1 Introduction
Let X be a smooth projective variety over an algebraically closed field k. A tuple
$(X, \phi , L)$
is called a polarized variety, or a variety X equipped with a polarized endomorphism
$(\phi , L),$
if L is an ample line bundle on X and
$\phi $
is a self-map of X such that
$\phi ^*(L) \cong L^{\otimes d}$
for some
$d>1$
. The study of periodic (and more generally preperiodic) points of such a dynamical system has gathered much interest in recent years, leading to progress on the Bogomolov conjecture and the dynamical Mordell–Lang conjecture. On the other hand, the study of geometric properties of such varieties is also of interest and has been pursued in several articles, including works by Nakayama [Reference Nakayama16], Nakayama and Zhang [Reference Nakayama and Zhang17], Meng and Zhang [Reference Meng and Zhang14], Ejiri and Yoshikawa [Reference Ejiri and Yoshikawa2], Yoshikawa [Reference Yoshikawa20], Kawakami and Totaro [Reference Kawakami and Totaro10], and others. The study of the latter kind is the subject of this article.
When X is of the nonnegative Kodaira dimension and k has characteristic 0, it is proved in [Reference Fakhruddin3, Theorem 4.2] that there exists an abelian variety A with a finite étale map to X. Furthermore, it was suggested in a footnote to Theorem 4.2 of loc. cit. that a positive characteristic analog should hold under an additional hypothesis on the self-map
$\phi $
. The following may be viewed as a result in this direction.
Theorem 1.1 Let X be a smooth projective variety of the nonnegative Kodaira dimension over an algebraically closed field k of characteristic
$p>0$
, L an ample line bundle on X, and
$\phi : X \rightarrow X$
a separable map such that
$\phi ^*L \cong L^{\otimes d}$
for an integer
$d>1$
. Suppose that the étale fundamental group of X contains a normal abelian subgroup of finite index. Then, there is an abelian variety A with a finite étale map to X.
The hypothesis on the fundamental group in the above theorem holds under the assumption that the variety X admits a lift to characteristic 0 as shown in Lemma 3.1. Under the lifting assumption on X, the arguments of the proof of Theorem 1.1 can be used to provide a proof of Theorem 1.1 that uses results of a more elementary nature (see Lemma 3.2). Finally, we prove appropriate analogs of [Reference Fakhruddin3, Theorem 4.2] and Theorem 1.1 over fields that are not necessarily algebraically closed. It is noteworthy that, over fields of characteristic 0, the assumption on fundamental groups is known to be true for any smooth projective variety with polarized self-maps, thanks to [Reference Greb, Kebekus and Peternell4, Theorem 1.21] and [Reference Nakayama and Zhang17, Corollary 4.6(2)].
The article is organized as follows. In Section 2.1, we state three lemmas that are required for the proof of Theorem 1.1. In Section 2.2.1, we complete the proof of the theorem under the assumption that
$k = \overline {\mathbb {F}}_p$
, and in Section 2.2.2, we reduce the proof of the theorem to the case of
$k=\overline {\mathbb {F}}_p$
. Section 3 consists of two lemmas whose contents have been described in the paragraph above. Section 4 generalizes (an appropriately modified version of) the above theorem to non-algebraically closed fields. At the end, we provide a remark on the generalizations of the results of this article to “int-amplified” self-maps.
2 Proof of Theorem 1.1
2.1 Preliminary lemmas
For any map
$\phi : X \rightarrow X$
, we write
$\phi ^r$
to denote the composite of
$\phi $
with itself r times.
Lemma 2.1 Let X be a smooth projective variety over k such that the étale fundamental group of X contains a normal abelian subgroup of finite index. Suppose L is an ample line bundle on X and
$\phi : X \rightarrow X$
is a finite étale map such that
$\phi ^*L \simeq L^{\otimes d}$
for an integer
$d>1$
. Then, there exists a connected finite étale cover Z of X, an ample line bundle M on Z, and a finite étale map
$\psi : Z \rightarrow Z$
such that
$\psi ^*M \simeq M^{\otimes e}$
for an integer
$e>1$
and the fundamental group of Z is abelian.
Proof Fix a point
$x_0 \in X$
, and let
$G \subset \pi ^{\acute{\rm e}\text {t}}_1(X, x_0)$
be a normal abelian subgroup of finite index. Let
$Y \rightarrow X$
be the Galois étale cover of X given by the normal subgroup G and let
$y_0 \in Y$
be a point lying over
$x_0$
. In particular,
$\pi ^{\acute{\rm e}\text {t}}_1(Y, y_0) \cong G$
is abelian. Consider the smooth projective (possibly disconnected) schemes
$Y^{(r)}$
defined by the Cartesian diagram
The left downward arrow
$Y^{(r)} \rightarrow X$
is finite étale and of the same degree as
$f = \deg (Y \rightarrow X)$
for all
$r \geq 1$
. Since
$\pi ^{\acute{\rm e}\text {t}}_1(X, x_0)$
is topologically finitely generatedFootnote
1
[Reference Grothendieck7, Exposé, Théorème 2.9], there are only finitely many homomorphisms
$\pi ^{\acute{\rm e}\text {t}}_1(X, x_0) \rightarrow H$
for any finite group H. Therefore, there are only finitely many étale covers of X of degrees
$\leq f$
; in particular, the collection of finite étale covers of X given by
$\lbrace Y^{(1)}, Y^{(2)}, \dots \rbrace $
is finite. Hence, there exists
$n> m$
such that
$Y^{(n)} \cong Y^{(m)}$
as étale covers of X and we get a Cartesian diagram
The map
$\psi $
is surjective by construction and therefore the map
$\pi _0(\psi ) : \pi _0(Y^{(m)}) \rightarrow \pi _0(Y^{(m)})$
induced on connected components is bijective. After possibly replacing
$\psi $
by
$\psi ^r$
for some integer r, we may assume that
$\psi $
fixes each connected component of
$Y^{(m)}$
. Let Z be a connected component of
$Y^{(m)}$
. Let M be the pullback of L to Z, and let
$\psi : Z \rightarrow Z$
be the induced map. We check that
$\psi ^*M \cong M^{\otimes e}$
, where
$e = d^{n-m}$
. This completes the proof after noting that
$\pi ^{\acute{\rm e}\text {t}}_1(Y)$
is abelian and
$\pi ^{\acute{\rm e}\text {t}}_1(Z)$
is a subgroup of
$\pi ^{\acute{\rm e}\text {t}}_1(Y)$
since
$Z \rightarrow Y$
is a finite étale map.
Lemma 2.2 [Reference Katz and Lang9, Section III, Lemma 5]
Let X be a smooth projective variety over k whose fundamental group is abelian. Let
$\mathrm {alb}_X : X \rightarrow \mathrm {Alb}_X$
be the Albanese map. Then, the kernel of the map
$(\mathrm {alb}_X)_*$
induced on fundamental groups is finite.
Lemma 2.3 [Reference Fakhruddin3, Lemma 4.1]
Let X be a smooth projective variety over an infinite field k,
$\phi : X \rightarrow X$
a morphism, and L an ample line bundle on X such that
$\phi ^*L \cong L^{\otimes d}$
for some
$d> 1$
. Assume further that
$\phi $
is separable and the Kodaira dimension of X is
$\geq 0$
. Then,
$\phi $
is étale, the class of
$\omega _X$
(canonical bundle of X) is torsion in
$\mathrm {Pic}(X)$
, and X does not contain any positive-dimensional subvarieties with the finite algebraic fundamental group.
2.2 Completing the proof of Theorem 1.1
We know from [Reference Fakhruddin3, Lemma 4.1] (which appears as Lemma 2.3 above) that
$\phi $
is étale. Without loss of generality, we may replace X by its finite étale cover. Using Lemma 2.1, we may, and do, further assume that the étale fundamental group of X is abelian.
2.2.1 Proof of the finiteness of the Albanese morphism for X under the assumption
$k = \overline {\mathbb {F}}_p$
Step 1. Show that the Albanese map for X is flat: Since
$k=\overline {\mathbb {F}}_p$
, any point of
$X(k)$
is preperiodic for any self-map
$\phi ,$
and hence there is a periodic point for
$\phi $
. Replacing
$\phi $
by
$\phi ^n$
, we may assume that there is a point
$x_0 \in X$
such that
$\phi (x_0) = x_0$
. With
$\mathrm {alb}_X(x_0)$
as a base point, we will henceforth assume that
$\mathrm {Alb}_X$
is an abelian variety with
$\mathrm {alb}_X(x_0)$
as the identity. It follows from the universal property of the Albanese morphism that there exists a morphism
$\phi _{\mathrm {Alb}_X} : \mathrm {Alb}_X \rightarrow \mathrm {Alb}_X$
making the diagram below commute
By construction,
$\phi _{\mathrm {Alb}_X}(\mathrm {alb}_X(x_0)) = \mathrm {alb}_X(\phi (x_0)) = \mathrm {alb}_X(x_0) = 0$
; therefore,
$\phi _{\mathrm {Alb}_X}(0) = 0$
. Hence,
$\phi _{\mathrm {Alb}_X}$
is a homomorphism of abelian varieties. Since
$\operatorname *{{\mathrm {im}}}(\mathrm {alb}_X)$
is not contained in any proper abelian subvariety, the commutative diagram above implies that
$\operatorname *{{\mathrm {im}}}(\phi _{\mathrm {Alb}_X})$
is not contained in a proper abelian subvariety. Thus,
$\phi _{\mathrm {Alb}_X}$
is a surjective homomorphism of abelian varieties of the same dimension. That is,
$\phi _{\mathrm {Alb}_X}$
is an isogeny, and hence flat.
Step 2. The map
$\phi _{\mathrm {Alb}_X}$
in the commutative diagram (2.1) is étale: A result of Igusa (see, for instance, [Reference Mehta and Srinivas13, Lemma 1.3]) says that
is injective. We have the following commutative diagram induced by the one above:
Since
$\phi $
is a surjective étale (hence separable) map, the top horizontal arrow is injective. The vertical arrows are injective as argued above. Therefore,
$\phi ^*_{\mathrm {Alb}_X}$
is injective, and hence an isomorphism (since
$\mathrm {H}^0(\mathrm {Alb}_X, \Omega ^1_{\mathrm {Alb}_X})$
is finite dimensional). Since
$\Omega ^1_{\mathrm {Alb}_X}$
is a free
$\mathcal {O}_{\mathrm {Alb}_X}$
-module and
$\Gamma (\mathrm {Alb}_X, \mathcal {O}_{\mathrm {Alb}_X}) = k$
, it follows that the kernel and cokernel of the map
$\phi ^*_{\mathrm {Alb}_X}\Omega ^1_{\mathrm {Alb}_X} \rightarrow \Omega ^1_{\mathrm {Alb}_X}$
are free
$\mathcal {O}_{\mathrm {Alb}_X}$
-modules. Applying
$\phi _{\mathrm {Alb}_X*}$
and using
$R^i\phi _{\mathrm {Alb}_X*} = 0$
for all
$i>0,$
we get the exact sequence below, where
$\Omega ^1_{\phi _{\mathrm {Alb}_X}}$
is the sheaf of relative differentials
Note that the argument above implies that
$\Omega ^1_{\phi _{\mathrm {Alb}_X}}$
is free. On the other hand, the induced map on global sections of the composite map
$\Omega ^1_{\mathrm {Alb}_X} \rightarrow \phi _{\mathrm {Alb}_X*}\phi ^*_{\mathrm {Alb}_X}\Omega ^1_{\mathrm {Alb}_X} \rightarrow \phi _{\mathrm {Alb}_X*}\Omega ^1_{\mathrm {Alb}_X}$
is an isomorphism, hence
$\Gamma (\mathrm {Alb}_X, \phi _{\mathrm {Alb}_X*}\Omega ^1_{\phi _{\mathrm {Alb}_X}}) = 0$
. Since
$\Omega ^1_{\phi _{\mathrm {Alb}_X}}$
is free (as observed above),
$\dim _k\Gamma (\mathrm {Alb}_X, \phi _{\mathrm {Alb}_X*}\Omega ^1_{\phi _{\mathrm {Alb}_X}}) = $
rank of
$\Omega ^1_{\phi _{\mathrm {Alb}_X}}$
, and therefore
$\Omega ^1_{\phi _{\mathrm {Alb}_X}} = 0$
. We have already seen that
$\phi _{\mathrm {Alb}_X}$
is flat, and hence
$\phi _{\mathrm {Alb}_X}$
is finite étale.
Step 3. Show that the locus of points in
$\mathrm {Alb}_X(k)$
where the fiber of
$\mathrm {alb}_X$
is not finite is empty: Consider
It is clear that Z is a Zariski closed subset, and we equip Z with the reduced induced subscheme structure. Moreover, the fact that
$\phi $
and
$\phi _{\mathrm {Alb}_X}$
are finite étale and the diagram (2.1) is commutative implies that
$\phi _{\mathrm {Alb}_X}(Z) = Z$
. Since
$\phi _{\mathrm {Alb}_X}$
is surjective, after possibly replacing
$\phi _{\mathrm {Alb}_X}$
by
$\phi ^n_{\mathrm {Alb}_X}$
(which can be achieved by replacing
$\phi $
by
$\phi ^n$
), we may assume that
$\phi _{\mathrm {Alb}_X}$
preserves the irreducible components of Z. Replacing Z by any irreducible component of Z, we get that Z is a variety over k and
$\phi _{\mathrm {Alb}_X}|_Z : Z \rightarrow Z$
is a finite surjective map (
$\phi _{\mathrm {Alb}_X}$
is finite). To prove that
$\mathrm {alb}_X$
is a finite map, it suffices to prove that
$Z = \emptyset $
since any proper quasi-finite map is finite. Suppose that Z is nonempty. Then (arguing as in the first paragraph of this section) there is a point
$p \in Z(k)$
that is periodic for
$\phi _{\mathrm {Alb}_X}$
of order n for some
$n>0$
. Replacing
$\phi $
by
$\phi ^n$
(and as a consequence replacing
$\phi _{\mathrm {Alb}_X}$
by
$\phi ^n_{\mathrm {Alb}_X}$
), we may assume that
$\phi _{\mathrm {Alb}_X}(p) = p$
. Let Y be a positive-dimensional connected component of
$\mathrm {alb}_X^{-1}(p)$
,
$y_0 \in Y$
any point, and
$i : Y \rightarrow X$
be the inclusion map. We now make the following.
Claim: Let
$i_* : \pi ^{\acute{\rm e}\text {t}}_1(Y, y_0) \rightarrow \pi ^{\acute{\rm e}\text {t}}_1(X, y_0)$
and
$G = \operatorname *{{\mathrm {im}}}(i_*)$
. Then, G is an infinite group.
Proof of claim.
Let
$X_p = \mathrm {alb}_X^{-1}(p)$
. Since
$\phi _{\mathrm {Alb}_X}(p) = p$
,
$\phi _{\mathrm {Alb}_X}$
is finite étale, and the diagram (2.1) is commutative, we get that
$\phi $
restricts to a finite étale map
Since there are finitely many connected components of
$X_p$
, we may replace
$\phi $
by
$\phi ^r$
and assume that
$\phi |_{X_p}$
fixes each connected component of
$X_p$
. Therefore, for any connected component Y of
$X_p$
, we get that
$\phi |_Y : Y \rightarrow Y$
is a finite étale map. It is also easy to see that
$(\phi |_Y)^*L \simeq (L|_Y)^{\otimes d}$
, and therefore
$\phi |_Y$
is of degree
$d^{\dim Y}$
. Since the restriction of
$\phi ^r$
to Y provides finite étale covers of Y of degree
$d^{r\dim Y}$
, we get that
$G = \operatorname *{{\mathrm {im}}}(i_*)$
must be infinite.
On the other hand, G lies in the kernel of the map
$(\mathrm {alb}_X)_* : \pi ^{\acute{\rm e}\text {t}}_1(X, y_0) \rightarrow \pi ^{\acute{\rm e}\text {t}}_1(\mathrm {Alb}_X, p)$
since
$\mathrm {alb}_X \circ i$
is constant. Lemma 2.2 implies that G must be finite. Thus, we get a contradiction to the assumption that Z is nonempty. This completes the proof of this step.
2.2.2 Proof of the finiteness of the Albanese morphism for X for arbitrary algebraically closed fields of positive characteristic
Any
$X, \phi $
, and L as above are defined over a finitely generated subfield l of k, so we may assume that l is of finite transcendence degree, say r, over
$\overline {\mathbb {F}}_p$
. We can also assume that
$X(l)\neq \emptyset $
and choose
$a \in X(l)$
. Note that
$X, \phi , L, a$
, and the isomorphism
$\phi ^*L \cong L^{\otimes d}$
are defined by finitely many equations; therefore, there exists a Noetherian ring R of finite type over
$\overline {\mathbb {F}}_p$
, and a triple
$(\textbf {X}, \Phi , \textbf {L})$
consisting of a projective variety
$\textbf {X}$
over
$\operatorname *{{\mathrm {Spec}}}(R)$
with a section
$\mathbf {a} \in \mathbf {X}(\operatorname *{{\mathrm {Spec}}}(R))$
,
$\Phi : \textbf {X} \rightarrow \textbf {X}$
a morphism over
$\operatorname *{{\mathrm {Spec}}}(R)$
, and a line bundle
$\textbf {L}$
on
$\textbf {X}$
such that
$\Phi ^*\textbf {L} \simeq \textbf {L}^{\otimes d}$
and over the geometric generic point of
$\operatorname *{{\mathrm {Spec}}}(R)$
, the triple coincides with
$(X, \phi , L)$
, and the section
$\mathbf {a}$
coincides with a. Using [Reference Grothendieck5, Lemme 8.10.5.2] and [Reference Grothendieck6, Proposition 17.7.11(c)], we may (after possibly localizing R at finitely many elements) furthermore assume that
$\textbf {X}$
is smooth over
$\operatorname *{{\mathrm {Spec}}}(R)$
,
$\textbf {L}$
is a relatively ample line bundle, and
$\Phi $
is finite étale. Since the generic fiber is smooth projective variety, the hypothesis of [Reference Laurent and Schröer12, Corollary 10.6] is satisfied; therefore, after possibly replacing S by a dense open subscheme, we get that there is a unique Albanese map
${\mathbf {alb}}_{\mathbf {X}_{/S}}: X \rightarrow {\mathbf {Alb}}_{\mathbf {X}_{/S}}$
that commutes with base change, where
$\mathbf {Alb}_{\mathbf {X}_{/S}}$
is an abelian scheme with the identity given by
${\mathbf {alb}}_{\mathbf {X}}\circ \mathbf {a}$
. Choose a prime ideal
$\mathfrak {p} \subset R$
of height 1. Finally, we may replace R by
$R_{\mathfrak {p}}$
to get a Noetherian local ring R whose residue field is a field of transcendence degree
$r-1$
. Let
$(X_0, \phi _0, L_0)$
be the triple obtained by restricting the triple
$(\textbf {X}, \Phi , \textbf {L})$
over
$\operatorname *{{\mathrm {Spec}}}(R)$
to a geometric point lying over the closed point in
$\operatorname *{{\mathrm {Spec}}}(R)$
. Then, it follows from the construction that
$(X_0, \phi _0, L_0)$
is a smooth projective polarized variety over an algebraically closed field of transcendence degree
$r-1$
, and
$\phi _0$
is finite étale. Let
$\overline {a}$
and
$\overline {a}_0$
be geometric points of X and
$X_0$
given by
$\mathbf {a}$
. Then, [Reference Grothendieck7, Exposé X, Corollary 2.4] implies that there is a specialization homomorphism
The specialization homomorphism is moreover surjective since
$\mathbf {X}$
is smooth proper, in particular separable proper, over
$\operatorname *{{\mathrm {Spec}}}(R)$
. By assumption (see Section 2.2),
$\pi ^{\acute{\rm e}\text {t}}_1(X)$
is abelian, and hence it follows that
$\pi ^{\acute{\rm e}\text {t}}_1(X_0)$
is also abelian.
Assume that the Albanese map is finite for all smooth projective polarized varieties
$(X, \phi , L)$
defined over an algebraically closed field of transcendence degree at most
$r-1$
with
$\phi $
finite étale and
$\pi ^{\acute{\rm e}\text {t}}_1(X)$
is abelian. Note that, for
$r=1$
, this assertion is known to be true from the argument in Section 2.2.1. Note that, over algebraically closed fields, the notion of Albanese variety and Albanese map considered in [Reference Laurent and Schröer12] coincides with the respective classical notions. Therefore,
$\mathbf {alb}_{\mathbf {X}_{/S}}$
over the geometric generic fiber coincides with
$\mathrm {alb}_X$
and over the geometric special fiber coincides with
$\mathrm {alb}_{X_0}$
. The induction hypothesis implies that
$\mathrm {alb}_{X_0}$
is a finite map. Since
$\mathrm {alb}_X$
is proper and
$\mathrm {alb}_{X_0}$
is finite, we get that
$\mathrm {alb}_X$
is finite (see, for instance, [19, Tag 02UP]).
2.2.3 Conclusion of the proof
For
$(X, \phi , L)$
as in the statement of the theorem, it follows from Section 2.2.2 that
$\mathrm {alb}_X$
is finite. Once again applying [Reference Fakhruddin3, Lemma 4.1] (which appears as Lemma 2.3 above), we get that the Kodaira dimension of X is 0. Therefore, using [Reference Hacon, Patakfalvi and Zhang8, Theorem 0.3(3)], we get that X is an abelian variety. This completes the proof of Theorem 1.1.
3 Two lemmas on the hypothesis of the theorem
Lemma 3.1 Let X be a smooth projective variety of the nonnegative Kodaira dimension over an algebraically closed field k, L an ample line bundle on X, and
$\phi : X \rightarrow X$
a separable map such that
$\phi ^*L \cong L^{\otimes d}$
for an integer
$d>1$
. If X admits a lift to characteristic zero, then the étale fundamental group of X contains a normal abelian subgroup of finite index.
Proof Let
$\omega _X$
denote the canonical bundle of X. Let
$\pi : \mathscr {X} \rightarrow \operatorname *{{\mathrm {Spec}}}(R)$
be a flat projective morphism, where R is a discrete valuation ring with residue field k. Let K be the fraction field of R. Since
$\pi $
is flat and the special fiber of
$\pi $
is smooth, it follows that the morphism
$\pi $
is smooth and proper. Let
$j : \widetilde {X} \rightarrow \mathscr {X}$
be the geometric generic fiber of
$\pi $
and
$i : X \rightarrow \mathscr {X}$
be the geometric special fiber of
$\pi $
. Using the functoriality of Chern classes, we get that
$j^*c_1(\omega _{\mathscr {X}/\operatorname *{{\mathrm {Spec}}}(R)}) = c_1(\omega _{\widetilde {X}})$
and
$i^*(c_1(\omega _{\mathscr {X}/\operatorname *{{\mathrm {Spec}}}(R)})) = c_1(\omega _X)$
. Now, [Reference Fakhruddin3, Lemma 4.1] implies that
$\omega _X$
is torsion and therefore its first Chern class
$c_1(\omega _X)$
in
$\ell $
-adic cohomology is 0. Since
$\pi $
is a smooth proper morphism, the maps
$i^*$
and
$j^*$
in
$\ell $
-adic étale cohomology are isomorphisms. Therefore,
$c_1(\omega _{\widetilde {X}}) = 0$
. Applying [Reference Beauville1, Théorème 2], we get that there exist an abelian variety A and a simply connected variety Z such that
$A\times Z \rightarrow \widetilde {X}$
is a finite étale cover. But
$\pi ^{\acute{\rm e}\text {t}}_1(A\times Z) \cong \pi ^{\acute{\rm e}\text {t}}_1(A)$
is abelian. Therefore,
$H = \pi ^{\acute{\rm e}\text {t}}_1(A\times Z)$
is an abelian subgroup of
$\pi ^{\acute{\rm e}\text {t}}_1(\widetilde {X})$
of finite index. Let
$f_1, f_2, \dots , f_r \in \pi ^{\acute{\rm e}\text {t}}_1(\widetilde {X})$
be the representatives of elements in
$\pi ^{\acute{\rm e}\text {t}}_1(\widetilde {X})/H$
. Then,
$G = \underset {i= 1}{\overset {r}{\cap }} f_iHf^{-1}_i$
is an abelian normal subgroup of
$\pi ^{\acute{\rm e}\text {t}}_1(\widetilde {X})$
of finite index. We may conclude using the fact that the specialization map
$\pi ^{\acute{\rm e}\text {t}}_1(\widetilde {X}) \rightarrow \pi ^{\acute{\rm e}\text {t}}_1(X)$
is surjective since
$\pi : \mathscr {X} \rightarrow \operatorname *{{\mathrm {Spec}}}(R)$
is smooth (in particular separable) proper.
Under the assumptions of the above lemma, we provide a proof of Theorem 1.1 that does not depend on the theorem of Hacon–Patakfalvi–Zhang.
Lemma 3.2 Let
$(X, \phi , L)$
be as in the lemma above. Then, there is an abelian variety with a finite étale map to X.
Proof In light of Lemmas 2.1 and 3.1, we may assume after replacing X by a finite étale cover that the fundamental group of X is abelian and the triple
$(X, \phi , L)$
continues to satisfy the assumptions of the theorem. Recall that X admits a lift to characteristic zero; that is, there is a smooth proper morphism
$\pi : \mathscr {X} \rightarrow \operatorname *{{\mathrm {Spec}}}(R)$
, where R is a complete discrete valuation ring with k as its residue field and K as its field of fractions. Let
$\widetilde {X}$
be the generic fiber of
$\pi $
, and let
$\widetilde {a} : \widetilde {X} \rightarrow \widetilde {A}$
be the Albanese map. As an application of a result of Beauville (as in Lemma 3.1), we get that there exist an abelian variety
$\widetilde {B}$
, a simply connected variety Z, and a finite étale map
$\widetilde {h} : \widetilde {B} \times Z \rightarrow \widetilde {X}$
. Moreover, since
$\mathrm {Alb}_Z = 0$
, the map
$\widetilde {a}\circ \widetilde {h} : \widetilde {B} \times Z \rightarrow \widetilde {A}$
factors through the projection
$\widetilde {B} \times Z \rightarrow \widetilde {B}$
. Therefore,
$\operatorname *{{\mathrm {im}}}(\widetilde {a}\circ \widetilde {h}) \subset \widetilde {A}$
is an abelian subvariety. Since
$\operatorname *{{\mathrm {im}}}(\widetilde {a})$
is not contained in any proper abelian subvariety of
$\widetilde {A}$
, we get that
$\operatorname *{{\mathrm {im}}}(\widetilde {a}\circ \widetilde {h}) = \widetilde {A}$
. Hence,
$\dim (\widetilde {B}) \geq \dim (\widetilde {A})$
. We get from the proof of the theorem that
$\dim (X) = \dim (\mathrm {Alb}_X)$
. This equality implies that
$\dim (\widetilde {X}) = \dim (X) = \dim (\mathrm {Alb}_X) = \dim (\widetilde {A})$
. Therefore, the étale cover
$\widetilde {B} \times Z \rightarrow \widetilde {X}$
considered in the previous paragraph satisfies
$\dim (\widetilde {B}) + \dim (Z) = \dim (\widetilde {A})$
. But
$\dim (\widetilde {B}) \geq \dim (\widetilde {A})$
due to the equality
$\operatorname *{{\mathrm {im}}}(\widetilde {a}\circ \widetilde {h}) = \widetilde {A}$
, and therefore we get
$\dim (Z) = 0$
and
$\dim (\widetilde {B}) = \dim (\widetilde {A})$
. Hence,
$\widetilde {h} : \widetilde {B} \rightarrow \widetilde {X}$
is a finite étale cover such that
$\widetilde {a}\circ \widetilde {h}$
is a surjective map between abelian varieties with
$\dim (\widetilde {B}) = \dim (\widetilde {A})$
. Hence, we may conclude that
$\widetilde {a} : \widetilde {X} \rightarrow \widetilde {A}$
is a finite étale map. Now, the Lang–Serre theorem implies that
$\widetilde {X}$
is also an abelian variety. Let
$\widetilde {0} \in \widetilde {X}(K)$
denote the identity element, and
$e : \operatorname *{{\mathrm {Spec}}}(R) \rightarrow \mathscr {X}$
be the arrow obtained in the diagram below using the valuative criterion of properness
Applying [Reference Mumford, Fogarty and Kirwan15, Theorem 6.14], we conclude that
$\mathscr {X} \rightarrow \operatorname *{{\mathrm {Spec}}}(R)$
is an abelian scheme. Therefore, its special fiber X is an abelian variety.
4 Results over non-algebraically closed fields
In this section, we prove analogs of the result [Reference Fakhruddin3, Theorem 4.2] and Theorem 1.1 over fields that are not necessarily algebraically closed.
4.1 Preliminary lemmas
Lemma 4.1 Let X be a geometrically connected algebraic variety over a field k, let
$\bar {x} \in X(\bar {k})$
be the geometric point lying over
$x \in X(k)$
, and let
$j_x$
be the splitting of the map
$\pi ^{\acute{\rm e}\text {t}}_1(X, \bar {x}) \rightarrow \mathrm {Gal}(\overline {k}/k)$
. Suppose that
$\pi ^{\acute{\rm e}\text {t}}_1(X_{\overline {k}}, \bar {x})$
has a characteristic subgroup H of finite index. Let
$Y \rightarrow X$
be the étale cover given by the subgroup
$H\cdot j_x(\mathrm {Gal}(\overline {k}/k))$
; then, the étale cover
$Y_{\overline {k}} \rightarrow X_{\overline {k}}$
obtained via base change is given by the subgroup
$H \subset \pi ^{\acute{\rm e}\text {t}}_1(X_{\overline {k}}, \bar {x})$
. Moreover,
$Y(k) \neq \emptyset $
and there is
$y \in Y(k)$
lying above
$x \in X(k)$
.
Proof The Galois group
$\mathrm {Gal}(\overline {k}/k)$
acts on the geometric fundamental group
$\pi ^{\acute{\rm e}\text {t}}_1(X_{\overline {k}}, \bar {x})$
by outer automorphisms. But any automorphism preserves the characteristic subgroup, and hence it preserves the subgroup H. Thus, the group generated by H and
$j_x(\mathrm {Gal}(\overline {k}/k))$
is a subgroup of
$\pi ^{\acute{\rm e}\text {t}}_1(X, \bar {x})$
of finite index. Let
$\pi : Y \rightarrow X$
be an étale cover given by
$H\cdot j_x(\mathrm {Gal}(\overline {k}/k))$
. Using the functoriality of the fundamental exact sequence of étale fundamental groups, it is easy to see that the map induced by
$\pi $
on the geometric fundamental group is the inclusion
$H \subset \pi ^{\acute{\rm e}\text {t}}_1(X_{\overline {k}}, \bar {x})$
.
It remains to prove the assertion concerning the k-rational point. Note that
$\mathrm {Gal}(\overline {k}/k)$
acts on the fiber
$Y_{\overline {x}}$
of
$\pi $
over
$\bar {x} \in X(\overline {k})$
. This action on the set
$Y_{\overline {x}}$
is given by restricting the action of
$\pi ^{\acute{\rm e}\text {t}}_1(X, \bar {x})$
on
$Y_{\overline {x}}$
to the subgroup
$j_x(\mathrm {Gal}(\overline {k}/k))$
. On the other hand, the étale cover
$\pi $
is constructed so that the set
$\pi ^{\acute{\rm e}\text {t}}_1(X, \bar {x})$
-set
$\pi ^{\acute{\rm e}\text {t}}_1(X, \bar {x})/(H\cdot j_x(\mathrm {Gal}(\overline {k}/k)))$
is isomorphic to the
$\pi ^{\acute{\rm e}\text {t}}_1(X, \bar {x})$
-set
$Y_{\overline {x}}$
. The identity coset is fixed by
$j_x(\mathrm {Gal}(\overline {k}/k)))$
; hence, it corresponds to a Galois-fixed point of
$Y_{\bar {x}}$
, and therefore to a k-rational point
$y \in Y(k)$
lying above x. This completes the proof of the lemma.
The following lemma is a corollary of [Reference Khukhro and Makarenko11, Theorem 1] and [Reference Nikolov and Segal18, Theorem 1.1].
Lemma 4.2 Any virtually abelian topologically finitely generated group contains a characteristic abelian open subgroup.
4.2 Char(k)=0
Theorem 4.3 Let X be a smooth projective variety of the nonnegative Kodaira dimension over a field k of characteristic 0, L an ample line bundle on X, and
$\phi : X \rightarrow X$
a self-map of X such that
$\phi ^*L \cong L^{\otimes d}$
for an integer
$d>1$
. If
$X(k) \neq \emptyset $
, then there is an abelian variety A with a finite étale map to X over k.
The assumption that
$X(k) \neq \emptyset $
is required on account of the existence of para-abelian varieties.
Proof Let
$x \in X(k)$
and
$\overline {x} \in X(\overline {k})$
be a geometric point lying over x. Recall from [Reference Fakhruddin3, Theorem 4.2] that
$X_{\overline {k}}$
admits a finite Galois étale cover by an abelian variety B. Thus, the image of the geometric fundamental group of B in
$\pi ^{\acute{\rm e}\text {t}}_1(X_{\overline {k}}, \overline {x})$
, say N, is an abelian normal subgroup. In particular,
$\pi ^{\acute{\rm e}\text {t}}_1(X_{\overline {k}}, \bar {x})$
is virtually abelian, and hence, by Lemma 4.2, contains a characteristic abelian open subgroup
$\Gamma $
necessarily of finite index. Moreover, it can be arranged that
$\Gamma \subset N$
, by replacing
$\Gamma $
by
$h\Gamma \subset \Gamma $
for an integer
$h = |\Gamma /(N\cap \Gamma )|$
. Consider the étale cover
$Z \rightarrow X_{\overline {k}}$
given by the subgroup
$\Gamma $
of
$\pi ^{\acute{\rm e}\text {t}}_1(X_{\overline {k}}, \overline {x})$
. Since
$\Gamma \subset N$
, Z is a finite étale cover of B and hence itself is an abelian variety by the Lang–Serre theorem. We may apply Lemma 4.1 to get that there exists a smooth projective variety A defined over k such that
$A \rightarrow X$
is a finite étale cover with a k-rational point, and
$A_{\overline {k}}$
is the étale cover of
$X_{\overline {k}}$
given by the subgroup
$\Gamma $
, which is known to be an abelian variety from the discussion above. Therefore, A itself must be an abelian variety. This completes the proof.
4.3 Char(k)
$>$
0
Theorem 4.4 Let X be a smooth projective variety of the nonnegative Kodaira dimension over a field k of positive characteristic, L an ample line bundle on X, and
$\phi : X \rightarrow X$
a separable map such that
$\phi ^*L \cong L^{\otimes d}$
for an integer
$d>1$
. If the geometric fundamental group of X is virtually abelian and
$X(k) \neq \emptyset $
, then there is an abelian variety A with a finite étale map to X over k.
The proof in this case follows verbatim as in the case of characteristic zero by applying Theorem 1.1 instead of [Reference Fakhruddin3, Theorem 4.2].
Remark 4.5 Let X be a smooth projective variety over an algebraically closed field k. A morphism
$f : X \rightarrow X$
is said to be int-amplified if there exists an ample line bundle L such that
$f^*L\otimes L^{-1}$
is an ample line bundle. It is easy to see that f is a finite map and hence is surjective in this case. Arguing as before, we may conclude that f is étale. Finally, note that f is not an automorphism and hence
$\deg (f)>1$
. The proof of Theorem 1.1 can be adapted using the observations made above to extend the theorem to the case of smooth projective varieties with int-amplified self-map f.
It may be reasonable to make the following.
Conjecture 4.6 The étale fundamental group of a smooth projective variety defined over an algebraically closed field that admits a separable self-map polarized by an ample line bundle must be virtually abelian.
Acknowledgments
The author thanks N. Fakhruddin and V. Srinivas for helpful discussions, comments, and suggestions.
