Algebraically hyperbolic manifolds have finite automorphism groups

Fedor Bogomolov, Ljudmila Kamenova111Partially supported by a grant from the
Simons Foundation/SFARI (522730, LK), Misha Verbitsky222Partially supported by
the Russian Academic Excellence Project ’5-100’, CNPq Process 313608/2017-2
and FAPERJ E-26/202.912/2018.

Abstract. A projective manifold $M$ is algebraically hyperbolic if there
exists a positive constant $A$ such that the degree of any curve of genus $g$
on $M$ is bounded from above by $A(g-1)$. A classical result is that Kobayashi
hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi
hyperbolic manifolds have finite automorphism groups. Here we prove that, more
generally, algebraically hyperbolic projective manifolds have finite
automorphism groups.

##  1 Introduction

J.-P. Demailly introduced an algebraic analogue for the analytic notions of
hyperbolicity in [De]. Using the classical Gauss-Bonnet formula he showed that
every compact complex Kobayashi hyperbolic manifold is also algebraically
hyperbolic. Demailly conjectured that for complex projective manifolds the two
notions, Kobayashi hyperbolicity and algebraic hyperbolicity, coincide.

Kobayashi proved that Kobayashi hyperbolic manifolds have only finite order
automorphisms, [Ko]. In [KV] L. Kamenova and M. Verbitsky proved that
algebraically hyperbolic projective hyperkähler manifolds have finite groups
of automorphisms. The general expectation is that all projective hyperkähler
manifolds are algebraically non-hyperbolic.

Theorem 1.1: (Kamenova, Verbitsky) Let $M$ be an algebraically hyperbolic
projective hyperkähler manifold. Then $M$ has finite group of automorphisms.

Here we generalize the result above and show that all algebraically hyperbolic
projective manifolds have finite groups of automorphisms.

Theorem 1.2: The group $\operatorname{Aut}(M)$ of automorphisms of an
algebraically hyperbolic manifold $M$ is finite.

The idea behind the proof of our main theorem is the following. We consider
the induced action of the automorphism group $\operatorname{Aut}(M)$ on the
cohomology $H^{1,1}(M,{\mathbb{R}})$. From the Hodge-Riemann relations it
follows that the image of $\operatorname{Aut}(M)$ in
$GL(H^{1,1}(M,{\mathbb{R}}))$ has to preserve some rational Kähler class.
Furthermore, we consider the case when the image of $\operatorname{Aut}(M)$ in
$\operatorname{Aut}(\operatorname{Pic}^{0}(M))$ is infinite. Using the
Albanese map we produce a subvariety of $M$ that admits self-isogenies of
arbitrarily high order, thus giving curves of constant genus and arbitrary
high degree in this subvariety, i.e., we obtain an algebraically non-
hyperbolic subvariety. This contradiction implies that the image of
$\operatorname{Aut}(M)$ in $\operatorname{Aut}(\operatorname{Pic}^{0}(M))$ is
finite. If we assume that $\operatorname{Aut}(M)$ is infinite, there would be
an infinite subgroup $\Gamma$ acting trivially on
$\operatorname{Aut}(\operatorname{Pic}(M))$. It fixes an ample line bundle
$L\in\operatorname{Pic}(M)$. The action of $\Gamma$ on
${\mathbb{P}}H^{0}(M,L)^{*}$ preserves the image of the projective embedding
$M\hookrightarrow{\mathbb{P}}H^{0}(M,L)^{*}$. Let $\overline{\Gamma}$ be the
Zariski closure of $\Gamma$ in $PGL(H^{0}(M,L)^{*})$. Then the orbits of
$\overline{\Gamma}$ are positive dimensional, which is a contradiction,
because $\operatorname{Aut}(M)$ is discrete when $M$ is algebraically
hyperbolic.

##  2 Brody curves, Kobayashi and algebraic hyperbolicity

In this section let $M$ be a compact complex manifold. We introduce some basic
hyperbolicity notions that can be found in [Ko] and [De]. Brody introduced the
Brody curves in [Br].

###  2.1 Kobayashi pseudometric and Brody curves

Definition 2.1: A pseudometric on $M$ is a function $d:\;M\times
M{\>\longrightarrow\>}{\mathbb{R}}^{\geqslant 0}$ which is symmetric:
$d(x,y)=d(y,x)$ and satisfies the triangle inequality $d(x,y)+d(y,z)\geqslant
d(x,z)$.

Remark 2.2: Let ${\mathfrak{D}}$ be a set of pseudometrics on $M$. Then
$d_{{\rm max}}(x,y):=\sup_{d\in{\mathfrak{D}}}d(x,y)$ is also a pseudometric.

Definition 2.3: The Kobayashi pseudometric $d_{M}$ on a complex manifold $M$
is $d_{M}:=d_{{\rm max}}$ for the set ${\mathfrak{D}}$ of all pseudometrics on
$M$ such that any holomorphic map from the Poincaré disk to $M$ is distance-
decreasing.

Remark 2.4: The definition above is equivalent to the standard definition that
the Kobayashi distance between points $x,y\in M$ is the infimum of the
Poincaré distance over all sets of Poincaré disks connecting $x$ to $y$.

Definition 2.5: A manifold $M$ is called Kobayashi hyperbolic if the Kobayashi
pseudometric $d_{M}$ is non-degenerate.

Definition 2.6: Let $M$ be a complex Hermitian manifold. A Brody curve is a
non-constant holomorphic map $f:\;{\mathbb{C}}{\>\longrightarrow\>}M$ such
that $|df|\leqslant C$ for some constant $C$. Here $|df|$ is understood as an
operator norm of $df:\;T_{z}{\mathbb{C}}{\>\longrightarrow\>}TM$, where
${\mathbb{C}}$ is equipped with the standard Euclidean metric.

Definition 2.7: Let $(\Delta_{r},g_{r})$ be a disk of radius $r$ in
${\mathbb{C}}$ with the Poincaré metric $g_{r}$, rescaled in such a way that
the unit tangent vector to $0$ has length $1$. A Brody map to a Hermitian
complex manifold $M$ is a map $f:\;\Delta_{r}{\>\longrightarrow\>}M$ such that
$|df|\leqslant 1$ (here the operator norm is taken with respect to the
Poincaré metric on $\Delta_{r}$) and $|df|(z)=1$ at $z=0$.

Lemma 2.8:  Let $f_{r}:\;\Delta_{r}{\>\longrightarrow\>}M$ be a sequence of
Brody maps with $r{\>\longrightarrow\>}\infty$. Then $\\{f_{r}\\}$ has a
subsequence which converges uniformly to a Brody curve $f$ satisfying
$|df|(z)=1$ at $z=0$.

Proof. Let $r_{1}<r_{2}$. The identity map
$\tau:\;(\Delta_{r_{1}},g_{1}){\>\longrightarrow\>}(\Delta_{r_{2}},g_{2})$ is
1-Lipschitz. Indeed, it is 1-Lipschitz with respect to the usual Poincaré
metric: $\tau^{*}(r_{2}^{-2}g_{2})\leqslant r_{1}^{-2}g_{1}$. Since
$r_{1}<r_{2}$, this gives $\tau^{*}g_{2}\leqslant g_{1}$. Restricted on any
disk $\Delta_{R}$, the family $\\{f_{r},r>R\\}$ is a normal family (since it
is Lipschitz), hence it has a subsequence which converges uniformly to a
Lipschitz map. Since a uniform limit of holomorphic maps is holomorphic, the
family
$\left\\{f_{r}{\left|{}_{{\phantom{|}\\!\\!}_{\Delta_{R}}}\right.},r>R\right\\}$
converges to a holomorhic map on
$\varphi_{R}:\;\Delta_{R}{\>\longrightarrow\>}M$. The map $\varphi_{R}$ is
Lipschitz with respect to all metrics $g_{r}$, $r>R$. Since
$\lim\limits_{r{\>\longrightarrow\>}\infty}g_{r}$ is the standard Euclidean
metric $g_{\infty}$, the map $\varphi_{R}$ is Lipschitz with respect to
$g_{\infty}$. The sequence
$\lim\limits_{R{\>\longrightarrow\>}\infty}\varphi_{R}$ converges to a
holomorphic Lipschitz map ${\mathbb{C}}{\>\longrightarrow\>}M$. Since all
$f_{r}$ and $\varphi_{R}$ satisfy $|d\varphi_{R}|(z)=1$ at $z=0$, the same is
true for the limit.

A classical result of Brody states that a compact complex manifold $M$ is
Kobayashi hyperbolic if and only if there are no non-constant holomorphic maps
from ${\mathbb{C}}$ to $M$.

Theorem 2.9: (Brody’s lemma) Let $M$ be a compact complex manifold which is
not Kobayashi hyperbolic. Then $M$ contains a Brody curve.

Proof: Let us equip $M$ with a Hermitian metric $h$. If $|df|(0)\leqslant C$
for any holomorphic map $(\Delta_{1},g_{1}){\>\longrightarrow\>}M$, then the
Kobayashi pseudometric satisfies $d_{K}\geqslant C^{-1}h$, and $M$ is
Kobayashi hyperbolic. If this quantity is non-bounded, we can always rescale
the disc to obtain a map $f_{r}:\;(\Delta_{r},g_{r}){\>\longrightarrow\>}M$
with $r=|df|(0)$, and then $|df_{r}|(0)=1$. Then Brody’s lemma follows from
2.1 and the following lemma.

Lemma 2.10: Let $M$ be a compact Hermitian manifold, and
$\psi_{r}:\;(\Delta_{r},g_{r}){\>\longrightarrow\>}M$ a sequence of
holomorphic maps satisfying $|d\psi_{r}|(0)\geqslant 1$,
$r{\>\longrightarrow\>}\infty$. Then there exists a sequence of Brody maps
$f_{s}:\;(\Delta_{s},g_{s}){\>\longrightarrow\>}M$, with
$s{\>\longrightarrow\>}\infty$.

Proof. We need to construct a sequence of Brody maps, which are 1-Lipschitz
maps $f_{s}:\;(\Delta_{s},g_{s}){\>\longrightarrow\>}M$, with $|df_{s}|(0)=1$.
The identity map

$\Psi_{r-\varepsilon,r}:\;(\Delta_{r-\varepsilon},g_{r-\varepsilon}){\>\longrightarrow\>}(\Delta_{r},g_{r})$

is 1-Lipschitz, and satisfies

$\lim_{z{\>\longrightarrow\>}\partial\Delta_{r-\varepsilon}}|d\Psi_{r-\varepsilon,r}|(z)=0.$

Let $u:=r-\varepsilon$ and
$\widetilde{f}_{u}:=\Psi_{r-\varepsilon,r}\circ\psi_{r}$ be a restriction of
$f_{r}$ to the disk $(\Delta_{r-\varepsilon},g_{r-\varepsilon})$. Then $f_{u}$
is also Lipschitz and $|d\widetilde{f}_{u}|$ reaches maximum at a point
$z_{u}$ somewhere inside the disk $\Delta_{u}$. Applying an appropriate
holomorphic isometry of $\Delta_{u}$, we may assume that
$|d\widetilde{f}_{u}|(z)$ takes maximum $C_{u}\geqslant 1$ for $z=0$.
Rescaling $\widetilde{f}_{u}$, and setting $s:=C_{u}u$, we obtain a map
$f_{s}:\;\Delta_{s}{\>\longrightarrow\>}M$ which is 1-Lipschitz and satisfies
$|df_{s}|\leqslant 1$, $|df_{s}|(0)=1$.

Definition 2.11: Let $M$ be a projective manifold. We say that $M$ is
algebraically hyperbolic if there exists a constant $A>0$ such that for any
curve $C\subset M$ of geometric genus $g$ one has $\deg C<A(g-1)$.

Remark 2.12: Algebraically hyperbolic manifolds contain no elliptic nor
rational curves.

Using Gauss-Bonnet in [De] Demailly proved that Kobayashi hyperbolicity
implies algebraic hyperbolicity. Here we’ll give a slightly different proof of
this well known result. The converse implication, “algebraically hyperbolicity
implies Kobayashi hyperbolicity”, was conjectured by J.-P. Demailly who
introduced the notion of algebraic hyperbolicity in [De].

Theorem 2.13: (Demailly, [De]) Kobayashi hyperbolicity implies algebraic
hyperbolicity.

Proof. Let $C\subset M$ be a curve in a Kobayashi hyperbolic manifold. The
genus $0$ and $1$ cases are ruled out by Brody’s Lemma, 2.1. Then genus
$g=g(C)$ is greater than $1$, and its universal cover is the disk $\Delta$.
Denote by $\varphi_{C}:\;\Delta{\>\longrightarrow\>}M$ the universal covering
map. The volume of $C$ with respect to the Fubini-Study metric on $M$ is $\deg
C$, and its volume with respect to the Poincaré metric is
$\int_{C}c_{1}(T^{*}C)=2\pi(2g-2)$. Therefore, $|\varphi_{C}|>\frac{\deg
C}{2\pi(2g-2)}$ somewhere on $\Delta$. Notice that $M$ is not algebraically
hyperbolic if and only if there is a sequence $\\{C_{i}\\}$ of curves in $M$
with $\lim\limits_{i{\>\longrightarrow\>}\infty}\frac{\deg
C_{i}}{2\pi(2g(C_{i})-2)}{\>\longrightarrow\>}\infty$. Suppose that $M$ is not
algebraically hyperbolic. This gives a sequence
$\varphi_{C_{i}}:\;\Delta_{1}{\>\longrightarrow\>}M$ with
$|\varphi_{C_{i}}|>\frac{\deg C_{i}}{2\pi(2g(C_{i})-2)}$ somewhere on
$\Delta$. Replacing $\Delta_{1}$ by $\Delta_{1-\varepsilon}$ as above if
necessary, we may assume that $|\varphi_{C_{i}}|$ reaches its maximum
somewhere on $\Delta_{1}$. Applying an isometry of $\Delta_{1}$, we may assume
that $|\varphi_{C_{i}}|$ reaches its maximum $R_{i}$ in $0\in\Delta_{1}$.
Rescaling $\varphi_{C_{i}}$ by $R_{i}$, we obtain a sequence of disks
$\widetilde{\varphi}_{C_{i}}(z)=\varphi_{C_{i}}(z/R_{i}):\;\Delta_{R_{i}}{\>\longrightarrow\>}M$,
giving a Brody curve by 2.1.

###  2.2 Algebraic Kobayashi metric

Definition 2.14: The Kobayashi pseudometric $d_{K}$ is defined as the infimum
of the path metric on connected chains of holomorphic disks equipped with the
Poincaré metrics. Define the algebraic Kobayashi pseudometric $d_{A}$ using
the Poincaré metric on connected chains of algebraic curves instead of the
disks.

Remark 2.15: This metric clearly satisfies $d_{A}\geqslant d_{K}$. In [DLS]
Demailly, Lempert and Shiffman prove that $d_{A}=d_{K}$ for any quasi-
projective manifold, i.e., one can compute the Kobayashi pseudodistance by
means of chains of algebraic curves. Also, algebraic hyperbolicity follows if
$d_{A}$ is non-degenerate; this is proven by reverse-engineering the argument
we used to prove that algebraic hyperbolicity is implied by Kobayashi
hyperbolicity. It is not clear, however, if algebraic hyperbolicity implies
the non-degeneracy of $d_{A}=d_{K}$.

Question 2.16:  Is $d_{A}$ always non-degenerate for algebraically hyperbolic
manifolds?

Remark 2.17: Note that the group $\operatorname{Aut}(M)$ of holomorphic
automorphisms of a complex manifold $M$ with $d_{A}$ non-degenerate is
necessarily compact. Indeed, any holomorphic automorphism of $M$ is an
isometry of $(M,d_{A})$, and the group of isometries of a compact metric space
is compact. On the other hand, a uniform limit of holomorphic maps is
holomorphic, hence compactness of the isometry group implies compactness of
$\operatorname{Aut}(M)$. Since an algebraically hyperbolic manifold has
discrete group of holomorphic automorphisms by 2.3, the group
$\operatorname{Aut}(M)$ is finite. Then an affirmative answer to 2.2 would
imply the main statement of this paper: finiteness of $\operatorname{Aut}(M)$
for algebraically hyperbolic $M$.

Remark 2.18: For any $g$ there are complex elliptic fibrations over
${\mathbb{P}}^{1}$ without multiple fibers where all horizontal curves (i.e.,
curves surjecting onto ${\mathbb{P}}^{1}$ under the projection) have genus
greater than $g$ (Remark 6.11 in [BT]). The Kobayashi pseudometric on such
surfaces is trivial but there are no elliptic or rational curves connecting
points in different fibers. This example illustrates the difference in
obtaining the Kobayashi pseudometric on the surface from mapping complex
surfaces and algebraic curves. We can connect any two points in the surface by
the image of a complex line, as shown in [BL]. However, we need to take a
limit over all chains of algebraic curves connecting points in different
fibers in order to obtain the Kobayashi pseudometric using only algebraic
curves.

###  2.3 Automorphisms of hyperbolic manifolds

In the rest of the section we summarize some facts about the automorphism
group of hyperbolic manifolds.

Proposition 2.19:  The group of automorphisms of an algebraically hyperbolic
manifold $M$ is discrete.

Proof. The group of automorphisms $G$ of a projective manifold $M$ is a
complex Lie group. If its connected component $G^{0}$ is non-trivial, this
gives a holomorphic map $\varphi:\;G^{0}{\>\longrightarrow\>}M$, depending on
the choice of a point. Any connected complex algebraic Lie group is an
extension of an affine group and an abelian variety. Affine algebraic groups
are rational varieties, hence the orbit of an affine algebraic group is
unirational and covered by rational curves. An abelian variety is not
algebraically hyperbolic, because an abelian variety of dimension $n$ admits a
self-isogeny of order $m^{n}$ mapping a curve $C$ of genus $g$ to a curve of
genus $g$ and degree $m^{n}\deg C$. The same argument shows that any positive-
dimensional orbit of a compact complex commutative Lie group is not
algebraically hyperbolic. Notice that the image of an algebraically non-
hyperbolic manifold is also not algebraically hyperbolic. Therefore, $M$
cannot be algebraically hyperbolic.

An even stronger statement is true for Kobayashi hyperbolic manifolds.

Proposition 2.20: (Kobayashi, [Ko]) The group of automorphisms of a compact
Kobayashi hyperbolic manifold $M$ is finite.

Proof. The group $G$ of automorphisms of $M$ is closed in its group of
isometries under the Kobayashi pseudometric. The group of isometries of a
compact metric space is compact, hence $G$ has only finitely many connected
components. Finally, $\dim G^{0}=0$ as shown in 2.3, i.e., $G$ is discrete and
compact, therefore finite.

##  3 Main Results

In this section $M$ is a compact complex projective manifold. For the
classical notions of the Picard scheme and the Albanese variety we refer the
readers to the survey manuscript [Kl].

Proposition 3.1:  Let $M$ be a complex projective manifold,
$\dim_{\mathbb{C}}M=n$. Suppose that the image of $\operatorname{Aut}(M)$ in
$GL(H^{1,1}(M,{\mathbb{R}}))$ does not preserve any rational Kähler class.
Then $M$ is not algebraically hyperbolic.

Proof. Assume $M$ is algebraically hyperbolic. Let $\omega$ be a rational
Kähler class which has an infinite $\operatorname{Aut}(M)$-orbit. Replacing
$\omega$ by $N\omega$, from the $(1,1)$-theorem we may assume that $\omega$ is
a class of a hyperplane section. Then $\omega^{n-1}=[C]$ is the fundamental
class of a smooth complex curve $C\subset M$ from Bertini’s theorem. Let
$f_{i}(\omega)$ be in the orbit of $\omega$, which is infinite by assumption.
Then

$\deg_{\omega}f_{i}(C)=\int_{M}\omega\wedge(f_{i}(\omega))^{n-1}=\int_{M}f_{i}^{-1}(\omega)\wedge\omega^{n-1}.$

Since the genus of $f_{i}(C)$ is constant, from algebraic hyperbolicity we
obtain $\int_{M}f_{i}(\omega)\wedge\omega^{n-1}<A$ for some constant $A>0$.

Let $|\cdot|$ denote the positive-definite Hodge-Riemann metric on
$H^{p,q}(M,{\mathbb{R}})$, that is, the metric defined as
$\eta,\eta^{\prime}{\>\longrightarrow\>}-(-1)^{l}(\sqrt{-1}\>)^{p-q}\int_{M}\eta\wedge\overline{\eta^{\prime}}\wedge\omega^{n-p-q}$,
where $\eta,\eta^{\prime}$ are $(p,q)$-forms which belong to a weight $l$
representation of $\mathfrak{sl}(2)$ associated with the Lefschetz
$\mathfrak{sl}(2)$-action. Replace the sequence
$\frac{f_{i}(\omega)}{|f_{i}(\omega)|}$ of points in the sphere $S\subset
H^{1,1}(M,{\mathbb{R}})$ by a converging subsequence, and let $R$ be its
limit. Since the sequence $f_{i}(\omega)$ is infinite, distinct and integral,
one has $\lim\limits_{i{\>\longrightarrow\>}\infty}|f_{i}(\omega)|=\infty$.
Then $\int_{M}f_{i}(\omega)\wedge\omega^{n-1}<A$ implies that
$\int_{M}R\wedge\omega^{n-1}=0$. By the Hodge-Riemann relations, this gives
$\int_{M}R\wedge R\wedge\omega^{n-2}=-|R|^{2}=-1$, hence

$\lim_{i}\frac{\int_{M}f_{i}(\omega)\wedge
f_{i}(\omega)\wedge\omega^{n-2}}{|f_{i}(\omega)|^{2}}=-1.$

This is a contradiction, because $f_{i}(\omega)$ is Kähler, hence
$\int_{M}f_{i}(\omega)\wedge f_{i}(\omega)\wedge\omega^{n-2}>0$.

Proposition 3.2:  Let $M$ be a complex projective manifold. Suppose that
$\operatorname{Aut}(M)$ is infinite, but the image of $\operatorname{Aut}(M)$
in $\operatorname{Aut}(\operatorname{Pic}(M))$ is finite. Then $M$ is not
algebraically hyperbolic.

Proof. From the assumption in the proposition we obtain that an infinite
subgroup $\Gamma\subset\operatorname{Aut}(M)$ acts trivially on
$\operatorname{Pic}(M)$. Then it fixes a very ample line bundle
$L\in\operatorname{Pic}(M)$. Therefore, $\Gamma$ acts on
${\mathbb{P}}H^{0}(M,L)^{*}$ preserving the image of the projective embedding
$M{\>\longrightarrow\>}{\mathbb{P}}H^{0}(M,L)^{*}$. Let $G=\overline{\Gamma}$
be the Zariski closure of $\Gamma$ in $PGL(H^{0}(M,L)^{*})$. Since $M$ is
Zariski closed and $\Gamma$-invariant, it is $G$-invariant. Since $\Gamma$
acts on $M$ with infinite orbits, the orbits of the $G$-action on $M$ are
positive-dimensional. This is impossible, because $\operatorname{Aut}(M)$ is
discrete as shown in 2.3.

Remark 3.3:  Notice that if the image ${\mathfrak{G}}$ of
$\operatorname{Aut}(M)$ in $GL(H^{1,1}(M,{\mathbb{R}}))$ does not preserve any
rational Kähler classes, then ${\mathfrak{G}}$ is infinite. Otherwise, take a
finite orbit of a Kähler class and notice that its geometric center is an
$\operatorname{Aut}(M)$-invariant Kähler class, because the convex hull of a
set of Kähler classes lies inside the Kähler cone.

Remark 3.4:  When $M$ is projective and the image ${\mathfrak{G}}$ of
$\operatorname{Aut}(M)$ in $GL(H^{1,1}(M,{\mathbb{R}}))$ is finite, we could
take a ${\mathfrak{G}}$-orbit of a rational Kähler class, and its geometric
center is also rational. Therefore, any projective manifold $M$ with
${\mathfrak{G}}$ finite admits an $\operatorname{Aut}(M)$-invariant integer
Kähler class.

Proposition 3.5:  Let $M$ be a complex projective manifold. Suppose that the
image of $\operatorname{Aut}(M)$ in $GL(H^{1,1}(M,{\mathbb{R}}))$ is finite,
but its image in $\operatorname{Aut}(\operatorname{Pic}^{0}(M))$ is infinite.
Then $M$ is not algebraically hyperbolic.

Proof. Consider the orbit $\operatorname{Aut}(M)\cdot\alpha$ of a Kähler class
$\alpha$. Its geometric center gives an $\operatorname{Aut}(M)$-invariant
Kähler class $\omega$, because the convex envelope of a set of Kähler classes
lies in the Kähler cone, as in 3.

The Albanese manifold
$\operatorname{Alb}(M)=H^{0}(M,\Omega^{1}_{M})^{*}/H_{1}(M,{\mathbb{Z}})$
admits a natural $\operatorname{Aut}(M)$-invariant flat Kähler metric induced
by the Hodge-Riemann form on $H^{1}(M,{\mathbb{C}})$. Since
$\operatorname{Aut}(M)$ acts on $\operatorname{Alb}(M)$ by isometries, the
image ${\mathfrak{T}}$ of $\operatorname{Aut}(M)$ in
$\operatorname{Aut}(\operatorname{Alb}(M))$ is compact. The connected
component of $\operatorname{Aut}(\operatorname{Alb}(M))$ is a subgroup
$\operatorname{Par}(\operatorname{Alb}(M))=\operatorname{Alb}(M)\subset\operatorname{Aut}(\operatorname{Alb}(M))$
acting on the torus $\operatorname{Alb}(M)$ by parallel transport. Since
${\mathfrak{T}}$ is compact, its image in the discrete group
$\frac{\operatorname{Aut}(\operatorname{Alb}(M))}{\operatorname{Par}(\operatorname{Alb}(M))}$
is finite. Let $\Gamma\subset{\mathfrak{T}}$ be the kernel of the map
${\mathfrak{T}}{\>\longrightarrow\>}\frac{\operatorname{Aut}(\operatorname{Alb}(M))}{\operatorname{Par}(\operatorname{Alb}(M))}$.
This is a finite index subgroup in ${\mathfrak{T}}$. Therefore, $\Gamma$ is
infinite.

Since $\Gamma$ is infinite, its closure
$T\subset\operatorname{Par}(\operatorname{Alb}M)$ is positive-dimensional.
Take a smooth fiber $\operatorname{Alb}^{-1}(x)$ over
$x\in\operatorname{Alb}(M)$. The general fiber of the real analytic map
$\operatorname{Alb}^{-1}(T\cdot
x)\stackrel{{\scriptstyle\pi}}{{{\>\longrightarrow\>}}}T\cdot x$ is smooth.
Since all fibers of $\pi$ exist in dense families, all fibers of $\pi$ are
smooth. Therefore, $\pi$ is a locally trivial fibration with isomorphic
fibers. Consider the Zariski closure $T_{1}$ of $T$, which is a compact
commutative Lie group, that is, the compact complex torus. This gives an
isotrivial fibration
$\operatorname{Alb}^{-1}(X)\stackrel{{\scriptstyle\pi_{1}}}{{{\>\longrightarrow\>}}}X$,
where $X$ is an orbit of $T_{1}$.

Isotrivial fibrations over $T_{1}$ with fiber $F$ are classified by
$H^{1}(T_{1},\operatorname{Aut}(F))$. The variety $F\subset M$ may be assumed
to be an algebraically hyperbolic manifold with $\dim F<\dim M$. Using
induction on dimension, we may assume that $\operatorname{Aut}(F)$ is finite.

The first cohomology of a torus with coefficients in a finite group $A$ is the
same as an $A$-valued local system. Therefore, it becomes trivial after an
appropriate finite covering. Then $\pi_{1}$ becomes a trivial fibration after
passing to a finite covering $Y{\>\longrightarrow\>}X$, giving a decomposition
$\widetilde{\operatorname{Alb}^{-1}(X)}=F\times Y$. This manifold admits self-
isogenies of arbitrary high order, giving curves of constant genus and
arbitrary high degree in $\widetilde{\operatorname{Alb}^{-1}(X)}$ and its
finite quotient $\operatorname{Alb}^{-1}(X)$. Therefore,
$\operatorname{Alb}^{-1}(X)\subset M$ is not algebraically hyperbolic, and $M$
is algebraically non-hyperbolic.

Using the above results we can prove the main result of this paper.

Theorem 3.6: The group $\operatorname{Aut}(M)$ of automorphisms of an
algebraically hyperbolic manifold $M$ is finite.

Proof. Assume $\operatorname{Aut}(M)$ is infinite. If the image
${\mathfrak{G}}$ of the group $\operatorname{Aut}(M)$ in
$GL(H^{1,1}(M,{\mathbb{R}}))$ does not preserve any rational Kähler class,
${\mathfrak{G}}$ is infinite by 3. Then we would get a contradiction by 3. If
the image of $\operatorname{Aut}(M)$ in
$\operatorname{Aut}(\operatorname{Pic}(M))$ is finite, we get a contradiction
by 3, and if its image in $\operatorname{Aut}(\operatorname{Pic}(M))$ is
infinite, we get a contradiction by 3.

To sum the argument up: we consider the image ${\mathfrak{G}}$ of the group
$\operatorname{Aut}(M)$ in $GL(H^{1,1}(M,{\mathbb{R}}))$. If it is finite, we
have an integer Kähler class which is ${\mathfrak{G}}$-invariant. If
$\operatorname{Aut}(M)$ acts with finite image on the Albanese variety of $M$,
this class gives an $\operatorname{Aut}(M)$-invariant polarization, hence
$\operatorname{Aut}(M)$ is a linear algebraic group; its connected part is
trivial because the orbits of a connected linear algebraic group are rational.

If ${\mathfrak{G}}$ is finite, but $\operatorname{Aut}(M)$ acts with infinite
image on the Albanese variety of $M$, the closure of an orbit of
$\operatorname{Aut}(M)$ contains a torus, also contradicting hyperbolicity.

Finally, if ${\mathfrak{G}}$ is infinite, be obtain curves of bounded genus
and arbitrary degree using the action of ${\mathfrak{G}}$ on cohomology.

Acknowledgments. Most of the results in this paper were finalized during the
Komplexe Analysis Oberwolfach Workshop in 2017. The second and third-named
authors are grateful to the Oberwolfach organizers and staff for their
hospitality. The first-named author has been funded by the Russian Academic
Excellence Project ’5-100’ and acknowledges support by Simons travel grant and
by the EPSRC program grant EP/M024830. We are thankful to the referee for many
valuable comments and corrections to the earlier version.

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Fedor A. Bogomolov
Department of Mathematics
Courant Institute, NYU
251 Mercer Street
New York, NY 10012, USA,
<EMAIL_ADDRESS>also:
National Research University, Higher School of Economics, Moscow, Russia,

Ljudmila Kamenova
Department of Mathematics, 3-115
Stony Brook University
Stony Brook, NY 11794-3651, USA,
<EMAIL_ADDRESS>

Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry,
National Research University Higher School of Economics,
Department of Mathematics, 6 Usacheva street, Moscow, Russia.


#  Vortex Creep Heating vs. Dark Matter Heating in Neutron Stars

Motoko Fujiwara<EMAIL_ADDRESS>Physik-Department, Technische
Universität, München, James-Franck-Straße, 85748 Garching, Germany  Koichi
Hamaguchi<EMAIL_ADDRESS>Kavli IPMU (WPI), University of
Tokyo, Kashiwa, Chiba 277–8583, Japan  Department of Physics, University of
Tokyo, Bunkyo-ku, Tokyo 113–0033, Japan  Natsumi Nagata  natsumi@hep-
th.phys.s.u-tokyo.ac.jp Department of Physics, University of Tokyo, Bunkyo-ku,
Tokyo 113–0033, Japan  Maura E. Ramirez-Quezada  me.quezada@hep-
th.phys.s.u-tokyo.ac.jp Department of Physics, University of Tokyo, Bunkyo-ku,
Tokyo 113–0033, Japan  Dual CP Institute of High Energy Physics, C.P. 28045,
Colima, México

###### Abstract

Dark matter particles captured in neutron stars deposit their energy as heat.
This DM heating effect can be observed only if it dominates over other
internal heating effects in NSs. In this work, as an example of such an
internal heating source, we consider the frictional heating caused by the
creep motion of neutron superfluid vortex lines in the NS crust. The
luminosity of this heating effect is controlled by the strength of the
interaction between the vortex lines and nuclei in the crust, which can be
estimated from the many-body calculation of a high-density nuclear system as
well as through the temperature observation of old NSs. We show that both the
temperature observation and theoretical calculation suggest that the vortex
creep heating dominates over the DM heating. The vortex-nuclei interaction
must be smaller than the estimated values by several orders of magnitude to
overturn this.

##  I Introduction

Dark Matter (DM) is a mysterious gravitational source that remains one of the
biggest enigmas in the universe. Among the potential candidates, Weakly
Interacting Massive Particles (WIMPs) are particularly intriguing. The thermal
history of WIMPs in the expanding universe offers a natural explanation for
the observed energy density of DM, as determined by observations of the Cosmic
Microwave Background (CMB) [1]. WIMPs interact with particles in the Standard
Model (SM), providing avenues to probe their properties. The different
approaches, such as direct detection, indirect detection, and collider
searches, provide valuable information on the properties of DM. However,
despite extensive efforts, the conclusive identification of DM particles
remains an ongoing challenge in the field of particle physics.

Recently, there has been growing interest in using Neutron Star (NS)
temperature observation as a means of searching for WIMPs [2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35]. By observing old NSs, we can explore the
unknown properties of DM through its interactions with stellar matter. NSs
possess a strong gravitational potential, causing WIMP particles to fall and
become gravitationally bound to them. During this process, WIMP particles
scatter off the stellar matter, losing some of their kinetic energy to the NS
[36]. The capture of DM can occur via interactions with nucleons, leptons, and
even hyperons. If WIMPs lose enough energy, they will be captured within a NS
core, where they thermalize and eventually annihilate into SM particles. The
capture, thermalization, and annihilation process inject energy into the NS,
serving as a late-time heating source of old isolated NSs [2]. The heating
effect typically predicts a surface temperature of a few thousand Kelvin [31],
which is anomalously hotter compared to the prediction from the standard
cooling theory for NSs [37, 38, 39, 40, 41, 42] that are older than
$10^{6}~{}\mathrm{yrs}$. Therefore, by observing NSs colder than this
predicted WIMP temperature, we may be able to place constraints on the WIMP-
nucleon cross section.

In the context of NS temperature observations to probe DM properties, it is
commonly assumed that the heating effects from DM dominate a late heating
source for the star. However, observations have revealed the existence of old
but still warm NSs with surface temperatures typically exceeding
$10^{5}~{}\mathrm{K}$ [43, 44, 45, 46, 47, 48, 49]. The presence of such high
temperatures in these compact objects suggests that additional internal
heating mechanisms must be at play. Since DM heating alone cannot explain the
observed temperatures, other heating sources must contribute to the thermal
energy of these objects. For a review of such a heating mechanism in NSs, see
Refs. [50, 51, 52]. If NSs are primarily heated by internal mechanisms that
predict a universal temperature as high as $10^{5}~{}\mathrm{K}$, as observed
recently, it would indeed pose challenges for probing WIMP through future
temperature observations [14].

The vortex creep heating [53, 54, 55, 56, 57, 58, 59, 60, 50, 61, 51, 62] is a
universal and internal heating mechanism. This heating is caused by friction
associated with the creep motion of vortex lines in the neutron superfluid of
the inner crust region. Its heating luminosity is proportional to the time
derivative of the angular velocity of the NS, and the proportional
coefficients (denoted by $J$ in this paper) are expected to be universal and
determined by the nuclear-vortex interaction in the inner crust region. We can
in principle evaluate this parameter $J$ by means of a nuclear many-body
calculation [63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]. It
is also possible to determine this from the NS surface temperature
observations by fitting the data with the prediction of the vortex creep
heating mechanism. In Ref. [62], we showed that the values of $J$ obtained
with these two approaches are consistent with each other. In the present
paper, we discuss the implications of this result for the WIMP DM search
through the NS surface temperature observations.

##  II Thermal history of neutron stars

We begin by reviewing the thermal history of NSs [37, 38, 39]. The
thermalisation process of NSs is typically completed by $t\lesssim
10^{2}~{}\mathrm{yrs}$ [78, 79], and the subsequent thermal evolution remains
largely independent of the initial conditions for each star. The following
equation describes the thermal evolution of NSs,

$\displaystyle\mathcal{C}^{\infty}(T^{\infty})\frac{dT^{\infty}}{dt}$
$\displaystyle=-L^{\infty}_{\nu}-L^{\infty}_{\gamma}+L^{\infty}_{\rm heat},$
(1)

where $\mathcal{C}^{\infty}(T^{\infty})$ is the heat capacity of the star. On
the right-hand side, the terms with a negative sign, $L_{\nu}$ and
$L_{\gamma}$, are redshifted luminosity for cooling by the neutrino and photon
emissions, respectively. The term with a positive sign, $L^{\infty}_{\rm
heat}$, denotes the redshifted luminosity for heating if any exists.

The early phase of cooling for $t\lesssim 10^{5}~{}\mathrm{yrs}$ occurs with
neutrino emission such as direct Urca [80], modified Urca [81, 82], and Pair-
breaking and formation (PBF) process [83, 84, 85]. The late phase of cooling
for $t\gtrsim 10^{5}~{}\mathrm{yrs}$ is associated with photon emission, and
the cooling luminosity is given by the Stefan-Boltzmann law. If NSs have a
heating source, the associated heating luminosity and the photon cooling
luminosity will eventually reach thermal equilibrium. Therefore, the late-time
surface temperature of NSs will be characterised by their luminosity through
the following relation.

$\displaystyle L^{\infty}_{\rm heat}\simeq L^{\infty}_{\gamma}=B(R_{\rm
NS})\times 4\pi R_{\rm NS}^{2}\sigma_{\rm SB}T_{\rm s}^{4},$ (2)

where $R_{\rm NS}$ is the NS radius, $T_{\rm s}$ is the surface temperature,
and $\sigma_{\rm SB}$ is the Stefan-Boltzmann constant. The factor of $B(r)$,
the time component of the Schwartzschild metric, represents the redshift
effect on luminosity. In the following, we will consider two types of heating
sources: DM heating and vortex creep heating.

##  III Dark matter heating

Dark matter particles accrete in NS cores and scatter with the stellar matter.
In particular, WIMPs with a mass of $1~{}\mathrm{GeV}\lesssim m_{\chi}\lesssim
1~{}\mathrm{PeV}$ will be efficiently captured gravitationally [7]. For
neutron targets, the threshold value of DM-neutron cross section for capture
is derived as $\sigma_{\chi n}\simeq 1.7\times
10^{-45}~{}\mathrm{cm}^{2}~{}(1.4\times 10^{-44}~{}\mathrm{cm}^{2})$ in Ref.
[21] (Ref. [24]).

The DM captured within a NS core will be thermalized and eventually
annihilated into SM particles. The capture, thermalization, and annihilation
processes will inject energy into the NS and increase its temperature [2]. The
general expression for the capture rate by nucleons and leptons in a NS is
given by [36, 2],

$C\\!=\\!\frac{4\pi}{v_{\star}}\frac{\rho_{\chi}}{m_{\chi}}{\rm
Erf}\\!\left(\\!\\!\sqrt{\frac{3}{2}}\frac{v_{\star}}{v_{d}}\right)\\!\\!\int_{0}^{R_{\rm
NS}}\\!\\!\\!\\!dr\frac{1-B(r)}{B(r)}r^{2}\Omega^{-}(r)\eta(r)~{},$ (3)

where $v_{\star}$ and $v_{d}$ are the NS and dispersion velocities,
respectively. Here $\rho_{\chi}$ is the DM density which, considering local
NSs, takes the value of $\rho_{\chi}\simeq 0.4~{}\mathrm{GeV/cm}^{3}$. We
introduce the function $\Omega^{-}(r)$ as the interaction rate between dark
matter and the target in the NS. This interaction rate corresponds to the case
when the DM is captured and depends on factors such as the DM-target cross
section and the target number density. Additionally, the function $\eta(r)$
takes into account the opacity of the star. After the capture and annihilation
of DM, and assuming that both processes are in equilibrium, we obtain the
following heating luminosity observed at the distance,

$L_{\chi}^{\infty}=B(R_{\rm
NS})\times\left[X+(\gamma-1)\right]m_{\chi}C(m_{\chi})~{},$ (4)

where $\gamma=B(R_{\rm NS})^{-1/2}$ is the Lorentz factor of the incoming DM
particle. We introduce $X$ to express the fraction of the annihilation energy
transferred to the heat of the NS, and the $(\gamma-1)$ factor accounts for
the contribution of the DM kinetic energy [7].

Assuming the heating luminosity is dominated by the DM effects expressed in
Eq. (4), we can read out the surface temperature of NSs in this scenario from
Eq. (2):

$T_{\rm s}|_{\rm
DM}\simeq\left(\frac{[X+(\gamma-1)]m_{\chi}\,C(m_{\chi})}{4\pi R_{\rm
NS}^{2}\sigma_{\rm SB}}\right)^{1/4}.$ (5)

The maximum value of $T_{\rm s}$ predicted, due to the heating by the capture
of DM and its annihilation within a NS core, is $T_{\rm s}\simeq 2600\,\rm K$
[31]. It is worth mentioning that the DM mass dependence in Eq. (5) is
cancelled by $C(m_{\chi})\propto m_{\chi}^{-1}$. Therefore, the predicted
surface temperature dominated by DM effects will be quite universal for a wide
range of DM mass, $1~{}\mathrm{GeV}\lesssim m_{\chi}\lesssim
1~{}\mathrm{PeV}$, if DM has a larger cross section with targets than the
threshold value.

##  IV Vortex creep heating

The key to understanding the vortex creep heating is the neutron ${}^{1}S_{0}$
superfluid that is expected to appear in the inner crust of a NS [86, 87, 88].
Except for this superfluid, the components, such as a rigid crust, a lattice
of nuclei, charged particles, and NS core (denoted as crust component), are
tightly coupled through electromagnetic force and are directly affected by the
external torque. On the other hand, the neutron ${}^{1}S_{0}$ superfluid
(denoted as superfluid component) is only indirectly affected through the
interaction with the crust component. This two-component description is
originally proposed in the context of NS glitch phenomena [89]. Under this
description, we need to follow two angular velocities, $\bm{\Omega}_{\rm c}$
and $\bm{\Omega}_{\rm s}$ for the crust and superfluid components,
respectively. The friction between these two components is crucial for the
vortex creep heating [53] as we will review below.

The interaction between the crust and superfluid components is mediated by the
vortex line of the neutron superfluid [90]. This string-shaped configuration
appears in a rotating superfluid system and has a nonzero vorticity quantized
in units of $\kappa\equiv h/(2m_{n})$, where $h$ is the Planck constant and
$m_{n}$ is the neutron mass. In a NS, the number of vortex lines saturates at
the point where its integrated vorticity corresponds to the rigid body
rotation of the superfluid component. In this saturation limit, the angular
velocity of the superfluid component at a certain position decreases only if
the number of vortex lines inside this position reduces. This is accomplished
by the radial motion of vortex lines, known as vortex creep. See Ref. [62] for
more detailed discussions on the vortex creep.

The vortex creep occurs as a consequence of two forces exerting on vortex
lines, pinning force and the Magnus force: The pinning force, $\bm{f}_{\rm
pin}$, originates from the interaction between the vortex and the lattice
nuclei and fixes a vortex line at some stable points in the crust. As a
result, the angular velocity of superfluid remains constant, while that of the
crust component slows down, leading to ${\Omega}_{\rm s}-{\Omega}_{\rm c}>0$.
This non-zero relative angular velocity will be a trigger for the Magnus force
as expressed below:

$\displaystyle\bm{f}_{\rm Mag}$ $\displaystyle=\rho\left[(\bm{\Omega}_{\rm
s}-\bm{\Omega}_{\rm c})\times\bm{r}\right]\times\bm{\kappa},$ (6)

where $\rho$ is the mass density of superfluid, and $\bm{r}$ denotes the
position vector from the NS center. We introduce the vorticity vector denoted
as $\bm{\kappa}$, which has the absolute value $|\bm{\kappa}|=\kappa$ and is
aligned to the rotational axis with right-handed screw direction. Equation (6)
shows that the direction of the Magnus force, and thus the direction of the
vortex creep motion, is also fixed radially outward.

Depending on the NS temperature, vortex lines come off the pinning positions
due to the thermal excitation or quantum tunnelling. According to the previous
estimation [91, 92, 61], the quantum tunnelling process is dominant for old
NSs (with $T_{\rm s}\lesssim 10^{6}~{}\mathrm{K}$), and the unpinning rate for
vortex lines is large enough for the system to be entered to the steady phase
where the superfluid and the crust component have the same spin-down rate,
$\dot{\Omega}_{\rm c}=\dot{\Omega}_{\rm s}\equiv\dot{\Omega}_{\infty}$. The
relative angular velocity at this steady motion is nearly the same as that for
the critical value where we have $f_{\rm pin}\simeq f_{\rm Mag}$. Using Eq.
(6), we obtain

$\displaystyle(\Omega_{\rm s}-\Omega_{c})_{\infty}\simeq\frac{f_{\rm
pin}}{\rho\kappa r},$ (7)

where $r$ is the distance from the rotational axis.

The rotational energy stored in the neutron superfluid, which rotates faster
than the crust component, is dissipated as heat via friction caused by the
vortex creep motion. The energy dissipation due to this steady vortex creep
motion gives the heating luminosity for NSs [53],

$\displaystyle L_{\rm VCH}^{\infty}\simeq B(R_{\rm NS})\times
J|\dot{\Omega}_{\infty}|,$ (8)

where the proportional constant $J$ is defined as

$\displaystyle J\equiv\int dI_{\rm pin}(\Omega_{\rm
s}-\Omega_{c})_{\infty}\simeq\int dI_{\rm pin}\frac{f_{\rm pin}}{\rho\kappa
r}.$ (9)

Here, $I_{\rm pin}$ denotes the moment of inertia for the NS crust where the
pinning force is relevant. The late-time surface temperature is given from Eq.
(2) once we assume the vortex creep heating dominates in a NS,

$\displaystyle T_{\rm s}|_{\rm
VCH}\simeq\left(\frac{J|\dot{\Omega}_{\infty}|}{4\pi R_{\rm NS}^{2}\sigma_{\rm
SB}}\right)^{\frac{1}{4}}.$ (10)

Note that the $J$ is only one parameter to characterize the predicted
temperature for each NS since we can input $|\dot{\Omega}_{\infty}|$ and
$R_{\mathrm{NS}}$ from observation.

The parameter $J$ is, in principle, determined by the pinning force that is
expected to be universal for all NSs. This pinning force can be evaluated
using a nuclear many-body calculation [67, 68, 69, 70, 71, 63, 64, 65, 72,
66], though the present theoretical calculation suffers from large
uncertainty. In Ref. [62], we estimated the parameter $J$ using the semi-
classical [72] and quantum [66] calculations of the pinning force and obtain
the following range of $J$:111We obtain larger values of $J$ if we use the
pinning force computed in Refs. [71, 65] (see Ref. [62]).

$\displaystyle
J\\!\in\\![3.9\\!\times\\!10^{40},1.9\\!\times\\!10^{43}]\,\mathrm{erg\cdot
s}\quad\text{(Semi-
classical~{}\cite[cite]{[\@@bibref{Number}{Seveso:2016}{}{}]})}~{},$ (11)
$\displaystyle
J\\!\in\\![1.7\\!\times\\!10^{40},2.7\\!\times\\!10^{42}]\,\mathrm{erg\cdot
s}\quad\text{(Quantum
\cite[cite]{[\@@bibref{Number}{Klausner:2023ggr}{}{}]})}~{}.$ (12)

On the other hand, we can also estimate the parameter $J$ from the temperature
observation of old NSs through Eq. (10). If the vortex creep heating dominates
the heating luminosity, we expect that the values of $J$ estimated from Eq.
(10) are almost universal over the NSs within the uncertainty coming from the
surface temperature observation (typically a factor of a few error in $T_{\rm
s}$ [93]) and from the NS structure (an $\mathcal{O}(1)$ factor in $J$ and
$R_{\mathrm{NS}}$). In Ref. [62], we showed that the current temperature
observations of old NSs indeed support this prediction, with $J$ in the range

$J\simeq
10^{42.9\text{--}43.8}~{}\mathrm{erg}\cdot\mathrm{s}\quad\text{(Observation~{}\cite[cite]{[\@@bibref{Number}{Fujiwara:2023tmr}{}{}]})}~{}.$
(13)

It is found that this observationally favoured range of $J$ can be consistent
with the theoretical estimate in Eq. (11), though much smaller values are also
allowed in the theoretical estimations.

##  V Results

Figure 1: Contours of $J$ $[\mathrm{erg}\cdot\mathrm{s}]$ corresponding to
$L_{\mathrm{VCH}}^{\infty}=L_{\chi}^{\infty}$ in the $t_{\rm
sd}$-$\dot{|{\Omega}|}$ plane (solid red lines). Grey dots represent spin-down
pulsar data from the ATNF pulsar catalogue [94, 95], and orange stars indicate
NSs used to test the quasi-universality of $J$ in vortex creep heating in Ref.
[62]. Green and red bands indicate the theoretical estimates (11), (12) and
the observationally favored range (13) of $J$, respectively. Blue dashed
contours show the temperatures predicted by the vortex creep heating for
$J=10^{42.9}~{}\mathrm{erg}\cdot\mathrm{s}$.

Now we discuss the implications of the vortex creep heating mechanism for the
prospects of the WIMP DM search via the NS surface temperature observation.
For the DM heating to have a visible effect on the evolution of the NS surface
temperature, its luminosity must dominate the heating luminosity. This
requires $L_{\mathrm{VCH}}^{\infty}<L_{\chi}^{\infty}$, imposing an upper
bound on $J$ in Eq. (8). In Fig. 1, we show contours of $J$ in units of
$\mathrm{erg}\cdot\mathrm{s}$ corresponding to
$L_{\mathrm{VCH}}^{\infty}=L_{\chi}^{\infty}$ in the $t_{\rm
sd}$-$\dot{|{\Omega}|}$ plane (red solid lines), where $t_{\rm sd}$ is the
spin-down age of the NS; in this evaluation, we take $T_{\rm
s}=2600~{}\mathrm{K}$ in Eq. (2) to estimate the most optimistic value of the
DM heating luminosity as estimated in Ref. [31]. We also show the spin-down
pulsar data taken from the Australian Telescope National Facility (ATNF)
pulsar catalogue [94, 95] as the grey dots. The orange stars indicate the NSs
used to test the vortex creep heating in Ref. [62]. The grey-shaded region
corresponds to NSs too young to probe DM heating ($t_{\rm
sd}<10^{6}~{}\mathrm{yrs}$). As we see, for the observed values of
$|\dot{\Omega}|$, we need
$J\lesssim\mathcal{O}(10^{36-39})~{}\mathrm{erg}\cdot\mathrm{s}$ in order for
DM heating to prevail over vortex heating. These values are smaller than both
the theoretical estimations (11), (12) (green band) and the observationally
favoured range (13) (red band) by orders of magnitude.

In other words, if the value of $J$ is in the observationally favoured range,
$J\sim 10^{43}\text{--}10^{44}~{}\mathrm{erg\cdot s}$, the surface temperature
of NSs becomes much higher than the maximal value predicted by the DM heating,
$T_{\rm s}\simeq 2600~{}\mathrm{K}$ [31]. To see this, we also show the
surface temperatures predicted by the vortex creep heating for
$J=10^{42.9}~{}\mathrm{erg}\cdot\mathrm{s}$ (the lower value in
observationally favoured range) in blue dashed contours in Fig. 1. We found
that for all of the known pulsars $T_{\rm s}\gg 2600$ K, implying that the
vortex heating effect always conceals the DM heating effect.

Figure 2: Time evolution of the surface temperature in the presence of both
the vortex creep and DM heating in the blue solid lines, where we set
$P\dot{P}=10^{-15}$ s, $P_{0}=10~{}\mathrm{ms}$,
$\rho_{\chi}=0.42~{}\mathrm{GeV}\cdot\mathrm{cm}^{-3}$,
$v_{d}=230~{}\mathrm{km}\cdot\mathrm{s}^{-1}$, $v_{\star}=0$, and $X=1$. The
blue dotted and black dashed lines show the cases for only the DM heating and
the standard cooling, respectively. The orange dots represent the temperature
data for ordinary pulsars considered in Ref. [62], where the bars show the
error in $T_{\rm s}$ and the arrows indicate the upper limits.

In Fig. 2, we show the time evolution of the surface temperature in the
presence of both the vortex creep and DM heating in the blue solid lines.222We
use the the Akmal-Pandharipande-Ravenhall (APR) [96] equation of state for a
NS mass of $1.4M_{\odot}$ to calculate the NS structure. For Cooper pairing
gap models, we use the SFB model [97] for the neutron singlet pairing, the
model “b” in Ref. [40] for the neutron triplet pairing, and the CCDK model
[98] for the proton singlet pairing. The late-time temperatures rarely depend
on the choice of these models. We take $J=10^{44}$, $10^{42}$, $10^{40}$,
$10^{38}$, and $10^{36}~{}\mathrm{erg}\cdot\mathrm{s}$ from top to bottom. In
the calculation of the vortex creep heating, we assume that the magnetic
dipole radiation causes the pulsar slowdown, and we set $P\dot{P}=10^{-15}$ s
and $P_{0}=10~{}\mathrm{ms}$, where $P$, $\dot{P}$, and $P_{0}$ are the
period, its time derivative, and initial period of the pulsar, respectively.
This choice corresponds to the surface magnetic field of $B_{s}\simeq 10^{12}$
G, which is a typical value for ordinary pulsars.333In millisecond pulsars, it
is likely that another heating mechanism called the rotochemical heating [99,
100, 101, 102, 103, 104, 105, 106, 107] (see also Refs. [108, 109]) operates
and its effect will hide the DM heating effect [14]. For this reason, we focus
on ordinary pulsars in the present analysis.  In the calculation of the DM
heating, we use $\rho_{\chi}=0.42~{}\mathrm{GeV}\cdot\mathrm{cm}^{-3}$ [110],
$v_{d}=230~{}\mathrm{km}\cdot\mathrm{s}^{-1}$, $v_{\star}=0$, and $X=1$, for
which we have $T_{\rm s}^{\infty}|_{\rm DM}\simeq 2\times 10^{3}$ K. The blue
dotted and black dashed lines show the cases for only the DM heating and the
standard cooling, respectively. The orange dots represent the temperature data
for ordinary pulsars considered in Ref. [62], where the bars show the error in
$T_{\rm s}^{\infty}$ and the arrows indicate the upper limits. This figure
shows that for observationally favoured values of $J$ in Eq. (13), the late-
time temperature evolution is governed by the vortex creep heating, and the
effect of the DM heating is completely invisible. The DM heating effect can be
comparable to the vortex creep heating effect only if the value of $J$ is
smaller than those values by orders of magnitude; for
$J=10^{36}~{}\mathrm{erg}\cdot\mathrm{s}$
($10^{38}~{}\mathrm{erg}\cdot\mathrm{s}$), the DM heating may be observable
for NSs older than $t\gtrsim 10^{7}$ years ($3\times 10^{8}$ years).

##  VI Conclusions & Discussion

We have studied the effect of the vortex creep heating on the search of WIMP
DM through the surface temperature observation of old NSs. We have found that
for values of $J$ that are favoured by both theory (Eqs. (11) and (12)) and
observation (Eq. (13)), the vortex creep heating effect is much stronger than
the DM heating effect, and thus it is challenging to detect the signature of
WIMP DM through the temperature observations of old NSs. For this to be
possible, $J$ should be smaller than these values by orders of magnitude, as
small as $J\lesssim 10^{38}~{}\mathrm{erg}\cdot\mathrm{s}$.

It is, of course, possible that future computations of the vortex-nuclei
interaction will indicate a much smaller pinning force than the present
calculations, leading to a sufficiently small value of $J$. Another
possibility to suppress the vortex creep heating is that the vortex creep rate
is much lower than the theoretical estimates [91, 92, 61] so that the spin-
down rate of the superfluid component does not follow that of the crust
component. In this case, the spin-down of the superfluid component does not
reach the steady phase, which is assumed in the derivation of Eq. (8), and the
vortex heating luminosity depends on $|\dot{\Omega}_{\mathrm{s}}|$. It is
found that $\dot{\Omega}_{\mathrm{s}}$ is proportional to the vortex creep
rate (see Ref. [62]), and therefore a very small creep rate could suppress the
vertex heating. To explore these possibilities, improvements in the
theoretical calculations of the vortex-nuclear interactions are highly
motivated.

Besides, it is also important to accumulate more data of the surface
temperatures of old NSs to observationally test the vortex creep heating
mechanism as discussed in Ref. [62]. We envision that this will be performed
in the near future through improved optical, UV, and X-ray observations.

###### Acknowledgements.

This work is supported in part by the Collaborative Research Center SFB1258
and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
under Germany’s Excellence Strategy - EXC-2094 - 390783311[MF], JSPS Core-to-
Core Program (No.JPJSCCA20200002 [MF]), the Grant-in-Aid for Innovative Areas
(No.19H05810 [KH and MRQ], No.19H05802 [KH], No.18H05542 [MF and NN]),
Scientific Research B (No.20H01897 [KH and NN]), Young Scientists (No.21K13916
[NN]).

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