Rigid manifolds of general type with non-contractible universal cover

For each n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 3$$\end{document} we give examples of infinitesimally rigid projective manifolds of general type of dimension n with non-contractible universal cover. We provide examples with projective and examples with non-projective universal cover.


Introduction
In [BC18] several notions of rigidity have been discussed, the relations among them have been studied and many questions and conjectures have been proposed.In particular the authors showed that a rigid compact complex surface has Kodaira dimension −∞ or 2, and observed that all known examples of rigid surfaces of general type are K(π, 1) spaces.Recall that a CW complex with fundamental group π is called K(π, 1) space if its universal cover is contractible, and that these spaces have the property that their homotopy type is uniquely determined by their fundamental group (cf.[Hat02, §1.B]).This implies that the topological invariants, such as homology and cohomology, are determined by π.In [BC18] the following natural question has been posed.

Question. Do there exist infinitesimally rigid surfaces of general type with non-contractible universal cover?
The aim of this paper is to give a positive answer for the analogous question in higher dimensions.More precisely, we construct for each n ≥ 3 an infinitesimally rigid manifold of general type of dimension n with non-contractible universal cover.For surfaces the question remains open.We recall now the notions of rigidity that are relevant for our purposes.

Definition 1.
Let X be a compact complex manifold of dimension n.
(1) A deformation of X is a proper smooth holomorphic map of pairs f : (X, X) → (B, b 0 ), where (B, b 0 ) is a connected (possibly not reduced) germ of a complex space.
(2) X is said to be rigid if for each deformation of X, f : (X, X) (3) X is said to be infinitesimally rigid if H 1 (X, Θ X ) = 0, where Θ X is the sheaf of holomorphic vector fields on X. (4) X is said to be (infinitesimally) étale rigid if all finite étale covers f : Y → X are (infinitesimally) rigid.
Remark 2. i) By Kodaira-Spencer-Kuranishi theory every infinitesimally rigid manifold is rigid.The converse does not hold in general as it was shown in [BP18] and [BGP20] (cf.also [MK71]).
Both the examples constructed in [BP18] and Beauville surfaces are product quotient varieties, i.e. (resolutions of singularities of) finite quotients of product of curves with respect to a holomorphic group action.In recent years, product quotients turned out to be a very fruitful source of examples of rigid complex manifolds with additional properties.Besides the examples above, we mention [BG20], where the authors construct the first examples of rigid complex manifolds with Kodaira dimension 1 in arbitrary dimension n ≥ 3, and [BG21] where they constructed new rigid three-and four-folds with Kodaira dimension 0. We refer to [CF18, FGP20, FG20, GPR18, LP16, LP20] for other interesting examples of product quotient varieties.
The manifolds we construct are also product quotients.More precisely, inspired by the construction in [BP18] in Section 1 we consider for each n ≥ 3 and d ≥ 4, even and not divisible by 3 the n-fold product C n of the Fermat curve C of degree d together with a suitable action of Z 2 d .The quotient X n,d := C n /Z 2 d is a normal projective variety with isolated cyclic quotient singularities of type 1 2 (1, . . ., 1), Kodaira dimension n and Blowing up the singular points, we obtain a resolution X n,d → X n,d such that ).Therefore, X n,d is an infinitesimally rigid projective manifold of general type.
In Section 2 we show that the universal cover U n,d of X n,d is non-contractible since it contains several P n−1 (see Propostion 10).We then discuss the finiteness of the fundamental group π 1 (X n,d ) = π 1 ( X n,d ).The crucial ingredient here is Armstrong's description of the fundamental group of a quotient space [Arm68] adapted to product quotients by [BCGP12].The finiteness of π 1 ( X n ) is equivalent to the finiteness of certain groups (Proposition 16: Finiteness criterion).This allows us to prove the following.
Theorem 3.For each n ≥ 3, d ≥ 4, even and not divisible by 3 there exists an infinitesimally rigid projective n-dimensional manifold of general type X n,d , whose universal cover U n,d is non-contractible.Moreover, the universal cover U n,d is projective if and only if d = 4.
The construction actually works also for n = 2: the surface X 2,4 is not rigid, whereas the surface X 2,d for d ≥ 8 is rigid but not infinitesimally rigid (see [BP18]), and its universal cover is non-contractible.
Notation.We work over the field of complex numbers, and we denote by Z n the cyclic group of order n and by ζ n a primitive n-th root of unity.The rest of the notation is standard in complex algebraic geometry.

The families
0} ⊂ P 2 be the Fermat curve of degree d.Consider the group action There are 3d points on C d with non-trivial stabilizer.They form three orbits of length d.A representative of each orbit and a generator of the corresponding stabilizer is given in the table below: Hence the quotient map f : is branched in (0 : 1), (1 : 0) and (1 : −1), each with branch index d.
1.1.The singular quotients X n,d .From now on we fix d ≥ 4, even and not divisible by 3, and denote C d simply by C. Let A be the automorphism of Z 2 d given by the matrix and let Remark 4. The diagonal action is not free, indeed Noting that φ 1|H = φ 2|H , we see that a point (z 1 , . . ., z n ) ∈ C n has a non-trivial stabilizer if and only if all its coordinates z i belong to one and only one of the three Z 2 d -orbits displayed in the table above.
Proposition 5.For n ≥ 3 the projective variety X n,d is infinitesimally rigid and of general type.The singular locus of X n,d consists of 6 • d n−2 cyclic quotient singularities of type 1 2 (1, . . ., 1).
Proof.By Remark 4 there are 3 • d n points on C n with non-trivial stabilizer, each generated by one of the order 2 elements in Z 2 d .Thus, X n,d has (3 According to Schlessinger [Sch71], isolated quotient singularities in dimension at least three are rigid, i.e.Ext 1 (Ω 1 X n,d , O X n,d ) = 0. Thus the local-to-global Ext spectral sequence yields Hence it suffices to verify that X n,d has no equisingular deformations.Since g(C) ≥ 3 we have H 0 (C, Θ C ) = 0, hence by Künneth formula we get Using the fact that the quotient map C n → X n,d is quasi-étale and the action is diagonal, we obtain The branch locus B of f : 1.2.Resolution of singularities of type 1 2 (1, . . ., 1).Proposition 6.A singularity U := C n /Z 2 of type 1 2 (1, . . ., 1) admits a resolution ρ : U → U by a single blow-up, with exceptional prime divisor For a proof we refer to [Sch71, proof of Theorem 4], see also [BG20, Corollary 5.9, Proposition 5.10].
Remark 7 (see [BG20, Remark 5.4]).Both properties are not obvious and in general even false.For any resolution ρ : Z ′ → Z of a normal variety Z, the direct image ρ * Θ Z ′ is a subsheaf of the reflexive sheaf Θ Z , and this inclusion is in general strict: e.g.take the blow-up of the origin of C 2 .
The vanishing of R 1 ρ * Θ Z ′ is also not automatic: take the resolution of an A 1 surface singularity (i.e. 1 2 (1, 1)) by a −2 curve, then R 1 ρ * Θ Z ′ is a skyscraper sheaf at the singular point with value Corollary 8. Let Z n be a projective variety of dimension n ≥ 3 with only singularities of type 1 2 (1, . . ., 1).Then there exists a resolution ρ : In particular, if Z n is infinitesimally rigid, so is Z n .
Proof.Since the singularities of Z n are isolated, we resolve them simultaneously using Proposition 6 and we get a resolution ρ : Z n → Z n having the same properties: By the corollary, for n ≥ 3 there exists a resolution X n,d → X n,d of the singularities of X n,d , which is infinitesimally rigid.By Remark 7, for n = 2 the minimal resolution X 2,d of X 2,d is not infinitesimally rigid, nevertheless the main theorem of [BP18] shows that X 2,d is rigid for d ≥ 8, whereas X 2,4 is a numerical Campedelli surface, whose Kuranishi family has dimension 6. 1.3.Non-étale infinitesimally rigidity.We conclude this section constructing an étale cover of X n,d which is not infinitesimally rigid, thus X n,d is not étale infinitesimally rigid.
Lemma 9. Let Y n,d := C n /H be the quotient with respect to the restricted diagonal action, then: (1) The natural morphism ψ : Y n,d → X n,d is an unramified Galois cover with group . By Remark 4 the stabilizer of a point z ∈ C n with respect to the Z 2 d -action is contained in H, whence the map ψ is unramified. ( arguing as in Proposition 5.

The universal cover of X n,d
In this section we prove that the universal cover U n,d of X n,d is non-contractible, and then we discuss whether it is projective or not.
Proposition 10.Let X be a compact Kähler manifold, containing a P m .Then the universal cover U of X is non-contractible.
Proof.Since P m is simply connected, the inclusion map i : P m ֒→ X lifts to a map f : P m → U .Looking for a contradiction, assume that U is contractible, then f is homotopic to a constant map, therefore the inclusion i is also homotopic to a constant map.In particular we see that the induced linear map i * : H 2 (X, C) → H 2 (P m , C) is the zero map.Now let [ω] be a Kähler class of X.Its restriction i * ([ω]) is a Kähler class of P m , whence non zero, contradiction.
Remark 12.By Lemma 9 the universal cover U n,d of X n,d is not infinitesimally rigid.

The Fundamental Group.
In this section we discuss the finiteness of the fundamental group π 1 ( X n,d ).In order to do this we use the main theorem of [Arm68] in the case of product quotient varieties following [BCGP12,DP12].We briefly recall their strategy and we refer to them for further details.
Let G be a finite group acting diagonally on a product Z := C 1 × . . .× C n of curves of genus at least 2, and consider the group G of all possible lifts of automorphisms induced by the action of G on Z to the universal cover u : H n → Z.The group G acts properly discontinuously on H n and u is equivariant with respect to the natural map G → G, hence we have an isomorphism H n /G ∼ = Z/G.Since H n is simply connected we can apply Armstrong's results (see [Arm68]) and get the following.
Proposition 13.Let Fix(G) be the normal subgroup of G generated by the elements having non-empty fixed locus.Then Assume that the G-action on Z restricts to a faithful action φ i on each factor C i .Let T i be the group of all possible lifts of automorphisms induced by the action of G on C i to the universal cover H of C i , and let ϕ i : T i → G be the natural map.In this setting, the above group G is the preimage of the diagonal subgroup ∆ G ⊂ G n under ϕ 1 × . . .× ϕ n : There is also a similar description of G in the non-faithful case, see [DP12, Proposition 3.3].
Remark 14. i) The group T i has a simple presentation (see also [Cat15, Example 29]): let g ′ be the genus of C i /G and m 1 , . . ., m r be the ramification indices of the branch points of the covering map ii) The group T(g ′ ; m 1 , . . ., m r ) is called the orbifold surface group of type [g ′ ; m 1 , . . ., m r ].
The non-trivial stabilizers of the T i -action on H are cyclic and generated by the conjugates of the elements c k .The restriction of ϕ i to each one of these subgroups is an isomorphism onto its image, which is the stabilizer of a point in C i .Conversely, all non-trivial stabilizers of the G-action on C i are of this form (see [BCGP12]).Definition 15.Let L i ⊂ T i be set of the elements c l j j ∈ T i such that ϕ i (c l j j ) ∈ G has non-empty fixed locus on Z = C 1 × . . .× C n , where j ∈ {1, . . ., r} and l j ∈ {1, . . ., m j − 1}.
We denote by L i T i the normal subgroup of T i generated by L i .
Proposition 16 (Finiteness criterion).The group π 1 (Z/G) = G/ Fix(G) is finite if and only if the groups T i / L i T i are finite.
Proof.According to [BCGP12, pag.1018-1019] the group G/ Fix(G) fits in an exact sequence where E is a finite group and H is a subgroup of finite index of the product Remark 17.Let X be a normal variety with only quotient singularities, and let ρ : X → X be a resolution of singularities.Then ρ * : π 1 ( X) → π 1 (X) is an isomorphism, by [Kol93,Theorem 7.8].
According to the description of X n,d given in the previous section its associated orbifold surface groups T i are all of type [0; d, d, d], and applying this discussion to our situation we get the following.
Note that the number of occurrences n i of the letter c i in the word w 2 k is even.Observe now, that in any group a product a • b can be written as b • (b −1 • a • b), hence we can write w 2 k as (2.1) for certain g i , . . ., h j ∈ T k .
Theorem 18.The universal cover U n,d of X n,d is projective if and only if d = 4. Proof.The universal cover U n,d of X n,d is projective if and only if the fundamental group π 1 ( X n,d ) is finite.Therefore, by Propositon 16 U n,d is projective if and only if the groups T i / L i T i are finite.Let k := d 2 .Since the elements in Z 2 d fixing points on C n are exactly the elements in H = (k, 0), (0, k) , by Remark 14 ii) we see that L