2 Background on $\mathrm {M}_d$ and its subvarieties

In this section, we provide important background on the moduli spaces $\mathrm {M}_d$ , and we look into properties of a maximally varying family $\mathcal {X} \to V$ of subvarieties of $\mathrm {M}_d$ .

2.1 The moduli space $\mathrm {M}_d$ and a critically-marked parameter space

For each degree $d \geq 2$ , the moduli space $\mathrm {M}_d$ is the space of conformal conjugacy classes of maps $f: \mathbb {P}^1\to \mathbb {P}^1$ defined over $\mathbb {C}$ . It is an affine algebraic variety of dimension $2d-2$ . Silverman showed that it exists as a geometric quotient over $\operatorname {Spec} \mathbb {Z}$ , and he introduced a projective compactification $\overline {M}_d$ via the tools of Geometric Invariant Theory which can be defined over $\mathbb {Q}$ [Reference SilvermanSi1]. We note that $\mathrm {M}_2 \simeq \mathbb {C}^2$ [Reference MilnorMi1] while $\mathrm {M}_d$ is singular in all degrees $d>2$ [Reference WestWe, Reference Miasnikov, Stout and WilliamsMSW]. Because of the existence of maps with automorphisms, this $\mathrm {M}_d$ is not a fine moduli space, but we can pass to a finite branched cover $\mathcal {M}_d^{cm} \to \mathrm {M}_d$ to build a family

(2.1) $$ \begin{align} f : \mathcal{M}_d^{cm} \times \mathbb{P}^1 \to \mathcal{M}_d^{cm} \times \mathbb{P}^1 \end{align} $$

of maps, and we may choose $\mathcal {M}_d^{cm}$ so that we also have functions

$$ \begin{align*}c_i: \mathcal{M}_d^{cm} \to \mathbb{P}^1\end{align*} $$

marking each of the critical points, $i = 1, \ldots , 2d-2$ .

2.3 The bifurcation current and its wedge powers

Following [Reference Dujardin and FavreDF], we can define a bifurcation current $T_i$ on $\mathcal {M}_d^{cm}$ associated to the pair $(f,c_i)$ for the map f of (2.1) and each critical point $c_i$ , as follows. We let $\omega $ denote a Fubini-Study form on $\mathbb {P}^1$ and let $\hat {\omega }$ be its pullback to $\mathcal {M}_d^{cm} \times \mathbb {P}^1$ by the projection map. Then the sequence of pullbacks $\frac {1}{d^n} (f^n)^* \hat \omega $ converge weakly to a current $\hat {T}_f$ on $\mathcal {M}_d^{cm} \times \mathbb {P}^1$ . We let

(2.2) $$ \begin{align} T_i = \pi_* \left( \hat{T}_f \wedge [\Gamma_{c_i}] \right), \end{align} $$

where $\pi : \mathcal {M}_d^{cm} \times \mathbb {P}^1 \to \mathcal {M}_d^{cm}$ is the projection and $[\Gamma _{c_i}]$ is the current of integration along the graph of $c_i$ . Then $T_i$ is a positive (1,1)-current with continuous potentials. The sum

(2.3) $$ \begin{align} T_{\mathrm{bif}} = \sum_{i = 1}^{2d-2} T_i \end{align} $$

descends to a well-defined positive $(1,1)$ -current on the moduli space $\mathrm {M}_d$ , and it agrees with the bifurcation current introduced in [Reference DeMarcoDe1]. Because of its continuous potentials, the wedge powers of $T_{\mathrm {bif}}$ are well defined. The bifurcation measure $T_{\mathrm {bif}}^{\wedge (2d-2)}$ was first introduced in [Reference Bassanelli and BertelootBB] and is non-zero on $\mathrm {M}_d$ .

If $X \subset \mathcal {M}_d^{cm}$ is an irreducible complex algebraic subvariety of dimension $r> 0$ whose image in $\mathrm {M}_d$ is not contained in the flexible Lattès locus $\mathcal {L}_d$ , then $T_{\mathrm {bif}} \not =0$ on X, as a consequence of [Reference McMullenMc1, Lemma 2.2] and [Reference DeMarcoDe1, Theorem 1.1]. Moreover, $T_i = 0$ on X if and only if the critical point $c_i$ is persistently preperiodic along X [Reference Dujardin and FavreDF, Theorem 2.5]. It was proved in [Reference Gauthier, Okuyama and VignyGOV, Lemma 6.8] that we also have $T_{\mathrm {bif}}^{\wedge r} \not =0$ on X; in other words, there always exists an r-tuple of critical points $c_{i_1}, \ldots , c_{i_r}$ so that $T_{i_1} \wedge \cdots \wedge T_{i_r} \not =0$ on X.

2.4 The critical height

For each $f: \mathbb {P}^1 \to \mathbb {P}^1$ of degree $d>1$ defined over $\overline {\mathbb {Q}}$ , recall that the canonical height on $\mathbb {P}^1(\overline {\mathbb {Q}})$ is defined by

$$ \begin{align*}\hat{h}_f(x) = \lim_{n\to\infty} \frac{1}{d^n} h(f^n(x)),\end{align*} $$

where h is a Weil height on $\mathbb {P}^1$ [Reference Call and SilvermanCS]. Counting the critical points with multiplicity, the sum

$$ \begin{align*}\hat{h}_{\mathrm{crit}}(f) := \sum_{\{c~: ~f'(c)=0\}} \hat{h}_f(c)\end{align*} $$

defines a function on $\mathcal {M}_d^{cm}$ , which Silverman called the critical height and conjectured that it is comparable with a Weil height on $\mathrm {M}_d \setminus \mathcal {L}_d$ ; see, for example, [Reference SilvermanSi2]. This comparability was proved by Ingram in [Reference IngramIn].

The construction of the critical height was revisited by Yuan and Zhang in [Reference Yuan and ZhangYZ, §6.3], where they show that $\hat {h}_{\mathrm {crit}}$ is the height associated to a non-degenerate adelically metrized line bundle on $\mathcal {M}^{cm}_d$ over $\mathbb {Q}$ . Indeed, they construct an f-invariant adelic line bundle $\overline {L}_f$ on $\mathcal {M}_d^{cm} \times \mathbb {P}^1$ for the f of (2.1). The critical height is then the height function associated to the “restriction” of $\overline {L}_f$ to the ramification divisor of f; see [Reference Yuan and ZhangYZ, §6.3]. Important for us is that the curvature distribution at the archimedean place is equal to the bifurcation current $T_{\mathrm {bif}}$ defined in §2.3. The non-degeneracy of this metrized line bundle follows because the bifurcation measure is non-trivial, as was proved in [Reference Bassanelli and BertelootBB].

Yuan and Zhang use their general equidistribution theorem [Reference Yuan and ZhangYZ, Theorem 5.4.3] to give an alternative proof of the arithmetic equidistribution of the points in $\mathrm {PCF}_d$ to the bifurcation measure $T_{\mathrm {bif}}^{\wedge 2d-2}$ in [Reference Yuan and ZhangYZ, §6.3], a result which was first obtained by Gauthier in [Reference GauthierGa3] (and was proved for the polynomial moduli spaces in [Reference Favre and GauthierFG1]).

2.5 The Chow variety and maximal variation

For general information about Chow varieties, we refer the reader to [Reference Gelfand, Kapranov and ZelevinskyGKZ] [Reference KollárKo]. Here we fix notation and state some basic properties that we need. We will apply the following discussion to $Y = \mathrm {M}_d$ or $Y = \mathcal {M}_d^{cm}$ .

Suppose that Y is an irreducible quasiprojective algebraic variety, equipped with a projective embedding of $Y \hookrightarrow \mathbb {P}^N$ defined over $\mathbb {Q}$ . We let $\overline {Y}$ denote the closure of Y in $\mathbb {P}^N$ . For each integer $0 \leq r < \dim Y$ , and with respect to this embedding, we first consider the Chow variety $\mathrm {Ch}(\overline {Y}, r, D)$ of algebraic cycles $X \subset \overline {Y}$ in dimension r with degree $\leq D$ . It is an algebraic subvariety, possibly reducible, of the Chow variety $\mathrm {Ch}(\mathbb {P}^N, r, D)$ , consisting of the cycles with dimension r and degree $\leq D$ supported in $\overline {Y}$ . As $\overline {Y}$ is defined over $\mathbb {Q}$ , the Chow variety $\mathrm {Ch}(\overline {Y}, r, D)$ is also defined over $\mathbb {Q}$ . We let $\mathrm {Ch}(Y, r, D)$ denote the Zariski-open subset consisting of cycles with support intersecting Y.

We will say that an irreducible subvariety $X \subset Y$ of dimension r has degree $\boldsymbol {\leq D}$ if its Zariski closure in $\overline {Y}$ is the support of a cycle represented by a point in $\mathrm {Ch}(Y, r, D)$ . Note that the degree depends on the choice of the embedding of Y into $\mathbb {P}^N$ , but the concept of bounded degree does not.

Suppose that $\pi : \mathcal {X}\to V$ is a family of r-dimensional varieties in Y, parametrized by $\lambda $ in an irreducible quasi-projective variety V of dimension $\ell \ge 1$ . Assume that the generic fiber of $\mathcal {X} \to V$ is geometrically irreducible. We say that $\mathcal {X}\to V$ is maximally varying if the natural map from V to the Chow variety $\mathrm {Ch}(Y, r, D)$ has finite fibers, where we send each irreducible fiber $X_\lambda := \pi ^{-1}(\lambda )$ to the point of $\mathrm {Ch}(Y, r, D)$ representing the cycle $[\overline {X}_\lambda ]$ on $\overline {Y}$ .

For any positive integer m, recall that $\mathcal {X}_V^m$ denotes the m-th fiber power of $\mathcal {X}$ over V. There is a natural map

$$ \begin{align*}\rho_m: \mathcal{X}_V^m\to Y^{m}\end{align*} $$

defined by $\rho _m(\lambda ,f_1,\ldots ,f_{m})=(f_1,\ldots ,f_{m})$ .

Proposition 2.1. Suppose that Y is an irreducible quasiprojective complex algebraic variety. Suppose that $\mathcal {X}\to V$ is a family of r-dimensional varieties $X_{\lambda }\subset Y$ , parametrized by $\lambda $ in a quasi-projective variety V of dimension $\ell \ge 1$ . Assume that V and the generic fiber of $\mathcal {X} \to V$ are irreducible. If $\mathcal {X} \to V$ is maximally varying, then the map

$$ \begin{align*}\rho_\ell: \mathcal{X}^\ell_V \to Y^\ell\end{align*} $$

is generically finite. In particular $\dim \rho _\ell (\mathcal {X}^{\ell }_V) = \dim \mathcal {X}_V^{\ell }$ .

Proof. We prove the result by induction on $\ell $ . For $\ell = 1$ , the result is clear, as $\dim \mathcal {X} = r+1$ and maximal variation implies that the image of $\mathcal {X}$ in Y is not contained in an r-dimensional subvariety, so $\dim \rho _1(\mathcal {X}) = \dim \mathcal {X} = r+1$ .

Now suppose $\ell> 1$ and that we know the result for maximally varying families with base dimension $< \ell $ . Note that $\dim \mathcal {X} = \ell + r$ and $\dim \mathcal {X}_V^\ell = \ell (r+1)$ . Choose any smooth point $x_0$ of $\rho _1(\mathcal {X})$ which is also smooth for a subvariety $X_{\lambda _0}$ it lies in, and which is regular for the maps $\rho _1: \mathcal {X} \to \rho _1(\mathcal {X})$ and $\pi _1: \rho _\ell (\mathcal {X}_V^\ell ) \to \rho _1(\mathcal {X})$ where the latter is the restriction of the projection $Y^\ell \to Y$ to the first coordinate. Here, “regular for the maps” means that the fiber over $x_0$ has the expected dimension and there is a point in the fiber over $x_0$ mapping surjectively on tangent spaces to that of $x_0$ .

Let $Y_1 \subset \rho _1(\mathcal {X}) \subset Y$ be a subvariety of codimension r in $\rho _1(\mathcal {X})$ containing $x_0$ , intersecting $X_{\lambda _0}$ (and therefore a general $X_\lambda $ ) in a finite set. Consider the projection $p_1: \mathcal {X}_V^{\ell } \to \mathcal {X}$ to the first factor and the projection $p_2: \mathcal {X}_V^{\ell } \to \mathcal {X}_V^{\ell -1}$ to the remaining $\ell -1$ factors. Note that $\rho _1^{-1}(Y_1) \subset \mathcal {X}$ projects dominantly to the base V, and $p_1^{-1}(\rho _1^{-1}(Y_1))$ projects dominantly and generically finitely by $p_2$ to $\mathcal {X}_V^{\ell -1}$ .

Now choose any $Y_1' \subset Y_1$ of codimension 1 containing $x_0$ , so of codimension $r+1$ in $\rho _1(\mathcal {X})$ . Then $\rho _1^{-1}(Y_1')$ has codimension $r+1$ in $\mathcal {X}$ , and its projection to the base V will be a subvariety $V_1$ of codimension 1. Let $\mathcal {X}|_{V_1} \to V_1$ denote the restricted family over the $(\ell -1)$ -dimensional base. Note that $p_1^{-1}(\rho _1^{-1}(Y_1'))$ is contained in $\mathcal {X}_{V_1}^\ell $ and projects dominantly and generically finitely by $p_2$ to $\mathcal {X}_{V_1}^{\ell -1}$ .

By induction we know that $\dim \rho _{\ell -1}(\mathcal {X}_{V_1}^{\ell -1}) = \dim \mathcal {X}_{V_1}^{\ell -1} = (\ell - 1)(r+1) = \ell (r+1) - (r+1)$ . Therefore

$$ \begin{align*}\dim \rho_\ell(p_1^{-1}(\rho_1^{-1}(Y_1'))) = \dim \mathcal{X}_{V_1}^{\ell-1} = \ell(r+1)-(r+1).\end{align*} $$

Now consider $\pi _1^{-1}(Y_1')$ in $\rho _\ell (\mathcal {X}_V^\ell )$ , which also has codimension $r+1$ by the choice of $x_0$ . But $\pi _1 \circ \rho _\ell = \rho _1\circ p_1$ , so that

$$ \begin{align*}\dim \pi_1^{-1}(Y_1') = \dim \rho_\ell(p_1^{-1}(\rho_1^{-1}(Y_1'))),\end{align*} $$

implying that

$$ \begin{align*}\dim \rho_\ell(\mathcal{X}_V^{\ell}) = \ell(r+1)\end{align*} $$

and the proof is complete.

3 Bifurcation measures on $\mathcal {X}_V^{\hspace{1pt}\ell} $ and $\mathcal {X}_V^{\hspace{1pt}\ell +1}$

Recall the definition of $\mathcal {M}_d^{cm}$ from §2.1. Throughout this section, we assume that $\mathcal {X}\to V$ is a maximally varying family of r-dimensional varieties $X_{\lambda }\subset \mathcal {M}_d^{cm}$ , parametrized by $\lambda $ in a quasi-projective variety V of dimension $\ell \ge 1$ . Assume that V and the generic fiber of $\mathcal {X} \to V$ are irreducible. For any positive integer m, recall from §1.2 that $\mathcal {X}_V^m$ denotes the m-th fiber power of $\mathcal {X}$ over V, and we denote elements of $\mathcal {X}_V^{m}$ by $(\lambda , f_1, \ldots , f_m)$ , with $f_i \in X_\lambda $ for each i. There is a natural map

(3.1) $$ \begin{align} \rho_m: \mathcal{X}_V^m\to (\mathcal{M}^{cm}_d)^m \end{align} $$

defined by $\rho _m(\lambda ,f_1,\ldots ,f_{m})=(f_1,\ldots ,f_{m})$ . From Proposition 2.1, we know that $\rho _m$ is generically finite for all $m \geq \ell $ .

In this section we construct nontrivial measures on $\mathcal {X}_V^{\ell }$ and $\mathcal {X}_V^{\ell +1}$ that are pullbacks of powers of the bifurcation currents in $\mathcal {M}_d^{cm}$ , defined in §2.3. The measure on $\mathcal {X}_V^{\ell }$ will be defined in terms of $T_{\mathrm {bif}}$ of (2.3), while two measures on $\mathcal {X}_V^{\ell +1}$ will additionally involve choices of r-tuples of marked critical points. The proofs build on the strong rigidity result that the power $T_{\mathrm {bif}}^{\wedge r}$ of the bifurcation current is nonzero on every r-dimensional subvariety of $\mathrm {M}_d\setminus \mathcal {L}_d$ , where $\mathcal {L}_d$ is the flexible Lattès locus; this is proved in [Reference Gauthier, Okuyama and VignyGOV, Lemma 6.8] for all r. (See the discussion in §2.3.)

We also appeal to descriptions of the support of powers of $T_{\mathrm {bif}}$ in $\mathrm {M}_d$ in terms of critical orbit behavior obtained by Buff-Epstein in [Reference Buff and EpsteinBE] and Dujardin in [Reference DujardinDu]. Precisely, suppose that $f_1, \ldots , f_m$ is a collection of holomorphic families of maps on $\mathbb {P}^1$ parameterized by $\lambda $ in a complex manifold $\Lambda $ , and let $\Phi := (f_1, \ldots , f_m): \Lambda \times (\mathbb {P}^1)^m \to \Lambda \times (\mathbb {P}^1)^m$ be the induced holomorphic map. We say that an m-tuple of marked points $\mathbf {a} = (a_1, \ldots , a_m): \Lambda \to (\mathbb {P}^1)^m$ is transversely (respectively, properly) prerepelling for $\boldsymbol {\Phi }$ at $\boldsymbol {\lambda _0}$ over $\boldsymbol {\Lambda }$ if there exist repelling periodic points $p_i$ of $f_{\lambda _0}$ and integers $n_i \ge 1$ such that $f_{\lambda _0}^{n_i}(a_i(\lambda _0)) = p_i$ for each $i = 1, \ldots , m$ , and the following holds: if $p_i: U \to \mathbb {P}^1$ denotes the holomorphic continuation of the repelling periodic point $p_i=p_i(\lambda _0)$ over some neighborhood $U \subset \Lambda $ of $\lambda _0$ where it remains repelling, then the graphs of the maps

$$\begin{align*}\lambda \longmapsto \big(f_{1,\lambda}^{n_1}(a_1(\lambda)), \ldots, f_{m,\lambda}^{n_m}(a_m(\lambda))\big) \quad \text{and} \quad \lambda \longmapsto (p_1(\lambda), \ldots, p_m(\lambda)) \end{align*}$$

intersect transversely (respectively, properly) in $U \times (\mathbb {P}^1)^m$ at $\lambda _0$ .

For positive integers $m \leq 2d-2$ , marked critical points $\mathbf {c} = (c_1, \ldots , c_m)$ , and $f_1 = \cdots = f_m := f$ , Dujardin proved that there is a dense set of parameters in $\operatorname {supp} T_1 \wedge \cdots \wedge T_m$ in $\mathcal {M}_d^{cm}$ at which $\mathbf {c}$ is transversely prerepelling over $\mathcal {M}_d^{cm}$ [Reference DujardinDu, Theorem 1.1]; recall that $T_i$ was defined in (2.2). And conversely, if the m-tuple of critical points is properly prerepelling for $(f,\ldots , f)$ at some parameter $\lambda _0$ over a parameter space of dimension $\geq m$ , then $T_1 \wedge \cdots \wedge T_m$ must be nonzero in that parameter space with $\lambda _0$ in its support, as first observed in [Reference Buff and EpsteinBE] (where it was proved in the case $m=2d-2$ and assuming transversely prerepelling) and revisited in [Reference GauthierGa1, Theorem 6.2], [Reference GauthierGa2, Theorem 2.3], and [Reference Berteloot, Bianchi and DupontBBD, Proposition 3.7] in more general contexts and without assuming transversality. See [Reference DeMarco and MavrakiDM1, Proposition 4.8] for the version we use, which implies that the latter converse result remains true with any m-tuple of maps $(f_1,\ldots , f_m)$ of degree d in place of $(f,\ldots ,f)$ .

3.1 Measure on $\mathcal {X}_V^{\hspace{1pt}\ell} $

Proposition 3.1. Let $\mathcal {X}\to V$ be a maximally varying family of dimension $r\ge 1$ varieties in $\mathcal {M}_d^{cm}$ with $\dim V = \ell \geq 1$ , and let $\rho _{\ell }$ be the map of (3.1) for $m = \ell $ . Set $\mathcal {Y}_\ell = \rho _\ell (\mathcal {X}_V^\ell )$ . Let $T_{\mathrm {bif}}$ denote the bifurcation current in $\mathcal {M}_d^{cm}$ and let $\pi _i: (\mathcal {M}_d^{cm})^{\ell }\to \mathcal {M}_d^{cm}$ be the projection to the i-th factor for $i=1,\ldots ,\ell $ . Then

$$ \begin{align*}\pi_1^{*}(T_{\mathrm{bif}}^{\wedge r+1})\wedge \cdots \wedge \pi_{\ell}^{*}(T_{\mathrm{bif}}^{\wedge r+1})\wedge [\mathcal{Y}_\ell]\neq 0\end{align*} $$

in $(\mathcal {M}_d^{cm})^\ell $ , so that

$$ \begin{align*}\mu_\ell := \rho_\ell^* \left(\pi_1^{*}(T_{\mathrm{bif}}^{\wedge r+1})\wedge \cdots \wedge \pi_{\ell}^{*}(T_{\mathrm{bif}}^{\wedge r+1})\right)\end{align*} $$

defines a positive measure on $\mathcal {X}_V^\ell $ . Moreover, there is a collection of marked critical points

$$ \begin{align*}\mathbf{c} = (c_{1,1}, \ldots, c_{1, r+1}, \ldots, c_{\ell, 1}, \ldots, c_{\ell, r+1})\end{align*} $$

defined on $\mathcal {X}_V^\ell $ which is transversely prerepelling for the family of product maps

$$ \begin{align*}\Phi := (f_1,\ldots, f_1, \ldots,f_\ell, \ldots, f_\ell)\end{align*} $$

at a parameter $x_0$ over $\mathcal {X}_V^\ell $ , where each $f_i$ is repeated $r+1$ times.

If $\mathbf {c}$ satisfies the conclusion of Proposition 3.1 for $\Phi $ , we say that it witnesses the nonvanishing of $\mu _\ell $ .

Proof. Let $\mathcal {Y} \subset \mathcal {M}_d^{cm}$ be the image of $\rho _1$ , and let k be the dimension of $\mathcal {Y}$ , so that

$$ \begin{align*}2 \leq r+1 \leq k \leq r+ \ell = \dim \mathcal{X}.\end{align*} $$

Since the projection $\mathcal {M}_d^{cm} \to \mathrm {M}_d$ has finite fibers, we know from [Reference Gauthier, Okuyama and VignyGOV, Lemma 6.8] that $T_{\mathrm {bif}}^{\wedge k} \not =0$ on $\mathcal {Y}$ .

If $\ell =1$ , then we are done, since $k = r+1$ and $T_{\mathrm {bif}}^{\wedge (r+1)} \wedge [\mathcal {Y}]$ in $\mathcal {M}_d^{cm}$ .

Assume $\ell> 1$ . Let

$$ \begin{align*}\Phi = (f_1,\ldots, f_1, \ldots, f_\ell, \ldots, f_\ell)\end{align*} $$

denote the family of product maps on $(\mathbb {P}^1)^{\ell (r+1)}$ parameterized by $\mathcal {Y}_\ell \subset (\mathcal {M}_d^{cm})^\ell $ , where each coordinate $f_i$ is repeated $r+1$ times.

Let $U\subset \mathcal {Y}$ be a Zariski-open set so that the fibers of the two maps $\rho _1: \mathcal {X} \to \mathcal {Y}$ and $\pi _1: \mathcal {Y}_\ell \to \mathcal {Y}$ to the first factor of $(\mathcal {M}_d^{cm})^\ell $ have the expected dimension over U, namely $r+\ell -k$ for fibers of $\rho _1$ and $\ell + r\ell - k$ for $\pi _1$ , and the maps are surjective on tangent spaces at some point of each fiber over elements of U. Note that, by the symmetry of $\mathcal {Y}_\ell $ , the fibers will have the same properties for projections $\pi _i$ for each i. From [Reference DujardinDu, Theorem 1.1], there exists a k-tuple of critical points $\{c_1, \ldots , c_k\}$ and a parameter $y_0 \in U$ in the support of $T_{\mathrm {bif}}^{\wedge k}$ at which these critical points are transversely prerepelling for the family of maps $f_1$ parameterized by U. We may assume that $y_0$ is a smooth point of $\mathcal {Y}$ and of the subvariety $X_{\lambda _0}$ for a smooth parameter $\lambda _0 \in V$ .

Now choose a collection of $r+1$ critical points $c_{1,1}, \ldots , c_{1,r+1} \in \{c_1, \ldots , c_k\}$ so that $(c_{1,1}, \ldots , c_{1,r})$ is transversely prerepelling at $y_0$ over $X_{\lambda _0}$ and the full $(r+1)$ -tuple is transversely prerepelling at $y_0$ over U. Let $Z_1 \subset U$ be an irreducible component of the codimension- $(r+1)$ subvariety containing $y_0$ where these critical points are persistently preperiodic. Note that $y_0$ is an isolated point of the intersection of $Z_1$ with $X_{\lambda _0}$ . Moreover, we know that $\pi _1^{-1}(Z_1)$ has codimension $r+1$ in $\mathcal {Y}_\ell $ .

Note that the preimage $\rho _\ell ^{-1}(\pi _1^{-1}(Z_1))$ in $\mathcal {X}_V^{\ell }$ projects to a subvariety $V_1$ of codimension 1 in the base V. In fact it maps with finite fibers to the $(\ell -1)$ -th fiber power of a subfamily

$$ \begin{align*}\mathcal{X}|_{V_1} \to V_1\end{align*} $$

where $\dim V_1 = \ell -1$ . Now let $\mathcal {Y}_1$ be the image of this subfamily $\mathcal {X}|_{V_1}$ in $\mathcal {M}_d^{cm}$ . We repeat the argument with $\mathcal {Y}_1$ in place of $\mathcal {Y}$ . By maximal variation of $\mathcal {X}$ , we know that

$$ \begin{align*}2 \leq r+1 \leq k_1 := \dim \mathcal{Y}_1 \leq r+ \ell -1\end{align*} $$

and $T_{\mathrm {bif}}^{\wedge k_1} \not =0$ on $\mathcal {Y}_1$ . We choose another collection of critical points $c_{2,1}, \ldots , c_{2,r+1} \in \{c_1, \ldots , c_k\}$ so that the first r are transversely prerepelling at $y_0$ over $X_{\lambda _0}$ and the full $(r+1)$ -tuple is transversely prerepelling at $y_0$ over $\mathcal {Y}_1$ . Note that we could take $c_{2,j} = c_{1,j}$ for $j = 1, \ldots , r$ . We let $Z_2 \subset \mathcal {Y}_1$ be the codimension- $(r+1)$ subvariety containing $y_0$ where these critical points are persistently preperiodic. Then $\pi _2^{-1}(Z_2) \cap \pi _1^{-1}(Z_1)$ has codimension $2(r+1)$ in $\mathcal {Y}_\ell $ .

In this way, we inductively construct a parameter $y = (y_0, \ldots , y_0) \in \mathcal {Y}_\ell $ and an $\ell (r+1)$ -tuple of marked critical points

$$ \begin{align*}\mathbf{c} = (c_{1,1}, \ldots, c_{1, r+1}, \ldots, c_{\ell, 1}, \ldots, c_{\ell, r+1})\end{align*} $$

which is transversely prerepelling for the map $\Phi $ at y over $\mathcal {Y}_\ell $ . In particular, in the language of [Reference DeMarco and MavrakiDM1, §4.3], the graph $\Gamma _{\mathbf {c}}$ of $\mathbf {c}$ in $\mathcal {Y}_\ell \times (\mathbb {P}^1)^{\ell (r+1)}$ defines a rigid repeller for the map $\Phi $ at this parameter, and we conclude from [Reference DeMarco and MavrakiDM1, Proposition 4.8] that

(3.2) $$ \begin{align} \hat{T}_\Phi^{\wedge \ell(r+1)} \wedge [\Gamma_{\mathbf{c}}] \not=0, \end{align} $$

where

$$ \begin{align*}\hat{T}_\Phi = \sum_{i=1}^\ell \sum_{j = 1}^{r+1} p_{i,j}^* \hat{T}_{f_i}\end{align*} $$

is the dynamical Green current for $\Phi $ on $\Lambda \times (\mathbb {P}^1)^{\ell (r+1)}$ , for the projections $p_{i,j}: \Lambda \times (\mathbb {P}^1)^{\ell (r+1)} \to \Lambda \times \mathbb {P}^1$ . Unwinding the definitions, we conclude that

$$ \begin{align*}\pi_1^{*}(T_{f_1, c_{1,1}}\wedge\cdots \wedge T_{f_1, c_{1, r+1}}) \wedge \cdots \wedge \pi_{\ell}^{*}(T_{f_\ell, c_{\ell,1}} \wedge \cdots \wedge T_{f_\ell, c_{\ell, r+1}})\wedge [\mathcal{Y}_\ell]\neq 0\end{align*} $$

and therefore that

$$ \begin{align*}\pi_1^{*}(T_{\mathrm{bif}}^{\wedge r+1})\wedge \cdots \wedge \pi_{\ell}^{*}(T_{\mathrm{bif}}^{\wedge r+1})\wedge [\mathcal{Y}_\ell]\neq 0\end{align*} $$

so that the measure $\mu _\ell \not =0$ . Here we used the fact that $\dim \mathcal {X}_V^{\ell }=\dim \mathcal {Y}_{\ell }$ from the maximal variation.

Finally, since the map from $\mathcal {X}_V^{\ell }$ to $\mathcal {Y}_\ell $ is generically finite by Proposition 2.1, our initial choice of $y_0$ guarantees that the marked critical points $\mathbf {c}$ are also transversely prerepelling at a preimage of y, over $\mathcal {X}_V^\ell $ .

3.2 Measures on $\mathcal {X}_V^{\ell +1}$

Let $\mu _\ell $ be the measure of Proposition 3.1, witnessed by the marked critical point $\mathbf {c}$ . We now use these critical points to construct nonvanishing measures on the next fiber power $\mathcal {X}_V^{\ell +1}$ . Let

$$ \begin{align*}\Phi = (f_1, \ldots, f_1, \ldots, f_\ell, \ldots, f_\ell)\end{align*} $$

be as in Proposition 3.1 parameterized by $\mathcal {X}_V^\ell $ via the map $\rho _{\ell }$ of (3.1), and let

$$ \begin{align*}\Phi_{\ell+1} = (f_1, \ldots, f_1, \ldots, f_\ell, \ldots, f_\ell, f_{\ell+1}, \ldots, f_{\ell+1})\end{align*} $$

be the family of maps on $(\mathbb {P}^1)^{\ell (r+1)+r}$ parameterized by $\mathcal {X}_V^{\ell +1}$ , where $f_1, \ldots , f_{\ell }$ are each repeated $r+1$ times, but the final coordinate $f_{\ell +1}$ is only repeated r times. Specializing at each $\lambda \in V$ we also have a family of maps

$$ \begin{align*}F^{(\lambda)}:=(f,\ldots,f): X_{\lambda}\times (\mathbb{P}^1)^r\to X_{\lambda}\times (\mathbb{P}^1)^r,\end{align*} $$

parametrized by $X_{\lambda } \subset \mathcal {M}_d^{cm}$ , and, focusing on the $\ell $ -th factor of $\mathcal {X}_V^\ell $ and associated components of $\mathbf {c}$ , we will use the marked critical points

$$ \begin{align*}c_{\ell,1}|_{X_\lambda}, \ldots, c_{\ell,r+1}|_{X_\lambda}: X_{\lambda}\to \mathbb{P}^1\end{align*} $$

to define two measures $\mu _\ell ^i$ and $\mu _\ell ^j$ on $\mathcal {X}_V^{\ell +1}$ . Conditions (1) and (2) of Proposition 3.2 will allow us to conclude that $\mu _\ell ^i$ and $\mu _\ell ^j$ are nonzero. Conditions (3) and (4) will allow us to prove in Section 4 that $\mu _\ell ^i$ and $\mu _\ell ^j$ are not proportional measures.

Proposition 3.2. Suppose the marked critical point

$$ \begin{align*}\mathbf{c} = (c_{1,1}, \ldots, c_{1, r+1}, \ldots, c_{\ell, 1}, \ldots, c_{\ell, r+1})\end{align*} $$

witnesses $\mu _\ell \not =0$ on $\mathcal {X}_V^\ell $ . Then there are parameters $x_0 = (\lambda _0, t_1, \ldots , t_\ell )$ and $x_1 = (\lambda _1, s_1, \ldots , s_\ell )$ in the support of $\mu _\ell $ in $\mathcal {X}_V^\ell $ and indices $i \not = j$ in $\{1, \ldots , r+1\}$ so that

1. 1. the r-tuple $(c_{\ell ,1}, \ldots , \widehat {c_{\ell , i}}, \ldots c_{\ell , r+1})$ is transversely prerepelling for $F^{(\lambda _0)}$ at $t_\ell $ over $X_{\lambda _0}$ ;

2. 2. the r-tuple $(c_{\ell ,1}, \ldots , \widehat {c_{\ell , j}}, \ldots c_{\ell , r+1})$ is properly prerepelling for $F^{(\lambda _0)}$ at $t_\ell $ over $X_{\lambda _0}$ ;

3. 3. the r-tuple $(c_{\ell ,1}, \ldots , \widehat {c_{\ell , i}}, \ldots c_{\ell , r+1})$ is transversely prerepelling for $F^{(\lambda _1)}$ at $s_\ell $ over $X_{\lambda _1}$ ; and

4. 4. the critical point $c_{\ell , i}$ is attracted to a parabolic periodic point of f at $s_\ell \in X_{\lambda _1}$ ,

where $\widehat {c_{\ell , i}}$ means that the i-th coordinate is omitted. Let $\Pi _1:\mathcal {X}^{\ell }_V \times _V \mathcal {X}\to \mathcal {X}_V^\ell $ and $\Pi _2:\mathcal {X}_V^{\ell }\times _V\mathcal {X}\to \mathcal {X}$ . Then

$$ \begin{align*}\mu_\ell^i := \Pi_1^{*}(\mu_{\ell}) \wedge \Pi_2^*(T_{c_{\ell,1}}\wedge \cdots \widehat{T_{c_{\ell, i}}} \cdots \wedge T_{c_{\ell,r+1}}) \neq 0\end{align*} $$

and

$$ \begin{align*}\mu_\ell^j := \Pi_1^{*}(\mu_{\ell}) \wedge \Pi_2^*(T_{c_{\ell,1}}\wedge \cdots \widehat{T_{c_{\ell, j}}} \cdots \wedge T_{c_{\ell,r+1}}) \neq 0\end{align*} $$

on $\mathcal {X}_V^{\ell +1}$ .

Proof. Via the map $\rho _\ell : \mathcal {X}_V^{\ell } \to (\mathcal {M}_d^{cm})^{\ell }$ , we view $\mathcal {X}_V^{\ell }$ as the parameter space for a family of product maps $\Phi = (f_1, \ldots , f_1, \ldots , f_\ell , \ldots , f_\ell )$ on $(\mathbb {P}^1)^{\ell (r+1)}$ . From Proposition 3.1, we know that $\mathbf {c}$ is transversely prerepelling for $\Phi $ at a parameter $x_0 = (\lambda _0, t_1, \ldots , t_\ell ) \in \operatorname {supp} \mu _\ell $ .

Let $S_{i,j}$ be the irreducible component of the hypersurface in $\mathcal {X}_V^\ell $ containing $x_0$ along which $c_{i,j}$ is persistently preperiodic for $f_i$ , $i = 1, \ldots , \ell $ , $j = 1, \ldots , r+1$ . Consider the component of the intersection

$$ \begin{align*}S_{1,1} \cap \cdots \cap S_{\ell-1, r+1}\end{align*} $$

containing $x_0$ in $\mathcal {X}_V^\ell $ . Similar to the proof of Proposition 2.1, for dimension reasons, it maps (generically finitely) to a family $\mathcal {X}_1 \to V_1$ of r-dimensional varieties over a 1-dimensional base, where these marked critical points are persistently preperiodic for $f_1, \ldots , f_{\ell -1}$ . We will continue to write $x_0 = (\lambda _0, t_\ell )$ for the point we started with, but now viewed in $\mathcal {X}_1$ .

By the transversality, we know that r of the remaining critical points, say $c_{\ell ,1}, \ldots , c_{\ell , r}$ , are transversely prerepelling at $t_\ell $ for $F^{(\lambda _0)} = (f, f, \ldots , f)$ over $X_{\lambda _0}$ .

We now aim to show that by modifying the parameter $x_0$ if necessary, we can arrange so that another r-tuple, say $c_{\ell , 2}, \ldots , c_{\ell , r+1}$ , is also properly prerepelling at $t_\ell $ for $F^{(\lambda _0)}$ over $X_{\lambda _0}$ . Note that transversality of the intersection $S_{1,1} \cap \cdots \cap S_{\ell -1, r+1}$ at the original $x_0$ persists in a neighborhood, so the first $(\ell -1)(r+1)$ critical points remain transversely prerepelling in a neighborhood of $x_0$ for $(f_1, \ldots , f_{\ell -1})$ .

Consider the intersection

$$ \begin{align*}S_{\ell,1} \cap \cdots \cap S_{\ell, r}\end{align*} $$

in $\mathcal {X}_1$ . These hypersurfaces intersect transversely at $x_0$ and so define a smooth holomorphic curve $\Gamma $ containing $x_0$ which is locally a section of $\mathcal {X}_1 \to V_1$ near $\lambda _0$ . The intersection of $\Gamma $ with $X_{\lambda _0}$ is $x_0 = (\lambda _0, t_\ell )$ . Note further that $c_{\ell , r+1}$ is not persistently preperiodic along $\Gamma $ (because of the original setup of transverse intersections), and so $c_{\ell , r+1}$ must be active along $\Gamma $ with $x_0$ in its bifurcation locus (because it lands at a repelling cycle there). The bifurcation locus for $c_{\ell ,r+1}$ in $\Gamma $ has no isolated points, so there must be an infinite collection of parameters $x \in \Gamma $ where $c_{\ell , r+1}$ is preperiodic to a repelling cycle for $f_\ell $ , dense in the bifurcation locus of $c_{\ell , r+1}$ in a neighborhood of $x_0$ .

Now omit $S_{\ell , 1}$ from the intersection, so that $S_{\ell ,2} \cap \cdots \cap S_{\ell , r}$ (in $\mathcal {X}_1$ ) defines a 1-parameter family of algebraic curves $\mathcal {C}$ , for all $\lambda $ in a small neighborhood of $\lambda _0\in V_1$ , by

$$ \begin{align*}C_\lambda = (S_{\ell,2} \cap \cdots \cap S_{\ell, r}) \cap X_\lambda.\end{align*} $$

We know that $S_{\ell , r+1}$ intersects the total space of this family $\mathcal {C}$ transversely at the given point $x_0$ . Suppose that it fails to intersect properly when restricted to $X_{\lambda _0}$ ; this would mean that $c_{\ell , r+1}$ is persistently preperiodic along the curve $C_{\lambda _0}$ in $X_{\lambda _0}$ . Suppose this happens at all parameters $x \in \Gamma $ where $c_{\ell , r+1}$ is preperiodic to a repelling cycle. This implies that the associated bifurcation current $T_{c_{\ell , r+1}}$ vanishes identically on each of these curves $C_x$ passing through each such x in the family $\mathcal {C}$ . By continuity of potentials of the bifurcation currents, we deduce that $T_{c_{\ell , r+1}}$ vanishes along $C_x$ for all x in $\operatorname {supp} T_{c_{\ell , r+1}}|_\Gamma $ near $x_0$ . But this means that $c_{\ell , r+1}$ is persistently preperiodic along each of these curves [Reference Dujardin and FavreDF, Theorem 2.5]. As they form an uncountable family, we see that $c_{\ell , r+1}$ must be preperiodic along the entire family $\mathcal {C}$ , which is a contradiction.

We conclude that we can adjust our parameter $\lambda _0$ slightly, if needed, so that points $c_{\ell , 1}, \ldots , c_{\ell , r}$ are transversely prerepelling at a new choice of $x_0 \in \Gamma $ along $X_{\lambda _0}$ while $c_{\ell , 2}, \ldots , c_{\ell , r+1}$ are properly prerepelling at $x_0$ along $X_{\lambda _0}$ . This completes the proofs of (1) and (2) of the proposition.

As $c_{\ell ,r+1}$ is bifurcating along the curve we called $\Gamma $ , the characterizations of J-stability (as in [Reference Mañé, Sad and SullivanMSS]) imply that there are parameters, everywhere dense in its bifurcation locus, for which the maps have a parabolic cycle; the theory of renormalization (see [Reference Douady and Hamal HubbardDH1]) implies that $c_{\ell , r+1}$ will lie in the basin of the parabolic cycle. We let $\lambda _1$ be a parameter close to $\lambda _0$ in the support of $\pi _* \mu _\ell $ , where $\pi $ is the projection to V, so that $c_{\ell ,r+1}$ is attracted to a parabolic cycle at $x_1 := \Gamma \cap X_{\lambda _1}$ , while $c_{\ell ,1}, \ldots , c_{\ell ,r}$ remain transversely prerepelling at $x_1$ along $X_{\lambda _1}$ . This completes the proof of properties (1)–(4).

To prove the non-vanishing of $\mu _\ell ^i$ and $\mu _\ell ^j$ , consider the map $\Phi _{\ell +1} = f_1^{\times (r+1)} \times \cdots \times f_{\ell }^{\times (r+1)}\times f_{\ell +1}^{\times r}$ on $(\mathbb {P}^1)^{\ell (r+1) + r}$ over $\mathcal {X}_V^{\ell +1}$ . The argument above, leading to properties (1) and (2), shows that we can label critical points of $f_1, \ldots , f_{\ell +1}$ so that the graphs of the marked points

$$ \begin{align*}\mathbf{c}_i = (c_{1,1}, \ldots, c_{1, r+1}, \ldots, c_{\ell, 1}, \ldots, c_{\ell, r+1}, c_{\ell+1,1}, \ldots \widehat{c_{\ell+1, i}} \ldots, c_{\ell+1, r+1})\end{align*} $$

and

$$ \begin{align*}\mathbf{c}_j = (c_{1,1}, \ldots, c_{1, r+1}, \ldots, c_{\ell, 1}, \ldots, c_{\ell, r+1}, c_{\ell+1,1}, \ldots \widehat{c_{\ell+1, j}} \ldots, c_{\ell+1, r+1})\end{align*} $$

each define a rigid repeller for the map $\Phi _{\ell +1}$ over a neighborhood of a parameter $(\lambda _0, t_1, \ldots , t_\ell , t_\ell )$ in $\mathcal {X}_V^{\ell +1}$ , in the sense of [Reference DeMarco and MavrakiDM1, §4.3], and so the measures are nonzero by [Reference DeMarco and MavrakiDM1, Proposition 4.8].

4 Proof of Theorem 1.8

Suppose that $\mathcal {X}\to V$ is a maximally varying family of r-dimensional varieties $X_{\lambda }\subset M_d$ parametrized by V of dimension $\ell \ge 1$ , defined over $\overline {\mathbb {Q}}$ . As in §2.1, we may pass to a finite branched cover $\mathcal {M}_d^{cm} \to \mathrm {M}_d$ over which we can define a family of maps $f:\mathbb {P}^1 \to \mathbb {P}^1$ with $2d-2$ marked critical points. Pulling back each $X_\lambda $ and possibly replacing V with a finite branched cover, we will continue to use the notation $\mathcal {X} \to V$ for the family of varieties in $\mathcal {M}_d^{cm}$ . Therefore, we may assume that we can define the family of product maps

$$ \begin{align*}\Phi= (f_1,\ldots, f_1, \ldots, f_\ell, \ldots, f_\ell),\end{align*} $$

on $(\mathbb {P}^1)^{(r+1)\ell }$ , parametrized by $\mathcal {X}^{\ell }_V$ and defined over $\overline {\mathbb {Q}}$ , where each coordinate $f_i$ is repeated $r+1$ times and a family

$$ \begin{align*}\Phi_{(\ell+1)}=(f_{\ell+1},\ldots,f_{\ell+1})\end{align*} $$

on $(\mathbb {P}^1)^{r}$ , parametrized by $\mathcal {X}$ , where each coordinate is repeated r times. We denote by $\overline {L}_{\Phi }$ and $\overline {L}_{\Phi _{(\ell +1)}}$ the $\Phi $ (respectively $\Phi _{(\ell +1)}$ )-invariant extension of the classical polarization as defined by [Reference Yuan and ZhangYZ, Theorem 6.1.1]. In particular, using the notation in [Reference Yuan and ZhangYZ], we have that $\overline {L}_{\Phi _{(\ell +1)}}\in \widehat {\mathrm {Pic}}(\mathcal {X}\times (\mathbb {P}^1)^r)_{\mathbb {Q},\mathrm {nef}}$ and $\overline {L}_{\Phi }\in \widehat {\mathrm {Pic}}(\mathcal {X}^{\ell }_V\times (\mathbb {P}^1)^{\dim \mathcal {X}^{\ell }_V})_{\mathbb {Q},\mathrm {nef}}$ .

In what follows it is convenient to work with the Deligne pairing (of relative dimension $0$ ), of the invariant metrized line bundles restricted to graphs of critical marked points (see [Reference Yuan and ZhangYZ, §4.2] for metrics of the Deligne pairing and [Reference Yuan and ZhangYZ, §6.2.2] for the “restriction” of the invariant metrized line bundle to a subvariety). By [Reference Yuan and ZhangYZ, Theorem 4.2.3], to each marked tuple of critical points $\mathbf {c}: \mathcal {X}^{\ell }_V\to (\mathbb {P}^1)^{\dim \mathcal {X}^{\ell }_V}$ we can associate a metrized line bundle $\langle \overline {L}_{\Phi }|_{\Gamma _{\mathbf {c}}}\rangle \in \widehat {\mathrm {Pic}}( \mathcal {X}^{\ell }_V)_{\mathbb {Q},\mathrm {nef}}$ , where $\Gamma _{\mathbf {c}}$ denotes the graph of $\mathbf {c}$ in $\mathcal {X}^{\ell }_V\times (\mathbb {P}^1)^{\dim \mathcal {X}^{\ell }_V}$ . Moreover, we let

$$ \begin{align*}\overline{L}:=\otimes _{\mathbf{c}} \langle\overline{L}_{\Phi}|_{\Gamma_{\mathbf{c}}}\rangle\in \widehat{\mathrm{Pic}}( \mathcal{X}^{\ell}_V)_{\mathbb{Q},\mathrm{nef}}\end{align*} $$

be the tensor product over all marked tuples of critical points $\mathbf {c}$ for the maps $f_1, \ldots , f_\ell $ .

Now let $\mathbf {c}=(c_{1,1},\ldots ,c_{\ell ,r+1}):\mathcal {X}^{\ell }_{V}\to (\mathbb {P}^1)^{(r+1)\ell }$ be a marked critical point for $\Phi $ that witnesses $\mu _{\ell } \not =0$ , as in Proposition 3.1. Reordering the critical points if needed, we can find $x_0,x_1\in \operatorname {supp}\mu _{\ell }$ as in Proposition 3.2, with $i=r+1$ and $j=1$ . We write $c_i:=c_{\ell ,i}:\mathcal {X}\to \mathbb {P}^1$ in what follows. As before, using [Reference Yuan and ZhangYZ, Theorem 4.2.3], we define two adelic metrized line bundles on $\mathcal {X}$ by

(4.1) $$ \begin{align} \begin{split} \overline{L}_1&:=\langle\overline{L}_{\Phi_{(\ell+1)}}|_{\Gamma_{(c_1,\ldots,c_r)}}\rangle\in \widehat{\mathrm{Pic}}( \mathcal{X})_{\mathbb{Q},\mathrm{nef}}\\ \overline{L}_2&:=\langle\overline{L}_{\Phi_{(\ell+1)}}|_{\Gamma_{(c_2,\ldots,c_{r+1})}}\rangle\in \widehat{\mathrm{Pic}}( \mathcal{X})_{\mathbb{Q},\mathrm{nef}}, \end{split} \end{align} $$

and note that their curvature forms (at an archimedean place) are the bifurcation currents

(4.2) $$ \begin{align} \begin{split} c_1(\overline{L}_1)&=T_{f_{\ell+1},c_1}+\cdots+T_{f_{\ell+1},c_r}\\ c_1(\overline{L}_2)&=T_{f_{\ell+1},c_2}+\cdots+T_{f_{\ell+1}, c_{r+1}}. \end{split} \end{align} $$

We can now define two metrized line bundles

$$ \begin{align*}\overline{M}_i=\Pi_1^{*}(\overline{L})+\Pi_2^{*}(\overline{L}_{i})\in \widehat{\mathrm{Pic}}( \mathcal{X}^{\ell+1}_V)_{\mathbb{Q},\mathrm{nef}}\end{align*} $$

on $\mathcal {X}^{\ell +1}_V$ for $i=1,2$ . By [Reference Yuan and ZhangYZ, Lemma 6.2.1, Theorem 4.1.3], the heights associated to $\overline {M}_i$ vanish at PCF points; see also [Reference Yuan and ZhangYZ, Lemma 6.3.2]. More precisely, for a point $(\lambda _0,t_1,\ldots ,t_{\ell +1})\in \mathcal {X}^{\ell +1}_V(\overline {\mathbb {Q}})$ we have

(4.3) $$ \begin{align} \begin{split} h_{\overline{M}_1}(\lambda_0,t_1,\ldots,t_{\ell+1})&= \sum_{i=1}^{\ell}h_{\mathrm{crit}}(f_{i}) + \sum_{j=1}^{r}\hat{h}_{f_{\lambda_0,t_{\ell+1}}}(c_{j}(\lambda_0,t_{\ell+1}))\\ h_{\overline{M}_2}(\lambda_0,t_1,\ldots,t_{\ell+1})&= \sum_{i=1}^{\ell}h_{\mathrm{crit}}(f_{i}) + \sum_{j=2}^{r+1}\hat{h}_{f_{\lambda_0,t_{\ell+1}}}(c_{j}(\lambda_0,t_{\ell+1})). \end{split} \end{align} $$

We will now show that $\overline {M}_i$ is “non-degenerate” for $i=1,2$ in the terminology of [Reference Yuan and ZhangYZ, §6.2.2]. Equivalently, by [Reference Yuan and ZhangYZ, Lemma 5.4], it suffices to show that

$$ \begin{align*}c_1(\overline{M_i})^{\wedge \ell(r+1)+r}\neq 0.\end{align*} $$

Recall that $T_{f_i,c_j}^{\wedge 2}=0$ for all $i,j$ and similarly $(\Pi ^*_2c_1(\overline {L}_i))^{\wedge r+1}= (\Pi _2)^*(c_1(\overline {L}_i)^{\wedge r+1})=0$ for $i=1,2$ , by the definition of the bifurcation current as limits of pullbacks, because the map $\Phi _{(\ell +1)}$ acts on the r-dimensional $(\mathbb {P}^1)^{r}$ . Moreover, since $\dim \mathcal {X}^{\ell }_V= \ell (r+1)$ we also have that $\Pi _1^{*}(c_1(\overline {L}))^{\wedge \ell (r+1)+1}=0$ . Thus, using also [Reference Yuan and ZhangYZ, Theorem 4.2.3] as before, we have

(4.4) $$ \begin{align} c_1(\langle\overline{L}\rangle)^{\wedge (r+1)\ell}=\alpha_1\,\mu_{\ell} \end{align} $$

on $\mathcal {X}_V^\ell $ , and

(4.5) $$ \begin{align} \begin{aligned} c_1(\overline{M_1})^{\wedge \ell(r+1)+r}&=\alpha_2\, \Pi_1^{*}(\mu_{\ell})\wedge \Pi_2^{*}(T_{f_{\ell+1},c_1}\wedge\cdots\wedge T_{f_{\ell+1},c_r})=\alpha_2\,\mu_{\ell}^{r+1}\neq 0,\\ c_1(\overline{M_2})^{\wedge \ell(r+1)+r}&=\alpha_3\,\Pi_1^{*}(\mu_{\ell})\wedge \Pi_2^{*}(T_{f_{\ell+1},c_2}\wedge\cdots\wedge T_{f_{\ell+1},c_{r+1}})=\alpha_3\,\mu^1_{\ell}\neq 0, \end{aligned} \end{align} $$

for positive constants $\alpha _1,\alpha _2,\alpha _3>0$ . Their non-vanishing follows by Proposition 3.2 (and our choice of critical points). Therefore, by the height inequality [Reference Yuan and ZhangYZ, Theorem 1.3.2] we infer that there are positive constants $\epsilon _1=\epsilon _1(\mathcal {X})>0,~\epsilon _2=\epsilon _2(\mathcal {X})>0$ such that

(4.6) $$ \begin{align} \sum_{i=1}^{\ell+1}\hat{h}_{\mathrm{crit}}(f_i)\ge \epsilon_1 h_{\mathrm{Ch}}(\lambda)-\epsilon_2, \end{align} $$

for all $(\lambda ,f_1,\ldots ,f_{\ell +1})\in U(\overline {\mathbb {Q}})$ for a Zariski open and dense subset $U\subset \mathcal {X}^{\ell +1}_V$ .

It remains to show that for some $\epsilon =\epsilon (\mathcal {X})$ , the set

$$ \begin{align*}\left\{(\lambda,f_1,\ldots,f_{\ell+1})\in \mathcal{X}_V^{\ell+1}(\overline{\mathbb{Q}})~:~ \sum_{i=1}^{\ell+1}\hat{h}_{\mathrm{crit}}(f_i)<\epsilon\right\}\end{align*} $$

is not Zariski dense in $\mathcal {X}_V^{\ell +1}$ . We prove this result by contradiction. Suppose that there is a generic Galois invariant sequence $\{x_n\}_n$ of critically small points in $\mathcal {X}_V^{\ell +1}$ , so that $h_{\overline {M}_i}(x_n)\to 0$ for $i=1,2$ . Since $\overline {M}_i$ is nef and non-degenerate, Yuan-Zhang’s fundamental inequality [Reference Yuan and ZhangYZ, Theorem 5.3.3] implies that $h_{\overline {M}_i}(\mathcal {X}_V^{\ell +1})=0$ for $i=1,2$ . We can use [Reference Yuan and ZhangYZ, Theorem 5.4.3] with the Galois-invariant critically small set to get that

$$ \begin{align*}\Pi_1^{*}(\mu_{\ell})\wedge \Pi_2^{*}(T_{f_{\ell+1},c_1}\wedge\cdots\wedge T_{f_{\ell+1},c_r})= \alpha \, \Pi_1^{*}(\mu_{\ell})\wedge \Pi_2^{*}(T_{f_{\ell+1},c_2}\wedge\cdots\wedge T_{f_{\ell+1},c_{r+1}})\neq 0,\end{align*} $$

on $\mathcal {X}_V^{\ell +1}$ for some constant $\alpha>0$ .

We now slice these measures with respect to the projection $\mathcal {X}_V^{\ell +1} \to \mathcal {X}_V^\ell $ ; see, for example [Reference Bassanelli and BertelootBB, Proposition 4.3]. This implies that

(4.7) $$ \begin{align} \int_{\mathcal{X}^{\ell}_V}\left( \int_{X_{\pi(x)}}\phi \cdot \bigwedge_{i=1}^r T_{f_{\ell+1},c_i}|_{X_{\pi(x)}}\right) d\mu_{\ell}(x) = \alpha \int_{\mathcal{X}^{\ell}_V}\left( \int_{X_{\pi(x)}}\phi \cdot \bigwedge_{i=2}^{r+1} T_{f_{\ell+1},c_i}|_{X_{\pi(x)}}\right) d\mu_{\ell}(x) \end{align} $$

for every continuous and compactly supported function $\phi $ on $\mathcal {X}^{\ell +1}_V$ . Here, $\pi : \mathcal {X}^{\ell }_V\to V$ denotes the projection to the base. (Strictly speaking, since the measure $\mu _\ell $ is not smooth, we should approximate $\mu _\ell $ by smooth and compactly-supported forms on $\mathcal {X}_V^\ell $ to deduce (4.7); compare the proof of [Reference Myrto Mavraki and SchmidtMaS, Theorem 1.8].)

For each $\lambda \in V$ , set

$$ \begin{align*}\mu^{r+1}_{\lambda}:=T_{f_{\ell+1},c_1}\wedge\cdots\wedge T_{f_{\ell+1},c_r}|_{X_{\lambda}}\end{align*} $$

and

$$ \begin{align*}\mu_\lambda^1=T_{f_{\ell+1},c_2}\wedge\cdots\wedge T_{f_{\ell+1},c_{r+1}}|_{X_{\lambda}}.\end{align*} $$

We infer from (4.7) that

(4.8) $$ \begin{align} \mu^{r+1}_{\lambda}=\alpha \, \mu^{1}_{\lambda} \end{align} $$

as measures on $X_{\lambda }$ , for some constant $\alpha>0$ and for all $\lambda $ in the support of $\pi _*\mu _\ell $ in V. Indeed, suppose there exists $b \in V(\mathbb {C})$ where (4.8) doesn’t hold, and so that $\pi _{*}\mu _{\ell }(U)>0$ for every open neighborhood U of b. Then we can find a continuous, non-negative, compactly supported function $\psi $ on $\mathcal {X}$ such that

$$ \begin{align*}\int_{X_b} \psi \, d\mu^{r+1}_{b}(x) \neq \alpha \int_{X_b} \psi \, d\mu^{1}_{b}(x).\end{align*} $$

By weak continuity of the measures $\mu ^{r+1}_{\lambda }$ and $\mu _\lambda ^1$ as $\lambda $ varies (since the bifurcation currents have continuous potentials), we find that

$$ \begin{align*}\int_{X_{\lambda}} \psi \, d\mu^{r+1}_{\lambda}(x) \neq \alpha\int_{X_{\lambda}} \psi \, d\mu^{1}_{\lambda}(x)\end{align*} $$

for all $\lambda $ in a small neighborhood U of b. Therefore, setting

$$ \begin{align*}\phi(\lambda, t_1, \ldots, t_{\ell},t_{\ell+1}) = h_b(\lambda) \psi(\lambda,t_{\ell+1})\end{align*} $$

on $\mathcal {X}^{\ell +1}_V$ for a non-negative function $h_b$ supported in a small neighborhood of b, the equality (4.7) will fail.

The remainder of the proof is devoted to deriving a contradiction from the equality (4.8). We first sketch the idea. Recall from Proposition 3.2 that we have parameter $x_1 = (\lambda _1, s_1, \ldots , s_\ell ) \in \operatorname {supp} \mu _\ell $ so that the tuple of marked critical points $(c_1, \ldots , c_r)$ is transversely prerepelling for $F^{(\lambda _1)} = (f,\ldots , f)$ at $s_\ell $ over $X_{\lambda _1}$ . This implies that $\mu _{\lambda _1}^{r+1}$ is nonzero on $X_{\lambda _1}$ , with $s_\ell $ in its support by [Reference Buff and EpsteinBE]. Moreover, as in [Reference Buff and EpsteinBE, Section 3] and the generalization appearing in [Reference GauthierGa2, Proposition 3.2], we know that if we zoom in on the measure $\mu ^{r+1}_{\lambda _1}$ on $X_{\lambda _1}$ at the parameter $s_\ell $ at an exponential rate in n (determined by the multipliers of the repelling cycles in the forward orbits of the marked critical points), and rescale the measure appropriately we get a nontrivial limit; see (4.11). On the other hand, we know from Proposition 3.2 that the critical point $c_{r+1}$ is attracted to a parabolic cycle at $s_{\ell }\in X_{\lambda _1}$ . Zooming into the measure $\mu _{\lambda _1}^1$ on $X_{\lambda _1}$ at $s_\ell $ at the same exponential rate (which is supposed to give a nonzero limit), we actually get 0 from the following lemma. This will produce our desired contradiction.

Lemma 4.1. Suppose that $f: \mathbb {D}^r \times \mathbb {P}^1\to \mathbb {D}^r \times \mathbb {P}^1$ is a holomorphic family of maps with degree $d\ge 2$ parametrized by $\mathbb {D}^r \subset \mathbb {C}^r$ with a marked critical point $c: \mathbb {D}^r \to \mathbb {P}^1$ . Assume that $c(0)$ is attracted to a parabolic fixed point at $z=0$ for $f_0$ . Then for any constant $\rho> 1$ , there exists $\delta>0$ so that the sequence of functions

$$ \begin{align*}\alpha_n(x) := f_{x/\rho^n}^n(c(x/\rho^n))\end{align*} $$

converges uniformly to 0 on the polydisk $(\mathbb {D}_\delta )^r$ , as $n\to \infty $ .

Proof. Let $\rho>1$ . Since $|f^{\prime }_0(0)| = 1$ , we can first fix a small $\varepsilon>0$ and $\rho _\varepsilon>1$ so that

$$ \begin{align*}\left|f_x'(z)\right| \leq \rho_\varepsilon < \rho,\end{align*} $$

for all $x\in \mathbb {D}^r_{\epsilon }$ and $z\in \mathbb {D}_\varepsilon $ . We replace c with an iterate so that $c(0)$ and all its forward iterates under $f_0$ are in the disk $\mathbb {D}_{\varepsilon /2}$ . Choose $M>0$ so that the gradient of f (with respect to parameter x) satisfies

$$ \begin{align*}\left|\nabla_x f_x(z) \right| \leq M\end{align*} $$

for all $x\in \mathbb {D}^r$ and $z\in \mathbb {D}_\varepsilon $ . Note that

$$ \begin{align*}\alpha_n(x)=f_{x/\rho^n}(\alpha_{n-1}(x/\rho)).\end{align*} $$

Taking derivatives, we have

(4.9) $$ \begin{align} |\nabla \alpha_n(x)|&=\left|\frac{1}{\rho^n}\cdot (\nabla_w f_w) (\alpha_{n-1}(x/\rho))+\frac{1}{\rho} f_{x/\rho^n}'(\alpha_{n-1}(x/\rho)) \nabla_{x/\rho} \alpha_{n-1}(x/\rho) \right| \nonumber \\ &\leq \frac{M}{\rho^n} +\frac{\rho_\varepsilon}{\rho}\, |\nabla_{x/\rho}\alpha_{n-1}(x/\rho)| \end{align} $$

as long as $\alpha _{n-1}(x/\rho )$ lies in $\mathbb {D}_\varepsilon $ .

Now choose $n_0$ so that

$$ \begin{align*}\frac{M}{\rho^{n_0}}<\frac{\rho_\varepsilon}{\rho}.\end{align*} $$

We take a very small $\delta _1$ so that

$$ \begin{align*}\alpha_{n_0-1}(x)\in \mathbb{D}_{\varepsilon}\end{align*} $$

for all $x \in \mathbb {D}_{\delta _1}^r$ , and set

$$ \begin{align*}M_1:=\max_{x\in \mathbb{D}_{\delta_1}^r}|\nabla\alpha_{n_0-1}(x)|.\end{align*} $$

From the inequality (4.9), we have

$$ \begin{align*}|\nabla\alpha_{n_0}(x)| \;\leq \; \frac{M}{\rho^{n_0}} + \frac{\rho_\varepsilon}{\rho}\, |\nabla_{x/\rho}\alpha_{n_0-1}(x/\rho)| \;\leq \; \frac{\rho_\varepsilon}{\rho}+ \frac{\rho_\varepsilon}{\rho} M_1\end{align*} $$

for each $x\in \mathbb {D}_{\delta _1}^r$ . Let

$$ \begin{align*}M_2:= \frac{\rho_\varepsilon}{\rho}+ \frac{\rho_\varepsilon}{\rho} M_1,\end{align*} $$

and define $M_i$ recursively by

$$ \begin{align*}M_{i+1}=\frac{\rho_\varepsilon}{\rho^i} + \frac{\rho_\varepsilon}{\rho}M_i\end{align*} $$

for $i\geq 2$ . Note that $M_i \to 0$ as $i\to \infty $ , because

$$ \begin{align*}M_i = \left(\frac{\rho_\varepsilon}{\rho}\right)^{i-1}(M_1+1) + \frac{1}{\rho^i} \sum_{j=1}^{i-2} \rho_\varepsilon^j.\end{align*} $$

Finally, we can show inductively that our derivative estimate (4.9) holds when applied to $\alpha _n$ for all $n> n_0$ on a sufficiently small disk near 0. This will imply that

$$ \begin{align*}|\nabla\alpha_{n_0+i}(x)|<\frac{M}{\rho^{n_0+i}}+\frac{\rho_\varepsilon}{\rho}\, |\nabla\alpha_{n_0+i-1}(x/\rho)| \leq \frac{\rho_\varepsilon}{\rho^{i+1}}+\frac{\rho_\varepsilon}{\rho}M_{i+1} = M_{i+2}\end{align*} $$

for all x in a small disk around 0. We know it holds for $i=0$ for all $|x| < \delta _1$ . We take $M_0:=\max _{i\geq 2} M_i$ , and set

$$ \begin{align*}\delta:=\min (\delta_1, \varepsilon/(2M_0), \varepsilon).\end{align*} $$

As $\alpha _{n_0}(0)\in \mathbb {D}_{\varepsilon /2}$ , the inequality (4.9) gives $|\nabla \alpha _{n_0}(x)|<M_2\leq M_0$ , so that $\alpha _{n_0}(x)$ is in $\mathbb {D}_{\varepsilon }$ for each $x\in \mathbb {D}^r_{\delta }$ . By induction, we see that if $\alpha _{n_0+i-1}(x)$ is in $\mathbb {D}_{\varepsilon }$ with $|\nabla \alpha _{n_0+i-1}|\leq M_{i+1}$ , then $\alpha _{n_0+i}(x)$ is in $\mathbb {D}_{\varepsilon }$ , because $\alpha _{n_0+i}(0)$ is in $\mathbb {D}_{\varepsilon /2}$ and $\delta <\varepsilon /(2M_0)$ . Since $M_i\to 0$ as i tends to infinity, we see that $\alpha _n$ converges to $0$ uniformly on $\mathbb {D}^r_\delta $ .

We now return to our proof. Abusing the notation slightly, we write $\{f_s\}_{s\in X_{\lambda _1}}$ to denote the restriction of the family of maps to $X_{\lambda _1}$ , $c_{r+1}$ for the restriction of $c_{r+1}$ to $X_{\lambda _1}$ , and $T_{c_{r+1}}$ its associated bifurcation current on $X_{\lambda _1}$ . We know from Lemma 4.1 that, zooming into $s_\ell $ in $X_{\lambda _1}$ at any exponential rate, the graphs of iterates of $c_{r+1}$ are converging to the graph of a constant function.

Let $p_0\in \mathbb {P}^1$ be the parabolic periodic point of $f_{s_\ell }$ ; replacing $f_{s_\ell }$ with an iterate, we have $f_{s_\ell }^n(c_{r+1}(s_\ell ))\to p_0$ as $n\to \infty $ . After a change of coordinates if necessary, we may assume that $c_{r+1}(s_\ell )$ and $p_0$ are not $\infty $ . Recalling the definition of the bifurcation current associated to $c_{r+1}$ , we can work in homogeneous coordinates on $\mathbb {P}^1$ over a neighborhood of $s_\ell $ in $X_{\lambda _1}$ . We define

$$ \begin{align*}G_s(X,Y) := \lim_{i\to \infty}\frac{1}{d^i}\log\|F_{s}^{i}(X,Y)\|,\end{align*} $$

with a choice of homogeneous lift $F_{s}$ for $f_{s}$ normalized so that $G_{s_\ell }(p_0,1)= 0$ . Note that $G_s(c_{r+1}(s), 1)$ is a potential of the bifurcation current $T_{c_{r+1}}$ on a neighborhood of $s_\ell \in X_{\lambda _1}$ . As long as $f^n_s(c_{r+1}(s))$ is not $\infty $ , which is always true when s is in $ \mathbb {D}(s_\ell , \delta /\rho ^n)$ for small $\delta>0$ , any $\rho>1$ and big n by Lemma 4.1, we define

$$ \begin{align*}U_n(s):=G_s(f^n_s(c_{r+1}(s)),1)\end{align*} $$

so that

$$ \begin{align*}dd^c U_n(s)=d^n\cdot dd^c G_s(c_{r+1}(s), 1) = d^n \, T_{c_{r+1}}\end{align*} $$

where defined, as $G_s(F^n_s(c_{r+1}(s),1)) = d^n\cdot G_s(c_{r+1}(s),1)$ differs from $U_n(s)$ by a harmonic function.

Let $r_0$ be any biholomorphic map from $\mathbb {D}^{r}$ to a neighborhood of $s_\ell \in X_{\lambda _1}$ , with $r_0(0)=s_\ell $ . From Lemma 4.1 and the choice of our lift, we deduce that, for any $\rho>1$ , the functions $U_n\circ r_0(y/\rho ^n)$ converge uniformly to $G_{s_{\ell }}(p_0,1)=0$ on a small neighborhood of $(0, \ldots , 0)\in \mathbb {D}^{r}$ . As a consequence, for any $\rho _i\in \mathbb {C}$ with $|\rho _i|>1$ , we can pick some $\rho>1$ with $\rho < \min _{1\leq i\leq {r}} (|\rho _i|)$ so that the sequence of functions

(4.10) $$ \begin{align} U_n\circ r_0(y_1/\rho_1^n, \ldots, y_{r}/\rho_{r}^n)=U_n\circ r_0(A_n(y)/\rho^n) \to 0 \end{align} $$

uniformly, as $A_n(y):=(y_1\cdot \rho ^n/\rho _1^n, \ldots , y_{r}\cdot \rho ^n/\rho _{r}^n)$ converges to zero uniformly for $n\to \infty $ .

Following the notation of [Reference GauthierGa2, Proposition 3.2], we now let $r_n: \mathbb {D}^r \to X_{\lambda _1}$ be the sequence of maps that define the zooms near the parameter $s_\ell $ for the measure $\mu ^{r+1}_{\lambda _1}$ corresponding to the transversely prerepelling critical point at $x_1$ , as described before Lemma 4.1. From [Reference GauthierGa2, Proposition 3.2] we know that there is a measure $\mu \neq 0$ (which can be described explicitly) such that

(4.11) $$ \begin{align} d^{nr}\,(r_n)^{*}(\mu^{r+1}_{\lambda_1})\to \mu, \end{align} $$

uniformly as $n\to \infty $ . On the other hand, it follows from (4.10) that the functions $U_n\circ r_n$ also converge uniformly to 0. We conclude that $d^n \, (r_n)^{*} T_{c_{r+1}} \to 0$ weakly, and therefore

$$ \begin{align*}d^{nr} \, (r_n)^{*} \mu^1_{\lambda_1} \to 0.\end{align*} $$

Therefore, $\mu ^1_{\lambda _1}$ cannot be proportional to $\mu ^{r+1}_{\lambda _1}$ , contradicting (4.8). This completes the proof of Theorem 1.8.

5 Proofs of Theorems 1.1, 1.2, 1.5, and 1.6

An elementary, but important, geometric observation is the following [Reference Gao, Ge and KühneGGK, Lemma 4.3]: If $\Sigma $ is a subset of a complex projective algebraic variety X, equipped with an ample line bundle L, and if the product $\Sigma ^M$ lies in a proper subvariety $Y \subset X^M$ for some integer $M\geq 1$ , then $\Sigma $ must be contained in a proper subvariety Z of X, where the number of components of Z and the degree of each are bounded by constants depending only on M, $\dim X$ , the degree of X, and the sum of the degrees of the components of Y. Here, we measure degree in $X^M$ with respect to the ample line bundle $p_1^*L \otimes \cdots \otimes p_M^*L$ on $X^M$ , where $p_i: X^M \to X$ is the projection to the i-th factor.

Fix degree $d\geq 2$ . For positive integers r and D, we will apply Theorem 1.8 to families $\mathcal {X} \to V$ over a subvariety $V \subset \mathrm {Ch}(\mathrm {M}_d, r,D)$ , all defined over $\overline {\mathbb {Q}}$ . We work inductively on the dimension r of the fibers in the family.

First suppose that $\mathcal {C} \to V$ is a family of curves in $\mathrm {M}_d$ of degree $\leq D$ , with irreducible $V \subset \mathrm {Ch}(\mathrm {M}_d, 1, D)$ defined over $\overline {\mathbb {Q}}$ and of dimension $\ell \geq 1$ , and assume that the generic fiber of $\mathcal {C} \to V$ is geometrically irreducible. From Theorem 1.8 applied to the fiber power $\mathcal {C}_V^{\ell +1}$ , we know that there exists an $\varepsilon = \varepsilon (\mathcal {C})$ so that points $(\lambda , f_1, \ldots , f_{\ell +1}) \in \mathcal {C}_V^{\ell +1}(\overline {\mathbb {Q}})$ with $\sum _i \hat {h}_{\mathrm {crit}}(f_i) < \varepsilon \max \{1,h_{\mathrm {Ch}}(\lambda )\}$ are not Zariski dense. We let $Z_\varepsilon $ denote the Zariski closure of this set in $\mathcal {C}_V^{\ell +1}$ , over $\overline {\mathbb {Q}}$ . It follows that there is a Zariski open $U \subset V$ (defined over $\overline {\mathbb {Q}}$ ) so that $Z_\varepsilon \cap C_\lambda ^{\ell +1}$ is a proper algebraic subvariety of $(C_\lambda )^{\ell +1}$ for all $\lambda \in U(\mathbb {C})$ , with its number of irreducible components bounded over $\lambda \in U(\mathbb {C})$ . It is worth noting that all points $(\lambda , f_1, \ldots , f_{\ell +1}) \in \mathcal {C}_V^{\ell +1}(\mathbb {C})$ where each $f_i$ is in $\mathrm {M}_d(\overline {\mathbb {Q}})$ with $\sum _i \hat {h}_{\mathrm {crit}}(f_i) < \varepsilon $ must lie in $Z_\varepsilon $ , also for $\lambda \in V(\mathbb {C}) \setminus V(\overline {\mathbb {Q}})$ ; indeed, any such $(\ell +1)$ -tuple must lie in a family of curves in V defined over $\overline {\mathbb {Q}}$ that includes this $C_\lambda $ .

We now apply [Reference Gao, Ge and KühneGGK, Lemma 4.3] to the set

$$ \begin{align*}\Sigma_{\lambda,\varepsilon} := \left\{f \in C_\lambda \cap \mathrm{M}_d(\overline{\mathbb{Q}}): \hat{h}_{\mathrm{crit}}(f) <\frac {\varepsilon}{\ell+1}\max\{1,h_{\mathrm{Ch}}(\lambda)\}\right\} \subset C_\lambda\end{align*} $$

for $\lambda \in U(\overline {\mathbb {Q}})$ , and to

$$ \begin{align*}\Sigma_{\lambda,\varepsilon} := \left\{f \in C_\lambda \cap \mathrm{M}_d(\overline{\mathbb{Q}}): \hat{h}_{\mathrm{crit}}(f) <\frac {\varepsilon}{\ell+1}\right\} \subset C_\lambda\end{align*} $$

for $\lambda \in U(\mathbb {C})\setminus U(\overline {\mathbb {Q}})$ , so that $\Sigma _{\lambda ,\varepsilon }^{\ell +1}\subset Z_{\varepsilon }\cap C^{\ell +1}_{\lambda }$ for all $\lambda \in U(\mathbb {C})$ . We deduce the existence of a constant N so that $|\Sigma _{\lambda ,\varepsilon }| \leq N$ for all $\lambda \in U(\mathbb {C})$ . Note here that we have used the fact that the number of components of $Z_{\varepsilon }\cap C^{\ell +1}_{\lambda }$ and the degree of each is uniformly bounded for $\lambda \in U(\mathbb {C})$ (controlled by the geometry of its generic fiber). Note also that if $0\le \varepsilon '<\varepsilon $ , then $\Sigma _{\lambda ,\varepsilon '}\subset \Sigma _{\lambda ,\varepsilon }$ and hence $|\Sigma _{\lambda ,\varepsilon '}|\leq N$ for each $\lambda \in U(\mathbb {C})$ .

We then repeat the argument with the families of curves over the positive-dimensional components in the complement of $V\setminus U$ ; we work with irreducible components so that the generic fibers of these families will be (geometrically) irreducible. Note that each family we treat is defined over $\overline {\mathbb {Q}}$ . We increase the bound N and decrease $\epsilon $ as needed. We are left with a finite set S in $V(\overline {\mathbb {Q}})$ containing all of the irreducible PCF-special curves with degree $\leq D$ as well as all curves in V with infinitely many points of critical height $< \frac {\varepsilon }{\ell +1}\max \{1,h_{\mathrm {Ch}}(\lambda )\}$ (or $< \frac {\varepsilon }{\ell +1}$ over complex parameters). This proves Theorems 1.2 and 1.6 for the curves in V. If for any of the curves $C_\lambda $ in this leftover finite set $S \subset V(\overline {\mathbb {Q}})$ , we have $N<|\Sigma _{\lambda ,\varepsilon }|<\infty $ , we just increase the bound N as needed so that Theorem 1.5 and 1.2 hold for all curves in V. We repeat the argument for all irreducible components V of $\mathrm {Ch}(\mathrm {M}_d, 1, D)$ with geometrically irreducible generic fiber and adjust the constants $\varepsilon ,~N$ as needed. This proves Theorems 1.1, 1.2, 1.5 and 1.6 for curves.

Now fix $r>1$ and suppose we know the conclusion of Theorems 1.1, 1.2, 1.5, and 1.6 for subvarieties in $\mathrm {M}_d$ of dimension $<r$ . Consider any family $\mathcal {X} \to V$ of r-dimensional varieties in $\mathrm {M}_d$ over an irreducible and quasiprojective $V \subset \mathrm {Ch}(\mathrm {M}_d, r, D)$ defined over $\overline {\mathbb {Q}}$ and of dimension $\ell \geq 1$ for which the generic fiber of $\mathcal {X} \to V$ is geometrically irreducible.

As in the case of curves, we infer from Theorem 1.8 that there exists $\varepsilon = \varepsilon (\mathcal {X})>0$ so that all points $(\lambda , f_1, \ldots , f_{\ell +1}) \in \mathcal {X}_V^{\ell +1}(\overline {\mathbb {Q}})$ where each $f_i$ is in $\mathrm {M}_d(\overline {\mathbb {Q}})$ with $\sum _i \hat {h}_{\mathrm {crit}}(f_i) < \varepsilon \max \{1,h_{\mathrm {Ch}}(\lambda )\}$ lie in a proper closed subvariety $Z_\varepsilon $ in $\mathcal {X}_V^{\ell +1}$ . Therefore, as before, from [Reference Gao, Ge and KühneGGK, Lemma 4.3], there is a Zariski-open subset $U\subset V$ (defined over $\overline {\mathbb {Q}}$ ) and constants $N_1$ and $B_1$ so that the points of

$$ \begin{align*}\Sigma^r_{\lambda,\varepsilon} := \left\{f \in X_\lambda \cap \mathrm{M}_d(\overline{\mathbb{Q}}): \hat{h}_{\mathrm{crit}}(f) < \frac{\varepsilon}{\ell+1}\max\{1,h_{\mathrm{Ch}}(f)\}\right\} \subset X_\lambda\end{align*} $$

for $\lambda \in U(\overline {\mathbb {Q}})$ , and of

$$ \begin{align*}\Sigma^r_{\lambda,\varepsilon} := \left\{f \in X_\lambda \cap \mathrm{M}_d(\overline{\mathbb{Q}}): \hat{h}_{\mathrm{crit}}(f) < \frac{\varepsilon}{\ell+1}\right\} \subset X_\lambda\end{align*} $$

for $\lambda \in U(\mathbb {C})\setminus U(\overline {\mathbb {Q}})$ , lie in a proper algebraic subvariety $Z_{(\lambda )} \subset X_\lambda $ which has $\leq N_1$ irreducible components, each with degree $\leq B_1$ , for all $\lambda \in U(\mathbb {C})$ . Note that $\Sigma ^r_{\lambda ,\epsilon }$ is not necessarily Zariski dense in $Z_{(\lambda )}$ , so we apply the induction hypotheses (specifically, the constants from each of the statements of Theorems 1.1, 1.2, 1.5, 1.6 for subvarieties of dimension $< r$ ) to each irreducible component $Y_{(\lambda )}$ of $Z_{(\lambda )}$ , for all $\lambda \in U(\mathbb {C})$ . Thus, by reducing $\varepsilon $ if needed, we conclude that the Zariski-closure of $\Sigma ^r_{\lambda ,\varepsilon }\cap Y_{(\lambda )}$ has uniformly bounded number of components, each with uniformly bounded degree for all $\lambda \in U(\mathbb {C})$ . Further, adjusting these uniform bounds if needed, using the induction hypothesis, we can arrange that they also bound the geometry of the Zariski closure of $\Sigma ^r_{\lambda , \varepsilon '}\cap Y_{(\lambda )}\subset \Sigma ^r_{\lambda , \varepsilon }\cap Y_{(\lambda )}$ for all $0\le \varepsilon '<\varepsilon $ (and in particular of $Y_{(\lambda )} \cap \mathrm {PCF}_d$ ). Combined with the initial bound of $N_1$ on the number of components of $Z_{(\lambda )}$ , repeating this procedure provides uniform bounds on the Zariski closures of $\Sigma ^r_{\lambda ,\varepsilon '}$ for all $\varepsilon '<\varepsilon $ and of $\overline {X_\lambda \cap \mathrm {PCF}_d}$ for all $\lambda \in U(\mathbb {C})$ .

We now repeat the argument over each of the irreducible components of the complement $V \setminus U$ corresponding to families with geometrically irreducible generic fibers, and we note that each family we treat is defined over $\overline {\mathbb {Q}}$ . Finally, we are left with a finite set S in $V(\overline {\mathbb {Q}})$ consisting of r-dimensional varieties that are either PCF-special or contain a Zariski-dense set of points in $\Sigma ^r_{\lambda , \varepsilon }$ . We repeat the arguments above for curves, reducing $\varepsilon $ and increasing B and N as needed, once again appealing to the induction hypotheses. We have thus completed the proofs of Theorems 1.1, 1.2, 1.5, and 1.6.