The classification of Hyperelliptic threefolds
aa r X i v : . [ m a t h . AG ] D ec THE CLASSIFICATION OF HYPERELLIPTICTHREEFOLDS
FABRIZIO CATANESE AND ANDREAS DEMLEITNER
Abstract.
We complete the classification of hyperelliptic threefolds,describing in an elementary way the hyperelliptic threefolds with group D . These are algebraic and form an irreducible 2-dimensional family. Introduction
A Generalized Hyperelliptic Manifold X is defined to be a quotient X = T /G of a complex torus T by the free action of a finite group G which containsno translations. We say that X is a Generalized Hyperelliptic Variety ifmoreover the torus T is projective, i.e., it is an Abelian variety A .The main purpose of the present paper is to complete the classification ofthe Generalized Hyperelliptic Manifolds of complex dimension three. Thecases where the group G is Abelian were classified by H. Lange in [La01],using work of Fujiki [Fu88] and the classification of the possible groups G given by Uchida and Yoshihara in [UY76]: the latter authors showed thatthe only possible non Abelian group is the dihedral group D of order 8.This case was first excluded but it was later found that it does indeed oc-cur (see [CD18] for an account of the story and of the role of the paper[DHS08]). Our paper is fully self-contained and show that the family de-scribed in [CD18] gives all the possible hyperelliptic threefolds with group D .Our main theorem is the following Theorem 0.1.
Let T be a complex torus of dimension admitting a fixedpoint free action of the dihedral group G := D := h r, s | r = 1 , s = 1 , ( rs ) = 1 i , such that G = D contains no translations.Then T is algebraic. More precisely, there are two elliptic curves E, E ′ suchthat:(I) T is a quotient T := T ′ /H, H ∼ = Z / , where T ′ := E × E × E ′ =: E × E × E ,H := h ω i , ω := ( h + k, h + k, ∈ T ′ [2] , and h, k are 2-torsion element h, k ∈ E [2] , such that h, k = 0 , h + k = 0 ; Date : December 27, 2018.AMS Classification: 14K99, 14D99, 32Q15The present work took place in the framework of the ERC Advanced grant n. 340258,‘TADMICAMT’ . (II) there is an element h ′ ∈ E ′ of order precisely , such that, for z =( z , z , z ) ∈ T ′ : r ( z ) = ( z , − z , z + h ′ ) = R ( z , z , z ) + (cid:0) , , h ′ (cid:1) ,s ( z ) = ( z + h, − z + k, − z ) = S ( z , z , z ) + ( h, k, . Conversely, the above formulae give a fixed point free action of the dihedralgroup G = D which contains no translations.In particular, we have the following normal form: E = C / ( Z + Z τ ) , E ′ = C / ( Z + Z τ ′ ) , τ, τ ′ ∈ H := { z ∈ C | Im ( z ) > } ,h = 1 / , k = τ / , h ′ = 1 / r ( z , z , z ) := ( z , − z , z + 1 / s ( z , z , z ) := ( z + 1 / , − z + τ / , − z ) . In particular, the Teichm¨uller space of hyperelliptic threefolds with group D is isomorphic to the product H of two upper halfplanes. Proof of the main theorem
We use the following notation: T = V /
Λ is a complex torus of dimension 3,which admits a free action of the group G = h r, s | r = s = ( rs ) = 1 i ∼ = D , such that the complex representation ρ : G → GL(3 , C ) is faithful.A first observation is that the complex representation ρ of G must containthe 2-dimensional irreducible representation V of G (else, ρ would be adirect sum of 1-dimensional representations: this, by the assumption on thefaithfulness of ρ , would imply that G is Abelian, a contradiction).Hence we have a splitting V = V ⊕ V , where V is 1-dimensional, and we can choose an appropriate basis so that,setting R := ρ ( r ) , S := ρ ( s ), we are left with the two cases Case 1: R = − , S = − − , Case 2: R = − , S = − . which are distinguished by the multiplicity of the eigenvalue 1 of S .Indeed R is necessarily of the form above, since the freeness of the G -actionimplies that ρ ( g ) must have eigenvalue 1 for every g ∈ G . Lemma 1.1.
In both Cases 1 and 2, the complex torus T = V / Λ is isogenousto a product of three elliptic curves, T ∼ isog. E × E × E , where E i ⊂ T ,for i = 1 , , and E and E are isomorphic elliptic curves. In other words,writing E j = W j / Λ j , the complex torus T is isomorphic to ( E × E × E ) /H, H = Λ / (Λ ⊕ Λ ⊕ Λ ) . YPERELLIPTIC THREEFOLDS 3
Proof.
Let I be the identity of T .In Case 1, we set E := ker( S − I ) = im( S + I ), E := ker( R − I ) and E := R ( E ) (here, the superscript zero denotes the connected componentof the identity). Then it is clear that E ∼ = E , and that T is isogenous to E × E × E .In Case 2, we define similarly E := ker( S + I ) = im( S − I ), E := ker( R − I ) and E := R ( E ). We obtain again E ∼ = E , and that T is isogenous to E × E × E . (cid:3) Lemma 1.2.
Writing E j = W j / Λ j , the following statements hold. (1) In Case 1, the lattice Λ is equal to W ∩ Λ . (2) In Case 2, the lattice Λ is equal to W ∩ Λ .Proof. (1) Obviously, E = R ( E ) = W /R (Λ ), i.e., Λ = R (Λ ) ⊂ W ∩ Λ.On the other hand, R ( W ∩ Λ) ⊂ W ∩ Λ = Λ , and applying the automor-phism R of Λ gives W ∩ Λ ⊂ R (Λ ) = Λ .(2) Here, E = R ( E ) = W /R (Λ ), i.e., Λ = R (Λ ) ⊂ W ∩ Λ. For theconverse inclusion, observe R ( W ∩ Λ) ⊂ W ∩ Λ = Λ , and applying R yieldsagain the result. (cid:3) We can now choose coordinates on V such that r is induced by a transfor-mation of the form r ( z , z , z ) = ( z , − z , z + c ) , by choosing as the origin in V a fixed point of the restriction of r to V .We can now view r, s as affine self maps of T induced by affine self maps of E × E × E of the form r ( z , z , z ) = ( z , − z , z + c ) ,s ( z , z , z ) := ( z + a , − z + a , ± z + a ) , and sending the subgroup H to itself. Lemma 1.3.
The freeness of the action of the powers of r is equivalent to: H contains no element with last coordinate equal to c , or c .Moreover, (0 , , c ) ∈ H .Proof. r ( z ) = z is equivalent to ( z − z , z + z , − c ) ∈ H . However, theendomorphism ( z , z ) ( z − z , z + z )of E × E is surjective, hence H cannot contain any element with lastcoordinate equal to c .Since r ( z ) = ( − z , − z , z +2 c ), r ( z ) = z is equivalent to ( − z , − z , c ) ∈ H , and we reach the similar conclusion that H cannot contain any elementwith last coordinate equal to 2 c .Finally, the condition that r is the identity is equivalent to (0 , , c ) ∈ H . (cid:3) Proposition 1.1.
Case 2 does not occur.
FABRIZIO CATANESE AND ANDREAS DEMLEITNER
Proof.
Since we assume that s ( z , z , z ) := ( z + a , − z + a , z + a ) , and that s is the identity, it must be(2 a , , a ) ∈ H. Consider now rs : rs ( z ) = ( − z + a , − z − a , z + a + c ) . The condition that ( rs ) is the identity is equivalent to:( a + a , − ( a + a ) , a + c )) ∈ H. This condition, plus the previous one, imply that( a − a , − ( a + a ) , c ) ∈ H, contradicting Lemma 1.3. (cid:3) Henceforth we shall assume that we are in Case 1, and we can choose theorigin in E so that s ( z , z , z ) := ( z + a , − z + a , − z ) . Lemma 1.4. If s ( z , z , z ) := ( z + a , − z + a , − z ) , then (2 a , , ∈ H and H contains no element of the form ( a , w , w ) . Proof.
The first condition is equivalent to s being the identity, while thesecond is equivalent to the condition that s acts freely, since s ( z ) = z isequivalent to ( a , − z + a , − z ) ∈ H . (cid:3) Proposition 1.2.
For each λ ∈ Λ there exist λ ′ ∈ Λ , λ ∈ Λ , λ ∈ Λ , λ ∈ Λ , , such that 2 λ = λ + λ ′ , λ ′ = λ + λ More precisely, we even have:Λ ⊂ (1 / + (1 / + (1 / . Proof.
Let λ ∈ Λ: we can write2 λ = ( I + S ) λ | {z } =: λ ∈ Λ + ( I − S ) λ | {z } =: λ ′ ∈ Λ . Furthermore, since λ ′ ∈ im( I − S ), we obtain2 λ ′ = ( I + R ) λ ′ | {z } =: λ ∈ Λ + ( I − R ) λ ′ | {z } =: λ ∈ Λ ∩ W =Λ . Hence, λ = λ + λ + λ for unique λ j ∈ Λ j . YPERELLIPTIC THREEFOLDS 5
Applying the automorphism R of Λ and the unicity of the λ j yields theresult, since R exchanges Λ and Λ . (cid:3) Proposition 1.3.
We haveΛ ⊂ (1 / + (1 / + (1 / . Proof.
For λ ∈ Λ we can write λ = λ + λ + λ for unique λ j ∈ Λ j .We now use the property E i ֒ → T ⇒ ∀ (0 , , d ) ∈ H, d = 0 . Indeed, 2 λ = λ + λ + λ , hence (0 , , [ λ ]) ∈ H and λ = 0 in E . Equiv-alently, there is an element λ ′ ∈ Λ with λ λ ′ . (cid:3) Lemma 1.5.
Consider the transformation rs : rs ( z ) = ( − z + a , − z − a , − z + c ) . The condition that its square is the identity amounts to ( a + a , − ( a + a ) , ∈ H, while the freeness of its action is equivalent to the fact that H contains noelement of the form ( w − a , w + a , w ) ⇔ ∀ ( d , d , d ) ∈ H : d + a = d − a . Proof.
The first condition is straighforward, while the freeness of the actionis equivalent to the non existence of ( z , z , z ) such that( z + z − a , z + z + a , z − c ) ∈ H. As usual, we observe that for each w , w there exist z , z , z with z + z = w , z − c = w . (cid:3) We put together the conclusions of Lemmas 1.3, 1.4, 1.5, • (i) (0 , , c ) ∈ H • (ii) (2 a , , ∈ H • (iii) ( a + a , − a − a , ∈ H , hence also ( a − a , a + a , ∈ H .(1) H contains no element of the form ( w , w , c ),(2) nor of the form ( w , w , c )(3) nor of the form ( a , w , w )(4) nor of the form ( w , w , w ) with w + a = w − a .It follows from (iii) and (3) that a = 0. While the condition that eachelement of H which has two coordinates equal to zero is indeed zero (since E i embeds in T !) imply 2 a = 0 , c = 0 . By conditions (1), (2), (3) the elements a , c have respective orders exactly2 ,
4. Moreover:
FABRIZIO CATANESE AND ANDREAS DEMLEITNER • (4) and (i) imply that a + a = 0 • (ii), (iii) and the fact that H has exponent 2 implies 2 a = 2 a = 0,2 a + 2 a = 0. Hence a = a are nontrivial 2-torsion elements.We have thus obtained the desired elements h := a , k := a , h ′ := c . It suffices to show that H is generated by ω := ( h + k, h + k,
0) = ( a + a , a + a , ω ∈ H , by condition (iii).Condition (4) implies that the first coordinate of an element of H must bea multiple of ( a + a ): since it cannot equal a , by condition (3), and if itequals a , we can add ω and obtain an element of H with first coordinate a . Using R , we infer that both coordinates must be a multiple of ( a + a ).Possibly adding ω , we may assume that w = 0: then by (4) we concludethat also w = 0. Finally, the condition that each element of H which hastwo coordinates equal to zero is indeed zero, show that H is then generatedby ω , as we wanted to show.The last assertions of the main theorem follow now in a straightforward way(see [CC17] concerning general properties of Teichm¨uller spaces of hyperel-liptic manifolds). References [CC17]
F. Catanese, P. Corvaja : Teichm¨uller spaces of generalized hyperellipticmanifolds. Complex and symplectic geometry, 39-49, Springer INdAM Ser., 21,Springer, Cham (2017).[CD18]
F. Catanese, A. Demleitner : Hyperelliptic Threefolds with group D , theDihedral group of order 8. Preprint (2018), arXiv:1805.01835.[DHS08] K. Dekimpe, M. Ha lenda, A. Szczepa´nski : K¨ahler flat manifolds. J. Math.Soc. Japan 61 (2009), no. 2, 363-377.[Fu88]
A. Fujiki : Finite automorphism groups of complex tori of dimension two. Publ.Res. Inst. Math. Sci., 24 (1988), 1-97.[La01]
H. Lange : Hyperelliptic varieties. Tohoku Math. J. (2) 53 (2001), no. 4, 491-510.[UY76]
K. Uchida, H. Yoshihara : Discontinuous groups of affine transformations of C . Tohoku Math. J. (2) 28 (1976), no. 1, 89-94. Lehrstuhl Mathematik VIII, Mathematisches Institut der Universit¨at Bayreuth,NW II, Universit¨atsstr. 30, 95447 Bayreuth
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