# Families of abelian varieties on the line (or more generally simply connected varieties)

I'm curious whether the following is true:

Question 1: Let $V/\mathbb{C}$ be a smooth connected variety such that $V^\text{an}$ is simply connected. Then, is every abelian scheme $f:\mathscr{A}\to V$ isotrivial?

Specifically I'm curious about the case when $V=\mathbb{A}^1_\mathbb{C}$.

I think the answer to Question 1 is yes by the following line of reasoning. Deligne shows in Theorie de Hodge II that $\mathscr{A}$ is characterized by $(R^1f^\text{an}_\ast\underline{\mathbb{Z}})^\vee$ as polarizable variation of Hodge structure of type $\{(-1,0),(0,-1)\}$. Now, in our case we know that $(R^1f_\ast^\text{an}\underline{\mathbb{Z}})^\vee$ is a $\mathbb{Z}$-local system and so trivial as such. But, of course, there is no a priori reason to believe that the Hodge filtration is constant.

That said, in this article (see the remarks following Theorem 11) any $\mathbb{Q}$-VHS on a simply connected compactifiable complex manifold is actually constant (as a VHS). In particular, this should imply (since every smooth variety is compactifiable by resolution of singularities), by Deligne's theorem that there is a constant family $\underline{A}$ on $V$ and an isogeny $\varphi:\underline{A}\to\mathscr{A}$. But, $\varphi$ must be $\underline{\mathbb{Z}/n\mathbb{Z}}$ and so $\mathscr{A}$ must be $\underline{A/(\mathbb{Z}/n\mathbb{Z})}$ as desired.

Is this correct?

It then leads to the following natural questions

Question 2: Is every $\mathbb{Z}$-VHS on a simply connected compactifiable (perhaps even algebraic) manifold constant?

and

Question 3: Let $V/\mathbb{C}$ be a smooth connected variety such that $\pi^1_{\acute{e}\text{t}}(V)=0$. Then, is every abelian scheme $f:\mathscr{A}\to V$ isotrivial?

I apologize if these are silly questions.

Also, of note, there must be something truly algebraic happening here as the 'universal family' $\mathscr{E}/\mathbb{H}$ shows.

Thanks!

• There is a notion, introduced I think by Kollar, of "large fundamental group". A variety has large $\pi_1$ if the image of the $\pi_1$ of every subvariety is infinite. He states a conjecture (attributed to Shafarevich) that a variety has large (topological) fundamental group iff the universal cover is a Stein space. I think that this conjecture should imply (modulo stacky issues) that the answer to Q1 (and maybe Q2, by looking at the corresponding period domain) is affirmative. Apr 29 '16 at 15:47
• @PiotrAchinger Hey Piotr! Thanks, I'll look into this. I'm having trouble locating the original paper. Do you have a clue what the title is? Thanks! Apr 29 '16 at 16:59
• J. Kollár, "Shafarevich maps and automorphic forms", MR1341589 ams.org/mathscinet-getitem?mr=1341589 Apr 29 '16 at 21:47
• NB I heard you like Serre-Tate, so you might appreciate this little fact as well: Nori and Srinivas in "Varieties in positive characteristic with trivial tangent bundle" prove that in characteristic p, every family of ordinary abelian varieties over a smooth projective curve becomes trivial after a finite etale cover. In particular, every family of ordinary abelian varieties over a proper variety (no assumptions on $\pi_1$) is isotrivial. Apr 30 '16 at 7:24
• @PiotrAchinger The isotriviality of families of ordinary abelian varieties is also proven in Moret-Bailly's 1985 asterisque 129 "Pinceaux de varietes abeliennes" in Thm. 5.1 (see also Thm. 5.2) on page 237 Chapter XI. Note that Moret-Bailly attributes Thm. 5.1 to Raynaud. I guess the paper of Nori-Srinivas came out at the same time. May 2 '16 at 8:42

Q2 should be yes for polarized VHS by a rigidity theorem of Schmid [theorem 7.24, Variations of Hodge structure, Inventiones 1973], which says roughly that a PVHS is determined by the Hodge structure at a fibre plus monodromy. Q1 should follow from this by Deligne's equivalence that you stated. Q3 is OK also. The point is that a group with trivial profinite completion will only have trivial representations into $GL_n(\mathbb{Z})$, since the target is residually finite.

Addendum Let me give an elementary argument that a family of princ. polarized abelian varieties over the affine line $\mathbb{C}$ is trivial. Given such a family, we get a holomorphic map $f$ from $\mathbb{C}$ to the Siegel upper half plane. This is a bounded domain, so you can separate points using bounded holomorphic functions. After pulling these back to $\mathbb{C}$, they must all be constant. Therefore $f$ is constant.

• Thanks Donu! This makes me very happy. One question: is Q2 obvious without using the cited fact that a $\mathbb{Q}$-VHS is constant? Namely, it's clear that this+Scmid's result imply Q2, but am I missing some obvious way of avoiding this result of Steenbrink and Peters? Apr 29 '16 at 17:08
• Thanks for reminding this simple argument in the addendum ! Naïve question: since $\mathscr A_{g,1}$ is a coarse moduli scheme, is it clear that this map $f$ is holomorphic, and that no covering should be needed? Also, is it true/known that the period spaces of VHS are hyperbolic as well?This would give another proof of Q2 along the same lines.
– ACL
Apr 29 '16 at 20:18
• Hey Donu, I was wondering if you could offer some insight into the following question concerning your addendum proof. Below in the comments of ACL's answer I give a proof for why there can be no non-isotrivial elliptic schemes over $\mathscr{E}/\mathbb{A}^1_\mathbb{C}$. Essentially $\mathscr{E}[N]$ must be constant for all $N$ (since $\mathbb{A}^1_\mathbb{C}$ is (étale) simply connected) and thus for some choice of $\alpha:(\mathbb{Z}/N\mathbb{Z})^2\mid_{/\mathbb{A}^1_\mathbb{C}}\cong \mathscr{E}[N]$ we get that the pair $(\mathscr{E},\alpha)$ defines a point of $Y(N)$. May 2 '16 at 12:49
• Of course, this corresponds to a map $\mathbb{A}^1_\mathbb{C}\to Y(N)$ which extends to a map $\mathbb{P}^1_\mathbb{C}\to X(N)$. But, for $N\gg 0$ we know that $g(X(N))>0$ and so any map $\mathbb{P}^1_\mathbb{C}\to X(N)$ must be constant which gives the isotriviality. Now, this argument is, according to ACL, spiritually the same as your addendum argument. But, there is one big difference that I can see. The method I described above extends to show, for example, that all elliptic schemes over $\mathbb{A}^1_{\overline{\mathbb{F}_p}}$ (or even $\mathbb{A}^1_{\mathbb{F}_p}$) are isotrivial. May 2 '16 at 12:51
• but the method in your addendum cannot apply since there are non-isotrivial abelian surfaces over $\mathbb{A}^1_{\overline{\mathbb{F}_p}}$. Your argument essentially relies on the complex differential geometry of $\mathfrak{h}_n$ opposed to, say, the algebraic properties of the quotient $\mathfrak{h}_n/\text{Sp}_{2n}(\mathbb{Z})$ (or further quotients of that). May 2 '16 at 12:53

Here is a partial answer to questions 1/3. Let be given an abelian scheme $\mathscr A/V$. Let us assume that it carries a principal polarization; in principle, we may reduce to this case by an appropriate isogeny but I do not know whether there is a reference in the litterature. For every integer $n\geq 1$, the $n$-torsion subscheme $\mathscr A_n$ is finite étale over $V$, hence is trivial because $\pi_1(V)=0$. The generic fiber of $\mathscr A/V$, an abelian variety over the field $\mathbf C(V)$ which admits level structures of all level. When $V$ is the affine line, this contradicts a theorem of Alan M. Nadel (“The Nonexistence of Certain Level Structures on Abelian Varieties over Complex Function Fields”, Annals of Mathematics, Vol. 129, No. 1 (Jan., 1989), pp. 161-178).

• Hey ACL, thanks for the information! Do you have a belief that $\mathscr{A}$ should be isogenous to a Jacobian (as in the case of fields)? And is your statement 'I don't know whether there is a reference in the literature' mean "I think it's true" or "It is true—no one's bothered to type it up yet"? Also, do you have any opinion about the 'proof'I described about above? Apr 29 '16 at 16:55
• Also, in this paper it's not clear to me whether the following is true. In the case of elliptic schemes the following works: choose a trivilization $\alpha:\mathscr{E}[n]\xrightarrow{\approx}(\mathbb{Z}/N\mathbb{Z})^2$. This then defines a map $\mathbb{A}^1_\mathbb{C}\to Y(N)$ which extends to a map $\mathbb{P}^1_\mathbb{C}\to X(N)$. For $N\gg 0$ the genus of $X(N)$ is positive and so this map must factor through a point—thus the family is isotrivial. Do you know if there is a way to proceed using the geometry of $\mathcal{A}_{g,1}$ (or $\mathcal{A}_{g,n}$, or its spaces with level structure)? Apr 29 '16 at 16:58
• I think it's true, but the litterature on abelian schemes is scarce. There are notes from a 1967-68 seminar in Orsay, math.u-psud.fr/~biblio/numerisation/docs/04_SEMINAIRE/pdf/….
– ACL
Apr 29 '16 at 20:13
• The argument in your second comment is more or less the (simple) one given by Donu Arapura. The complex definition of $\mathcal A_{g,n}$, as a quotient of a bounded symmetric domain by the free action of a discrete group, shows that it is hyperbolic. Consequently, it does not receive non-constant holomorphic maps from the affine line.
– ACL
Apr 29 '16 at 20:15
• Thanks so much! I'm going to accept Donu's answer because it answers all three of my questions, but I greatly appreciate your help. Apr 29 '16 at 20:18

Here's a bit of a long comment which I hope will help the OP. I also answer some of the questions that arose in the comments.

Firstly, any smooth complex algebraic quasi-projective variety carrying an immersive period map is known to be "hyperbolic" in the sense that

i) it is Brody hyperbolic,

ii) all its subvarieties are of log-general type, and

iii) the fundamental group of $X$ is infinite (and more...).

The hardest of these to show is ii) which follows from a theorem of Kang Zuo; see Theorem 0.1 in https://www.researchgate.net/publication/252994947_On_the_negativity_of_kernels_of_Kodaira-Spencer_maps_on_Hodge_bundles_and_applications

or the more general

These results are used in Lemma 6.3 of http://arxiv.org/pdf/1505.02249v1.pdf . The statement of Lemma 6.3 is a consequence of the statement in Brunebarbe as any PVHS coming from geometry is unipotent (up to finite etale base-change of the base). The result of Zuo is also used in the proof of Proposition 3.1 in Abromovich--Varilly-Alvarado's http://arxiv.org/pdf/1601.02483v2.pdf . I mention these statements as they provide a more direct link to your questions.

To prove iii) you can use the argument given by Arapura or use Theorem 3.1 in Voisin's second book on Hodge theory (which is the same argument as in Arapura's answer).

The above results of Zuo (and Brunebarbe) tell us that all horizontal (smooth) algebraic subvarieties of period domains of PVHS's are hyperbolic. This is I think also the best one could hope for: the period domains of PVHS's are not in general "hyperbolic".

Finally, the argument in Arapura's answer is applied to the stack $\mathcal {A}_{g,1}^{\textrm{an}}$ or, if stackiness bothers you, to the fine moduli space of $g$-dimensional ppav's with full level $3$-structure. It is, in my opinion, unnatural to consider the coarse moduli space (which is not hyperbolic, as it contains copies of the $j$-line) when studying hyperbolicity of parameter spaces of ppav's (or smooth proper curves of genus at least two, or polarized K3 surfaces, polarized Calabi-Yau manifolds, etc.). Note that Siegel upper half-space $\mathbb H_g$ is the universal cover of the stack $\mathcal {A}_{g,1}$. The holomorphicity of a lift $\mathbb C\to \mathbb H_g$ for a holomorphic map $\mathbb C\to \mathcal A_{g,1}^{\textrm{an}}$ is therefore immediate.

I can't give you an intuitive reason as to why $\mathcal A_{g,1, \overline{\mathbb F_p}}$ is no longer hyperbolic when $g\geq 2$. If $g=2$ the non-hyperbolicity comes from the supersingular locus. Does that help?

• Dear Ariyan, thank you so much for your kind answer! I just sent you an email with some questions. May 2 '16 at 14:26