vol\,}}}(\alpha^{\prime}+t\gamma^{\prime})\geq\beta^{n}+nt\beta^{n-1}\cdot\gamma^{\prime}-8^{n}\omega^{n}t^{2}$.
Taking the supremum over all $X^{\prime}$ and all nef
$\beta\leq\alpha^{\prime}$, we find, using Theorem 9.10 and the definition of
$\psi$:

${\mathop{\operatorname{\rm
vol\,}}}(\alpha+t\gamma)\geq{\mathop{\operatorname{\rm
vol\,}}}(\alpha)+nt\psi(\alpha)\cdot\gamma-8^{n}\omega^{n}t^{2}$

Applying this to $\alpha^{*}=\alpha+t\gamma$ ($t$ small enough so that this is
still big), $\gamma^{*}=-\gamma$, so $\alpha=\alpha^{*}+t\gamma^{*}$,

${\mathop{\operatorname{\rm vol\,}}}(\alpha)\geq{\mathop{\operatorname{\rm
vol\,}}}(\alpha+t\gamma)-nt\psi(\alpha^{*})\cdot\gamma-8^{n}\omega^{n}t^{2}$

As $\psi$ is continuous at $\alpha$, this implies that
$\frac{{\mathop{\operatorname{\rm
vol\,}}}(\alpha+t\gamma)-{\mathop{\operatorname{\rm vol\,}}}(\alpha)}{t}$ is
bounded on both sides by functions that approach $n\psi(\alpha)$ as $t\to 0$.
∎

Let $N_{1}^{+}(X)$ be the closed dual cone to $N^{1}_{+}(X)$; thus $c\in
N_{1}^{+}(X)$ iff for every effective $e\in N^{1}(X)$ we have $c\cdot e\geq
0$.

######  Definition 10.8.

A class $c\in N_{1}(X)$ is called movable if for any hypersurface $H\subset
X$, there exists an irreducible curve $C$ not lying on $H$, and some
$\alpha>0$ with $c=\alpha[C]$. Call a class in $N_{1}(X)$ ‘strongly movable’
if it is a pushforward, of the intersection product of $n-1$ ample rational
classes in some blowup of $X$; if these are the same ample class, call it
‘very strongly movable’. In either case, note that such a class is a positive
scalar multiple of the class of a curve, a complete intersection of very ample
divisors.

######  Theorem 10.9.

Let $\dim(X)=n>1$. Then

$N_{1}^{+}(X)^{o}\subseteq\psi(N^{1}_{+}(X)^{o})\subseteq N_{1}^{+}(X)$

###### Proof.

By definition of $\psi$ as a positive intersection product, any element of
$\psi(N^{1}_{+}(X)^{o})$ is the supremum of a family of very strongly movable
classes, and in particular the limit of a sequence of such classes. Now if $a$
is ample in $X^{\prime}$ and $f:X^{\prime}\to X$ is a birational morphism,
then for any effective $e$ on $X$ we have $f_{*}(a^{n-1})\cdot e=a^{n-1}\cdot
f^{*}(e)\geq 0$. Thus any very strongly movable class lies in $N_{1}^{+}(X)$
and so $\psi(N^{1}_{+}(X)^{o})\subseteq N_{1}^{+}(X)$.

To show the inclusion $N_{1}^{+}(X)^{o}\subseteq\psi(N^{1}_{+}(X)^{o})$ we
verify the conditions of Theorem 5.2, with $U=N^{1}_{+}(X)^{o}$ the big cone,
$\phi(x)={\mathop{\operatorname{\rm vol\,}}}(x)^{1/n}$,
$F={\mathop{\operatorname{\rm vol\,}}}^{-\frac{n-1}{n}}\psi$,
$C=N_{1}^{+}(X)$. The volume function is continuous and non-negative and hence
so is $\phi$. The concavity (1) follows from Corollary 8.8 (or from the nef
case along with Theorem 9.10, by continuity).

The positive homogeneity (2) is clear, and (3) comes from the definition of
the big cone.

Note that ${\mathbb{R}}^{\geq 1}F(U)\subseteq{\mathbb{R}}^{>0}\psi(U)=\psi(U)$
(this last equality, by the $n-1$-positive homogeneity of $\psi$). By
Corollary 5.3, $N_{1}^{+}(X)^{o}\subseteq\psi(U)$.

∎

######  Corollary 10.10.

[2] The dual of the pseudo-effective cone, $N_{1}^{+}(X)$, is the closed
convex cone generated by the (very) strongly movable classes, or by all
movable classes.

For our purposes it is important to know that the image of $\psi$ is dense,
without taking convex closure.

######  Corollary 10.11.

Let $c\in N_{1}^{+}(X)$. Then $c$ can be approximated by (very strongly)
movable classes.

Indeed $[C]$ can be taken to have the form $f_{*}(D_{1}\cap\cdots\cap
D_{n-1})$ where $D_{i}$ are Bertini / Weil representatives of an ample class
on $X^{\prime}$, $f:X^{\prime}\to X$ birational.

######  Question 10.12.

Is any ${\mathbb{Q}}$-Cartier divisor in $N_{1}^{+}(X)^{o}$ movable?

######  Remarks 10.13.

  1. (1)

${\psi}$ is injective on $Nef^{1}(X)\cap N^{1}_{+}(X)^{o}$.

  2. (2)

If $x,y\in Nef^{1}(X)\cap N^{1}_{+}(X)^{o}$ and ${\psi}(x),{\psi}(y)$ are
proportional then so are $x,y$.

  3. (3)

${\psi}$ is a homeomorphism on the ample cone, and continuous on the nef cone.

###### Proof.

(1,2) ${\mathop{\operatorname{\rm vol\,}}}^{1/n}$ is strictly concave on
$Nef^{1}(X)\cap N^{1}_{+}(X)^{o}$ by [3] Corollary E. Hence
$\log{\mathop{\operatorname{\rm vol\,}}}$ is also strictly concave there. By
Lemma 5.1, the differential $-n{\mathop{\operatorname{\rm vol\,}}}^{-1}{\psi}$
is therefore strictly increasing, and hence injective.

If ${\psi}(x),{\psi}(y)$ are proportional, say ${\psi}(x)=c{\psi}(y)$. Note
$c>0$ since $N_{1}^{+}(X)^{o}$ is a strict cone. Let $x^{\prime}=\lambda x$
with $\lambda=c{\mathop{\operatorname{\rm
vol\,}}}(y)/{\mathop{\operatorname{\rm vol\,}}}(x)$. Then
${\mathop{\operatorname{\rm
vol\,}}}(x^{\prime})^{-1}{\psi}(x^{\prime})=\lambda^{-1}{\mathop{\operatorname{\rm
vol\,}}}(x)^{-1}{\psi}(x)={\mathop{\operatorname{\rm
vol\,}}}(y)^{-1}{\psi}(y)$. By injectivity of ${\mathop{\operatorname{\rm
vol\,}}}^{-1}{\psi}$ we have $y=\lambda x$. In case $c=1$, by homogeneity of
${\psi}$ we have $\lambda^{n-1}=1$ so $\lambda=1$, and $x=y$.

(3) In fact $\psi$ is polynomial on the nef cone, and injective on the ample
cone (by (1)).

∎

##  11\. Quantifier-free types over a one-dimensional base

We will work over a constant field $k=k^{alg}$, and describe the quantifier-
free GVF types over $k$ geometrically. We will see later how to deduce a
description of types over the non-constant GVF $k(t)^{alg}$.

When $X=SpecA$ is affine, the Berkovich space of $X$ over $k$ can be defined
as the set of all ${\mathbb{R}}$-valued semi-valuations of $A$; these are just
the ${\mathbb{R}}$-valued valuations of the fraction field of $A/P$, for any
prime ideal $P$ of $A$. The topology on ${\widehat{X}}$ is induced from the
Tychonoff topology on the functions from $K$to ${\mathbb{R}}\cup\\{\infty\\}$.
For general $X$, the Berkovich space ${\widehat{X}}$ can be defined by glueing
over an affine covering.

Let $X$ be an irreducible, normal projective variety over $k$, $K=k(X)$. Then
${\widehat{X}}$ is compact.

Let ${\widehat{K}}$ be the space of ${\mathbb{R}}$-valued valuations of $K/k$.
So ${\widehat{K}}={\widehat{X}}\smallsetminus\cup_{Y}\widehat{Y}$, the union
ranging over proper subvarieties of $X$.

We define a pairing $\beta:{\widehat{X}}\times Pic(X)\to{\mathbb{R}}_{\infty}$
as follows: given $v\in hX$ and a Cartier divisor $D$ pick a Zariski open set
$U$ of $X$ such that ${\mathcal{O}}_{X}(U)\subset{\mathcal{O}}_{v}$ and $D$ is
represented on $U$ by $f\in K$; and let $\beta(v,f)=v(f)$. It is easy to check
that this is well-defined and linear in the $Pic(X)$\- coordinate. We also
have $\beta(v,(f))=v(f)$ where $(f)$ is the principal Cartier divisor
represented everywhere by $f$.

This restricts to

$\beta:{\widehat{K}}\times Pic(X)\to{\mathbb{R}}$

Extend it to an ${\mathbb{R}}$-linear map

$\beta:{\widehat{K}}\times Pic_{\mathbb{R}}(X)\to{\mathbb{R}}$

Let ${\mathcal{I}}_{X}$ be the set of birational morphisms $\pi:Y\to X$ (over
$k$). Given $\pi,\pi^{\prime}$ there exists at most one
$j_{Y^{\prime},Y}:Y^{\prime}\to Y$ with $\pi^{\prime}=\pi\circ f$; in this
case write $\pi\leq\pi^{\prime}$; we obtain a directed partially ordered set.
Let $\widehat{Pic}(K)$ be the direct limit of $Pic_{\mathbb{R}}(Y)$ over
${\mathcal{I}}_{X}$. By going to the limit we obtain:

$\beta:{\widehat{K}}\times\widehat{Pic}_{\mathbb{R}}(K)\to{\mathbb{R}}$

Finally, let $C({\widehat{K}})$ be the space of continuous functions
${\widehat{K}}\to{\mathbb{R}}$, and define $\beta_{*}:Pic_{\mathbb{R}}(K)\to
C({\widehat{K}})$ by ${\beta_{*}}(D)(v)=\beta(v,D)$.

######  Lemma 11.1.

The image $\beta_{*}(Pic_{\mathbb{R}}(K))$ separates points on
${\widehat{K}}$, and is closed under addition, scalar multiplication and
$\min$. It separates points on ${\widehat{K}}$.

###### Proof.

Closure under addition and scalar multiplication follows from linearity of
$\beta$ in the second variable.

Let $v\neq v^{\prime}\in{\widehat{X}}$. Then $v(f)\neq v^{\prime}(f)$ for some
$f\in K$. So $\beta(v,(f))\neq\beta(v^{\prime},(f))$ for the associated
principal divisor $(f)$. Thus $\beta_{*}$ separates points.

Closure under $\min$ follows from the existence of stable meets, in an
appropriate blowup. By adding a large ample divisor, it suffices to prove this
for two effective divisors; we can assume they come from divisors
$W_{1},W_{2}$ on a blowup $Y$ of $X$, with ideal sheaves $I_{1},I_{2}$.
Further by blowing up $I_{1}+I_{2}$ we may assume it is principal, and hence
determines a Cartier divisor $W$. In this case, it is easy to see that
$\beta_{*}(W)=\min\beta_{*}(W_{1}),\beta_{*}(W_{2})$, as required.

∎

If $Z\subset{\widehat{X}}$ is compact, let $C(Z)$ be the space of continuous
functions $Z\to{\mathbb{R}}$, and define $\beta_{Z}$ to be the composition of
$\beta_{*}$ with the restriction function $C({\widehat{K}})\to C(Z)$. We endow
$C(Z)$ with the topology of uniform convergence.

######  Lemma 11.2.

Let $Z\subset{\widehat{K}}$ be compact, and assume $v(f_{0})=1$ for some
$f_{0}$ and for all $v\in Z$. Then $\beta_{Z}(Pic_{\mathbb{R}}(K))$ is dense
in $C(Z)$.

###### Proof.

By Lemma 5.13, it suffices to show that the image of $\beta_{Z}$ separates
points on $Z$, and is closed under addition, scalar multiplication and $\min$,
as well as containing the constant function $1$. The constant function $1$ is
provided by $v(f_{0})$. The rest was proved in Lemma 11.1. ∎

######  Remark 11.3.

We can similarly define $\beta:{\widehat{X}}\times
Pic_{\mathbb{R}}(K)\to{\mathbb{R}}_{\infty}$, and hence
$\beta_{*}:Pic_{\mathbb{R}}(K)\to C({\widehat{X}},{\mathbb{R}}_{\infty})$. Let
$Z\subset{\widehat{X}}$ be compact, and $\beta_{Z}(y)=\beta_{*}(y)|Z$. Let
$B=\beta_{*}(Pic_{\mathbb{R}}(K))\cap C(Z)$, be the set of functions of the
form $\beta_{*}(y)|Z$ that take only finite real values, and $\bar{B}$ the
uniform closure of $B$. Then $\bar{B}=C(Z)$, provided that $1\in\bar{B}$.

The proof is the same as of 11.2. For separation of points, if $z\neq
z^{\prime}\in Z$, we find $y\in Pic_{\mathbb{R}}(K)$ as above with
$\beta(z,y)\neq\beta(z^{\prime},y)$, then replace $y$ by
$\max(-r,\min(\beta(y,r)))$ where $-r\leq\beta(z,y),\beta(z^{\prime},y)\leq
r$.

Let $M_{K}$ be the set of regular Borel measures $\mu$ on ${\widehat{K}}$,
such that for any $f\in K$, $v\mapsto v(f)$ is integrable. It follows that for
any $D\in\widehat{Pic}(K)$, $v\mapsto\beta(v,D)$ is $\mu$ integrable. Thus we
can naturally extend $\beta$ to a map

$\beta:M_{K}\times\widehat{Pic}(K)\to{\mathbb{R}}$

######  Corollary 11.4.

Let $\mu,\mu^{\prime}\in M_{K}$ and suppose they define different GVF
structures on $K$, i.e. $\mu,\mu^{\prime}$ are distinct even up to
renormalization. Then for some $D\in\widehat{Pic}(K)$,
$\beta(\mu,D)\neq\beta(\mu^{\prime},D)$.

###### Proof.

We may assume $k$ is countable 111For this we use the equivalence of GVF
structures and globalizing measures up to renormalization. We have
${\widehat{K}}\smallsetminus\\{v_{triv}\\}=\cup_{f\in
K\smallsetminus(0)}W_{f}$ where $W_{f}=\\{v:v(f)>0\\}$. Thus for some $f$, the
measures $\mu,\mu^{\prime}$ are distinct (even up to normalization) when
restricted to $W=\\{v:v(f)>0\\}$. By renormalizing we may assume
$\mu,\mu^{\prime}$ concentrate on $\\{v:v(f)=1\\}$. Moreover these are regular
measures, approximated by their values on compact sets, hence
$\mu,\mu^{\prime}$ have distinct restrictions to some compact
$Z\subset\\{v:v(f)=1\\}$. Now the statement follows from Lemma 11.2 (with
$f_{0}=f$). ∎

######  Proposition 11.5.

Let $\mu\in M_{K}$ satisfy the product formula:

$\hbox{for all }f\in K\smallsetminus(0),\ \ \int v(f)d\mu(v)=0$

Then $\beta(\mu,\ ):Pic_{\mathbb{R}}(Y)\to{\mathbb{R}}$ factors through the
quotient map $Pic_{\mathbb{R}}(Y)\to NS(Y)$.

###### Proof.

We have a normal, irreducible variety $Y$, and a homomorphism $\xi=\beta(\mu,\
):Pic(Y)\to{\mathbb{R}}$. Let $H$ be an ample divisor on $Y$. Let $Pic^{0}(Y)$
be the group of divisors algebraically equivalent to $0$. It is known ([21],
Proposition 5.3 or Theorem 5.4, and Proposition 5.10) that $Pic^{0}(Y)$ is an
algebraic variety, in particular constructible. For any $a\in Pic^{0}(Y)$, (as
for any element of $Pic(Y)$), for large enough $m$, the line bundle
corresponding to $mH-a$ has a nonzero section. So $-a+mH\geq 0$. Since
$Pic^{0}(Y)$ is constructible, by compactness, there exists $m$ such that for
all $a\in Pic^{0}(Y)$, $-a+mH\geq 0$. So $\xi(a)\leq m\xi(H)$. Thus
$\xi|Pic^{0}(Y)$ is a homomorphism into ${\mathbb{R}}$ with bounded image; so
it must be zero. Since ${\mathbb{R}}$ is torsion-free, if $mb\in Pic^{0}(Y)$
then $\xi(b)=0$. Hence $\xi$ vanishes on the divisors that have integral
multiples algebraically equivalent to zero. But by [21], Theorem 6.3, these
are the same as the divisors numerically equivalent to $0$. So $\xi$ induces a
homomorphism $NS(Y)_{\mathbb{R}}\to{\mathbb{R}}$. and factors through that
homomorphism.∎

Note that the homomorphism $\beta_{\mu}:Pic_{\mathbb{R}}(Y)\to{\mathbb{R}}$
obtained above is order-preserving with respect to the effective cone; i.e. an
effective divisor has non-negative image. This is because by definition,
$\beta(v,D)=v(g)\geq 0$ where $D$ is represented by $(g)$ on some affine open
set $Y^{\prime}$ with ${\mathcal{O}}_{Y^{\prime}}\subset{\mathcal{O}}_{v}$,
and as $D$ is effective, $g\in{\mathcal{O}}_{Y^{\prime}}$ so $v(g)\geq 0$.
Taking into account that $\mu$ is a non-negative measure, the induced
homomorphism $NS(Y)_{\mathbb{R}}\to{\mathbb{R}}$ is order-preserving with
respect to the to the pseudo-effective cone.

Let $S_{K}$ be the space of GVF structures on $K$ that are trivial on $k$; it
can be viewed as the space of quantifier-free GVF types whose restriction to
ACF is the generic type of $X$. If we pick any transcendence basis
$f_{1},\ldots,f_{n}$ for $K$ over $k$, and $r\in{\mathbb{R}}$, then the
subspace of types with $ht(f_{i})\leq r$ is compact. Each element $p$ of
$S_{K}$ is induced by a measure $\mu\in M_{K}$ satisfying the product formula:
this can be proved using a homogeneous variant of the Riesz representation
theorem, but will also follow from the proof of Theorem 11.6 below.

It is clear that $\beta(\mu,D)$ depends only on $p$ and not on the choice of
$\mu$; for instance this may be checked for very ample $D$, with a section
$f$, so $\beta(\mu,D)=\int\min(v(f),0)d\mu(v)$. We thus write $\beta(p,D)$ for
$\beta(\mu,D)$.

Let $N^{1}(K)$ be the direct limit of the ${\mathbb{R}}$-spaces $N^{1}(Y)$
along ${\mathcal{I}}_{X}$; let $N_{1}(K)$ be the dual space, the inverse limit
of the spaces $N_{1}(Y)$. (Cf. [3].) Note that when $j:Y^{\prime}\to Y$ is
birational, $j^{*}N_{1}^{+}(Y)\subset N_{1}^{+}(Y^{\prime})$. Let
$N_{1}^{+}(K)$ be the inverse limit of the cones $N_{1}(Y)^{+}$, dual to the
effective cones of $N^{1}(Y)$.

We define a map $\alpha:S_{K}\to N_{1}^{+}(K)$ via the pairing
$\beta:S_{K}\times N^{1}(K)\to{\mathbb{R}}$. So $\alpha=\lim\alpha_{Y}$ where
$Y$ is similarly defined via $\beta:S_{Y}\times N^{1}(Y)\to{\mathbb{R}}$, with
$Y$ a birational extension of $X$.

######  Theorem 11.6.

$\alpha:S_{K}\to N_{1}^{+}(K)$ is a homeomorphism.

###### Proof.

Injectivity: Let $p,p^{\prime}$ be distinct elements of $S_{K}$. They are
represented by measures $\mu,\mu^{\prime}$ that are distinct even up to
renormalization. By Lemma 11.4, for some $D$ we have
$\beta(\mu,D)\neq\beta(\mu^{\prime},D)$. Thus
$\alpha(p)\neq\alpha(p^{\prime})$.

Continuity: It suffices to prove that $\alpha:S_{K}\to N_{1}(Y)$ is
continuous. Now the topology on $N_{1}(Y)$ is generated by
$O_{\beta,\gamma,D}=\\{a:\beta<|a\cdot D|<\gamma\\}$, where $\beta,\gamma$ are
reals and $D$ is a very ample divisor on $Y$. Let $s_{1},\ldots,s_{n}$ be
global sections of $D$. Then $\alpha^{-1}(O_{\beta,\gamma,D})$ is cut out by
$\beta<ht(s_{1},\ldots,s_{n})<\gamma$.

A closed map:

Let $f_{1},\ldots,f_{n}$ be a transcendence basis for $K/k$; let $A$ be an
ample divisor, such that each $f_{i}$ has poles at most on $A$. Let $D_{i}$ be
the divisor of poles of $f_{i}$. Let $C$ be a closed subset of $S_{X}$. For
$t>0$ let $R_{t}=\\{b\in N_{1}^{+}(K):(b,D_{i})\leq t\\}$. Then $R_{t}$ is
contained in the interior of $R_{t^{\prime}}$ for $t<t^{\prime}$, and
$\cup_{t}R_{t}=N_{1}^{+}(K)$. Thus to show that a set is closed, it suffices
to show that the intersection with each $R_{t}$ is closed. As
$\alpha^{-1}(R_{t})$ is closed, to show that $\alpha$ is a closed map, it
suffices to show that $\alpha(C)$ is closed when $C$ is a closed subset of
$\alpha^{-1}R_{t}$ for some $t$. But such a set is compact as $ht(f_{i})$ is
bounded on it.

In particular, the image of $\alpha$ is closed. Hence to show that $\alpha$ is
surjective, it suffices to prove that $\alpha$ is dense. For this, it suffices
to show that each $\alpha_{Y}$ is surjective. Using de Jong, find a smooth
projective variety ${\widetilde{Y}}$ and a generically finite morphism
$\phi:{\widetilde{Y}}\to Y$. By Lemma 7.8, $N_{1}^{+}({\widetilde{Y}})\to
N_{1}^{+}(Y)$ is surjective. This reduces the statement to the smooth
projective case; here it follows from Lemma 11.9. ∎

######  Remark 11.7.

The fact that continuity was proved using the open sets

$\beta<ht(f_{1},\ldots,f_{n})<\gamma$

alone shows that the projective height functions, along with the field
operations, generates the entire language of GVFs.

###  11.8. Discrete globalizing measures

Let $X$ be a smooth projective variety. Let $\mathcal{W}$ be the set of
irreducible hypersurfaces of $X$.

Let $a$ be a class in $N_{1}(X)^{+}$.

We let $\mu_{a}$ be the measure, supported on divisorial valuations $v_{D}$ of
$K$ for each irreducible hypersurface $D$ of $X$, and with
$\mu_{a}(v_{D})=a\cdot[D]$.

The same definition extends additively to Cartier divisors, in such a way that
principle divisors map to $0$. This amounts to the fact that a rational
function on a curve has as many zeroes as poles.

From this it is clear that for $f\in K$, letting $(f)=\sum m_{i}D_{i}$ be the
divisor of zeroes and poles of $f$, we have $\sum m_{i}\mu(D_{i})=0$. Indeed
let $p\in C_{t}\cap D_{i}$; if $D_{i}$ is cut out by $(g)$ locally at $p$ then
$(g^{m_{i}})=(f)$ on this local ring, so $m_{i}i(C_{t},D_{i})$ is the order of
zero (or pole) of $f$ at $p$ on $C_{t}$, and the sum of these is zero. Thus
$\mu_{a}$ is a GVF measure. In particular, it defines an element of $S_{X}$,
depending only on the numerical equivalence class of $C$, that we denote
$\delta_{X}([C])$. This gives a map

$\delta_{X}:N_{1}^{+}(X)\to S_{K}$

######  Lemma 11.9.

$\alpha_{X}\circ\delta_{X}=Id_{N_{1}^{+}(X)}$.

###### Proof.

This amounts to the relations: $v_{D}(D)=1$, $v_{D}(D^{\prime})=0$ when
$D,D^{\prime}\in\mathcal{W}$ are distinct, and $v_{D}$ is the valuation
corresponding to $D$, evaluated on $D,D^{\prime}$ as Cartier divisors. Both
are straightforward. ∎

######  Remark 11.10.

It is not true that $\delta_{X}\circ\alpha_{X}$ is the identity on $S_{K}$;
but $\delta_{X}\circ\alpha_{X}(p)$ and $p$ do give the same value to a formula
$\int t(vf_{1},\ldots,vf_{m})dv$ if $f_{1},\ldots,f_{m}$ extend to morphisms
$X\to{\mathbb{P}}^{1}$, $t$ is a term formed out of $+,-,\min$, and for any
minimum occuring in $t(vf_{1},\ldots,vf_{m})$, the corresponding minimum of
Cartier divisors $(f_{1}),\ldots,(f_{m})$ is is a stable meet on $X$.

In particular, the same height is computed for rational functions $f$
extending to a morphism $X\to{\mathbb{P}}^{1}$.

This makes it possible to define $\alpha_{K}$ as the limit of $\alpha_{Y}$
over all blowups $Y$, and prove injectivity in a more explicit way.

######  Remark 11.11.

Let $a\in N_{1}^{+}(X)$; assume $a\cdot b=0$ for some effective $b$ with
nonzero section growth. Then there exists a nonconstant rational function $g$
on $X$ such that $F(g)$ is a constant subfield of the GVF $F(X)_{a}$. The
converse is also true.

###### Proof.

By assumption there exists a nonconstant rational function $g$ with poles only
on (a representative of) $b$. So the height of $g$ in $F(X)_{a}$ is zero. ∎

###  11.12. Quantifier-free types over the function field of a curve

Here the picture is very similar. Let $k$ be a constant field, and let
$k_{1}=k(U)$ for a curve $U$. Let $K=k_{1}(X)$ be a finitely generated field
extension of $k_{1}$. Let $\mu_{1}$ be a nontrivial GVF structure on
$k_{1}/k$. Then $N_{1}^{+}(k_{1})={\mathbb{R}}^{\geq 0}$, with $\mu_{1}$
corresponding to $1$; and we have a natural map $\pi_{*}:N_{1}^{+}(K)\to
N_{1}^{+}(k_{1})$; it is obtained as the limit of the projections
$\pi_{*}:N_{1}(X)\to N_{1}(U)$ at the level of varieties. Let
$N_{1}^{+}(K/\mu_{1})$ be the pullback of $1$. We have an
${\mathbb{R}}^{>0}$-action on $N_{1}(X)$ and on $N_{1}(U)$ respecting
$\pi_{*}|N_{1}(X)$; so $N_{1}^{+}(K/\mu_{1})$ can also be identified with the
set of elements of $N_{1}^{+}(K)$ with nontrivial images in $N_{1}^{+}(U)$,
modulo the ${\mathbb{R}}^{>0}$-action.

Letting $S_{X}$ denote the space of quantifier-free types over
$(k_{1},\mu_{1})$, we immediately obtain from Theorem 11.6, by restriction:

######  Theorem 11.13.

$\alpha:S_{X}\to N_{1}^{+}(K/\mu_{1})$ is a homeomorphism.

##  12\. Existential closedness of $k(x)^{alg}[r]$

Let $k=k^{alg}$.

Let $\mu$ be a globalizing measure for $k(V)$ over $k$. Let $D$ be an ample
divisor on $V$.

######  Lemma 12.1 (Artin-Whaples).

Let $C$ be a curve of genus $g$ over $k$. Let $x\in k(C)\smallsetminus k$, and
let $r>0$. Then $k(C)$ admits a unique GVF structure over $k$ with $ht(x)=r$.

###### Proof.

The nontrivial valuations of $k(C)$ over $k$ can be identified with the points
of $C(k)$. They form a discrete set; the compactification includes also the
trivial valuation, which plays no role here. We have to show that if $\mu$ is
a measure on $C(k)$ satisfying the product formula, then $\mu$ gives equal
weight to each element of $k(C)$. Let $a,b$ be two points. Then $(n+g)a-nb$ is
effective, so $(n+g)\mu(a)\geq n\mu(b)$. Thus $(1+g/n)\mu(a)\geq\mu(b)$.
Letting $n\to\infty$, $\mu(a)\geq\mu(b)$. So $\mu(a)=\mu(b)$ for all $a,b\in
k(C)$. ∎

A $c$-twisting automorphism $\sigma$ of $K$ is an isomorphism from $K$ to
$K^{\prime}$, where $K^{\prime}$ is the renormalization of $K$ moving from
$\mu$ to $c\mu$.

######  Corollary 12.2.

There is a unique $GVF$ structure $k(x)^{alg}[r]$ on $k(x)^{alg}$ over $k$,
with $ht(x)=r$. Moreover if $F$ is a finitely generated extension field of $k$
of transcendence degree $1$, and $g,g^{\prime}:F\to k(x)^{alg}$ are two field
embeddings, and $a\in F\smallsetminus k$, there exists an
$ht(f^{\prime}(a))/ht(f(a))$-twisting automorphism $\sigma$ of $k(x)^{alg}$
with $g^{\prime}=\sigma\circ g$.

######  Remark 12.3.

If $r/r^{\prime}\in{\mathbb{Q}}$ then $k(x)^{alg}[r]\cong
k(x)^{alg}[r^{\prime}]$. Let $m\in{\mathbb{N}}$ and let $\sigma$ be an
automorphism of the field $k(x)^{alg}$ with $\sigma(x)=x^{m}$. Then $\sigma$
is $m$-twisting so it gives an isomorphism $k(x)^{alg}[r]\to k(x)^{alg}[mr]$.
Similarly find an $m^{\prime}$-twisting $\sigma^{\prime}$, with
$m/m^{\prime}=r/r^{\prime}$. Then $(\sigma^{\prime})^{-1}(\sigma)$ is an
isomorphism $k(x)^{alg}[r]\to k(x)^{alg}[r^{\prime}]$.

######  Corollary 12.4.

Let $K^{*}$ be an ultrapower of $K=k(x)^{alg}$. Let $F$ be finitely generated
over $k$, of transcendence degree $1$, and let $f,f^{\prime}:F\to K^{*}$ be
two embeddings, with $ht(f(a))=ht(f^{\prime}(a))>0$ for some $a\in F$. Then
there exists an automorphism $\sigma^{*}$ of $K^{*}$ with $\sigma\circ
f=f^{\prime}$.

This also holds if $K^{*}$ is an ultraproduct of GVF fields $k_{i}(x)^{alg}$
of increasing positive characteristics, and $F$ is finitely generated over
${\mathbb{Q}}$, of transcendence degree $1$.

###### Proof.

$F$ is the field of fractions of a finitely generated $k$-algebra $D$, with
$a\in D$. $f|D,f^{\prime}|D$ can be represented as ultrapowers of
homomorphisms $f_{i},f_{i}^{\prime}:F^{\prime}\to K$. We have $ht(f_{i}(a))\to
ht(f(a))$ and $ht(f_{i}^{\prime}(a))\to ht(f^{\prime}(a))$ along the
ultrafilter $u$. So $r_{i}=ht(f_{i}(a))/ht(f_{i}^{\prime}(a))$ approaches $1$
along $u$. Note that $f_{i},f_{i}^{\prime}$ extend to field embeddings (first
on $k(a)$ since $f_{i}(a),f_{i}^{\prime}(a)\notin k=k^{alg}$; then on $F$
since it is a finite extension of $k(a)$). Using Corollary 12.2 let
$\sigma_{i}$ be an $r_{i}$-twisting automorphism of $K$ with
$f_{i}^{\prime}=\sigma\circ f_{i}$. Let $\sigma$ be the ultraproduct of the
$\sigma_{i}$. Then $f^{\prime}=\sigma\circ f$ and $\sigma$ is a $1$-twisting
automorphism, i.e. simply an automorphism. The statement with ultraproducts is
proved similarly. ∎

###### Proof of Theorem 2.1.

We first show that any quantifier-free type $q$ over $k$ can be approximately
realized in $K=k(t)^{alg}[1]$. Consider first a smooth projective variety $X$
over $k$, and a family of curves $C_{t}$ on $X$ as in § 11.8, with class
$[C]\in N_{1}^{+}(X)$. We have the corresponding quantifier-free type
$\delta_{X}(C)$. Now the inclusion of $C_{t}$ in $X$ corresponds to an element
$a_{t}\in X(k(C_{t}))$. It is easy to see that any approximation to
$\delta_{X}(C)$ involving finitely many divisors is realized by $a_{t}$, for
almost all $t$ (namely as soon as $C_{t}$ is not contained in the support of
these finitely many divisors). Thus $\delta_{X}([C])$ is approximately
realized in $K$. Since $K\cong K[m]$ for integral $m$ by Remark 12.3, we aso
have $\delta_{X}(\frac{1}{m}[C])$ approximately realized in $K$. Now by
Corollary 10.11, Theorem 11.6 and Lemma 10.9, the types
$\delta_{Y}(\frac{1}{m}[C])$ for $Y$ a blow-up of $X$, and $C$ as above on
$Y$, are dense on $S_{X}$. Thus all types in $S_{X}$ are approximately
realized in $K$.

Now we must also consider types over $K$ and not just over $k$. Let $K\leq L$,
$L$ finitely generated over $K$, with a GVF structure. We have to find a $K$\-
embedding of GVF’s $L\to K^{*}$, where $K^{*}$ is an ultrapower of $K$. By the
above we do have a $k$-embedding of GVFs $h:L\to K^{*}$. Let $g$ be the
inclusion of $K$ in $K^{*}$, and $g^{\prime}=h|K$. Using Lemma 12.4, find
$\sigma\in Aut(K^{*})$ with $\sigma\circ g^{\prime}=g$. Then $\sigma\circ h$
is an embedding of $L$ to $K^{*}$ over $K$, as required.

∎

##  13\. A non-archimedean Yau theorem

Theorem 10.9 can be viewed as an absolute version of the non-archimedean Yau
theorem of [4] and [22]. To bring this out, we give a proof along the same
lines of a version of the relative theorem. (Our proof is valid in any
characteristic, and addresses to some extent the ‘regularity’ issue raised in
[4] in the model case, in that we show that the metric is determined by a big
divisor on the same model.)

We are given not only a smooth projective $X$, but also a morphism to $U$. In
the relative (Yau) version, the data is a measure on the vertical divisors, as
well as an ample line bundle $L$ on the generic fiber of $X$; the theorem
yields a big line bundle on $X$ compatible with the measure (vertically) and
with $L$ (generically.) By comparison in Theorem 10.9, the data is a measure
on all divisors of $X$, including the horizontal ones and the outcome is again
big divisor compatible with both vertical and horizontal measures.

###  13.1. The relative setting

Assume now that $\dim(X)=n+1$, and we are given not only $X$ but also a
morphism $\pi:X\to U$ and a point $0\in U$, where $U$ is a curve. Let
$K=k(U)$, and let $X_{K}$ be the generic fiber, and $X_{0}=f^{-1}(0)$ the
special fiber. Let $I$ be the set of irreducible components of $X_{0}$. Within
$N^{1}(X)$, we have the subspace $V_{0}$ of vertical divisors generated by
$I$. Note that $[X_{0}]^{2}=0$ and moreover $[X_{0}]\cdot[E]=0$ for any $E\in
I$, since $X_{0}$ can be moved to any other fiber. By Lemma 6.14 (applied to
$V_{0}$ and the vector $f=[X_{0}]$, and taking into account the Grothendieck
connectedness theorem), $V_{0}/(f)$ is negative definite and $f=0$ is the
unique relation on the generators $I$; as $f\neq 0$, in fact $I$ is a basis
for $V_{0}$.

Any ample $D\in N^{1}(X)$ defines, on the one hand, an ample line bundle $L$
on $X_{K}$, obtained by restriction of $D$ to the generic fiber. On the other
hand, the point $0\in U$ determines a valuation $v_{0}$ on $K$, and by
extending rational sections of $L$ to sections of $D$ we obtain a metrization
of $L$ over the valued field $(K,v_{0})$. (Actually for any big $D\in
N^{1}(X)$ with fixed ample generic part $L$, we can define a metrization of
$L$ over $(K,v_{0})$, the limit of the metrics associated to ample divisors
$D-E$ with $E$ effective and vertical and $D-E$ ample, over all base change
morphisms $X^{\prime}\to X$ that are isomorphisms away from $X_{0}$.)

In Arakelov theory, this is viewed as analogous to a smooth metrization of
$L$, assuming given an embedding of $K$ into ${\mathbb{C}}$ in place of the
valuation $v_{0}$.

In the Kähler case, a metrized line bundle gives rise to a measure on $X$,
obtained by taking the $n$’th power of the curvature $2$-form. A non-
archimedean analogue of this construction was defined in [6], following work
of Gillet-Soulé, Gubler, Zhang and others. In the basic geometric case we are
considering, the measure is very simple to define. It concentrates on $I$; any
$c\in I$ is given weight $m_{c}c\cdot D^{n}$, where $m_{c}$ is the
multiplicity with which $c$ occurs in $X_{0}$.

More generally, if $D$ is a divisor on a blowup $b:X^{\prime}\to X$ we can
still define the measure by $\mu_{D}(c)=m_{c}c\cdot
b_{*}(D^{n})=m_{c}b^{*}(c)D^{n}$.

For us it is natural to extend the definition to any big Cartier
${\mathbb{R}}$\- divisor $D$ (with generic part $L$):

$\hat{\mu}_{D}(c)=m_{c}c\cdot\psi(D)$

######  Lemma 13.2.

Any big line bundle $L$ of $X_{K}$ extends to a big divisor ${\mathbf{L}}$ of
$X$.

###### Proof.

We can represent $L$ by an effective divisor $D_{K}$ on $X_{K}$. There exist
$g_{1},\ldots,g_{n}\in k(X)=K(X_{K})$ that are algebraically independent over
$K$, and whose poles are at most $D_{K}$. We can view the $g_{i}$ as elements
of $k(X)$, and by adding some vertical divisors, can find a divisor $D$ on $X$
with generic part $D_{K}$, such that the poles of the $g_{i}$ are bounded by
$D$. On the other hand we can find $f_{1},\ldots,f_{r}\in K=k(U)$,
$r=tr.deg._{k}K=\dim(U)$, algebraically independent over $k$; by adding to $D$
the pullback of the polar divisors of $f_{1},\ldots,f_{r}$ to $D$ we may
assume each $f_{i}\in L(D)$ as well. Finally since $D$ is effective, we have
$1\in L(D)$. It follows that $D$ is big. ∎

######  Theorem 13.3.

Let $L$ be a big line bundle on $X$, represented by an irreducible Cartier
divisor. Let $I_{t}$ be the set of irreducible components of $X_{t}$,
$I=\cup_{t}I_{t}$. Let $\mu$ be a nowhere vanishing positive measure on the
vertical divisors, such that the total mass of each fiber $X_{t}$ is
$\mu(I_{t})=D^{n}=\deg(D)$. Then there exists a big ${\mathbb{R}}$-Cartier
divisor $B$ on $X$ with generic part $L$, $\hat{\mu}_{B}=\mu$.

Note that for all but finitely many $t\in U$, $I_{t}$ has a single point, so
$\mu$ is fully constrained on $I_{t}$.

######  Remark 13.4.

If only given a measure on $I_{0}$ of total mass $L^{n}$, it can be extended
to a nowhere vanishing positive measure on $I$, as the proof below shows; thus
Theorem 13.3 includes the local version usually considered.

###### Proof.

Let $D_{K}$ be an irreducible Cartier divisor on $X_{K}$, representing $L$.

Let $\gamma:Pic_{\mathbb{R}}(X)\to Pic_{\mathbb{R}}(X_{K})$ be ‘restriction to
the generic fiber’. We will also write $\gamma$ for restriction of
${\mathbb{R}}$-Cartier divisors. Fix an ${\mathbb{R}}$-Cartier divisor $D$
with generic part $D_{K}$.

Let $E\leq Pic_{\mathbb{R}}(X)$ the the vector space generated by the class of
$D$ and by the vertical divisors. Let $E^{+}$ be the cone in $E$ generated by
effective divisors.

Consider the codimension one subspace $E_{0}:=\ker(\gamma)$ of $E$. Let
$E_{0}^{+}$ be the cone of effective Cartier divisors in $E_{0}$. $E$ is
generated by the finite set $I$, using Remark 6.15; so it is finite
dimensional, and closed; the natural homomorphism $Pic_{\mathbb{R}}(X)\to
N^{1}(X)$ is, again by Remark 6.15, injective on $E_{0}$, and $E_{0}^{+}$ is
the pulback of the effective cone in $N^{1}(X)$.

Let $U$ be the intersection of $E$ with the pullback to $Pic(X)$ of the cone
of big divisors. By Lemma 13.2, this intersection is nonempty, so it is an
open cone on $E$. The hypotheses (1)-(3) of Theorem 5.2 are satisfied for the
function ${\mathop{\operatorname{\rm vol\,}}}^{\frac{1}{n+1}}|E$, since this
is shown for $N^{1}(X)$ in Theorem 10.9, and they remain true upon restriction
to the image of $E$. For $B\in U$, write $\psi(B)$ for $\psi$ of the image of
$B$ in $N^{1}(X)$.

Now $\mu$ defines a linear map on $E_{0}$, positive on $E_{0}^{+}$. By the
strict positivity assumption, $\mu$ lies in the interior of the dual of
$E_{0}^{+}$ by Lemma 5.7, by Lemma 5.8 it lifts to an element $\mu^{\prime}$
in the interior of the dual of $E^{+}$.

By Corollary 5.3, there exists $B^{\prime}\in E\cap U$ such that
$\psi(B^{\prime})\cdot c=\mu^{\prime}(c)$ for any $c\in I$. We may represent
$B^{\prime}$ by an ${\mathbb{R}}$-Cartier divisor
$B^{\prime}=\alpha^{\prime}D+F^{\prime}$.

It remains to show that $\alpha^{\prime}$ may be chosen to be $1$. We do not
claim that it necessarily equals $1$, but rather that there exists an
${\mathbb{R}}$-Cartier $B\leq B^{\prime}$ with the same sections, with
$\psi(B)=\psi(B^{\prime})$, so that $B=D+F$. Indeed take $B$, as an
${\mathbb{R}}$-Weil divisor, be $\sup\\{(f)/n:(f)\leq nB,n=1,2,\cdots\\}$.
Then by definition $B,B^{\prime}$ have the same sections, hence same Okounkov
bodies, the same volume, and by Proposition 10.4, $\psi(B)=\psi(B^{\prime})$.

Restricting to the generic fiber, and viewing $B,B^{\prime}$ as
${\mathbb{R}}$-Weil divisors, we have
$\gamma(B)\leq\gamma(B^{\prime})=\alpha^{\prime}D_{K}$ so using the
irreducibility of $D_{K}$, $\gamma(B^{\prime})=\alpha D_{K}$ for some
$\alpha\leq\alpha^{\prime}$.

By definition, $\psi(B)$ is the limit of $H^{n}$ over nef divisors $H\leq
b^{*}(B)$, where $b:X^{\prime}\to X$ is a blowup of $X$. Working in
$X^{\prime}$, let $D^{\prime}=b^{*}D,F^{\prime}=m^{*}F$,
$\gamma^{\prime}:N_{1}(X^{\prime})\to N_{1}(X^{\prime}_{K})$ the restriction.
Let $f$ denote the class of a generic fiber of $\pi$. Note $f\cdot F=0$ for
any vertical divisor $F$.

$H^{n}\cdot b^{*}(f)\leq(\alpha D+F)^{n}\cdot
f=\alpha^{n}L^{n}=\alpha^{n}\mu(f)$. Thus taking the limit,
$\mu(f)\leq\alpha^{n}\mu(f)$ so $\alpha\geq 1$.

On the other hand if $\alpha>1$ then some section witnesses this; on some
blowup $b:X^{\prime}\to X$ there exists a semi-free (hence nef)
${\mathbb{Q}}$\- divisor $H\leq\alpha D+F$ with generic part $H_{K}\geq\beta
D$, for some $\beta>1$. But in this case for some vertical $F^{\prime}$ we
have $H\geq\beta D+F^{\prime}$, so $\mu(f)=\psi(B)\cdot f\geq H^{n}\cdot
b^{*}(f)\geq(\beta D+F^{\prime})^{n}\cdot f=\beta^{n}D^{n}\cdot
f=\beta^{n}\mu(f)$, a contradiction. Thus $\alpha=1$. So $\hat{\mu}_{B}=\mu$.
∎

######  Question 13.5.

(and remarks). Clarify Calabi uniqueness in this setting. For semi-positive
metrics on an ample line bundle, supremum of the nefs on blowups dominated by
$B$, is proved in [27]. In the global case, uniquenesss follows from a the
strict log-concavity of volume in non-radial directions, [3], [11].

As far as I could see nothing is known outside the nef cone, and it is
possible that the collapse from $B$ to $B^{\prime}$ above is responsible.

Consider $B$ with generic part $D$, $\hat{\mu}_{B}=\mu$.

  * •

The freedom in the choice of $\mu^{\prime}$ corresponds to additive
translation of $B^{\prime}$ by a scalar multiple of $f$. At least in the
multilinear domain, $B^{\prime}+\alpha f$ represents the same measure on $I$.
Thus we can only expect to determine the image of $B$ in $N^{1}(X)/N^{1}(U)$.
This is analogous to the fact that in the usual Calabi theorem, uniqueness is
up to a scalar multiple.

  * •

In the non-multi-linear region of $\psi$, is addition of $f$ still the same as
addition of a constant to the Weil function, after collapse? (Seems so.)

  * •

An interesting way to prove uniqueness of the asymptotic section spaces: use
the strict convexity in Brunn-Minkowski. What can be concluded about
$B,B^{\prime}$ if the Okounkov body is the same for every choice of $v$?

  * •

Is this image determined by the associated metrization
($v\mapsto\lim_{n}1/n\min v(sections)$).

  * •

Is it the case that if $\hat{\mu}_{B}=\hat{\mu}_{B^{\prime}}$ and
$B|D=B^{\prime}|D$ is ample then the nef divisors on blowups below $B$ are
cofinal with those below $B^{\prime}$ ? (issue of f).

  * •

Assuming this cofinality, we have $B=B^{\prime}$. For every section of $B$ is
a section of some free $N$ on a blowup, $N\leq\pi^{*}B$; hence of
$\pi^{*}B^{\prime}$ hence of $B^{\prime}$. (issue of f.)

  * •

Assuming this and that the limit is uniform, so that Yuan-Zhang Calabi
applies, we do have $B=B^{\prime}$.(issue of f.)

  * •

Does a big divisor defines the same metrization as a limit over blow-up of nef
divisors approaching it? This apparently does not imply that it is nef itself.
Fujita says yes in terms of volume. Pointwise, for maximal rank $v$, the
answer is also yes: Okounkov bodies must approach if volume does.

######  Question 13.6.

What about higher dimensional $U$? Analogue of splitting divisors is clear.

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# Hydrogen-Atmosphere White Dwarfs Are Less Likely To Be Found with Wide-
Binary Companions

Jeremy Heyl1 0000-0001-9739-367X 1University of British Columbia, Vancouver,
BC, Canada

###### Abstract

The fraction of white dwarfs found in wide binaries is estimated by cross
referencing a catalogue of wide binaries with catalogues photometrically
determined white-dwarf candidates and spectroscopically confirmed white dwarf
stars. The wide-binary fraction of white dwarfs with hydrogen-dominated
atmospheres is about 5%, but the fraction of white dwarfs with helium or
carbon-dominated atmospheres is significantly larger. Using spectroscopic
classifications, the binary fraction of DA white dwarfs is determined to be
$0.063\pm 0.002$ and for non-DA white dwarfs is larger at $0.080\pm 0.004$.

###### keywords:

binaries: visual – white dwarfs – stars: atmospheres

##  1 Introduction

Most stars are members of multiple stellar systems, and most stars will become
white dwarfs when they exhaust their nuclear fuel, so it is quite natural to
examine the fraction of white dwarfs in multiple stellar systems. Additionally
the bulk of white dwarfs have substantial hydrogen atmospheres while a smaller
fraction have helium or carbon atmospheres. Noor et al. (2024) has examined
the correlation of binarity with the metal abundance within white dwarf
atmospheres to seek hints of the role of companions in the metal pollution.
However, perhaps a more basic question is: what fraction of white dwarfs are
actually in binaries and whether white dwarfs with different atmospheric
compositions have different binary fractions?

The Gaia satellite (Gaia Collaboration et al., 2016a) and its successive data
releases (Gaia Collaboration et al., 2016b, 2018, 2021, 2023) have driven a
revolution of our understanding of stellar astrophysics and especially of
white dwarfs. Gentile Fusillo et al. (2021, GF) has developed a catalogue of
white dwarf candidates from Gaia EDR3 that provides a tentative classification
of the white dwarfs as having hydrogen or helium atmospheres. Furthermore, El-
Badry et al. (2021) created a catalogue of high-probability wide binaries from
Gaia EDR3. The Montreal White Dwarf Database (Dufour et al., 2017, MWDDB)
provides a complementary census of the white-dwarf population with many white
dwarfs classified by their spectral type which yields more precise
determination of the atmospheric composition. Cross referencing the El-Badry
et al. (2021) binary catalogue with these two white-dwarf catalogues yields a
catalogue of likely and confirmed white-dwarf stars that are very likely to
have companions. Furthermore, these resultant catalogues yield estimates of
the binary fraction of white dwarfs as a function of atmospheric composition,
spectral type and mass.

##  2 Data

The El-Badry et al. (2021) binary catalogue applies astrometric and
photometric cuts on the entire Gaia EDR3 catalogue before seeking wide binary
companions from spatial and kinematic coincidences among the sources in the
catalogue. In particular, the constraints are that the source must have a
measured Gaia $G$-band magnitude, its parallax must exceed 1 mas, the error in
the parallax must be less than 2 mas, and the ratio of the measured parallax
to the error in the parallax must exceed five. To estimate the binary fraction
of the white dwarfs, these constrains are also applied to the GF and MWDDB
catalogues. Finally, both the GF catalogue and the MWDDB are cross referenced
with the primary and secondary stars listed in the El-Badry et al. (2021)
catalogue using the Gaia EDR3 source id that all three catalogues use in
common to yielding a catalogue of candidate wide binaries containing white-
dwarf stars.

##  3 Results

The two white dwarf catalogues trade completeness against accuracy. The GF
catalogue yields a probability that a particular Gaia EDR3 source is a white
dwarf and estimates of the mass and atmospheric composition obtained from
fitting photometric data to model spectral energy distributions. The portion
of the MWDD used in this study gives spectral classifications of the white
dwarfs and mass estimates obtained by fitting spectroscopic data to model
spectra.

###  3.1 Photometric Classification

From the GF catalogue, a sample of probable white-dwarf stars is obtained by
selecting only those sources where the white-dwarf probability,
$P_{\textrm{\scriptsize WD}}$, is greater than 0.99. From this sample,
selecting those sources where the hydrogen model yields a significantly better
fit than the helium model yields a subsample of white dwarfs likely to have
hydrogen atmospheres. Conversely, a sample of helium atmosphere (or non-
hydrogen atmosphere) white dwarfs is built by selecting sources where the
helium model yields a significantly better fit than the hydrogen model. In
both cases, objects where a mixed model is preferred are excluded. Each of
these subsamples are cross-referenced against the binary catalogue, and
sources with chance coincidences greater than 10% ($R_{\textrm{\scriptsize
chance}}$) are excluded. Table 1 outlines the down-selections to create the
photometric catalogues.

Table 1: Results of Photometric Classification Number of objects in GF main catalogue | 1280266
---|---
with $P_{\textrm{\scriptsize WD}}>0.99$ | 192613
Hydrogen Atmospheres
with $\chi^{2}_{\textrm{\scriptsize H}}<\chi^{2}_{\textrm{\scriptsize He}}-1$ | 53364
meet binary catalogue astrometric criteria | 50950
with $\chi^{2}_{\textrm{\scriptsize H}}<\chi^{2}_{\textrm{mix}}$ | 50871
with a wide binary companion | 3210
with $R_{\textrm{\scriptsize chance}}<0.1$ | 2381
binary fraction | $0.0468\pm 0.0009$
Helium Atmospheres
with $\chi^{2}_{\textrm{\scriptsize He}}<\chi^{2}_{\textrm{\scriptsize H}}-1$ | 7662
meet binary catalogue astrometric criteria | 7158
with $\chi^{2}_{\textrm{\scriptsize He}}<\chi^{2}_{\textrm{mix}}$ | 5351
with a wide binary companion | 753
with $R_{\textrm{\scriptsize chance}}<0.1$ | 640
binary fraction | $0.120\pm 0.004$
difference | $+21\sigma$

The ratio of white dwarfs whose photometry is best fit by hydrogen models to
those best fit by helium models is about eleven to one among the entire
sample. This is much larger than the ratio of DA white dwarfs to non-DAs (i.e.
DB, DC, DO and DQ) which is about three or four to one (Kilic et al., 2020,
and § 3.2; c.f. Guo et al. 2022 found a ten-to-one ratio); therefore, possibly
many of objects classified as hydrogen atmosphere white dwarfs actually do not
have hydrogen atmospheres. Looking more closely, the binary fraction of the
helium-atmosphere white dwarfs is about 12% compared to less than 5% for those
white dwarf best fit by hydrogen atmospheres. Given the sizes of the samples,
this difference is significant at the $21-\sigma$ level (the significance is
calculated directly from the binomial distribution, not the Poisson or normal
approximations). However, does this indeed imply that white dwarfs with
hydrogen atmospheres are nearly three times less likely to be in a wide binary
than those with helium atmospheres?

Figure 1 depicts the cumulative mass distribution (using the masses determined
from the photometric fits) of the white dwarfs with best-fitting hydrogen and
helium models and within those two classes the objects with wide-binary
companions. The stars best-fitted with hydrogen atmospheres typically have
larger masses than those best-fitted with helium atmospheres. This contradicts
what one expects from spectroscopic determinations of the white-dwarf
atmospheres where DB white dwarfs typically have larger masses than DA
(Genest-Beaulieu & Bergeron, 2019; Kilic et al., 2020, and§ 3.2). Furthermore,
for both categories, the masses of those stars with companions are typically
smaller than those without companions. A second diagnostic is to examine the
distribution of projected separations between the white dwarfs and their
companions (Fig. 2). The companions of those white dwarfs best fitted by
helium atmospheres are typically closer than those best-fitted with hydrogen
atmospheres. The median angular distance between a white dwarf fitted with a
hydrogen atmosphere is 11 arcseconds, and for a helium atmosphere the
corresponding distance is just 3.5 arcseconds. These results indicate that
perhaps the photometric fitting is affected by the presence of a companion:
the presence of a companion may increase the likelihood that a star is better
fitted by a helium atmosphere and by a lower mass model. Although the details
are somewhat different, these distributions are similar to the distributions
for subsamples within 100 pc and for subsamples with $u-$band photometry,
which may allow for more precise determination of the atmospheric parameters
from photometry; consequently, these trends bear further scrutiny through the
examination of a sample of spectroscopically classified white dwarfs.

Figure 1: The Cumulative Distribution of Masses of Hydrogen and Helium
Atmosphere White Dwarfs Obtained by Photometric Modelling. The median mass of
each population is presented in the legend. Figure 2: The Cumulative
Distribution of Projected Binary Separations of Hydrogen and Helium Atmosphere
White Dwarfs Identified by Photometric Modelling

###  3.2 Spectroscopic Classification

The MWDDB provides spectroscopic classifications of white dwarf atmospheres as
well as estimates of their masses. The database contains 732 unique spectral
types. Among those white dwarfs that meet the astrometric criteria of the
binary catalogue (43,324), more than half (21,963) fall within the following
spectral types: DA, DB, DC, DO and DQ. The catalogue contains subtypes of
these classes such as (DA1-2, DA3.0 and DB1.8) which are included in the
corresponding major type such as DA, but stars with other non-numerical
subtype designations (such as DAH or DAZ) and spectroscopic binaries are
excluded from this analysis, resulting in the numbers of white dwarfs of
various classes shown in Table 2. Again for all of the non-DA classes of white
dwarfs, the binary fraction is significantly larger 8.0% compared to 6.3% for
DA stars, a nearly five-sigma difference. The result of a binary fraction of
6.3% for DA stars agrees with the result of Noor et al. (2024) of
$6.8^{+3.1}_{-2.4}\%$ for a sample of 82 stars. Of course, this is much less
dramatic than the result for the photometric classification.

Table 2: Results of Spectroscopic Classification Number of objects in MWDDB | 70681
---|---
meet binary catalogue astrometric criteria | 43324
DA classification | 17692
with a wide binary companion | 1370
with $R_{\textrm{\scriptsize chance}}<0.1$ | 1109
binary fraction | $0.063\pm 0.002$
DB classification | 1277
with a wide binary companion | 115
with $R_{\textrm{\scriptsize chance}}<0.1$ | 89
binary fraction | $0.070\pm 0.007$
difference | $+1.1\sigma$
DC classification | 2533
with a wide binary companion | 233
with $R_{\textrm{\scriptsize chance}}<0.1$ | 211
binary fraction | $0.083\pm 0.005$
difference | $+4.1\sigma$
DO classification | 42
with a wide binary companion | 3
with $R_{\textrm{\scriptsize chance}}<0.1$ | 1
binary fraction | $0.02\pm 0.02$
difference | $-0.7\sigma$
DQ classification | 419
with a wide binary companion | 42
with $R_{\textrm{\scriptsize chance}}<0.1$ | 42
binary fraction | $0.10\pm 0.01$
difference | $+3.0\sigma$
Subtotal of Non-DA classifications above | 4271
with a wide binary companion | 393
with $R_{\textrm{\scriptsize chance}}<0.1$ | 343
binary fraction | $0.080\pm 0.004$
difference | $+4.6\sigma$

An examination of the distribution of the masses of the white dwarfs in each
class compared to the distribution on those within binaries (Fig. 3) does not
uncover potential systematic effects. The typical masses of DB, DC and DO
stars are larger than those of the DA and DQ stars; furthermore, the typical
masses of the DA, DB and DQ stars in binaries are similar to the entire
population. The typical masses of DC stars in binaries are slightly less than
the general population of DC stars. Both the distribution of DA and DQ stars
in binaries are slightly more peaked than the general population. The masses
of DB stars in binaries are consistent with being drawn from the general
population.

Figure 3: The Cumulative Distribution of Masses of White Dwarfs of Various
Spectral Types Obtained by Spectroscopic Modelling. The median mass of each
population is presented in the legend. Only a single DO was found in a binary,
so no cumulative distribution is depicted.

Focusing on those white-dwarf stars with companions, Figure 4 depicts the
distribution of the projected separations of the white dwarfs from their
companions. The DA stars have a median separation from their companions of
about 1700 AU while the DBs are typically a bit larger at 2300 AU and the DCs
and DQs are smaller at 1300 AU. All of the distributions are approximately
log-normal with a width of about a factor of four. The median angular distance
between the white dwarfs and their companions ranges from 10 to 14 arcseconds,
sufficiently distant that it would be difficult to argue that the companion
affects the spectroscopic classification of the white dwarfs. In fact the DC
white dwarfs which are likely to be the closest to their companions actually
appear further in angular separation.

Figure 4: The Cumulative Distribution of Projected Binary Separations of White
Dwarfs of Various Spectral Types. Upper: in physical distance, Lower: angular
sepration.

##  4 Conclusions

The paper examines the simple question of whether the spectral type of a white
dwarf correlates with its presence in a binary system. Surprisingly, the
evidence indicates that white dwarfs with hydrogen features in their spectra
(DA white dwarfs) are nearly 25% less likely (at 6.3%) to have a companion
than those white dwarfs that lack hydrogen features (DB, DC, DO and DQ white
dwarfs, at 8.0%). The subclass of DQ white dwarfs (those that exhibit carbon
features) are the most likely (among these classes) to have companions at 10%.
So what does binarity have to do with the formation of white dwarfs that lack
hydrogen atmospheres? Perhaps nothing. The DB and DC white dwarfs both in
binaries and in general typically are more massive that the DA white dwarfs
indicating that they likely were born from more massive stars which are more
likely to have companions. The modest increase in typical mass from
$0.57\textrm{M}_{\odot}$ for DAs to $0.62\textrm{M}_{\odot}$ for DBs and DCs,
corresponds to an increase in the mass of the progenitor from
$1.0\textrm{M}_{\odot}$ to $1.6\textrm{M}_{\odot}$ (Cummings et al., 2018),
and a corresponding increase in the binary fraction, for a similar range of
separations, of the progenitors from 7.5% to 9% (Moe & Di Stefano, 2017). This
explanation leaves out the DQ stars which exhibit the largest binary fraction
at 10%, the closest separations and the smallest typical mass at
$0.55\textrm{M}_{\odot}$. In the case of these stars, perhaps the presence of
their progenitors in a binary system led to their manifestation as DQ white
dwarfs, or perhaps their masses have been underestimated systematically and
the DQ stars originate from even more massive stars than the DB and DC stars
with correspondingly larger binary fractions.

## Acknowledgements

Simon Blouin posed brought up the question of the binary fraction of white
dwarfs after Hiba tu Noor’s presentation at EuroWD24. JH acknowledges support
from the Natural Sciences and Engineering Research Council of Canada (NSERC)
through a Discovery Grant. This work presents results from the European Space
Agency space mission Gaia, where data are processed by the Gaia Data
Processing and Analysis Consortium.

## Data Availability

The catalogue of candidate binaries identified by El-Badry et al. (2021) is
available at https://zenodo.org/records/4435257. The Gentile Fusillo et al.
(2021) catalogues is accessible at https://cdsarc.cds.unistra.fr/viz-
bin/cat/J/MNRAS/508/3877. The Montreal White Dwarf Database (Dufour et al.,
2017) can be downloaded from https://www.montrealwhitedwarfdatabase.org/. The
catalogues created for this work can be downloaded at
https://zenodo.org/records/12803292.

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