$ and, more generally, the real locus of a real
algebraic hypersurface of $\mathbf{C}\mathbf{P}^{n}$ is a lagrangian of its
complex locus. As an application of Theorem 1.3, we obtain an easy proof of
the following result by D. Gayet.

######  Theorem 1.4.

[6, Theorem 1.1] For any smooth closed hypersurface $\Sigma$ of
$\mathbf{R}^{n}$, there exists $c>0$ and $m_{0}\in\mathbf{N}$ such that, for
any $m\geq m_{0}$, any degree $m$ smooth complex hypersurface of
$\mathbf{C}\mathbf{P}^{n}$ contains at least $cm^{n}$ pairwise disjoint
lagrangians diffeomorphic to $\Sigma$.

###### Proof.

Theorem 1.3 provides smooth complex hypersurfaces of degree $m$ with at least
$cm^{n}$ pairwise disjoint lagrangians diffeomorphic to $\Sigma$ (as the real
locus of a real algebraic hypersurface is lagrangian). As, from a symplectic
point of view, all degree $m$ smooth complex hypersurfaces of
$\mathbf{C}\mathbf{P}^{n}$ are isomorphic, this implies that any degree $m$
smooth complex hypersurface contains at least $cm^{n}$ pairwise disjoint
lagrangians diffeomorphic to $\Sigma$. ∎

Actually, the same result and argument works for any Kähler $(X,\omega)$
equipped with an ample divisor $D$ with $\omega\in c_{1}(D)$, that _can be
defined over $\mathbf{R}$_. Although our proof is simpler than the one in [6]
(because we do not use any quantitative Moser-type argument), it should be
noted that in [6] this result is proved for any complex projective variety
$X$, and not just for those that can be defined over $\mathbf{R}$.

It is worth noticing that the order $m^{n}$ appearing in Theorem 1.4 is
optimal when the Euler characteristic of $\Sigma$ is nonzero. Indeed, as
remarked in [6, Corollary 1.2], if $\chi(\Sigma)\neq 0$ and $Z_{m}$ denotes a
smooth degree $m$ complex hypersurface of $\mathbf{C}\mathbf{P}^{n}$, then
(the homology class of) the disjoint langrangians diffeomorphic to $\Sigma$
are linearly independent in $H_{n-1}(Z_{m},\mathbf{Z})$ and the dimension of
$H_{n-1}(Z_{m},\mathbf{Z})$ grows exactly as $m^{n}$ when
$m\rightarrow\infty$.

###  1.4. Existence of complete intersections with prescribed topology

We can actually prove all the previous results not only for hypersurfaces, but
more generally for complete intersections. For this, recall that, as for the
case of hypersurfaces, the topology of the complex locus of a complete
intersection $Z_{1}\cap\dots\cap Z_{r}$, with $Z_{i}\in\absolutevalue{mD}$,
does not depend on the choice of the hypersurfaces $Z_{i}$, if they are chosen
generically. One can then compute the asymptotics for the total Betti number
of $Z_{1}(\mathbf{C})\cap\dots\cap Z_{r}(\mathbf{C})$ for generic
$Z_{i}\in\absolutevalue{mD}$ and get

$b(Z_{1}(\mathbf{C})\cap\dots\cap
Z_{r}(\mathbf{C}))=\binom{n-1}{r-1}D^{n}m^{n}+O(m^{n-1})$

as $m\rightarrow\infty$, see [2, Proposition 2.2]. In this setting, we have
the following result.

######  Theorem 1.5.

Let $X$ be a real algebraic variety of dimension $n$ and $D$ be a real very
ample divisor of $X$. For any $r$-codimensional closed submanifold $\Sigma$ of
$\mathbf{R}^{n}$ with trivial normal bundle, there exists $c>0$ and
$m_{0}\in\mathbf{N}$ such that for any $m\geq m_{0}$ there exists
$Z_{1},\dots,Z_{r}\in\absolutevalue{mD}$ with

$\mathcal{N}_{\Sigma}\left(Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R})\right)\geq cm^{n}.$

Theorem 1.3 is exacly Theorem 1.5 for $r=1$. Remark indeed that any closed
hypersurface of $\mathbf{R}^{n}$ has trivial normal bundle. On the contrary,
it is worth noting that Theorem 1.5 is not a consequence of Theorem 1.3 by
some induction, for example by considering $Z_{1}\cap Z_{2}$ as a hypersurface
of $Z_{1}$ and then applying Theorem 1.3 to $Z_{1}$. Indeed the hypersurfaces
$Z_{1},Z_{2}\in\absolutevalue{mD}$ have degree that varies with $m$, and thus
no induction can be applied since Theorem 1.3 requires a fixed ambient
variety. Even if one tried to get around this problem, fixed once for all one
of the hypersurfaces, say $Z_{1}$, and considered $Z_{1}\cap Z_{2}$ as a
hypersurface of $Z_{1}$, then applying Theorem 1.3 to $Z_{1}$ one would get
only $cm^{n-1}$ connected components diffeomorphic to $\Sigma$, and not
$cm^{n}$.

######  Remark 1.6.

Analogues of Theorems 1.1 and 1.4 can then be translated for complete
intersections and are direct consequences of the Theorem 1.5, in the same way
that Theorems 1.1 and 1.4 are consequences of the Theorem 1.3.

###  1.5. Organization of the paper

The paper is organized as follows.
In Section 2, we define a probability measure on the linear system
$\absolutevalue{mD}$ and prove Theorem 1.5 admitting a result, namely
Proposition 2.4. In Section 3, we prove Proposition 2.4. In Section 4 we make
some comments on this paper and on related works and we adress some questions
arising from this work.

### Acknowledgments

This paper was written while I was postdoc at the Institut de Recherche
Mathématique Avancée, Université de Strasbourg. I thank the IRMA for the
excellent conditions given to me during my postdoc.

##  2\. Probability measures on the linear systems and proof of the main
result

In this section we equip the linear system $\absolutevalue{mD}$ with a
probability measure. Here $D$ is a very ample divisor of a real algebraic
variety $X$. With respect to this probability measure, we estimate the
expected value of $\mathcal{N}_{\Sigma}(Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R}))$, for $Z_{i}\in\absolutevalue{mD}$. This is the content of
the Theorem 2.3, which directly implies Theorem 1.5.

###  2.1. Linear systems and hyperplane sections

The very ample divisor $D$ embeds $X$ into $\mathbf{P}^{N}$, for some
$N\in\mathbf{N}$. We fix once for all such an embedding and identify $X$ with
its image in $\mathbf{P}^{N}$. A real algebraic hypersurface
$Z\in\absolutevalue{D}$ is then obtained as a intersection $H\cap X$ of $X$
with an hyperplane $H\in\absolutevalue{\mathcal{O}(1)}$ defined over
$\mathbf{R}$. More generally, given a degree $m$ real algebraic hypersurfaces
$H$ in $\mathbf{P}^{N}$, its intersection with $X$ defines a real algebraic
hypersurface $Z:=H\cap X$ of $X$ in the linear system $\absolutevalue{mD}$.
Note however that not all hypersurfaces lying in $\absolutevalue{mD}$ are of
the form $H\cap X$, for $H\in\absolutevalue{\mathcal{O}(m)}$, that is, the map
$H\in\absolutevalue{\mathcal{O}(m)}\mapsto H\cap X\in\absolutevalue{mD}$ is
not surjective in general, when $m>1$.

###  2.2. Probability measure on $\absolutevalue{mD}$

Let $\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]$ be the space of real homogeneous
polynomials of degree $m$ in $N+1$ variables. This space coincides with the
space of global algebraic sections of the line bundle $\mathcal{O}(m)$ on
$\mathbf{P}^{N}$, which are defined over $\mathbf{R}$. We endow the space
$\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]$ with a Gaussian probability measure
$\mu_{m}$ as follows. Let $h_{m}$ be the Fubini-Study metric on the line
bundle $\mathcal{O}(m)$ and let $g_{FS}$ be the Riemannian Fubini-Study metric
on $\mathbf{P}^{N}(\mathbf{R})$. Then, for any pair of real polynomials
$P_{1},P_{2}\in\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]$ we define

(1) $\langle
P_{1},P_{2}\rangle_{L^{2}}=\int_{\mathbf{P}^{N}(\mathbf{R})}h_{m}(P_{1},P_{2})d\mathrm{vol}_{FS}.$

This scalar product induces a Gaussian measure $\mu_{m}$ on
$\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]$ defined by

(2)
$\mu_{m}(U)=\frac{1}{\sqrt{\pi}^{d_{m}}}\int_{U}e^{-\norm{P}_{L^{2}}^{2}}dP$

for any open set $U\subset\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]$, where
$d_{m}=\dim\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]$.

For any polynomial $P\in\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]$, we denote by
$V_{P}=\\{P=0\\}\subset\mathbf{P}^{N}$ the real algebraic hypersurface defined
by $P$ and by $Z_{P}:=V_{P}\cap X$ the real algebraic hypersurface of $X$
defined by $P_{\mid X}$. As said Section 2.1, the hypersurface $Z_{P}$ is an
element of the linear system $\absolutevalue{mD}$.

######  Definition 2.1.

The map $\varphi:P\in\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]\mapsto
Z_{P}\in\absolutevalue{mD}$ induces a probability measure
$\mathrm{Prob}_{m}:=\varphi_{*}\mu_{m}$ on $\absolutevalue{mD}$. For any
$r\in\\{1,\dots,n\\}$ we equip the $r$-fold cartesian products
$\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]^{r}$ and $\absolutevalue{mD}^{r}$
with the product measures, that we still denote respectively by $\mu_{m}$ and
$\mathrm{Prob}_{m}$.

######  Remark 2.2.

By what we said in Section 2.1, the probability measure $\mathrm{Prob}_{m}$ is
degenerate, that is, it is supported in a proper subspace of
$\absolutevalue{mD}^{r}$.

###  2.3. Expected value of $\mathcal{N}_{\Sigma}(Z(\mathbf{R}))$

Let $\Sigma$ be a smooth closed $r$-codimensional submanifold of
$\mathbf{R}^{n}$ with trivial normal bundle. We are interested in the random
variable

$(Z_{1},\dots,Z_{r})\in\absolutevalue{mD}^{r}\mapsto\mathcal{N}_{\Sigma}\left(Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R})\right),$

where $\mathcal{N}_{\Sigma}$ denotes the number of connected components that
are diffeomorphic to $\Sigma$, see Definition 1.2. The next theorem estimates
from below the expectation

$\mathbf{E}\left(\mathcal{N}_{\Sigma}(Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R}))\right)\\\
=\displaystyle\int_{(Z_{1},\dots,Z_{r})\in\absolutevalue{mD}^{r}}\mathcal{N}_{\Sigma}\left(Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R})\right)d\mathrm{Prob}_{m}(Z)$

where $\mathrm{Prob}_{m}$ is the probability measure on
$\absolutevalue{mD}^{r}$ defined in Definition 2.1.

######  Theorem 2.3.

There exists $c>0$ and $m_{0}$ such that for any $m\geq m_{0}$ we have

$\mathbf{E}\left(\mathcal{N}_{\Sigma}(Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R}))\right)\geq cm^{n}.$

Theorem 2.3 directly implies Theorem 1.5, since we have the trivial bound

$\sup_{(Z_{1},\dots,Z_{r})\in\absolutevalue{mD}^{r}}\mathcal{N}_{\Sigma}\big{(}Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R})\big{)}\geq\mathbf{E}\left(\mathcal{N}_{\Sigma}(Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R}))\right).$

In the rest of the section we prove Theorem 2.3. For this, remark that the
real locus $X(\mathbf{R})$ of $X$ inherits from $\mathbf{P}^{N}(\mathbf{R})$ a
Riemannian metric $g:=g_{FS\mid X(\mathbf{R})}$. For any point $x\in
X(\mathbf{R})$, we denote by $B(x,R)$ the geodesic ball of radius $R$ around
$x$.

Theorem 2.3 will be a direct consequence of the following proposition.

######  Proposition 2.4.

There exists $R,c_{\Sigma}>0$ and $m_{0}$ such that for any point $x\in
X(\mathbf{R})$ and any $m\geq m_{0}$ we have

$\mathrm{Prob}_{m}\bigg{\\{}(Z_{1},\dots,Z_{r})\in\absolutevalue{mD}^{r},\hskip
2.84526pt\mathcal{N}_{\Sigma}\left(Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R})\cap B\big{(}x,\frac{R}{m}\big{)}\right)\geq 1\bigg{\\}}\geq
c_{\Sigma}.$

We postpone the proof of this proposition to the next section. We now show how
this implies Theorem 2.3.

###### Proof of Theorem 2.3.

Let $R$ be the positive constant given by Proposition 2.4. For any
$m\in\mathbf{N}$, we fix a set of points $\\{x_{i}\\}_{i\in I_{m}}$ of
$X(\mathbf{R})$ with the property that the balls $B(x_{i},\frac{R}{m})$ are
all disjoint and the balls $B\left(x_{i},\frac{2R}{m}\right)$ cover the whole
$X(\mathbf{R})$. In particular, we get

$\textrm{Vol}\left(X(\mathbf{R})\right)\leq\sum_{i\in
I_{m}}\textrm{Vol}\left(B\left(x_{i},\frac{2R}{m}\right)\right)\leq\absolutevalue{I_{m}}\textrm{Vol}\left(B_{\mathbf{R}^{n}}\left(0,\frac{3R}{m}\right)\right)$

for $m$ large enough. This implies that for $m$ large enough we have

(3)
$\absolutevalue{I_{m}}\geq\frac{\textrm{Vol}\left(X(\mathbf{R})\right)}{\textrm{Vol}\big{(}B_{\mathbf{R}^{n}}\left(0,\frac{3R}{m}\right)\big{)}}=c^{\prime}\textrm{Vol}\left(X(\mathbf{R})\right)m^{n}.$

We can now estimate the expected value of
$\mathcal{N}_{\Sigma}(Z_{1}(\mathbf{R})\cap\dots\cap Z_{r}(\mathbf{R}))$, for
$Z\in\absolutevalue{mD}$. We have the lower bound

(4) $\mathbf{E}\left(\mathcal{N}_{\Sigma}(Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R}))\right)\geq\\\ \sum_{i\in
I_{m}}\mathrm{Prob}_{m}\bigg{\\{}(Z_{1},\dots,Z_{r})\in\absolutevalue{mD}^{r},\hskip
2.84526pt\mathcal{N}_{\Sigma}\left(Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R})\cap B\big{(}x_{i},\frac{R}{m}\big{)}\right)\geq
1\bigg{\\}}.$

By Proposition 2.4, the latter is bigger or equal than
$c_{\Sigma}\absolutevalue{I_{m}}$ which, by (3), is bigger or equal than
$c^{\prime}\textrm{Vol}\left(X(\mathbf{R})\right)m^{n}=:cm^{n}$. Hence the
result. ∎

##  3\. Proof of Proposition 2.4

In this section we prove Proposition 2.4. Following [9], the idea is to find
one complete intersection $Z_{1}\cap\cdots\cap Z_{r}$ of degree $m\gg 1$ such
that

$\mathcal{N}_{\Sigma}\left(Z_{1}(\mathbf{R})\cap\dots\cap
Z_{r}(\mathbf{R})\cap B\big{(}x,\frac{R}{m}\big{)}\right)\geq 1$

and then prove that this complete intersection can perturbed in a precise
quantitative way so that this property happens with positive probability.
Actually, still keeping in mind this idea, we will use a slightly different,
more flexible, point of view, developped recently in [16].

The formalism developped in [16] works better when one considers functions
rather than hypersurfaces or sections of line bundles. We then consider
$S^{N}$ the double covering of $\mathbf{P}^{N}(\mathbf{R})$ and $M\subset
S^{N}$ the double covering of $X(\mathbf{R})$. We fix on $S^{N}$ the standard
round metric $g_{S^{N}}$ (so that the Fubini-Study metric $g_{FS}$ is the
quotient of $g_{S^{N}}$ by the antipodal map). For any $r$-tuple of real
polynomials
$\underline{P}=(P_{1},\dots,P_{r})\in\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]^{r}$
we denote by $V_{\underline{P}}$ the complete intersection in $\mathbf{P}^{N}$
defined by $\\{P_{1}=0\\}\cap\dots\cap\\{P_{r}=0\\}$ and by
$Z_{\underline{P}}$ the complete intersection in $X$ defined by
$V_{\underline{P}}\cap X$. Fix any point $x\in M$. To prove Proposition 2.4,
it is sufficient to show that

(5)
$\mu_{m}\left\\{\underline{P}\in\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]^{r},\mathcal{N}_{\Sigma}\left(Z_{\underline{P}}(\mathbf{R})\cap
B\big{(}x,\frac{R}{m}\big{)}\right)\geq 1\right\\}\geq c_{\Sigma}.$

where the probability measure $\mu_{m}$ is defined in (2).

Remark that the scalar product (1) can be also defined by

$\langle
P_{1},P_{2}\rangle_{L^{2}}=\int_{S^{N}}P_{1}P_{2}d\mathrm{vol}_{S^{N}}.$

A random polynomial $P\in\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]$ with respect
to the Gaussian measure $\mu_{m}$ can be then written as

$P=\sum_{i=1}^{d_{N}}a_{i}P_{i}$

where $d_{N}=\dim\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]$, the family
$\\{P_{i}\\}_{i}$ is an orthonormal basis of
$\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]$ and $(a_{i})_{i}$ are independent
standard Gaussian variables. A random
$\underline{P}\in\mathbf{R}^{hom}_{m}[X_{0},\dots,X_{N}]^{r}$ is then a
$r$-tuple $\underline{P}=(P_{1},\dots,P_{r})$ of independent random
polynomials $P_{i}$.

The covariance function $\mathcal{K}_{m}(x,y):=\mathbf{E}(P(x)P(y))$ of the
random polynomial $P$ is known to have a universal limit at scale $m^{-1}$
(see [12, Theorem 4.4]). This means that for any fixed $R>0$, any point $x$
and any $u,v\in B_{\mathbf{R}^{N}}(0,R)\subset\mathbf{R}^{N}\simeq T_{x}S^{N}$
(where $T_{x}S^{N}\simeq\mathbf{R}^{N}$ is given by any isometry) one has that

$K_{m,x}(u,v):=m^{-n}\mathcal{K}_{m}\left(\exp_{x}\left(\frac{u}{m}\right),\exp_{x}\left(\frac{v}{m}\right)\right)$

converges in the $\mathscr{C}^{\infty}$-topology to a function
$K:B_{\mathbf{R}^{N}}(0,R)\times B_{\mathbf{R}^{N}}(0,R)\rightarrow\mathbf{R}$
which is independent of $x$. The function $K$ is explicit and given by

$K(u,v)=\int_{B_{\mathbf{R}^{N}}(0,1)}e^{i\langle u-v,\xi\rangle}d\xi.$

By restriction, we then have that the covariance function of the random
polynomial $P$ restricted to $M$ also has a universal limit at scale $m^{-1}$.
This universal limit is nothing but the restriction of $K$ to
$B_{\mathbf{R}^{n}}(0,R)\times B_{\mathbf{R}^{n}}(0,R)$, where we see
$T_{x}M\subset T_{x}S^{N}$ as the subspace
$\mathbf{R}^{n}\subset\mathbf{R}^{N}$ given by the last $N-n$ coordinates
equal to $0$.

The restriction of the function $K$ to $B_{\mathbf{R}^{n}}(0,R)\times
B_{\mathbf{R}^{n}}(0,R)$ is the covariance function of a Gaussian field
$F:B_{\mathbf{R}^{n}}(0,R)\rightarrow\mathbf{R}$ (such Gaussian field is
precisely characterized by the property $K(u,v)=\mathbf{E}(F(u)F(v))$).

Consider $r$ independent copies $F_{1},\dots,F_{r}$ of the Gaussian field $F$,
and denote by $\underline{F}=(F_{1},\dots,F_{r})$, which is a Gaussian field
taking values in
$\mathscr{C}^{\infty}\left(B_{\mathbf{R}^{n}}(0,R),\mathbf{R}^{r}\right)$. We
now need three lemmas about the geometry of this Gaussian field
$\underline{F}$.

######  Lemma 3.1.

The restriction to $B_{\mathbf{R}^{n}}(0,R)$ of the rescaled $r$-tuple of
random polynomials

$\underline{P_{m}}(\cdot):=\left(m^{-n/2}P_{1}\left(\exp_{x}\left(\frac{\cdot}{m}\right)\right),\dots,m^{-n/2}P_{r}\left(\exp\left(\frac{\cdot}{m}\right)\right)\right)$

converges in probability to $\underline{F}$ as $m\rightarrow\infty$.

###### Proof.

By construction, we have that the covariance function of the rescaled random
polynomial $m^{-n/2}P\left(\exp_{x}\left(\frac{\cdot}{m}\right)\right)$
converges (in the $\mathscr{C}^{\infty}$-topology) to the covariance function
of $F$. This implies that the covariance function of $\underline{P_{m}}$
converges to the covariance function of $\underline{F}$. By [16, Theorem 5],
this implies the result. ∎

######  Lemma 3.2.

For any smooth function
$f\in\mathscr{C}^{\infty}\left(\bar{B}_{\mathbf{R}^{n}}(0,R),\mathbf{R}^{r}\right)$
and any neighborhood $U$ of $f$ in
$\mathscr{C}^{\infty}\left(\bar{B}_{\mathbf{R}^{n}}(0,R),\mathbf{R}^{r}\right)$,
the probability that $\underline{F}$ takes value in $U$ is strictly positive.
That is, the support of $\underline{F}$ contains the space
$\mathscr{C}^{\infty}\left(\bar{B}_{\mathbf{R}^{n}}(0,R),\mathbf{R}^{r}\right)$.
Here, $\bar{B}_{\mathbf{R}^{n}}(0,R)$ denotes the closed ball of radius $R$
around $0$.

###### Proof.

It is enough to prove that the support of each components of $\underline{F}$
includes
$\mathscr{C}^{\infty}\left(\bar{B}_{\mathbf{R}^{n}}(0,R),\mathbf{R}\right)$.
By the density of polynomials inside
$\mathscr{C}^{\infty}\left(\bar{B}_{\mathbf{R}^{n}}(0,R),\mathbf{R}\right)$,
it is actually enough to prove that any monomial $x_{1}^{k_{1}}\cdots
x_{n}^{k_{n}}$ is in the support of each components of $\underline{F}$.

By definition, a component of $\underline{F}$ is a copy of the Gaussian field
$F$ whose covariance function is

$K(u,v)=\int_{B_{\mathbf{R}^{N}}(0,1)}e^{i\langle u-v,\xi\rangle}d\xi$

and whose spectral measure $\sigma$ is the Lebesgue measure on
$B_{\mathbf{R}^{N}}(0,1)$.

The support of $F$ is known to be equal to the closure (in the
$\mathscr{C}^{\infty}(\bar{B}(0,R),\mathbf{R})$ topology) of the following
space of function

$\mathcal{H}_{F}=\mathrm{Span}\\{K_{v},v\in\mathbf{R}^{n}\\}$

where $K_{v}(x):=\int_{B_{\mathbf{R}^{N}}(0,1)}e^{i\langle
x-v,\xi\rangle}d\xi$, see [16, Theorem 6]. It is then enough to prove that any
monomial $x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}$ can be approximated (uniformly
together with its derivatives) by elements of the space $\mathcal{H}_{F}$.

In order to prove this, let
$\varphi:\mathbf{R}^{N}\rightarrow[0,1]\subset\mathbf{R}$ be an even smooth
function, which equals $0$ outside $B_{\mathbf{R}^{N}}(0,1)$ and such that
$\int_{\mathbf{R}^{N}}\varphi(\xi)\xi=1$. Let us consider the function
$\varphi_{t}(\xi)=\frac{1}{t^{N}}\varphi(\frac{\xi}{t})$, for any $0<t\leq 1$.
Given $k_{1},\dots,k_{n}\in\mathbf{N}$, with $\sum_{i=1}^{n}k_{i}=k$, consider
the function

$\frac{\partial^{k}}{\partial\xi_{1}^{k_{1}}\cdots\partial\xi_{n}^{k_{n}}}\varphi_{t}.$

By construction, the value at $x\in B_{\mathbf{R}^{n}}(0,R)$ of its inverse
Fourier transform equals

$\int_{\mathbf{R}^{N}}\frac{\partial^{k}}{\partial\xi_{1}^{k_{1}}\cdots\partial\xi_{n}^{k_{n}}}\varphi_{t}(\xi)e^{i\langle\xi,x\rangle}d\xi=(-1)^{k}x_{1}^{k_{1}}\cdots
x_{n}^{k_{n}}\int_{\mathbf{R}^{N}}\varphi_{t}(\xi)e^{i\langle\xi,x\rangle}d\xi$

where
$\int_{\mathbf{R}^{N}}\varphi_{t}(\xi)e^{-i\langle\xi,x\rangle}d\xi\xrightarrow{t\rightarrow
0}1$ in $\mathscr{C}^{\infty}(\bar{B}_{\mathbf{R}^{n}}(0,R),\mathbf{R})$.
Remark also that the support of $\varphi_{t}$ is included in
$\bar{B}_{\mathbf{R}^{N}}(0,1)$ for any $t\leq 1$. Denote by $f_{t}$ the
inverse Fourier transform of
$(-1)^{k}\frac{\partial^{k}}{\partial\xi_{1}^{k_{1}}\cdots\partial\xi_{n}^{k_{n}}}\varphi_{t}.$
We then have that the function

$g_{t}(x):=\int_{v\in\mathbf{R}^{N}}\int_{\xi\in\bar{B}_{\mathbf{R}^{N}}(0,1)}f_{t}(v)e^{i\langle
x-v,\xi\rangle}dvd\xi$

converges to $x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}$ in the
$\mathscr{C}^{\infty}(\bar{B}_{\mathbf{R}^{n}}(0,R),\mathbf{R})$ topology as
$t\rightarrow 0$. On the other hand, the functions $g_{t}$ can be approximated
in the $\mathscr{C}^{\infty}(\bar{B}_{\mathbf{R}^{n}}(0,R),\mathbf{R})$
topology by elements of $\mathcal{H}_{F}$, by taking Riemann sums. This
implies that the monomial $x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}$ can be
approximated (uniformly together with its derivatives) by elements of
$\mathcal{H}_{F}$. Hence the result. ∎

######  Lemma 3.3.

The probability that the zero locus of $\underline{F}$ is diffeomorphic to
$\Sigma$ is bigger than some strictly positive number $c_{\Sigma}$.

###### Proof.

Let
$f_{\Sigma}\in\mathscr{C}^{\infty}\left(\bar{B}_{\mathbf{R}^{n}}(0,R),\mathbf{R}^{r}\right)$
be a function vanishing transversally and such that $\\{f_{\Sigma}=0\\}$ is
diffeomorphic to $\Sigma$. Remark that such function exists exactly by the
hypothesis that the normal bundle of $\Sigma$ is trivial. By Thom isotopy
lemma, the zero set of any other function $f$ lying in a small neighborhood
$U_{\Sigma}$ of $f_{\Sigma}$ is still diffeomorphic to $\Sigma$. By Lemma 3.2,
$f_{\Sigma}$ is in the support of $\underline{F}$, and then the probability
that $\underline{F}$ takes value in $U_{\Sigma}$ is strictly positive, which
proves the lemma. ∎

We are now able to finish the proof of Proposition 2.4. By Lemmas 3.1 and 3.3
and by [16, Theorem 5], we obtain that, for $m$ large enough, the probability
that $Z_{\underline{P}}(\mathbf{R})\cap B(x,\frac{R}{m})$, for
$\underline{P}\in\mathbf{R}_{m}^{hom}[X_{0},\dots,X_{N}]^{r}$, has a connected
component diffeomorphic to $\Sigma$ is bigger than $c_{\Sigma}$. This is
exactly (5), which finishes the proof of Proposition 2.4. ∎

##  4\. Further comments and related results

###  4.1. Some questions

Let $D$ be a divisor on a real algebraic variety of dimension $n$. For any
$i\in\\{0,\dots,n-1\\}$ we define

$v_{i}(D)=\displaystyle\limsup_{m\rightarrow\infty}\frac{1}{m^{n}}\sup_{Z\in\absolutevalue{mD}}b_{i}(Z(\mathbf{R}))\hskip
11.38109pt\mathrm{and}\hskip
11.38109ptv(D)=\displaystyle\limsup_{m\rightarrow\infty}\frac{1}{m^{n}}\sup_{Z\in\absolutevalue{mD}}b(Z(\mathbf{R})).$

One always has the trivial inequality $v_{i}(D)\leq v(D)$. The present paper
implies that $v_{i}(D)>0$ if $D$ is ample. In this case, the existence of
asymptotically maximal hypersurfaces in $\absolutevalue{mD},m\gg 0$ is
equivalent to the equality $v(D)=D^{n}$, where $D^{n}$ denotes the top self-
intersection number of $D$. What about the other numbers $v_{i}(D)$? Are there
examples of $(X,D)$ for which $v_{i}(D)<v(D)$, for some $i$? Does there exist
relations between the numbers $v_{i}(D)$, apart from the trivial equality
$v_{i}(D)=v_{n-i-1}(D)$? In real algebraic geometry, these numbers may refine
the concept of volume of divisors. Recall that, if $D$ is a divisor of a
complex manifold $X$ then the volume of $D$ is defined by the formula

$\textrm{Vol}(D)=\displaystyle\limsup_{m\rightarrow\infty}\frac{n!}{m^{n}}\dim
H^{0}(X,\mathcal{O}(mD)).$

The volume of a divisor encodes some informations about it. For example if $D$
is ample, then $\textrm{Vol}(D)=D^{n}$. A divisor $D$ is big precisely when
$\textrm{Vol}(D)>0$. If now we consider $X$ and $D$ to be defined over
$\mathbf{R}$, can one caracterize the bigness of $D$ in terms of $v_{i}(D)$?
If $D$ is a big, are the numbers $v_{i}(D)$ positive for any
$i\in\\{0,\dots,n-1\\}$?

###  4.2. Some comments on the probability measures on $\absolutevalue{mD}$

Let $D$ be an ample divisor over a real algebraic variety $X$ and denote by
$H^{0}(X,\mathcal{O}(mD))$ the space over global algebraic sections of
$\mathcal{O}(mD)$ defined over $\mathbf{R}$. From the algebro-geometric point
of view, the probability measures on $H^{0}(X,\mathcal{O}(mD))$ that seem more
natural are the so-called complex Fubini-Study measures. These measures are
the Gaussian measure on $H^{0}(X,\mathcal{O}(mD))$ induced by the scalar
product defined by

(6) $\langle s_{1},s_{2}\rangle=\int_{X(\mathbf{C})}h^{\otimes
m}(s_{1},s_{2})\frac{\omega^{\wedge n}}{n!}$

where $h$ is a hermitian metric on $\mathcal{O}(D)$ with positive curvature
$\omega$. It should be noted that the Kodaira embeddings
$\Phi_{m}:X(\mathbf{C})\rightarrow\mathbf{P}^{d_{m}}(\mathbf{C})$ induced by
the Hermitian products (6) are asymptotically isometries [19, 4], in the sense
that $\frac{1}{m}\Phi^{*}_{m}\omega_{FS}\rightarrow\omega$, as
$m\rightarrow\infty$. With respect to the probability measure induced by the
Hermitian product (6), asymptotically maximal hypersurfaces are exponentially
rare [1, 5, 7]. This reflects the fact that it is very difficult in practice
to construct such hypersurfaces: if one randomly writes down an equation, it
is unlikely to define a (almost) maximal hypersurface. For these probability
measures, the expected value
$\mathbf{E}\left(b_{i}\left(Z(\mathbf{R})\right)\right)$ of each Betti number
of a random real hypersurface $Z\in\absolutevalue{mD}$ is of order $m^{n/2}$
(see [8, 9]) and any closed hypersurface $\Sigma\subset\mathbf{R}^{n}$ appears
with positive probability in any ball of radius $\sim m^{-1/2}$ (see [9]).

In contrast, the probability measure used in this paper seems less natural
from an algebraic geometry point of view and produces, because of this, more
extreme effects: the average of the Betti numbers of a random hypersurface is
of maximal order (for $X=\mathbf{P}^{n}$ and for the number of connected
components, this was proved in [15, 17]) and this implies precisely the
existence of hypersurfaces with rich topologies. Note that for the sphere
$S^{n}$ the existence of real algebraic hypersurfaces with many components
diffeomorphic to a given closed hypersurface $\Sigma\subset\mathbf{R}^{n}$ can
also be obtain from [10], by remarking that linear combinations of the
Laplacian eigenfunctions with eigenvalues smaller than $L$ coincide with
homogenous polynomials of degree $\sim\sqrt{L}$.

We recall that for the projective space, and more generally for toric
varieties, the existence of asymptotically maximal hypersurfaces is known. In
these cases, where more combinatorial tools are accessible, it would be
interesting to construct probability measures in which such hypersurfaces
appear with high probability.

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