Let Γ⊂PU(2,1) be a lattice which is not co-ompact, of finite covolume with respect to the Bergman metric and acting freely on the open unit ball
B⊂ℂ2. Then
the toroidal compactification X=Γ\B¯ is a projective smooth surface with elliptic compactification divisor D=X\(Γ\B). In this short note we
discover a new class of unramifed ball quotients X. We consider ball quotients
X with kod(X)=1 and h1(X,𝒪X)=1. We prove that each minimal surface
with finite Mordell-Weil group in the class described admits an étale covering
which is a pull-back of X6(6). Here X6(6) denotes the elliptic modular surface
parametrizing elliptic curves E with 6-torsion points x,y which generate E[6].

1. Introduction

Let the symbol 𝒯 denote the class of complex projective smooth surfaces X which contain pairwise disjoint elliptic curves D1,…,DhX such that U=X∖⋃Di admits the open unit ball B⊂ℂ2 as universal holomorphic covering. As explained in [1], 𝒯 forms the “generic” class of compactified ball-quotient surfaces. There are several motivations to study surfaces in 𝒯 without assuming that the fundamental group π1(U,*) with its Poincaré action on B is an arithmetic lattice of PU(2,1); we refer to [2] or to the introduction of [1]. Since the discovery of blown-up abelian surfaces in 𝒯 by Hirzebruch and Holzapfel some years ago (cf. [3]) there have been no further examples of surfaces of special type in 𝒯. In this short note we present new examples of modular surfaces X∈𝒯 of Kodaira dimension kod(X)=1.

In what follows we only consider complex projective smooth surfaces. An elliptic surface is an elliptic fibration π:X→C of a surface X over a smooth curve C. If π is an elliptic fibration, then a smooth fiber F of π is an elliptic curve and hence isomorphic to ℂ/(ℤ+τℤ) for a τ in the upper half plane. The j-invariant of an elliptic surface π:X→C is the unique morphism from X to the projective line which maps a smooth fiber F of π to j(F)=j(τ). An elliptic surface with finite Mordell-Weil group MW(X) of sections is called extremal if the rank ρ(X) of the Néron-Severi group of X equals h1,1(X). Particular examples of elliptic surfaces arise in the following way. To each pair of positive integers (m,n)∉{(1,1),(1,2),(2,2),(1,3),(1,4),(2,4)}
there exists a modular surface πn(m):Xn(m)→Cn(m) in the sense of Shioda [4] which is extremal, has no multiple fibers and nonconstant j-invariant. The fibration πn(m) has the following properties.

The Mordell-Weil group MW(Xn(m)) is isomorphic to ℤ/mℤ×ℤ/nℤ.

Cn(m) is the (compactified) curve Γn(m)∖ℍ¯ where Γn(m)⊂Sl2(ℤ) is the group
{(abcd);(abcd)≡(1*01)modm,b≡0modn}.

The curve Cn(m) parametrizes triples (E,x,y) of elliptic curves E and points x∈E[m],y∈E[n] such that |ℤx+ℤy|=mn.

There are sections σ1, σ2 of order m, respectively, n generating the Mordell-Weil group, such that a point c∈Γn(m)∖ℍ corresponds to the triple
E=fiberoverc,x=E∩σ1,y=E∩σ2.

All singular fibers of πn(m) are of type Ik in Kodaira's notation and they lie over the cusps of c∈Cn(m). A representant of c in ℚ∪{∞} is stabilized by a matrix γ∈Γ which is an Sl2(ℤ)-conjugate of
(1k01).

By [5] each extremal elliptic surface π:X→C with nonconstant j-invariant, no multiple fibers and Mordell-Weil group MW(X̃) isomorphic to ℤ/mℤ×ℤ/nℤ, where (m,n) is as above, allows a cartesian diagram of finite maps

With this perspective we formulate our main result. A complex projective smooth surface X is irregular if h1(X,𝒪X)>0. An irregular surface of Kodaira dimension kod(X)=1 admits an up to isomorphism unique elliptic fibration to a curve of genus h1(X,𝒪X). If h1(X,𝒪X)=1, then this elliptic fibration coincides with the Albanese morphism. Finally, we remark that h1(X6(6),𝒪X6(6))=1 and that C6(6) is an elliptic curve.

Theorem 1.1.

Let X be an irregular minimal surface with kod(X)=1, elliptic fibration π:X→C and empty or finite Mordell-Weil group.

The surface X is in 𝒯 if and only if the following holds. The curve C is elliptic and there exists an isogeny ν:C̃→C of elliptic curves such that X̃=X×C̃ν is isomorphic to a pull-back X6(6)×C̃μ arising from an isogeny μ:C̃→C6(6) with the property that degν=6degμ/χ(X)≤36.

Assume that X is a surface in 𝒯 isomorphic to a pull-back X6(6)×C̃μ. Then the compactification divisor D of X consists of the 36 sections of π; each section has self-intersection number -χ(X)=-6(degμ); the fibration π admits 12(degμ) singular fibers of type I6, and each component of an I6 intersects D in precisely 6 points; one has ρ(X)=60(degμ)+2 and X is extremal.

2. Some Basic Properties of Surfaces in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M112"><mml:mrow><mml:mi>𝒯</mml:mi></mml:mrow></mml:math></inline-formula>

We cite two results on ball-quotient surfaces which will be needed for the proof of the theorem. The first result is essentially [6, Theorem 3.1] restricted to dimX=2 with attention to sign conventions, except the assertion on semistability. The latter assertion follows from [7]. A reduced effective divisor on a surface X is called semistable if it has normal crossings and if every rational smooth prime component intersects the remaining components in more than one point. If T,D are divisors on X, we say that T is ample modulo D if T2>0 and if the intersection number CT is positive for each curve C on X not supported in D.

Theorem 2.1 (see [<xref ref-type="bibr" rid="B10">6</xref>, <xref ref-type="bibr" rid="B6">7</xref>]).

Let X be a smooth projective surface and D⊂X a divisor with normal crossings. Suppose that KX+D is big and ample modulo D. Then
c12(ΩX1(logD))≤3c2(ΩX1(logD)),
with equality if and only if X∖D is an unramified ball quotient Γ∖B and D is semistable.

There is a canonical exact sequence 0⟶ΩX1⟶ΩX1(logD)→resOD⟶0,
where res is the Poincaré residue map. With this one proves that c1(ΩX1(logD))=[D]-c1(X)∈H2(X,ℂ) and c2(ΩX1(logD))=c2(X)-(c1(X),[D])+([D],[D])∈H4(X,ℂ). Therefore, c12(ΩX1(logD))=(KX+D)2.
If Γ′⊂Γ is a neat normal subgroup with finite index in Γ, then Γ′∖B is compactified by a smooth elliptic divisor, and Γ∖B is compactified by a divisor D. As D is the quotient D′/G, where G=Γ/Γ′, it is a normal curve. Hence, D is smooth and consists of elliptic curves, for rational curves cannot appear because of semistability. Thus, if equality holds in the theorem, then D is smooth. Moreover, one can show that KX+D is big and ample modulo D.

Lemma 2.2.

Let X be in 𝒯 with compactification divisor D and consider an irreducible curve L⊂X. If L is smooth rational, then |L∩D|≥3. And if L is a smooth elliptic curve, then |L∩D|≥1.

Proof.

If the statement is wrong for some rational curve L, then the induced holomorphic map of universal coverings L∖D̃→B=X∖D̃ yields a contradiction to Liouville's theorem.

3. Proof of the Theorem

We begin by proving (1) and suppose that X is an irregular minimal surface of Kodaira dimension kod(X)=1 in 𝒯. Then X is an elliptic surface in a unique way. We denote by π:X→C the elliptic fibration and assume that its Mordell-Weil group is empty or finite. As above, D is the compactification divisor of X. Moreover, we let F be the numerical class of a fiber of π. Since KX+D is ample modulo D, it follows that F has positive intersection with D. Thus, a component of D dominates C. The theorem of Hurwitz implies that C is an elliptic curve and h1(X,𝒪X)=1. Moreover, after transition to an etale cover ν:C̃→C and performing a base change X̃=X×C̃ν, we achieve that every D̃i is a section as soon as it dominates C̃ [1, Lemma 3.3]. We will assume for the time being that X=X̃; once we have shown that (1) is true if X=X̃, it will be easy to obtain (1) in the general case. If X=X̃, then the curves Di are all sections, because they are disjoint. In this case we have the following.

Lemma 3.1.

The identities 36χ(X)=DF·χ(X)=-D2 and DF=36 hold.

Proof.

The canonical bundle formula implies that KX=π*(𝔠) with a Weil divisor 𝔠∈Div(C). Moreover, h0(X,mKX)=h0(C,m𝔠). The theorem of Riemann-Roch yields h0(X,KX)=deg𝔠>0. It results from the adjunction formula that
Di2=-degc=-h0(X,KX)=-χ(X).
Hence, -D2=-∑Di2=DFχ(X). Furthermore, 12χ(X)=c2(X) by Noether's formula. So, Theorem 2.1 yields the remaining identities.

We consider the Mordell-Weil group MW(X)=MW(X̃). By assumption, MW(X) is finite and equals MWtor(X). It follows from the previous lemma that |MWtor(X)|≥DF=36. We next prove the following lemma of general interest.

Lemma 3.2.

Let π:X→C be a minimal elliptic surface over an elliptic curve C. Assume that kod(X)≥1 and that each rational curve L⊂X intersects at least one section of π. Suppose moreover that |MWtor(X)|≥33. Then the following assertions hold.

MW(X) is a torsion group isomorphic to ℤ/6ℤ×ℤ/6ℤ.

All singular fibers of π are semistable of type I6 and each rational curve L⊂In intersects 6 distinct sections of π.

X has 2χ(X) singular fibers.

ρ(X)=h1,1(X)=10χ(X)+2.

Proof.

Statement (1) follows directly from [8, equation (4.8)]. If MW(X) is a torsion group, then its sections do not intersect each other. So, their sum is a smooth divisor D. Moreover, [8, Lemma 1.1] implies that all singular fibers are of type In for some n>0. If Hn⊂M(X) is the nontrivial isotropy group of a node x∈In then Hn and MWtor(X)/Hn are cyclic by [8, Lemma 2.2]. Because of (1) we thus have |Hn|=6 for all isotropy groups Hn. Let S∈MW(X) be the neutral section. By the proof of [8, Lemma 2.2], Hn consists of those sections which intersect the prime component L⊂In containing S∩In. However, since by assumption we may take any section to be the neutral element of MW(X), we have LD=6 for each component L⊂In. As DIn=36, we get n=6. This yields (2). Recalling that ∑Inn=c2(X), we find for the number t of singular fibers of πt=2χ(X)=2g(C)-2+rankMW(X)+2χ(X).

We receive (3). Finally, according to [8, Proposition 1.6] the last equality holds if and only if ρ(X)=h1,1(X), so that X is extremal. An easy calculation shows now (4).

As explained in Section 1, X is isomorphic to a pull-back X6(6)×Cμ. This shows (1) in the theorem if X=X̃. Next we withdraw the additional assumption that X=X̃ from the beginning of the proof and let ν:C̃→C be an isogeny of minimal degree such that X̃ is a pull-back X6(6)×Cμ. We are left to show that degν≤36 and χ(X)·degν=6degμ. The former estimate is clear, because over a curve Di⊂X there lie ≤36 curves D̃i⊂X̃. The latter equality holds, because χ(X)=6degμ/degν by the lemma below. This yields (1) in the general case. Statement (2) in the theorem results from Lemma 3.2, the fact that μ is étale and the following lemma.

Lemma 3.3.

The modular surface X6(6) has invariant χ(X6(6))=6.

Proof.

This is a consequence of the formulae in [9, page 77f].

MomotA.Irregular ball-quotient surfaces with non-positive Kodaira dimensionHolzapfelR.-P.HolzapfelR.-P.Jacobi theta embedding of a hyperbolic 4-space with cuspsShiodaT.On elliptic modular surfacesKloostermanR.Extremal elliptic surfaces and infinitesimal TorelliTianG.YauS.-T.Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometryMiyaokaY.The maximal number of quotient singularities on surfaces with given numerical invariantsMirandaR.PerssonU.Torsion groups of elliptic surfacesBarthW.HulekK.Projective models of Shioda modular surfaces