2.1 Convergence of measures

Let us start by recalling a few classical facts in measure theory which are used mainly in § 3.

Let $S$ be an analytic subset of dimension $d$ of a manifold $M$ whose smooth locus is oriented, and let $\omega$ be a smooth form of degree $d$ on $M$. Then the restriction of $\omega$ to the smooth locus of $S$ defines a signed Radon measure on $S$. This measure is regular in the sense that:

1. (i) for every open subset $U$ of $S$ and every sequence of compact sets $(C_n)_{n\in \mathbb {N}}$ with $C_n \subset \mathring {C}_{n+1}$ and $\bigcup _{n\in \mathbb {N}} C_n = U$, we have

   \[ \int_U \omega = \lim_{n\to +\infty} \int_{C_n} \omega; \]

2. (ii) for every compact subset $C\subset S$ and every sequence of open sets $(U_n)_{n\in \mathbb {N}}$ with $\bar U_{n+1} \subset U_n$ and $\bigcap _{n\in \mathbb {N}} U_n=C$, we have

   \[ \int_C \omega = \lim_{n\to +\infty} \int_{U_n} \omega. \]

In the absence of any precision, we say that a set $A\subset S$ has measure zero if the intersection of $A$ with the smooth locus of $S$ has Lebesgue measure zero in any coordinate chart. This implies that its measure with respect to $\omega$ is zero.

Given a sequence of signed Radon measures $\mu _n$, we have the following equivalent characterizations of the convergence of $\mu _n$ to $\omega$:

1. (i) for every continuous function $f:S\to \mathbb {R}$ with compact support,

   \[ \mu_n(f) \underset{n\to +\infty}{\longrightarrow} \int_S f \omega; \]

2. (ii) for every relatively compact open subset $\Omega$ of $S$ with boundary of measure zero,

   \[ \mu_n(\Omega) \underset{n\to +\infty}{\longrightarrow} \int_\Omega\omega. \]

We then say that $\mu _n$ converges weakly to $\omega$ and we write

\[ \mu_n\underset{n\to +\infty}{\rightharpoonup}\omega. \]

We say that $\mu _n$ converges weakly to $\omega$ on an open subset $U$ if the restriction of $\mu _n$ to $U$ converges weakly to the restriction of $\omega$. Equivalently, $\mu _n$ converges weakly to $\omega$ on $U$ if property (i) holds for any function with compact support inside $U$.

The following standard facts will be useful in proving weak convergence of measures.

Here, $\vert \mu _n\vert = \mu _n^+ + \mu _n^-$ denotes the total variation measure of $\mu$.

2.2 Currents on manifolds

For more details on this section, we refer to [Reference Griffiths and HarrisGH94, Chapter 3, § 1].

Let $M$ be a real manifold of dimension $n$. For $k\geq 0$, let $\Omega _{c}^{k}(M)$ be the vector space of $C^{\infty }$ differential forms on $M$ of degree $k$ with compact support. It is endowed with its natural topological space structure making it a Fréchet space.

Example 2.4 If $N\hookrightarrow M$ is an oriented properly immersed submanifold of codimension $k$ of $M$, the integration current $T_N\in \mathcal {D}^k(M)$ is defined by

\[ T_N(\beta) = \int_N \beta, \quad \beta\in \Omega_{c}^{n-k}(M). \]

Example 2.5 A differential $k$-form $\alpha$ induces a $k$-dimensional current $T_\alpha$ defined by

\[ T_\alpha(\beta) = \int_M \alpha \wedge \beta, \quad \beta\in \Omega_{c}^{n-k}(M). \]

The exterior derivative $d$ on differential forms induces a map

\[ d:\mathcal{D}^k(M)\rightarrow \mathcal{D}^{k+1}(M) \]

defined by

\[ dT(\phi)=(-1)^{k+1}T(d\phi),\quad \phi\in\Omega_{c}^{n-k-1}(M). \]

A current $T$ is closed if $dT=0$.

The exterior derivative defines a cochain complex structure on $(\mathcal {D}^\bullet (M))$, and Example 2.5 gives a morphism of cochain complexes

\[ (\Omega^\bullet(M),d)\rightarrow (\mathcal{D}^\bullet(M),d). \]

The previous morphism is, in fact, a quasi-isomorphism (i.e. it induces isomorphisms at the level of cohomology groups, see [Reference Griffiths and HarrisGH94, p. 382]).

The space $\mathcal {D}^k(M)$ is naturally a topological vector space when equipped with the weak topology: a sequence $T_n$ of degree $k$ currents converges weakly to a current $T$ (which we write $T_n \rightharpoonup T$) if

\[ T_n(\beta)\underset{n\to +\infty}{\longrightarrow} T(\beta) \]

for all $\beta \in \Omega _c^{n-k}(M)$.

Assume now that $M$ is a complex manifold of complex dimension $n$. Then the complex $\mathcal {D}^\bullet (M)$ admits a bigrading

\[ \mathcal{D}^k(M) = \bigoplus_{p+q = k} \mathcal{D}^{p,q}(M), \]

where $\mathcal {D}^{p,q}(M)$ is the topological dual of the complex vector space $\Omega _c^{n-p,n-q}(M)$.

In particular, if $Z\subseteq M$ is a closed complex analytic subvariety of complex codimension $k$, we can similarly define a closed integration current $T_Z\in D^{k,k}(M)$ by integrating over (the smooth locus of) $Z$.

2.3 Homogeneous spaces, orientations, and volume forms

In this section, we introduce the notation that we use throughout the paper and recall some facts on volume forms on Lie groups.

Let $G$ denote a real algebraic semi-simple Lie group. Suppose we are given the following subgroups of $G$:

1. (i) a lattice $\Gamma$;

2. (ii) a semi-simple Lie subgroup $H$ without compact factor;

3. (iii) a compact subgroup $K$.

Let $L$ be the intersection of $K$ and $H$. This is a compact subgroup of $H$. The group $H$ acts on the right on the quotient $\Gamma \backslash G$, and we are interested in the next section in equidistribution properties of orbits of this action and their projection to $\Gamma \backslash G/K$.

Let $\mathfrak {g}$, $\mathfrak {h}$, $\mathfrak {k}$, and $\mathfrak {l}$ denote the Lie algebras of $G$, $H$, $K$, and $L$, respectively. Up to taking subgroups of index two, we can assume that the adjoint actions of $G$, $H$, $K$, and $L$ have determinant one. We then fix once and for all some orientation of $\mathfrak {g}$, $\mathfrak {h}$, $\mathfrak {k}$, and $\mathfrak {l}$ and orient accordingly the quotient spaces $\mathfrak {g}/\mathfrak {k}$, $\mathfrak {g}/\mathfrak {h}$, $\mathfrak {g}/\mathfrak {l}$, $\mathfrak {k}/\mathfrak {l}$, and $\mathfrak {h}/\mathfrak {l}$. Those orientations induce orientations on $G/K$, $G/H$, $G/L$, $K/L$, and $H/L$, respectively.

Recall that the Lie algebra $\mathfrak {g}$ carries a natural symmetric bilinear form called the Killing form, which is invariant under the adjoint action and non-degenerate because $G$ is semi-simple. Its restriction to $\mathfrak {h}$, $\mathfrak {k}$, or $\mathfrak {l}$ is still non-degenerate. We denote by $\omega _G$, $\omega _H$, $\omega _K$, and $\omega _L$ the volume forms associated with the restricted Killing metric and the prescribed orientations on $\mathfrak {g}$, $\mathfrak {h}$, $\mathfrak {k}$, and $\mathfrak {l}$, respectively, as well as the induced bi-invariant volume forms on $G$, $H$, $K$, and $L$. Finally, we denote by $\omega _{G/K}$, $\omega _{G/H}$, $\omega _{G/L}$, $\omega _{K/L}$, and $\omega _{H/L}$ the invariant volume forms on the corresponding homogeneous spaces induced by $\omega _G$, $\omega _H$, $\omega _K$, and $\omega _L$.

The volume forms $\omega _G$ and $\omega _{G/K}$ factor to volume forms on $\Gamma \backslash G$ and $\Gamma \backslash G/K$, respectively, and we define

\[ \operatorname{Vol}(\Gamma \backslash G) = \int_{\Gamma \backslash G}\omega_G,\quad\operatorname{Vol}(K) = \int_{K}\omega_K,\quad \operatorname{Vol}(\Gamma \backslash G/K) = \int_{\Gamma \backslash G/K}\omega_{G/K}, \]

respectively. The compatibility of the volume forms gives the following identity:

\[ \operatorname{Vol}(\Gamma \backslash G) = \operatorname{Vol}(\Gamma\backslash G/K)\cdot \operatorname{Vol}(K). \]

Similarly, if $xH$ is a closed $H$-orbit in $\Gamma \backslash G$, then $\omega _H$, $\omega _L$, and $\omega _{H/L}$ induce volume forms on $xH$, $L$, and $xH/L$. We denote by $\operatorname {Vol}(xH)$, $\operatorname {Vol}(L)$, and $\operatorname {Vol}(xH/L)$ their total masses, respectively, and we have the following identity:

\[ \operatorname{Vol}(xH) = \operatorname{Vol}(xH/L)\cdot \operatorname{Vol}(L). \]

2.4 Equidistribution in terms of currents

In this section, we reformulate equidistribution results of sequences of $H$-orbits in terms of convergence of currents. As before, for all $n\in \mathbb {N}$, let $\mathcal {O}_n$ be a finite union of closed $H$-orbits in $\Gamma \backslash G$. The starting point of our work is the remark that the equidistribution of a sequence $\{\mathcal {O}_n\subset \Gamma \backslash G,n\in \mathbb {N}\}$ can be reformulated as an equidistribution of the currents of integration over $\mathcal {O}_n$.

To be more precise, let $p$ denote the projection from $G/L$ to $G/H$ and $\pi$ the projection from $G/L$ to $G/K$.

Lemma 2.6 Assume that the sequence $(\mathcal {O}_n)_{n\in \mathbb {N}}$ is equidistributed in $\Gamma \backslash G$ and let $\widehat {T}_{\mathcal {O}_n/L}$ denote the integration current on $\mathcal {O}_n/L \subset \Gamma \backslash G/L$. Then

\[ \frac{1}{\operatorname{Vol}(\mathcal{O}_n/L)}\widehat{T}_{\mathcal{O}_n/L} \rightharpoonup \frac{1}{\operatorname{Vol}(K/L)\operatorname{Vol}(\Gamma \backslash G/K)} p^*\omega_{G/H}. \]

The form $p^*\omega _{G/H}$ can be seen as a transverse volume form to the foliation of $G/L$ by translates of $H/L$. Note that, by applying the lemma to $L= \{\mathbf {1}_G\}$, we obtain the following corollary.

Corollary 2.7 Assume that the sequence $(\mathcal {O}_n)_{n\in \mathbb {N}}$ is equidistributed in $\Gamma \backslash G$ and let $T_{\mathcal {O}_n}$ denote the integration current on $\mathcal {O}_n$. Then

\[ \frac{1}{\operatorname{Vol}(\mathcal{O}_n)}T_{\mathcal{O}_n} \rightharpoonup \frac{1}{\operatorname{Vol}(\Gamma \backslash G)}\widehat{p}^*\omega_{G/H}, \]

where $\widehat {p}$ is the projection from $G$ to $\Gamma \backslash G$.

Finally, one can push this equidistribution forward by the fibration map from $G/L$ to $G/K$. Recall that the map $\pi$ induces a push-forward map

\[ \pi_*: \Omega^\bullet(G/L) \to \Omega^{\bullet-\dim(K/L)} (G/K) \]

which is, by definition, Poincaré dual to the pull-back map $\pi ^*$, i.e.

\[ \int_{G/K}(\pi_*\alpha) \wedge \beta = \int_{G/L} \alpha \wedge (\pi^*\beta) \]

for all $\beta$ with compact support (for more details, see § 4.1 or [Reference Bott and TuBT82, p. 37]). We still denote by $\pi$ and $\pi _*$ the factorization of those maps by the left action of $\Gamma$.

Theorem 2.8 Assume the sequence $(\mathcal {O}_n)_{n\in \mathbb {N}}$ equidistributes in $\Gamma \backslash G$ and let $T_{\mathcal {O}_n/L}$ denote the integration current on $\mathcal {O}_n/L \subset \Gamma \backslash G/K$. Then

\[ \frac{1}{\operatorname{Vol}(\mathcal{O}_n/L)}T_{\mathcal{O}_n/L} \rightharpoonup \frac{1}{\operatorname{Vol}(K/L)\operatorname{Vol}(\Gamma \backslash G/K)}\pi_* p^*\omega_{G/H}. \]

Proof. Let $\mathcal {H}$ denote the foliation of $\Gamma \backslash G/L$ by the left translates of $H/L$ and $T\mathcal {H} \subset T(\Gamma \backslash G/L)$ the tangent distribution to this foliation. The volume form $\omega _{H/L}$ on $H/L$ defines a smooth section $\omega _{\mathcal {H}}$ of $\Lambda ^{\max}T^*\mathcal {H}$.

Let $\alpha$ be a form of degree $\dim (G)-\dim (H)$ with compact support on $\Gamma \backslash G/L$, and let $f$ be the smooth function with compact support such that $\alpha _{\vert T\mathcal {H}} = f \omega _{\mathcal {H}}$. The compatibility between the various volume forms gives

\[ \alpha \wedge p^*\omega_{G/H} = f \omega_{G/L}. \]

Let $p_0$ denote the projection from $\Gamma \backslash G$ to $\Gamma \backslash G/L$. We have

\begin{align*} \frac{1}{\operatorname{Vol}(\mathcal{O}_n/L)}\int_{\mathcal{O}_n/L} \alpha & = \frac{1}{\operatorname{Vol}(\mathcal{O}_n/L)}\int_{\mathcal{O}_n/L} f \omega_{\mathcal{H}}\\ & = \frac{1}{\operatorname{Vol}(\mathcal{O}_n)}\int_{\mathcal{O}_n} f\circ p_0 \omega_H \\ &\quad\underset{n\to +\infty}{\longrightarrow} \frac{1}{\operatorname{Vol}(\Gamma \backslash G)}\int_{\Gamma \backslash G} f\circ p_0 \omega_G \end{align*}

and

\begin{align*} \frac{1}{\operatorname{Vol}(\Gamma \backslash G)}\int_{\Gamma \backslash G} f\circ p_0 \omega_G &= \frac{1}{\operatorname{Vol}(\Gamma \backslash G/L)}\int_{\Gamma \backslash G/L} f \omega_{G/L}\\ &=\frac{1}{\operatorname{Vol}(\Gamma \backslash G/L)}\int_{\Gamma \backslash G} \alpha \wedge p^*\omega_{G/H}. \end{align*}

This shows that $({1}/{\operatorname {Vol}(\mathcal {O}_n/L)})\widehat {T}_{\mathcal {O}_n/L}$ converges weakly to $({1}/{\operatorname {Vol}(\Gamma \backslash G/L)}) \omega _{G/H}$.

Proof of Theorem 2.8 We push forward the equidistribution of Lemma 2.6 to $\Gamma \backslash G/K$. Note that we have $T_{\mathcal {O}_n/L} = \pi _* \widehat {T}_{\mathcal {O}_n/L}$.

Let $\alpha$ be a form of degree $\dim (H/L)$ with compact support on $\Gamma \backslash G/K$. We then have

\begin{align*} \frac{1}{\operatorname{Vol}(\mathcal{O}_n/L)}\langle T_{\mathcal{O}_n/L}, \alpha\rangle & = \frac{1}{\operatorname{Vol}(\mathcal{O}_n/L)} \langle \widehat{T}_{\mathcal{O}_n/L}, \pi^*\alpha \rangle\\ &\to \frac{1}{\operatorname{Vol}(\Gamma \backslash G/L)}\int_{\Gamma \backslash G/L} \pi^* \alpha \wedge p^* \omega_{G/H}\\ &= \frac{1}{\operatorname{Vol}(\Gamma \backslash G/L)} \int_{\Gamma \backslash G/K} \alpha \wedge \pi_* p^* \omega_{G/H} \text{ (by definition of $\pi_*$).}\quad \end{align*}

For $n\in \mathbb {N}$, the integration current $T_{\mathcal {O}_n/L}$ is closed because $\mathcal {O}_n/L$ has empty boundary. It thus has a well-defined cohomology class $[\mathcal {O}_n/L]\in H^{d_H}(\Gamma \backslash G/K,\mathbb {R})$ where $d_H$ is the codimension of $H/L$ in $G/K$.

Corollary 2.9 Assume that the sequence $(\mathcal {O}_n)$ is equidistributed in $\Gamma \backslash G$ and let $[\mathcal {O}_n/L]$ denote the corresponding cohomology class in $H^{d_H}(\Gamma \backslash G/K,\mathbb {R})$. Then

\[ \frac{1}{\operatorname{Vol}(\mathcal{O}_n/L)}[\mathcal{O}_n/L] \underset{n\rightarrow \infty}{\rightarrow} \frac{1}{\operatorname{Vol}(K/L)\operatorname{Vol}(\Gamma \backslash G/K)}[\pi_* p^*\omega_{G/H}]. \]

2.5 Ratner theory and its consequences

In this final section, we recall some equidistribution results à la Ratner of homogeneous orbits in locally homogeneous spaces. These results will be used when studying the refined Noether–Lefschetz locus in § 6.1.

We place ourselves in an arithmetic setting. Though this is not a requirement of Ratner theory, it is the source of its most remarkable consequences and will be sufficient for our applications.

We thus assume that we are given an inclusion $\mathbf {H}\subset \mathbf {G}$ of semi-simple algebraic groups over $\mathbb {Q}$ such that:

1. (i) $G$ is a subgroup of $\mathbf {G}_\mathbb {R}$ containing $\mathbf {G}_\mathbb {R}^0$;

2. (ii) $H= G\cap \mathbf {H}_\mathbb {R}$;

3. (iii) $\Gamma$ is commensurable to $\rho ^{-1}(\operatorname {GL}(n,\mathbb {Z}))$, for some faithful representation $\phi : \mathbf {G} \to \operatorname {GL}(n)$ over $\mathbb {Q}$.Footnote ⁴

By a lemma of Chevalley (see [Reference BenoistBen08, Proposition 4.6]), we can find a free $\mathbb {Z}$-module $\mathbf {V}_\mathbb {Z}$ and an embedding $\mathbf {G}\hookrightarrow \operatorname {GL}(\mathbf {V}_\mathbb {Q})$ such that $\mathbf {H}$ is the stabilizer in $\mathbf {G}$ of a vector $v_0\in \mathbf {V}_\mathbb {Z}$. We can moreover assume that the $G$-orbit of $v_0$ is Zariski closed and that $\Gamma$ preserves $\mathbf {V}_\mathbb {Z}$. For every $\lambda \in \mathbb {R}_{>0}$, we can now identify the homogeneous space $G/H$ with the $G$-orbit of $\lambda v_0$ in $\mathbf {V}_\mathbb {R}$.

The following classical result of homogeneous dynamics establishes the equivalence between closed $H$-orbits in $\Gamma \backslash G$ and discrete $\Gamma$-orbits in $G/H$.

Now let $V_n$ be a finite union of discrete $\Gamma$-orbits in $G/H$ and let $\mathcal {O}_n$ be the corresponding finite union of closed $H$-orbits in $\Gamma \backslash G$. The volume form $\omega _H$ induces an $H$-invariant measure $\nu _n$ on $\Gamma \backslash G$, supported by $\mathcal {O}_n$, whose total mass $\operatorname {Vol}(\mathcal {O}_n)$ is finite.

Definition 2.11 We say that the sequence $(\mathcal {O}_n)_{n\in \mathbb {N}}$ is equidistributed in $\Gamma \backslash G$ if the sequence of probability measures

\[ \frac{1}{\operatorname{Vol}(\mathcal{O}_n)} \nu_n \]

converges weakly to

\[ \frac{1}{\operatorname{Vol}(\Gamma \backslash G)}\omega_G. \]

We say that the sequence $(V_n)_{n\in \mathbb {N}}$ is equidistributed in $G/H$ if the discrete measure

\[ \frac{1}{\operatorname{Vol}(\mathcal{O}_n)} \sum_{x\in V_n} \delta_x \]

converges weakly to $\omega _{G/H}$.

Recall the following classical lemma from [Reference Eskin and OhEO06b, Proposition 2.2].

In [Reference Eskin and OhEO06b], Eskin and Oh gave an equidistribution criterion for finite unions of closed orbits which relies on Ratner's groundbreaking work on unipotent dynamics as well as further developments by Mozes and Shah [Reference Mozes and ShahMS95] and Dani and Margulis [Reference Dani and MargulisDM93, DM93].

Let $\mathcal {O}_n = \bigcup _{i=1}^{l_n} x_{i,n} H$ be a finite union of closed $H$-orbits in $\Gamma \backslash G$ and let $V_n = \bigcup _{i=1}^{l_n} \Gamma v_{i,n}$ be the corresponding union of discrete $\Gamma$-orbits in $G/H$.

Definition 2.13 We say that the sequence $(\mathcal {O}_n)_{n\in N}$ has no loss of mass if for all compact subsets $C$ of $G/H$,

\[ \frac{1}{\operatorname{Vol}(\mathcal{O}_n)}\Biggl(\sum_{\underset{x_{i,n}H\cap C = \emptyset}{i}} \operatorname{Vol}(x_{i,n}H)\Biggr)\underset{n\to +\infty}{\longrightarrow} 0. \]

Definition 2.14 The sequence $(\mathcal {O}_n)$ is called focused if there exists $g\in G$ and a subgroup $H'$ of $G$ containing $gHg^{-1}$ and defined over $\mathbb {Q}$ such that

\[ \limsup_{n\to +\infty} \frac{1}{\operatorname{Vol}(\mathcal{O}_n)}\Biggl(\sum_{\underset{\Gamma v_{i,n}\subset \Gamma H' g Z(H) v_0}{i}} \operatorname{Vol}(x_{i,n}H)\Biggr)> 0. \]

It is called non-focused otherwise.

Note that a sequence of closed $H$-orbits of $\Gamma \backslash G$ leaving every compact subset can only exist if $H$ is contained in a proper parabolic subgroup of $G$. We thus have the following proposition which results from Propositions 3.2 and 3.4 in [Reference Eskin and OhEO06b].

3.1 Moderate geometry of locally symmetric spaces

We recall in this section notions from o-minimal geometry in the context of locally symmetric spaces following [Reference Bakker, Klingler and TsimermanBKT20] and the structure of definable maps. For a general introduction to o-minimal structures, we refer to [Reference van den DriesvdD98].

A structure $\mathcal {S}$ on $\mathbb {R}$ expanding the real field $\mathbb {R}$ is by definition a collection $(\mathcal {S}_n)_{n\in \mathbb {N}^\times }$ where each $\mathcal {S}_n$ is a set of subsets of $\mathbb {R}^n$, called the definable sets, which is a Boolean subalgebra of the subsets of $\mathbb {R}^n$ containing all the algebraic subsets and which satisfy the following properties:

1. (i) if $A\in \mathcal {S}_n$, $B\in \mathcal {S}_m$, then $A\times B\in \mathcal {S}_{n+m}$;

2. (ii) if $p:\mathbb {R}^{n+1}\rightarrow \mathbb {R}^n$ is the projection on the first $n$-coordinates, $A\in \mathcal {S}_{n+1}$, then $p(A)\in \mathcal {S}_n$.

The structure is called o-minimal Footnote ⁵ if any element of $\mathcal {S}_1$ is a finite union of points and intervals. Given an o-minimal structure, one can define the following notions:

1. (i) a map $f:A\rightarrow B$ between two definable sets is definable if its graph $\Gamma _f\subset A\times B$ is definable;

2. (ii) a $\mathcal {S}$-definable manifold is a manifold having a finite atlas of charts $(\phi _i:U_i\rightarrow \mathbb {R}^n)_{i\in I}$ such that the intersections $\phi _i(U_i\cap U_j)\subset \mathbb {R}^n$ are definable and the change of coordinates maps $\phi _i\circ \phi _j^{-1}:\phi _j(U_i\cap U_j)\rightarrow \phi _i(U_i\cap U_j)$ are $\mathcal {S}$-definable maps.

Intuitively, definable manifolds in an o-minimal structure have reasonable geometry locally and at infinity, complex algebraic varieties being an example of definable manifolds. The first example of an o-minimal structure is that given by semi-algebraic subsets denoted by $\mathbb {R}_{\rm alg}$. More examples of o-minimal structures have been studied during recent years and the ones relevant to Hodge theory are:

1. (i) $\mathbb {R}_{\rm an}$, the smallest o-minimal structure expanding $\mathbb {R}_{\rm alg}$ and for which restricted analytic functions are definable (see [Reference GabrièlovGab68]);

2. (ii) $\mathbb {R}_{\rm exp}$, the smallest o-minimal structure expanding $\mathbb {R}_{\rm alg}$ and for which the real exponential map is definable [Reference WilkieWil96];

3. (iii) $\mathbb {R}_{\rm an,exp}$, the smallest o-minimal structure expanding the two previous structures [Reference van den Dries and MillervdDM94, Reference van den Dries and MillervdDM96].

One of the main theorems of [Reference Bakker, Klingler and TsimermanBKT20, Theorem 1.1] asserts that locally symmetric spaces can be endowed with a semi-algebraic structure which is compatible in $\mathbb {R}_{\rm an}$ with the analytic structure on the Borel–Serre compactification, see [Reference Bakker, Klingler and TsimermanBKT20] for more details. More precisely, let $G$ be a real connected semi-simple group which has a $\mathbb {Q}$-structure, $\Gamma$ a torsion-free arithmetic subgroup of $G$ and $K\subset G$ a compact sub-group.

One last ingredient we need from o-minimal geometry is the structure of definable maps: a definable map $f:M\rightarrow N$ between definable sets has a definable trivialization if there exists a pair $(F,\lambda )$ where $F$ is a definable set and $\lambda : M\rightarrow F$ is a definable map which induces a definable homeomorphism $M\rightarrow F\times N$ compatible with maps to $N$. The following theorem is from [Reference van den DriesvdD98, Chapter 9, Theorem 1.2].

An easy corollary is that the number of connected components of the fibers of a definable map is finite and uniformly bounded.

3.2 Counting intersection points

In this section, we explain our conventions for counting points of intersection of varieties mapping to the quotient $\Gamma \backslash G/K$ with closed $H$-orbits. We adopt the same notation as in § 2.3 to which we refer. From now on, $\mathcal {S}$ will be a fixed o-minimal structure which extends $\mathbb {R}_{\rm alg}$.

Let $S\subset \Gamma \backslash G/K$ be a definable real analytic subvariety of dimension $d_H = \dim (G/K)-\dim (H/L)$, and assume that the smooth locus of $S$ is oriented. Let $\mathcal {O}$ be a finite union of closed $H$-orbits in $\Gamma \backslash G$. Recall that we denote by $\mathcal {O}/L$ its projection to $\Gamma \backslash G/L$ and by $\pi (\mathcal {O}/L)$ its projection to $\Gamma \backslash G/K$.

In general, $\pi _{\vert \mathcal {O}/L}$ is not an embedding, which is why it is more convenient to count intersection at the level of $\Gamma \backslash G/L$, where $\mathcal {O}/L$ is a finite union of closed leaves of the foliation $\mathcal {H}$ introduced in § 2.4.

Let $(\Gamma \backslash G/L)_S$ be the preimage of $S$ by the fibration $\pi : \Gamma \backslash G/L\rightarrow \Gamma \backslash G/K$. By assumption on $S$, $(\Gamma \backslash G/L)_S$ and $\mathcal {O}/L$ have complementary dimension in $\Gamma \backslash G/L$. A point $y\in (\Gamma \backslash G/L)_S \cap \mathcal {O}/L$ is a transverse intersection point if $(\Gamma \backslash G/L)_S$ is smooth at $y$ (equivalently, if $S$ is smooth at $\pi (y)$) and

(3.2.1)\begin{equation} T_y(\Gamma \backslash G/L)_S \oplus T_y (\mathcal{O}/L) = T_y(\Gamma \backslash G/L). \end{equation}

For $y\in (\Gamma \backslash G/L)_S \cap \mathcal {O}$, we set $\epsilon (y) = 0$ if $y$ is not a transverse intersection point, and $\epsilon (y) =1$ if $y$ is a transverse intersection point such that the direct sum (3.2.1) is compatible with orientations, and $\epsilon (y) = -1$ otherwise.

We have the following distribution on $(\Gamma \backslash G/L)_S$ given by summing over transverse intersection points:

(3.2.2)\begin{equation} \widehat{T}_\mathcal{O}^S = \sum_{x\in (\Gamma\backslash G/L)_S\cap\mathcal{O}/L}\epsilon(x)\delta_x. \end{equation}

Note that, because $(\Gamma \backslash G/L)_S\cap \mathcal {O}/L$ is definable, it has a finite number of connected components. Moreover, components of dimension at least one consist only of non-transverse intersections points, for which $\epsilon (y) = 0$. Hence, $\widehat {T}_\mathcal {O}^S$ is a finite signed measure. We can finally state the following definition.

Definition 3.3 The transverse intersection measure between $S$ and $\mathcal {O}/L$ is the signed measure

\[ T^S_{\mathcal{O}} \overset{\text{def}}= \pi_* \widehat{T}^S_{\mathcal{O}}. \]

Informally, $T^S_{\mathcal {O}}$, counts the transverse intersection points of $S$ and $\pi (\mathcal {O}/L)$, with a multiplicity corresponding to the (signed) number of branches of $\pi (\mathcal {O}/L)$ meeting $S$ at a smooth point $x$. The asymptotic behavior on $S$ of the distributions $T_{\mathcal {O}_n}^S$ for an equidistributing sequence $\mathcal {O}_n$ is discussed in the next section. In particular, we prove that they are equidistributed with respect to the form $\pi _*p^*\omega _{G/H}$.

3.3 Equidistribution of intersection points

We keep the notation from the previous section, namely, $G$ is a semi-simple $\mathbb {Q}$-group, $\Gamma \subset G$ is a torsion-free arithmetic subgroup, and $H\subset G$ is a semi-simple subgroup without compact factor. Let $\mathcal {D}=G/K$ where $K\subset G$ is a compact subgroup and $L=K\cap H$. Let $d_H$ be as before the codimension of $H/L$ in $\mathcal {D}$. Let $S$ be an analytic subvariety of $\Gamma \backslash \mathcal {D}$ of dimension $d_H$ and consider an equidistributing sequence $(\mathcal {O}_n)_{n\in \mathbb {N}}$ of finite unions of closed $H$-orbits in $\Gamma \backslash G$. In this section, we refine Theorem 2.8 into an equidistribution theorem for the measures $T^S_{\mathcal {O}_n}$ introduced in the previous paragraph, which will imply Theorem 1.1.

Theorem 3.6 Let $S$ be an analytic subspace of $\Gamma \backslash \mathcal {D}$ of dimension $d_H$ and let $(\mathcal {O}_n)_{n\in \mathbb {N}}$ be an equidistributing sequence of finite unions of closed $H$-orbits in $\Gamma \backslash G$. Then

\[ \frac{1}{\operatorname{Vol}(\mathcal{O}_n)}T_{\mathcal{O}_n}^S \underset{n\to +\infty}{\rightharpoonup} \pi_*p^*\omega_{G/H}. \]

Proof. Recall that it is enough to prove the weak convergence on every open subset of a covering of $S$ by Proposition 2.1. We can thus restrict ourselves to an open relatively compact definable subset $\Omega \subset S$ that lifts to $\mathcal {D}$. We still denote this lift by $\Omega$. We want to prove the equidistribution of the transverse intersection of $\Omega$ with $\pi (p^{-1}(V_n))$, where $V_n$ is the finite union of discrete $\Gamma$-orbits in $G/H$ such that $\mathcal {O}_n = \Gamma \backslash \pi ^{-1}(V_n)$.

Denote by $(G/L)_\Omega$ the preimage of $\Omega$ by $\pi$, and consider the following open subsets of $(G/L)_\Omega$:

1. (i) the domain $U_{\rm smooth}$ where $(G/L)_\Omega$ is smooth;

2. (ii) the domain $U_{\rm sub}$ where $(G/L)_\Omega$ is smooth and $p_{\vert \Omega }$ is a submersion (hence, a local diffeomorphism by equality of the dimensions).

By definition, the distribution $\widehat {T}_{\mathcal {O}_n}^\Omega$ is supported by $U_{\rm sub}$. We first prove that $({1}/{\operatorname {Vol}(\mathcal {O}_n)}) \widehat {T}_{\mathcal {O}_n}^\Omega$ converges weakly to $p^*\omega _{G/H}$ on $U_{\rm sub}$, then extend the weak convergence successively to $U_{\rm smooth}$ and $(G/L)_\Omega$ using Proposition 2.2. After pushing forward by $\pi$, we get the desired result.

To prove the weak convergence on $U_{\rm sub}$, it is enough to prove it on every open relatively compact subset $U'$ of $U_{\rm sub}$ such that $p_{\vert U'}$ is a diffeomorphism onto its image (because those open sets cover $U_{\rm sub}$). Thus, let $f$ be a continuous function with compact support on such $U'$. Let $\epsilon (U')$ be $1$ if $p_{\vert U'}$ preserves orientation and $-1$ otherwise. Then

\begin{align*} \frac{1}{\operatorname{Vol}(\mathcal{O}_n)}\widehat{T}_{\mathcal{O}_n}^{\Omega}(f) &= \frac{\epsilon(U')}{\operatorname{Vol}(\mathcal{O}_n)} \sum_{x\in V_n\cap p(U')} f\circ p^{-1}(x)\\ &\quad \underset{n\to+\infty}{\longrightarrow} \epsilon(U') \int_{p(U')} f\circ p^{-1}(x) \omega_{G/H}(x)= \int_{U'} f p^*\omega_{G/H} \end{align*}

because $V_n$ is equidistributed in $G/H$.

This shows the weak convergence of $({1}/{\operatorname {Vol}(\mathcal {O}_n)})\widehat {T}_{\mathcal {O}_n}^{\Omega }$ on every $U'$, hence on $U_{\rm sub}$.

Let us now extend the weak convergence to $(G/L)_\Omega$. As the projection map $p_{\vert (G/L)_\Omega }: (G/L)_\Omega \to G/H$ is definable in the o-minimal structure $\mathbb {R}_{\rm an,exp}$, it follows from Theorem 3.2 that the number of connected components of its fibers is uniformly bounded by some number $N$.

First, let $C$ be a compact subset of $U_{\rm smooth}\setminus U_{\rm sub}$. Then, by Sard's lemma, $p(C)$ has measure $0$. Hence, for every $\epsilon >0$, there exists an open neighborhood $U'$ of $C$ in $U_{\rm smooth}$ such that

\[ \biggl \vert\int_{p(U')} \omega_{G/H}\biggr \vert \leq \epsilon. \]

As for all $x\in p(U')\cap V_n$ the set $p^{-1}(x)\cap U'$ has at most $N$ isolated points, we get that

\[ \frac{1}{\operatorname{Vol}(\mathcal{O}_n)} \vert \widehat{T}_{\mathcal{O}_n}^\Omega\vert(U') \leq \frac{N}{\operatorname{Vol}(\mathcal{O}_n)} \vert p(U')\cap V_n\vert , \]

hence

\[ \frac{1}{\operatorname{Vol}(\mathcal{O}_n)}\vert\widehat{T}_{\mathcal{O}_n}^\Omega\vert(U') \leq \vert\int_{p(U')} \omega_{G/H}\vert \leq N\epsilon. \]

We conclude that $({1}/{\operatorname {Vol}(\mathcal {O}_n)})\widehat {T}_{\mathcal {O}_n}^{\Omega }$ converges weakly on $U_{\rm smooth}$ by Proposition 2.2.

Finally, the complement of $U_{\rm smooth}$ in $(G/L)_\Omega$ is the singular locus, which has dimension less than $d_H$. As $p$ is definable, its image has measure zero, and one can reproduce the previous argument to show that $({1}/{\operatorname {Vol}(\mathcal {O}_n)})\widehat {T}_{\mathcal {O}_n}^{\Omega }$ converges weakly to $p^*\omega _{G/H}$ on $(G/L)_\Omega$.

In the previous theorem, we chose to restrict to $\dim (S)= d_H$ for simplicity. We indicate briefly without proof how these statements should be adapted when $\dim (S)> d_H$: we define the transverse intersection current $T_S^{\mathcal {O}_n}$ as the integration current over the transverse intersection locus of $S\cap \mathcal {O}_n$ with the sign normalizations as in § 3.2. Intersecting further with submanifolds of dimension $d_H$ and applying our theorem, one would obtain the convergence of these intersection currents (after normalization) to the current ${\pi _*p^*\omega _{G/H}}_{\vert S}$. We thus get the following theorem.

Theorem 3.7 Let $S$ be an analytic subspace of dimension $d\geq d_H$ and let $(\mathcal {O}_n)_{n\in \mathbb {N}}$ be an equidistributing sequence of finite unions of closed $H$-orbits in $\Gamma \backslash G$. Then for every $\beta \in \Omega _{c}^{d-d_H}(S)$, we have

\[ \frac{1}{\operatorname{Vol}(\mathcal{O}_n)}T_{\mathcal{O}_n}^S(\beta) \underset{n\to +\infty}{\rightarrow} \pi_*p^*\omega_{G/H}\wedge\beta. \]

4.1 Push-forward of forms

Let us first recall without any proof the construction of the push-forward of a form $\alpha$ under a smooth fibration $\pi : M\to B$ with compact oriented fibers. This construction can be found for instance in [Reference Bott and TuBT82, p. 37].

Let $r$ be the dimension of the fibers of $\pi$. Let $x$ be a point in $B$ and denote by $F$ the fiber of $\pi$ over $x$. Choose $\omega$ a volume form on $F$ compatible with the orientation of the fiber, and let

\[ X\in \Lambda^r(TF)\subset \Lambda^r TM \]

be the multivector such that $\omega (X) = 1$.

Now let $\alpha$ be a smooth $p$-form on $M$. The contraction $\iota _X\alpha$ is a $(p-r)$ form with kernel $TF$, which can thus be seen as a section of $\Lambda ^{p-r}(NF^\vee )$, where $NF = TM/TF$ is the normal bundle to $F$ and $NF^\vee = \{\varphi \in T^*M \mid \varphi _{\vert TF} = 0\}$ is its dual.

Finally, the differential of $\pi$ along the fiber $F$ induces an isomorphism $NF \simeq F\times T_xB$ and therefore $\Lambda ^{p-r}(NF^\vee ) = \Lambda ^{p-r}(T_xB^\vee )$. With these identifications, we can now state the following definition.

Definition 4.1 The push-forward of the form $\alpha$ is the $(p-r)$-form on $B$ given at $x$ by

\[ (\pi_*\alpha)_x = \int_{y\in F} \iota_X(\alpha)_y \omega. \]

Let $p+q$ be the dimension of $M$. Then we have the following result.

Proposition 4.2 The form $\pi _*\alpha$ is the unique $(p-r)$ form on $B$ such that for any $q$-form $\beta$ on $B$ with compact support,

\[ \int_B \pi_*\alpha \wedge \beta = \int_M \alpha \wedge \pi^*\beta. \]

Moreover, the push-forward operation commutes with the exterior derivative. In particular, the push-forward of a closed form is closed.

4.2 A formula for the pull–push form

Let us now apply the previous general considerations in order to give a formula for the pull–push form $\pi _*p^*\omega _{G/H}$ at the Lie algebra level.

Let $G/H$ be a $G$-homogeneous space and $o$ denote the basepoint of $G/H$ (that is, the projection of the unit element of $G$). Then the tangent space $T_o G/H$ with the induced linear action of $H$ identifies canonically with $\mathfrak {g}/\mathfrak {h}$ endowed with the adjoint action of $H$. This identification induces an isomorphism between the differential algebra $\Omega ^\bullet (G/H, \mathbb {C})^G$ of smooth $G$-invariant forms on $G/H$ with complex coefficients and the differential algebra $\Lambda ^\bullet ((\mathfrak {g}/\mathfrak {h})_\mathbb {C}^\vee )^H$ of $H$-invariant exterior forms on $\mathfrak {g}/\mathfrak {h}$, with derivative given by

\[ \mathrm{d} \alpha(x_1,\ldots, x_{k+1}) = \sum_{i< j} (-1)^{i+j}\alpha([x_i,x_j],x_1,\ldots, \widehat{x}_i, \ldots, \widehat{x}_j, \ldots, x_{k+1}), \]

for $\alpha \in \Lambda ^k((\mathfrak {g}/\mathfrak {h})_\mathbb {C}^\vee )^H$ and $x_1,\ldots,x_{k+1}\in \mathfrak {g}/\mathfrak {h}$.

Let us now come back to the setting of the previous section, where $H$ is a semi-simple subgroup of $G$, $K$ a compact subgroup of $G$, and $L=G/H$. At the Lie algebra level, the pull-back homomorphism $p^*$ (respectively, $\pi ^*$) identifies exterior forms on $\mathfrak {g}/\mathfrak {h}$ (respectively, $\mathfrak {g}/\mathfrak {k}$) to those exterior forms on $\mathfrak {g}/\mathfrak {l}$ having $\mathfrak {h}/\mathfrak {l}$ (respectively, $\mathfrak {k}/\mathfrak {l}$) in their kernel.

Reinterpreting Definition 4.1 for $G$-invariant forms on $G/L$, we obtain the following result.

Proposition 4.3 Let $\alpha$ be a $G$-invariant form on $G/L$. Then the form $\pi _*\alpha$ on $G/K$ is $G$-invariant and corresponds on $\mathfrak {g}/\mathfrak {k}$ to the $\operatorname {Ad}_K$-invariant form

\[ \int_{k\in K/L} {\operatorname{Ad}_k}^* (\iota_u\alpha) \omega_{K/L}, \]

where $u \in \Lambda ^{\max} (\mathfrak {k}/\mathfrak {l})$ is such that $\omega _{K/L}(u) = 1$.

Corollary 4.5 Let $(\alpha _1\ldots, \alpha _{p_r})$ be an oriented orthonormal basis of $\{\phi \in (\mathfrak {g}/\mathfrak {k})^\vee \mid \phi _{\vert \mathfrak {h}/\mathfrak {l}} = 0\}$. Then there exists a positive constant $\lambda$ such that

\[ \pi_*p^*\omega_{G/H} = \lambda \int_{k\in K/L} {\operatorname{Ad}_k^*}(\alpha_1\wedge \ldots \wedge \alpha_{p-r}) \omega_{K/L}(k). \]

If, moreover, $\mathfrak {k}/\mathfrak {l}$ is orthogonal to $\mathfrak {h}/\mathfrak {l}$ in $\mathfrak {g}/\mathfrak {l}$, then $\lambda = 1$.

Proof of Proposition 4.3 The $G$-invariance of $\pi _*\alpha$ easily follows from Proposition 4.2, so we only need to compute $\pi _*\alpha$ at the basepoint of $G/K$.

Let $o$ denote the basepoint of $G/L$ and $v_1,\ldots, v_{p-r}$ be $p-r$ vectors in $T_o G/L = \mathfrak {g}/\mathfrak {l}$. Let $k$ be an element of $K$. At the point $k\cdot o$, we have

\[ \iota_u\alpha(k\cdot v_1,\ldots, k\cdot v_{p-r}) = \iota_u \alpha(v_1,\ldots, v_{p-r}) \]

by left invariance of $\iota _u \alpha$.

Now, the differential of $\pi$ maps a vector $k\cdot v \in T_{k\cdot o}G/L$ to the vector $\operatorname {Ad}_{k}(v) \in T_{\pi (o)} G/K = \mathfrak {g}/\mathfrak {k}$. Applying Definition 4.1 to $\pi _*\alpha$ gives the required formula.

Proof of Corollary 4.5 Complete $(\alpha _1,\ldots, \alpha _{p-r})$ into an orthonormal basis $(\beta _1,\ldots, \beta _r, \alpha _1, \ldots, \alpha _{p-r})$ of $\{\phi \in (\mathfrak {g}/\mathfrak {l})^* \mid \phi _{\vert \mathfrak {h}/\mathfrak {l}}=0\}$. We then have

\[ p^*\omega_{G/H} = \bigwedge_{i=1}^r \beta_i \wedge \bigwedge_{i=1}^{p-r} \alpha_i, \]

hence

\[ \iota_up^*\omega_{G/H} = \lambda \bigwedge_{i=1}^{p-r} \alpha_i, \]

where $\lambda = \bigwedge _{i=1}^r \beta _i(u)$, because all the $\alpha _i$ vanish on $\mathfrak {k}/\mathfrak {l}$.

If, furthermore, $\mathfrak {k}/\mathfrak {l}$ is orthogonal to $\mathfrak {h}/\mathfrak {l}$, then $(\beta _1, \ldots, \beta _r)$ can be chosen as an orthonormal basis of $(\mathfrak {k}/\mathfrak {l})^\vee$, so that $\lambda =1$.

Proposition 4.3 now concludes the proof.

In practice, using this integral formula to compute the pull–push form quickly leads to rather involved computations. In [Reference TholozanTho15], the second author used this formula to give a sufficient vanishing criterion: if there exists $k\in K$ such that $\operatorname {Ad}_k$ preserves $\mathfrak {h}/\mathfrak {l}$ and reverses its orientation, then ${\operatorname {Ad}_k}_* (\iota _u \alpha ) = - \iota _u \alpha$, and Proposition 4.3 shows that $\pi _*p^*\omega _{G/H}$ vanishes.

In terms of our equidistribution result, this vanishing means that the positive and negative part of the intersection measure cancel out asymptotically, because a ‘random’ translate of $H/L$ in $G/K$ can intersect a submanifold $S$ with two opposite equiprobable orientations.

In contrast, we can prove the following non-vanishing criterion.

Proof. Let $\alpha _1,\ldots, \alpha _{({p-r})/{2}}$ be a complex basis of $\{\phi \in (\mathfrak {g}/\mathfrak {k})^\vee \mid \phi _{\vert \mathfrak {h}/\mathfrak {l}}=0\}$. Then $\iota _u p^*\omega _{G/H}$ is proportional to

\[ \bigwedge_{i=1}^{({p-r})/{2}} \alpha_i\wedge \bar\alpha_i,\]

which is non-negative on every complex subspace of dimension $({p-r})/{2}$. By invariance of the complex structure, the same holds for

\[ {\operatorname{Ad}_k}^*\Biggl(\bigwedge_{i=1}^{({p-r})/{2}} \alpha_i\wedge \bar\alpha_i\Biggr). \]

Finally, $\bigwedge _{i=1}^{({p-r})/{2}} \alpha _i\wedge \bar \alpha _i$ is positive on a complex complement of $\mathfrak {h}/\mathfrak {l}$, and so is

\[ \int_{k\in K/L} {\operatorname{Ad}_k}^*\Biggl(\bigwedge_{i=1}^{({p-r})/{2}} \alpha_i\wedge \bar\alpha_i\Biggr) \omega_{K/L}.\quad \]

4.3 Compact duality

In [Reference TholozanTho15], the second author investigated further the pull–push form in the case where $G/K$ and $H/L$ are symmetric spaces. There, he proved that $\pi _*p^*\omega _{G/H}$ is in some sense ‘Poincaré dual’ to the inclusion $H/L\hookrightarrow G/K$, a statement which is made precise by passing to the compact dual of the symmetric space. As cohomology classes of compact symmetric spaces are represented by a unique invariant form, this argument completely characterizes the pull–push forms and provides an efficient way to compute it in practice. The goal of this section is to extend these results to more general compact subgroups $K\subset G$.

Recall that a Cartan involution of $G$ is an involutive automorphism whose fixed subgroup is a maximal compact subgroup. All the Cartan involutions of $G$ are conjugated, and every compact subgroup of $G$ is fixed by a Cartan involution.

In this section, we make the assumption (verified in all the examples we consider in this paper) that there exists a Cartan involution $\theta$ of $G$ fixing $K$ and preserving $H$. We denote by $G^\theta$ and $H^\theta$ the compact subgroups of $G$ and $H$ fixed by $\theta$. The Lie algebras $\mathfrak {g}$ and $\mathfrak {h}$ decompose respectively as

\[ \mathfrak{g} = \mathfrak{g}^\theta \oplus {\mathfrak{g}^\theta}^\perp \]

and

\[ \mathfrak{h} = \mathfrak{h}^\theta \oplus {\mathfrak{h}^\theta}^\perp. \]

We now introduce the dual Lie algebras

\[ \mathfrak{g}_U = \mathfrak{g}^\theta \oplus i {\mathfrak{g}^\theta}^\perp \subset \mathfrak{g}_\mathbb{C} \]

and

\[ \mathfrak{h}_U = \mathfrak{h}^\theta \oplus i {\mathfrak{h}^\theta}^\perp \subset \mathfrak{h}_\mathbb{C}. \]

These are the Lie algebras of compact real forms $G_U$ and $H_U$ of $G_\mathbb {C}$ and $H_\mathbb {C}$, respectively, called the compact duals of $G$ and $H$.

From $G_U$ and $H_U$, one can define compact duals to the homogeneous spaces $G/H$, $G/L$, and $G/K$, given by $G_U/H_U$, $G_U/L$, and $G_U/K$, respectively. The various inclusions between these groups give the following commutative diagram.

Now, the inclusion of $G/H$ into $G_\mathbb {C}/H_\mathbb {C}$ induces an isomorphism of differential algebras

\[ \Omega^\bullet(G/H,\mathbb{C})^G = \Lambda^\bullet((\mathfrak{g}/\mathfrak{h})^\vee)^H\otimes_\mathbb{R} \mathbb{C} \simeq \Lambda^\bullet_\mathbb{C}((\mathfrak{g}_\mathbb{C}/\mathfrak{h}_\mathbb{C})^\vee)^{H_\mathbb{C}} = \Omega_\mathbb{C}^\bullet(G_\mathbb{C}/H_\mathbb{C})^{G_\mathbb{C}}, \]

where $\Lambda ^\bullet _\mathbb {C}$ and $\Omega ^\bullet _\mathbb {C}$ denote the complex of $\mathbb {C}$-multilinear forms. The same holds for all the inclusions in the above diagram. In other words, the differential algebra of real invariant forms on a homogeneous space and its compact dual are two distinct real forms of the same complex differential algebra.

Proposition 4.7 We have the following commutative diagram of differential complexes.

Now, $\omega _{G/H}$ and $\omega _{G_U/H_U}$ are both generators of $\Lambda ^{\max}_\mathbb {C}(\mathfrak {g}_\mathbb {C}/\mathfrak {h}_\mathbb {C})$, so they are complex multiples one of the other. More precisely, we can identify

\[ \mathfrak{g}/\mathfrak{h} = \mathfrak{g}^\theta /\mathfrak{h}^\theta \oplus {\mathfrak{g}^\theta}^\perp/{\mathfrak{h}^\theta}^\perp \]

with

\[ \mathfrak{g}_U/\mathfrak{h}_U = \mathfrak{g}^\theta /\mathfrak{h}^\theta \oplus i{\mathfrak{g}^\theta}^\perp/{\mathfrak{h}^\theta}^\perp \]

via the morphism

\[ \phi:(u,v)\mapsto (u,iv) \]

and normalize $\omega _{G_U/H_U}$ so that

\[ \phi^*\omega_{G_U/H_U} = \omega_{G/H}. \]

With this normalization, we have the following equality in $\Lambda ^{\max}_\mathbb {C}(\mathfrak {g}_\mathbb {C}/\mathfrak {h}_\mathbb {C})$

\[ \omega_{G/H} = i^{\dim({\mathfrak{g}^\theta}^\perp/{\mathfrak{h}^\theta}^\perp)} \omega_{G_U/H_U}. \]

Finally, applying Proposition 4.7, we conclude as follows.

Corollary 4.8 The following equality holds in $\Lambda ^\bullet _\mathbb {C}(G_\mathbb {C}/K_\mathbb {C})^{K_\mathbb {C}}$:

\[ \pi_* p^*\omega_{G/H} = i^{\dim({\mathfrak{g}^\theta}^\perp/{\mathfrak{h}^\theta}^\perp)} {\pi_U}_*p_U^*\omega_{G_U/H_U}. \]

What we gained by switching to the compact dual space $G_U/K$ is that we can now talk about the cohomology class of the pull–push form. The following theorem was proven in [Reference TholozanTho15] under the assumption that $K=G^\theta$, but the proof easily adapts to our more general context.

Lemma 4.9 The de Rham cohomology class of the pull–push form $({1}/{\operatorname {Vol}(G_U/H_U)}) {\pi _U}_* p_U^*\omega _{G_U/H_U}$ is Poincaré dual to the homology class of $H_U/L \subset G_U/K$; that is, for every closed form $\beta$ on $G_U/K$ of degree $\dim (H_U/L)$, we have

\[ \int_{H_U/L} \beta = \frac{1}{\operatorname{Vol}(G_U/H_U)} \int_{G_U/K} {\pi_U}_* p_U^*(\omega_{G_U/H_U})\wedge \beta. \]

Proof. This is essentially formal. Denote by $\iota _1$ and $\iota _2$ the inclusions of $H_U/L$ in $G_U/L$ and $G_U/K$, respectively, so that we have $\iota _2 = \pi _U\circ \iota _1$ and $\iota _1(H_U/L) = p_U^{-1}(o)$, where $o$ is the basepoint of $G_U/H_U$.

Now, the form $({1}/{\operatorname {Vol}(G_U/H_U)})\omega _{G_U/H_U}$ is Poincaré dual to $[o]$ in $\mathrm H_{dR}^\bullet (G_U/H_U,\mathbb {R})$, so its pull-back under $p_U$ is Poincaré dual to $[{p}_{U}^{-1}(o)] = \iota _{1*}[H_U/L]$. Finally, let $\beta$ be a closed form of degree $\dim (H_U/L)$ on $G_U/K$. We then have

\begin{align*} \int_{H_U/L} \iota_2^* \beta &= \int_{\iota_1(H_U/L)} \pi_U^*\beta \\ &= \int_{G_U/L}p_U^*\omega_{G_U/H_U}\wedge \pi_U^*\beta\\ &= \int_{G_U/K} {\pi_U}_*p_U^* \omega_{G_U/H_U} \wedge \beta.\quad \end{align*}

4.3.1 The symmetric case

When $K= G^\theta$ is a maximal compact subgroup of $G$, a theorem of Cartan [Reference CartanCart30] states that all $G$-invariant forms on $G/K$ are closed. Hence, the exterior derivative is trivial on $\Omega ^\bullet (G/K)^G$ and we have isomorphisms:

\[ \Omega^\bullet(G/K,\mathbb{C})^G \simeq \Omega^\bullet(G_U/K, \mathbb{C})^{G_U} \simeq H_{dR}^\bullet(G_U/K,\mathbb{C}). \]

In other words, a $G$-invariant form on $G/K$ is completely characterized by the cohomology class of the corresponding form on $G_U/K$.

This property does not hold for more general homogeneous spaces. In the next section we show however that it remains true on Mumford–Tate domains when restricting to a variation of Hodge structure.

5.1 Variations of Hodge structure and their period domains

Let us first recall some definitions of Hodge theory, merely to fix notation.

Let $V$ be a free $\mathbb {Z}$-module of finite rank $d\in \mathbb {N}$ endowed with a bilinear form $B: V\times V \to \mathbb {Z}$. Given a field $\mathbb {K}$, we write $V_\mathbb {K}= V\otimes _\mathbb {Z}\mathbb {K}$ and still denote by $B$ the natural $\mathbb {K}$-bilinear extension of $B$ to $V_\mathbb {K}$.

A Hodge structure of weight $k$ on $V$ polarized by $B$ is the data of a filtration of complex vector spaces

\[ 0\subseteq F^{k}\subseteq\cdots\subseteq F^0= \mathbb{V}_\mathbb{C} \]

such that for all $0\leq p\leq k$:

1. (i) $V_\mathbb {C}=F^{p}\oplus \overline {F}^{k-p+1}$;

2. (ii) $B(u, v) = 0$ for all $(u,v)\in F^p \times F^{k-p+1}$;

3. (iii) $i^{p-q}B(v, \bar v)>0$ for all $v\in (F^p \cap \overline {F}^{q})\backslash \{0\}$ with $p+q=k$.

For $p+q = k$, define $V^{p,q} = F^p \cap \overline F^{q}$. Then $V^{p,q}$ is a complement of $F^{p+1}$ in $F^p$. In particular, we have a decomposition

\[ V_\mathbb{C} = \bigoplus_{p+q=k} V^{p,q},\quad\text{with}\quad \overline{V}^{p,q}=V^{q,p}. \]

The Hodge numbers of the Hodge structure are the numbers $h^{p,q} \overset {\text {def}}= \dim _\mathbb {C}(V^{p,q})$.

Let $\mathbb {S}=\mathrm {Res}_{\mathbb {C}/\mathbb {R}}\mathbb {G}_m$ denote the Deligne torus, i.e. the restriction of scalars of the multiplicative group $\mathbb {G}_m$ from $\mathbb {C}$ to $\mathbb {R}$. Then $\mathbb {S}(\mathbb {R})=\mathbb {C}^\times$ seen as an algebraic group over $\mathbb {R}$. Every Hodge structure on $(V_\mathbb {Z}, B)$ induces a representation $\varphi : \mathbb {S}(\mathbb {R}) \to \operatorname {GL}(V_\mathbb {R})$ given by

\[ z\cdot u = z^{-p} \bar z^{-q} u \]

on $u\in V^{p,q}$.

Let $k\in \mathbb {Z}$ and $\underline {h} = (h_{p,q})_{p+q = k} \in \mathbb {N}^{k+1}$ be such that $h_{p,q} = h_{q,p}$ and $\sum _{p=0}^k h^{p,q} = d$. The period domain of Hodge structure of weight $k$ and Hodge numbers $(h_{p,q})_{p+q = k}$ is the set $\mathcal {D}$ of all filtrations $(F^{p})_{0\leq p\leq k}$ which define a Hodge structure of weight $k$ and Hodge numbers $h^{p,q}$ on $(V_\mathbb {Z},B)$. It is a complex manifold homogeneous under the action of the group $\operatorname {Aut}_\mathbb {R}(B)$, and the stabilizer of a point is a compact subgroup of $\operatorname {Aut}_\mathbb {R}(B)$.

The period domain $\mathcal {D}$ is an open subset of the compactified period domain $\widehat {\mathcal {D}}$ of complex flags $0\subseteq F^{k}\subseteq \cdots \subseteq F^0 = V_\mathbb {C}$ such that $F^{k-p} = {F^p}^\bot$ and $\dim _\mathbb {C}(F^p/F^{p+1}) = h^{p,k-p}$ for all $p$. The compactified period domain is a flag variety of the group $\operatorname {Aut}_\mathbb {C}(B)$ (i.e. a quotient of $\operatorname {Aut}_\mathbb {C}(B)$ by a parabolic subgroup).

Let $\mathcal {U}$ denote the trivial complex vector bundle $\mathcal {D} \times V_\mathbb {C}$ equipped with the action of $\operatorname {Aut}_\mathbb {R}(B)$ given by the tautological linear action in the fibers. By construction, this bundle admits a $\operatorname {Aut}_\mathbb {R}(B)$-invariant real structure and a complex bilinear pairing $B$ as well as a universal Hodge decomposition, i.e. a smooth decomposition as a direct sum of $G_\mathbb {R}$-equivariant complex vector bundles $\mathcal {U}^{p,q}$ such that, at a point $x$, the induced decomposition of $\mathcal {U}_x = V\otimes _\mathbb {Z} \mathbb {C}$ is the Hodge decomposition associated with $x$.

Now let $X$ be a complex analytic variety. A (polarized) variation of Hodge structure of weight $k$ over $X$ is the data of:

1. (i) a local system $\mathbb {V}_\mathbb {Z}$ of free $\mathbb {Z}$-modules of finite rank $d$ with a flat bilinear pairing $B:\mathbb {V}_\mathbb {Z}\otimes \mathbb {V}_\mathbb {Z}\rightarrow \underline {\mathbb {Z}}_X$;

2. (ii) a decreasing filtration $\mathcal {F}^{\bullet }\mathcal {V}$ on $\mathcal {V}=\mathbb {V}_\mathbb {Z}\otimes _{\underline {\mathbb {Z}}_X} \mathcal {O}_X$ by holomorphic sub-vector bundles $0\subseteq \mathcal {F}^{k}\mathcal {V}\subseteq \cdots \subseteq \mathcal {F}^0\mathcal {V}=\mathcal {V}$;

which satisfy the following conditions.

1. (a) Hodge property: for every $x\in X$, the flag $0\subseteq \mathcal {F}^{k}\mathcal {V}_x\subseteq \cdots \subseteq \mathcal {F}^0\mathcal {V}_x\overset {\text {def}}= \mathcal {V}_x$ is a Hodge structure on ${\mathbb {V}_{\mathbb {Z},x}}$.

2. (b) Griffiths’ transversality: the flat connection $\nabla$ associated on $\mathbb {V}_\mathbb {Z}\otimes \mathcal {O}_X$ satisfies

   \[ \nabla(\mathcal{F}^{p}\mathcal{V})\subseteq \mathcal{F}^{p-1} \mathcal{V}\otimes \Omega^{1}_X \text{ for } 0\leq p\leq k. \]

Let $\{\mathbb {V}_\mathbb {Z},\mathcal {F}^\bullet \mathcal {V},B\}$ be a variation of Hodge structure of weight $k$ over $X$. Its Hodge decomposition is the ($C^{\infty }$) decomposition

\[ \mathcal{V} = \bigoplus_{p+q= k} \mathcal{V}^{p,q}, \]

where $\mathcal {V}^{p,q} = \mathcal {F}^p\mathcal {V} \cap \overline {\mathcal {F}}^{k-p} \mathcal {V}$, and its Hodge numbers are

\[ h^{p,q} = \dim_\mathbb{C}(\mathcal{V}^{p,q}), p+q = k. \]

Now let $\pi :\widetilde {X}\to X$ be the universal cover of $X$ and $x$ and arbitrary point in $\widetilde {X}$. The local system $\pi ^*\mathbb {V}_\mathbb {Z}$ is trivial, and one obtains a map

\[ \widetilde{f}: \widetilde{X} \to \mathcal{D} \]

such that $\widetilde {f}(y)$ is the Hodge structure $\mathcal {F}^\bullet \mathcal {V}_y$ on $(\pi ^*{\mathbb {V}_{\mathbb {Z}}})_y = {\mathbb {V}_{\mathbb {Z},x}}$. This map is equivariant with respect to the monodromy $\rho :\pi _1(X) \to G_\mathbb {Z} = \operatorname {Aut}({\mathbb {V}_{\mathbb {Z},x}})$ of the local system and thus factors to a map

\[ f:X \to G_\mathbb{Z} \backslash \mathcal{D} \]

called the period map of the variation of Hodge structure. There are canonical isomorphisms

\[ \mathcal{V}^{p,q} \simeq f^*\mathcal{U}^{p,q}. \]

In terms of the period map, Griffiths’ transversality condition admits the following interpretation. Let $x$ be a point in $\mathcal {D}$ and let $\varphi : \mathbb {S} \to G_\mathbb {R}$ be the associated representation of the Deligne torus. Then the Lie algebra $\mathfrak {g}_\mathbb {C}$ decomposes under the adjoint action as

\[ \mathfrak{g}_\mathbb{C}=\bigoplus_p\mathfrak{g}^{p,-p}, \]

where

\[ \mathfrak{g}^{p,-p}=\{\xi\in\mathfrak{g},\xi\cdot V^{r,s}\subset V^{r+p,s-p}\}. \]

The subalgebra $\mathfrak {g}^{0,0}$ is the Lie algebra of the stabilizer of $x$, and its complement identifies with the complexified tangent space to $\mathcal {D}$ at $x$. The eigenspace of the complex structure on $T_x \mathcal {D}$ for $i$ is the subspace $\bigoplus _{p<0} \mathfrak {g}^{p,-p}$.