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Comptes Rendus
Mathématique
Mohamed Saïdi
On sections of arithmetic fundamental groups of open p-adic annuli
Volume 360 (2022), p. 291-296
<https://doi.org/10.5802/crmath.303>
© Académie des sciences, Paris and the authors, 2022.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
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Comptes Rendus
Mathématique
2022, 360, p. 291-296
https://doi.org/10.5802/crmath.303
Number theory / Théorie des nombres
On sections of arithmetic fundamental groups
of open p-adic annuli
Mohamed Saïdia
a College of Engineering, Mathematics, and Physical Sciences, University of Exeter,
Harrison Building, North Park Road, EXETER EX4 4QF, United Kingdom
E-mail: M.Saidi@exeter.ac.uk
Abstract. We show the non-existence of sections of arithmetic fundamental groups of open p-adic annuli of
small radii. This implies the non-existence of sections of arithmetic fundamental groups of formal boundaries
of formal germs of p-adic curves.
Manuscript received 20th March 2021, accepted 25th November 2021.
1. Introduction/Statement of the Main Result
Let p ≥ 2 be a prime integer, k a p-adic local field (i.e., k/Qp is a finite extension), with ring of
integers Ok , uniformiser π, and residue field F . Thus, F is a finite field of characteristic p.
Let X → SpecOk be a flat, proper, relative Ok -curve, with X normal, and Xk def= X ×SpecOk Speck
geometrically connected. Assume X (F ) 6= ;. Let x ∈ X cl(F ) be a closed point, OX ,x the local ring
at x, ÔX ,x its completion, and E
def= ÔX ,x ⊗Ok k = ÔX ,x
[ 1
π
]
. Write X
def= SpecE , which we assume to
be geometrically connected. We shall refer to X as the formal germ of X at x.
Let η be a geometric point of X with values in its generic point. Thus, η determines an
algebraic closure k of k, and a geometric pointηof Xk̄
def= X×Speck Speck. There exists a canonical
exact sequence of profinite groups (cf. [3, Exposé IX, Théorème 6.1])
1 −→π1(Xk̄ ,η) −→π1(X ,η) −→Gk −→ 1. (1)
Here, π1(X ,η) denotes the arithmetic étale fundamental group of X with base point η, π1(Xk̄ ,η)
the étale fundamental group of Xk̄ with base point η, and Gk
def= Gal(k/k) the absolute Galois
group of k.
The sequence (1) splits if X (k) 6= ;. This is for example the case if the morphism X → SpecOk
is smooth at x. If X (k) =;; for instance if X is stable and regular, and x is an ordinary F -rational
double point of XF
def= X ×SpecOk F , the existence of sections s : Gk → π1(X ,η) of the projection
π1(X ,η)  Gk would provide examples of sections of the projection π1(Xk ,η)  Gk which are
non-geometric (η induces a geometric point of Xk , denoted also η, via the morphism Xk → Xk ),
i.e., which do not arise from rational points. These in turn will provide counter-examples to
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
292 Mohamed Saïdi
the p-adic version of the Grothendieck anabelian section conjecture. This prompts the following
question.
Question 1. With the above notations, assume X (k) =;. Does the exact sequence (1) split?
In this note we investigate the case where X is a p-adic open annulus. Let
A
def= OkS, B def= A⊗Ok k = A
[
1
π
]
,
D
def= Spf A is the formal standard open disc, and D def= Dk = SpecB its “generic fibre” which is
the standard open disc centred at the point “S = 0”. Let A1k = Speck[S], Yk = P1k its smooth
compactification with function field k(S), and Y = P1
Ok
the smooth compactification of A1
Ok
=
SpecOk [S]. We shall identify A with the completion of the local ring of Y at the closed point
“S = 0”. We have a natural morphism D →P1k , which induces an identification between the set of
closed points of D and the set {
x ∈P1k : |S(x)| < 1
}
.
For an integer n ≥ 1, let
An
def= OkS,T 
(SnT −π) , Bn
def= An ⊗Ok k, and Cn
def= SpecBn .
The natural embedding Cn ,→D induces an identification between the set of closed points of Cn
and the open annulus {
x ∈D : |π| 1n < |S(x)| < 1
}
.
Further, let P
def= A(π) be the localisation of A at the ideal (π), and P̂ the completion of P , which
is a complete discrete valuation ring isomorphic to
OkS{S−1} def=
{ ∞∑
i=−∞
ai S
i : ai ∈Ok , −→
i→−∞
lim |ai | = 0
}
,
where | · | is a normalised absolute value of Ok (cf. [2, §2, 5]). Let L def= Fr(P̂ ) be the fraction field of
P̂ , and C∞
def= SpecL. We shall refer to C∞ as a formal boundary of the formal germs D, and Ci
for i ≥ 1. We have natural scheme morphisms
C∞ −→ ·· · −→Cn+1 −→Cn −→ ·· · −→C1 −→D −→P1k .
Let η be a geometric point of C∞, which induces a geometric point (denoted also η) of Cn for
n ≥ 1. For i ∈N∪ {∞}, we have an exact sequence of arithmetic fundamental groups
1 −→π1(Ci ,k̄ ,η) −→π1(Ci ,η) −→Gk −→ 1, (2)
where π1(Ci ,η) denotes the arithmetic étale fundamental group of Ci with base point η,
π1(Ci ,k̄ ,η) the étale fundamental group of Ci ,k̄
def= Ci ×Speck Speck with base point η; which is
induced by η, and Gk
def= Gal(k/k) the absolute Galois group of k. Here k is the algebraic closure
of k determined by η.
Our main result in this note is the following.
Theorem 2. We use notations as above. There exists an integer N ≥ 1, such that for every integer
n ≥ N , the projection π1(Cn ,η)Gk doesn’t split.
The author ignores, for the time being, if the projection π1(C1,η)Gk splits or not.
As a corollary of Theorem 2, we obtain the following.
Theorem 3. The projection π1(C∞,η)Gk doesn’t split.
C. R. Mathématique — 2022, 360, 291-296
Mohamed Saïdi 293
One of the consequences of Theorems 2 and 3 is that one can not produce examples of
sections of hyperbolic curves over p-adic local fields, which arise from sections of arithmetic
fundamental groups of boundaries of formal fibres, or open annuli with small radii. Those
sections would be non-geometric, hence would provide counter-examples to the p-adic version
of the Grothendieck anabelian section conjecture.
Finally we observe the following. For i ∈ N∪ {∞}, let π1(Ci ,k̄ ,η)ab be the maximal abelian
quotient of π1(Ci ,k̄ ,η), and consider the push-out diagram
1 −−−−−→ π1(Ci ,k̄ ,η) −−−−−→ π1(Ci ,η) −−−−−→ Gk −−−−−→ 1y y ∥∥∥
1 −−−−−→ π1(Ci ,k̄ ,η)ab −−−−−→ π1(Ci ,η)(ab) −−−−−→ Gk −−−−−→ 1
(3)
Thus, π1(Ci ,η)(ab) is the geometrically abelian quotient of π1(Ci ,η).
Proposition 4. The projection π1(Ci ,η)(ab) Gk splits, ∀i ∈N.
The author ignores, for the time being, if the projection π1(C∞,η)(ab) Gk splits or not.
2. Proof of Theorem 2
In this section we shall prove Theorem 2. We use the notations used in Section 1. We argue by
contradiction, and assume that the projection π1(Cn ,η)Gk splits, ∀n ≥ 1.
Proposition 5. There exists a relative curve X → SpecOk with the following properties.
(i) The morphism X → SpecOk is flat, proper, stable, and Xk def= X ×SpecOk Speck is geometri-
cally connected.
(ii) X is regular.
(iii) The set of singular points X singF of the special fibre XF
def= X ×SpecOk SpecF of X consists of
F -rational ordinary double points, U
def= XF \ X singF is F -smooth, and U (F ) =; holds.
(iv) X (k) =; holds.
Proof. First, assume p 6= 2. Let C̃ def= P1F with function field k(C̃ ). Thus, Card(C̃ (F )) = CardF + 1
is even. Arrange the set C̃ (F ) in pairs of F -rational points: C̃ (F ) = {(xi , yi )}1≤i≤ CardF+12 . One can
identify in C̃ the points xi and yi ; 1 ≤ i ≤ CardF+12 , to construct a stable proper F -curve C which
is geometrically connected and geometrically reduced, with normalisation C̃ →C . Moreover, the
set of singular points C sing = {ci }1≤i≤ CardF+12 consists of F -rational ordinary double points, and the
pre-image of ci in C̃ consists of the two F -rational points {xi , yi }. In particular, C (F ) = C sing =
{ci }1≤i≤ CardF+12 . More precisely, for 1 ≤ i ≤
CardF+1
2 , let Õi
def= OC̃ ,xi ∩OC̃ ,yi ⊂ k(C̃ ), Nxi
def= mxi ∩ Õi ,
and Nyi
def= myi ∩ Õi , where mxi (resp. myi ) is the maximal ideal of OC̃ ,xi (resp. OC̃ ,yi ). Define
Oci
def= F +Nxi Nyi ⊂ Õi . Then Oci is a local ring (with maximal ideal Nxi Nyi , and residue field F )
whose integral closure is Oci (cf. [1, Proposition 3.1, Theorem 3.4, and the references therein] for
the properties of Oci , as well as the existence of C with the required properties).
In case p = 2. Consider the affine F -curve Spec( F [s,t ](st ) ), and C̃ its smooth compactification.
Thus, C̃ consists of two F -smooth irreducible components C̃ 1 =P1F , and C̃ 2 =P1F , which intersect
at the F -rational ordinary double point c = (s, t ) ∈ Spec( F [s,t ](st ) ). On each irreducible component C̃ i
of C̃ ; 1 ≤ i ≤ 2, the set of F -rational points of C̃ i \{c} is non-empty and comes into pairs of rational
points {(xi , j , yi , j )}1≤ j≤ CardF2 . As above we can identify each of those pairs of F -rational points
(xi , j , yi , j ) into an F -rational ordinary double point ci , j to construct a reducible and geometrically
connected stable curve F -curve C such that the set of singular points C sing consists of F -rational
C. R. Mathématique — 2022, 360, 291-296
294 Mohamed Saïdi
ordinary double points, a double point ci , j lies on a unique irreducible component of C , and
C sing =C (F ) (the local ring at ci is defined as above; the case p 6= 2. See [7, §4], for a discussion of
this procedure and the existence of such a curve C in the case of reducible curves).
Now the stable F -curve C can be deformed to a semi-stable Ok -curve X → SpecOk with special
fibre XF = C satisfying (i) and (ii) (cf. [9, Proposition 7.10, Corollary 7.11 and its proof]). By our
construction (iii) holds also. If x ∈ X (k), then X specialises in a point x ∈ C (F ) which is a regular
point of C and lies on a unique irreducible component of C (cf. [5, Corollary 9.1.32]). Thus, (iv)
follows from (iii). 
Let X → SpecOK be a regular, proper, flat, and stable Ok -curve as in Proposition 5. Let
y ∈ XF (F ) be an F -rational point, which is an ordinary double point and a regular point of X
(cf. Proposition 5(ii) and (iii)). We fix an isomorphism ρ : ÔX ,y
∼→ RS,T /(ST −π), and identify
X
def= Spec(ÔX ,y ⊗Ok k) with C1 via the isomorphism ρk : ÔX ,y ⊗Ok k ∼→ RS,T (ST−π) ⊗Ok k induced by ρ.
Thus, we have scheme morphisms
C∞ −→ ·· · −→Cn+1 −→Cn −→ ·· · −→C1 −→ Xk .
For n ≥ 1, write
π1(Xk \Sn ,η)
def= lim←−−
Sn⊂Xk \Cn
π1(Xk \ Sn ,η),
where the projective limit is over all finite sets of closed points Sn ⊂ Xk \Cn , and π1(Xk \ Sn ,η) is
the arithmetic fundamental group of the affine curve Xk \Sn with base point η. (Here, we identify
the set of closed points of Cn ; n ≥ 1, with a subset of the set of closed points of Xk which specialise
in y .) There is a natural projection π1(Xk \Sn ,η)Gk , and we have a commutative diagram
π1(Cn ,η) −−−−−→ Gky ∥∥∥
π1(Xk \Sn ,η) −−−−−→ Gk
(4)
where the left vertical map is induced by the morphism Cn → Xk .
Further, we have a natural map
lim←−−
n≥1
π1(Cn ,η) −→ lim←−−
n≥1
π1(Xk \Sn ,η),
and lim←−−n≥1π1(Xk \ Sn ,η) is naturally identified with the absolute Galois group Gk(X )
def=
Gal
(
k(X )sep/k(X )
)
, where k(X )sep is the separable closure of the function field k(X ) of X deter-
mined by the geometric point η.
Lemma 6. The projection Gk(X ) Gk splits.
Proof. First, our assumption that the projectionπ1(Cn ,η)Gk splits, implies that the projection
π1(Xk \Sn ,η) →Gk splits, ∀n ≥ 1 (cf. diagram (4)).
Let (Hi )i∈I be a projective system of quotients Gk(X )  Hi , where Hi sits in an exact sequence
1 → Fi → Hi → Gk → 1 with Fi finite, and Gk(X ) = lim←−−i∈I Hi . [More precisely, write Gk(X ) as a
projective limit of finite groups {H̃ i }i∈I . Then H̃ i fits in an exact sequence 1 → Fi → H̃ i →Gi → 1,
where Gi is a quotient of Gk , and Fi a quotient of Gal(k(X )
sep/k(X )k). Let 1 → Fi → Hi →Gk → 1
be the pull-back of the group extension 1 → Fi → H̃ i  Gi → 1 by Gk  Gi . Then Gk(X ) =
lim←−−i∈I Hi ]. The set Sect(Gk ,Gk(X )) of group-theoretic sections of the projection Gk(X )  Gk is
naturally identified with the projective limit lim←−−i∈I Sect(Gk , Hi ) of the sets Sect(Gk , Hi ) of group-
theoretic sections of the projection Hi  Gk . For each i ∈ I , the set Sect(Gk , Hi ) is non-empty.
Indeed, Hi (being a quotient of Gk(X )) is a quotient of π1(Xk \ Sn ,η) for some n ≥ 1, this
quotientπ1(Xk \Sn ,η) Hi commutes with the projections onto Gk , and we know the projection
C. R. Mathématique — 2022, 360, 291-296
Mohamed Saïdi 295
π1(Xk \Sn ,η) →Gk splits. Hence the projection Hi Gk splits. Moreover, the set Sect(Gk , Hi ) is,
up to conjugation by the elements of Fi , a torsor under the group H 1(Gk ,Fi ) which is finite since
k is a p-adic local field (cf. [6, (7.1.8) Theorem (iii)]). Thus, Sect(Gk , Hi ) is a nonempty finite set.
Hence the set Sect(Gk ,Gk(X )) is nonempty being the projective limit of nonempty finite sets. This
finishes the proof of Lemma 6. (See also [8, the proof of Proposition 1.5] for similar arguments in
a slightly different context.) 
Let s : Gk →Gk(X ) be a section of the projection Gk(X ) Gk (cf. Lemma 6).
Lemma 7. The section s is geometric, i.e., s(Gk ) ⊂ Dx , where Dx ⊂Gk(X ) is a decomposition group
associated to a (unique) rational point x ∈ X (k). In particular, X (k) 6= ;.
Proof. This follows from [4, Proposition 2.4(2)]. 
Now the conclusion of Lemma 7 that X (k) 6= ; contradicts the assertion (iv) in Proposition 5
that X (k) = ;. This is a contradiction. Thus, our assumption that the projection π1(Cn ,η)  Gk
splits, ∀n ≥ 1, can not hold. Let N ≥ 1 be such that the projection π1(CN ,η) Gk doesn’t splits.
Then the projection π1(Cn ,η) Gk doesn’t splits, ∀n ≥ N as required. Indeed, this follows from
the fact that for n ≥ N we have a natural homomorphism π1(Cn ,η) →π1(CN ,η) which commutes
with the projections onto Gk . Hence if the projection π1(Cn ,η)  Gk splits then the projection
π1(CN ,η)Gk .
This finishes the proof of Theorem 2. 
3. Proof of Theorem 3
Next, we explain how Theorem 3 can be derived from Theorem 2. We have,∀n ≥ 1, a commutative
diagram
π1(C∞,η) −−−−−→ Gky ∥∥∥
π1(Cn ,η) −−−−−→ Gk
where the horizontal maps are the natural projections, and the left vertical map is induced by the
morphism C∞ →Cn .
Now assume that the projection π1(C∞,η)  Gk splits. Then the projection π1(Cn ,η)  Gk
splits, ∀n ≥ 1, by the above diagram. But this contradicts Theorem 2.
This finishes the proof of Theorem 3. 
4. Proof of Proposition 4
Let n ≥ 1 be an integer, and `1, `2, distinct prime integers such that `1 ≥ 2n, and `2 ≥ 2n. Let O1,
and O2, be totally ramified extensions of Ok of degree `1, and `2, with fraction fields L1 = Fr(O1),
and L2 = Fr(O2); respectively. Thus, the extensions L1/k and L2/k, are disjoint and Cn(Li ) 6= ;,
for i ∈ {1,2}. A restriction-corestriction argument shows that the class [π1(Cn ,η)(ab)] of the group
extension π1(Cn ,η)(ab) in H 2(Gk ,π1(Cn,k̄ ,η)
ab) is trivial. Thus the group extension π1(Cn ,η)(ab)
splits.
This finishes the proof of Proposition 4. 
C. R. Mathématique — 2022, 360, 291-296
296 Mohamed Saïdi
References
[1] Y. Aubry, A. Iezzi, “On the Maximum Number of Rational Points on Singular Curves over Finite Fields”, Mosc. Math. J.
15 (2015), no. 4, p. 615-627.
[2] N. Bourbaki, Eléments de mathématique. Algèbre commutative. Chapitre 9: Anneaux locaux noethériens complets,
Masson, 1983.
[3] A. Grothendieck (ed.), Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, vol. 224, Springer,
1971.
[4] J. Koenigsmann, “On the ‘Section Conjecture’ in anabelian geometry”, J. Reine Angew. Math. 588 (2005), p. 221-235.
[5] Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University
Press, 2002.
[6] J. Neukirch, A. Schmidt, K. Wingberg, “Cohomology of number fields”, 2000.
[7] M. Rosenlicht, “Equivalence relations on algebraic curves”, Ann. Math. 56 (1952), no. 1, p. 169-191.
[8] M. Saïdi, “On the existence of non-geometric sections of arithmetic fundamental groups”, Math. Z. 277 (2014), no. 1-2,
p. 361-372.
[9] M. Talpo, A. Vistoli, “Deformation theory from the point of view of fibered categories”, in Handbook of moduli. Volume
III, Advanced Lectures in Mathematics (ALM), vol. 26, International Press, 2013, p. 281-397.
C. R. Mathématique — 2022, 360, 291-296


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Yiyu Wang

Comptes Rendus
Mathématique
Yiyu Wang
Ramified cover of varieties with nef cotangent bundle
Volume 360 (2022), p. 929-932
https://doi.org/10.5802/crmath.365
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2022, Vol. 360, p. 929-932
https://doi.org/10.5802/crmath.365
Algebraic geometry / Géométrie algébrique
Ramified cover of varieties with nef cotangent
bundle
YiyuWanga
a Department of Mathematics, University of Wisconsin - Madison, 480 Lincoln Drive,
213 Van Vleck Hall, Madison, WI 53706, USA
E-mail: yiyu.wang@wisc.edu
Abstract. We construct examples to show that having nef cotangent bundle is not preserved under finite
ramified covers. Our examples also show that a projective manifold with Stein universal cover may not have
nef cotangent bundle, disproving a conjecture of Liu–Maxim–Wang [7].
Manuscript received 19 February 2022, revised and accepted 6 April 2022.
1. Introduction
The notion of nefness of the cotangent bundle is closely related to various notions of hyperbolic-
ity of a projective manifold (or more precisely, non-ellipticness), and it entails some topological
obstructions. For example, any perverse sheaf on a projective manifold with nef cotangent bun-
dle has nonnegative Euler characteristic [7, Proposition 3.6]. In [7], inspired by the Singer–Hopf
conjecture and the Shafarevich conjecture, the authors made the following conjecture.
Conjecture ([7, Conjecture 6.3]). Let Y be a projective manifold. If the universal cover of Y is a
Stein manifold, then the cotangent bundle of Y is nef.
Since having Stein universal cover is preserved under finite ramified covering, one may expect
the property of having nef cotangent bundle is also preserved under finite ramified covering.
However, we show that it is not the case in this paper.
Theorem 1. For any positive integer n ≥ 2, there exists a finite surjective map between projective
smooth n-folds f : X → Y , such that Y has nef cotangent bundle, while X does not.
As a corollary, we give a counterexample to the conjecture above.
Corollary 2. There is a smooth projective variety X whose universal cover is Stein while its
cotangent bundle is not nef.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
930 Yiyu Wang
By a theorem of Kratz [4, Theorem 2], any compact quotient of bounded domain in Cn (or any
Stein manifold) has nef cotangent bundle. Theorem 1 shows that boundedness in Kratz’s theorem
is necessary.
There are other hyperbolic-type properties that are preserved under finite surjective mor-
phism, for example the property of having large fundamental group. A normal variety is said to
have large fundamental group, if its universal cover does not contain any positive dimensional
proper complex subspaces (see for example [3]). If a variety has Stein universal cover, then it has
large fundamental group ([7, Proposition 6.7]). Having large fundamental group is also related to
various notions of hyperbolicity, as the Shafarevich conjecture predicts varieties with nonpositive
sectional curvature have Stein universal covers (also see [7]).
Corollary 3. There is a projective variety X which has large fundamental group while its cotangent
bundle is not nef.
The example in Theorem 1 is constructed using the cyclic covering trick. We produce ramified
covering map with smooth branched locus. Using a lemma of Sommese, we prove the ramified
cover cannot have nef cotangent bundle.
Acknowledgements.
The author would like to thank his advisor Botong Wang for suggesting this problem and helpful
discussions. He also would like to thank Conner Simpson for polishing the first draft.
2. A Lemma of Sommese
A line bundle L on a projective variety is called nef if for every irreducible curve C , we have
L ·C ≥ 0. A vector bundle E is called nef if the tautological line bundle OP(E)(1) is nef on P(E)
(the space of hyperplanes in E).
The next lemma is due to Sommese [9, Lemma 1.9].
Lemma 4. Suppose f : X → Y is a double covering between n-dimensional smooth projective
varieties over C, n ≥ 2, with smooth ramification locus R ⊂ X and branched locus B ⊂ Y . Sup-
pose furthermore at any point p ∈ R, f can be written (analytical) locally as (u1,u2, . . . ,un) →
(u21,u2, . . . ,un). If X has nef cotangent bundle, then R has nef conormal bundle.
Proof. Consider the natural exact sequence
f ∗Ω1Y →Ω1X →Ω1f → 0,
and restrict it to R,
Ω1X |R →Ω1f |R → 0,
We claim that Ω1f |R ' OR (−R). Assuming this is true, since Ω1X |R is nef and any quotient bundle
of a nef bundle is also nef, we must have OR (−R) is nef.
To prove the factΩ1f |R 'OR (−R), consider the natural short exact sequence
0 →IR /I 2R →Ω1X |R →Ω1R → 0. (1)
Let ϕ be the composition OR (−R) ' IR /I 2R → Ω1X |R → Ω1f |R , and we want to prove ϕ is an
isomorphism. This can be proved locally. In the coordinate patch U that satisfies the second
condition of this lemma, we know thatΩ1f can be locally written as⊕n
i=1 OU {dui }
OU {u1du1}
⊕n
i=2 OU {dui }
'OU /u1OU {du1},
Yiyu Wang 931
which is a line bundle over R ∩U , generated by du. Moreover, the equivalence class of u
which generates IR /I 2R maps to the class of du, which generates Ω
1
f |R . So ϕ must be an iso-
morphism. 
Corollary 5. Under the assumption of the above lemma, for any irreducible curve C ⊂ R, we must
have C ·R ≤ 0. In particular, if X is of dimension 2, then R ·R ≤ 0.
Proof. For such a curve C , R ·C = degC (OX (R)) = degC (OR (R)) ≤ 0 by OR (−R) is nef. 
Remark. What we actually proved here is the short exact sequence (1) splits. This is clear
geometrically: the tangent bundle T X of X restricted to R has two natural subbundles, one is
the tangent bundle of R, and the other is the kernel bundle of d f |R : T X |R → T Y |B . Moreover,
the assumption “analytic locally” can be replaced by “formal locally”, since for checking a is an
isomorphism, we only need to verify it stalk by stalk formally. Therefore the argument works in
all characteristics not 2.
3. The Cyclic Covering
We recall the construction of cyclic covering.
Proposition 6 ([5, Proposition 4.1.6]). Let Y be an algebraic variety, and L a line bundle on Y .
Suppose given a positive integer m ≥ 1 plus a non-zero section
s ∈ Γ(Y ,Lm)
defining a divisor D ⊂ Y . Then there exists a finite flat covering π : X → Y , where X is a scheme
having the property that the pullback L′ = π∗L of L carries a section s′ ∈ Γ(X ,L′) with (s′)m = π∗s.
The divisor D ′ = di v(s′) maps isomorphically to D. Moreover, if Y and D are non-singular, then so
too are X and D ′.
The construction of X can be described in detail locally. On an affine open subset U of Y
that Lm is trivial, s can be viewed as a regular function on U . Then π−1(U ) ⊂ Y ×A1 is defined
by the equation t m − s = 0. In particular, if m = 2, and D is nonsingular, then the covering we
construct here satisfies the second condition of Lemma 4. The reason is that analytic locally, the
map π can be written as (x1, x2, . . . , xn , t ) → (x1, x2, . . . , xn). Since D is nonsingular, at any given
point p ∈ D , there exists some i such that (x1, x2, . . . , x̂i , . . . , xn , s) is a local coordinate of Y . Then
(x1, . . . , x̂i , . . . , xn , t ) → (x1, . . . , x̂i , . . . , xn , s) is the desired coordinate presentation of π.
Now, we can prove the claimed results in the introduction.
Proof of Theorem 1. We pick any smooth n-fold Y with nef cotangent bundle, a very ample line
bundle H , and any smooth section D ∈ |2H |. By the construction above, we get a double covering
map f : X → Y , branched over D , satisfying all the conditions of Lemma 4. In particular, if X
has nef cotangent bundle, then for any irreducible curve C ⊂ D , D ′ · f ∗C = 2D ′ · f −1(C ) ≤ 0 by
Corollary 5 ( f −1(C ) is still irreducible since f maps D ′ isomorphically to D). On the other hand,
D ′ · f ∗C = 2D ·C = 4H ·C > 0 by the assumption H is very ample. Therefore X cannot have nef
cotangent bundle. 
Proof of Corollary 2. Let Y in the above construction be an abelian variety. Then Y has universal
cover Cn . Taking the fiber product in the category of complex spaces, we have the Cartesian
diagram
X ′ Cn
X Y
932 Yiyu Wang
where X ′ is an infinite unramified cover of X . Since X → Y is finite, X ′ → Cn is also finite, by [2,
Theorem V.1.1] X ′ is Stein. Therefore, the universal cover of X is Stein because any unramified
covering (not necessarily finite) of a Stein manifold is Stein (see [10]). But the cotangent bundle
of X is not nef. 
Proof of Corollary 3. For a normal variety X , having large fundamental group is equal to say for
any positive dimensional cycle w : W → X , the image of π1(W ) → π1(X ) is infinite (see [3]). It’s
obvious that having large fundamental group is preserved under finite surjective morphism using
this definition. Take Y to be an abelian variety, X as in the proof of Theorem 1, then X has large
fundamental group while its cotangent bundle is not nef. 
Kobayashi hyperbolicity is an important example of hyperbolicity of complex manifolds. If a
projective variety has ample cotangent bundle, then it also has Kobayashi hyperbolicity (see [6,
Theorem 6.3.26]). However, the above construction shows
Corollary 7. There is a projective variety X which is Kobayashi hyperbolic while its cotangent
bundle is not nef.
Proof. Take Y to be any projective Kobayashi hyperbolic variety in the above construction. Then
f : X → Y is finite surjective but X does not have nef cotangent bundle. The variety X is also
Kobayashi hyperbolic, because if there is any nonconstant holomorphic map g :C→ X , then f ◦g
is constant since Y is Kobayashi hyperbolic. Therefore g (C) is contained in a fiber of f , which is
a finite set, so g must be constant, and X is also Kobayashi hyperbolic. 
Remark. Such examples were already known. A smooth hypersurface in Pn cannot have nef
cotangent bundle if n ≥ 3 (see for example [6, Example 7.2.3]), while there are Kobayashi hyper-
bolic hypersurfaces in all dimensions (see for example [8]).
All these examples show that, the nefness of cotangent bundle is too strong as a hyperbolicity
condition. Inspired by [1], one possible adjustment is to look at the class of projective manifolds
that admit a finite morphism to a manifold with nef cotangent bundle. Since the pushforward
of a perverse sheaf under a finite morphism is still perverse, by [7, Proposition 3.6] any perverse
sheaf on such manifold has nonnegative Euler characteristic. In [1], the authors proved the same
property holds for projective varieties that admit a complex variation of Hodge structures with
finite period map. This result also indicates that admitting a finite morphism to a projective
manifold with nef cotangent bundle is a more appropriate hyperbolicity condition.
References
[1] D. Arapura, B. Wang, “Perverse sheaves on varieties with large fundamental groups”, 2021, https://arxiv.org/abs/
2109.07887.
[2] H. Grauert, R. Remmert, “Applications of Theorems A and B”, p. 125-185, Springer, 2004.
[3] J. Kollár, “Shafarevich maps and plurigenera of algebraic varieties”, Invent. Math. 113 (1993), no. 1, p. 177-215.
[4] H. Kratz, “Compact complex manifolds with numerically effective cotangent bundles”, Doc. Math. 2 (1997), p. 183-
193.
[5] R. Lazarsfeld, Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series, Springer, 2004.
[6] ——— , Positivity in Algebraic Geometry II: Positivity for Vector Bundles, and Multiplier Ideals, Springer, 2004.
[7] Y. Liu, L. Maxim, B. Wang, “Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár”, J. Reine
Angew. Math. 781 (2021), p. 1-18.
[8] B. Shiffman, M. Zaidenberg, “Hyperbolic hypersurfaces in Pn of Fermat–Waring type”, Proc. Am. Math. Soc. 130
(2002), no. 7, p. 2031-2035.
[9] A. J. Sommese, “On the density of ratios of Chern numbers of algebraic surfaces”, Math. Ann. 268 (1984), no. 2, p. 207-
221.
[10] K. Stein, “Überlagerungen holomorph-vollständiger komplexer Räume”, Arch. Math. 7 (1956), no. 5, p. 354-361.


Ramified cover of varieties with nef cotangent bundle

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