aa r X i v : . [ m a t h . AG ] D ec On Shimura curves in the Torelli locus of curves
Xin Lu Kang Zuo
Abstract
Oort has conjectured that there do not exist Shimura curves lying generically in theTorelli locus of curves of genus g ≥
8. We show that there do not exist one-dimensionalShimura families of semi-stable curves of genus g ≥ g ≥
8. The first result proves a slightly weaker form of theconjecture for the case of Shimura curves of Mumford type. The second result provesthe conjecture for the Torelli locus of hyperelliptic curves. We also present examples ofShimura curves contained generically in the Torelli locus of curves of genus 3 and 4.
1. Introduction
Let M g be the moduli space of smooth projective curves of genus g ≥ C , and A g be the moduli space of g -dimensional principally polarized abelian varieties over C . Thereis a natural morphism called the Torelli morphism , j : M g → A g , which sends a curve to its canonically principally polarized Jacobian. The image T o g iscalled the open Torelli locus , and its Zariski closure T g ⊆ A g is called the Torelli locus .According to a conjecture of Coleman, for a fixed genus g ≥
4, there are only finitelymany CM-points in M g . This conjecture is known to be false for 4 ≤ g ≤
7, by the factthat there exist Shimura subvarieties Z of positive dimension contained generically in theTorelli locus , i.e., Z ⊆ T g and Z ∩ T o g = ∅ . We refer to [26] for a beautiful discussion of this topic. Combining with the conjecture ofAndr´e-Oort, which says that a Shimura variety is characterized by having a dense subsetof CM-points, one has the following expectation (cf. [30, § § Conjecture 1.1 (Oort) . For large g (in any case g ≥ ), there does not exist a Shimurasubvariety of positive dimension contained generically in the Torelli locus. We study here the problem for the case of Shimura curves, and in general Kuga curves.Recall that a closed subvariety
Z ֒ → A g = Sp g ( Z ) \ Sp g ( R ) /U ( g ) This work is supported by SFB/Transregio 45 Periods, Moduli Spaces and Arithmetic of AlgebraicVarieties of the DFG (Deutsche Forschungsgemeinschaft), and partially supported by National Key BasicResearch Program of China (Grant No. 2013CB834202). n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo is called a
Kuga subvariety if the inclusion is induced by a homomorphism G → Sp g forsome algebraic group G (cf. [28]). Moreover Z is a Shimura subvariety if Z is Kuga andcontains a CM-point. A Kuga (resp.
Shimura ) curve U is a one-dimensional connectedKuga (resp. Shimura) variety. The corresponding universe family of abelian varieties overa Kuga (resp. Shimura) curve is called a Kuga (resp.
Shimura ) family of abelian varieties .Let h : X → B be a semi-stable family of g -dimensional abelian varieties over a smoothprojective curve B , with singular fibres Υ nc / ∆ nc . Let U := B − ∆ nc and V := h − ( U ).Then h : V → U is an abelian scheme and the direct image sheaf R h ∗ C V is a local systemon U which underlies a variation of polarized Hodge structure V of weight one. Let (cid:0) E , ⊕ E , , θ (cid:1) := (cid:16) h ∗ Ω X/B (log Υ nc ) ⊕ R h ∗ O X , θ (cid:17) denote the logarithmic Higgs bundle by taking the grading of the Hodge filtration on theDeligne’s extension of the de Rham bundle ( R h ∗ C V ⊗ O U , ∇ ) , where the Higgs field θ isgiven by the edge morphism of the tautological sequence0 −→ f ∗ Ω B (log ∆ nc ) −→ Ω X (log Υ nc ) −→ Ω X/B (log Υ nc ) −→ . By [8] or [17], E is decomposed as a direct sum( E, θ ) = (cid:0) A , ⊕ A , , θ | A , (cid:1) ⊕ (cid:0) F , ⊕ F , , (cid:1) of Higgs bundles, where A , is an ample vector bundle, F , and F , are vector bundlesunderlying unitary local subsystems F , ⊕ F , ⊂ V . Following [36], the Higgs field is called to be maximal if θ | A , : A , → A , ⊗ Ω B (log ∆ nc )is an isomorphism, and strictly maximal if furthermore F = 0.The Arakelov inequality (cf. [7] or [14]) says thatdeg A , ≤ rank A , · deg Ω B (log ∆ nc ) . We say that the family of abelian varieties X → B reaches the Arakelov bound if the aboveinequality becomes an equality. It is shown in [36] that this property is equivalent to themaximality of the Higgs field for A , i.e., θ | A , is an isomorphism.It is proved in [37] by Viehweg and the second author that if h : V → U has a strictlymaximal Higgs field, then h : V → U is a universal family over a Shimura curve of Mumfordtype , which means that if ∆ nc = ∅ , then V is isogenous over U to a g -tuple self-product ofa universal family of elliptic curves; if ∆ nc = ∅ , then h is derived from the corestriction of aquaternion division algebra over a totally real number field with all infinite places ramifiedexcept one (see [37] for more details). Moreover, M¨oller showed in [23] that the conversealso holds. In general, it is showed in [24] that h : V → U is a Kuga family if and only if ithas a maximal Higgs field.In [9], Hain studied locally symmetric families of compact Jacobians satisfying someadditional conditions. Based on his methods, de Jong and Zhang ([13]) proved that certainShimura subvarieties parameterizing abelian varieties with real multiplication do not liegenerically in T g for g ≥
4. In [12], de Jong and Noot developed a method based on a—2— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo criterion due to Dwork-Ogus using p − adic Hodge theory ([6]) and proved that the basevarieties of some specific universal families of curves arising from cyclic covers of P arenot contained generically in T g . Extending this to some general case, recently Moonen([25]) proved that there are exactly twenty families of curves coming from cyclic coversof P such that the base varieties lie generically in T g with g ≤
7, which implies thatConjecture 1.1 holds if the corresponding families arising from a universe cyclic cover of P .In [18], Kukulies showed that a given rational Shimura curve with strictly maximal Higgsfield in A g cannot be contained generically in T g for g sufficiently large.Our first result is to exclude certain Shimura curves arising from families of curves withstrictly maximal Higgs field. We prove an effective bound on the genus g for which thereexists a Shimura family of curves of genus g with strictly maximal Higgs field.Let f : S → B be a family of semi-stable curves over a smooth projective curve B andlet ∆ nc ⊂ B denote those points corresponding to fibres of f with non-compact Jacobians.Put U = B \ ∆ nc and S := f − ( U ). Then the relative Jacobian jac ( f ) : J ac ( S /U ) → U is an abelian scheme over U . We call f to be a Kuga (resp.
Shimura ) family of curves , if jac ( f ) is a Kuga (resp. Shimura) family of abelian varieties. The family f is called to bewith strictly maximal Higgs field, if the Higgs field associated to jac ( f ) is strictly maximal,or equivalently if jac ( f ) is a universal family over a Shimura curve U of Mumford type by[37]. Theorem 1.2.
For g ≥ , there does not exist a Shimura family f : S → B of genus- g curves with strictly maximal Higgs field. Our next result is regarding Kuga and Shimura curves arising from families of hyperel-liptic curves, without the assumption on the strictly maximality of Higgs field. Let H g ⊂ M g denote the moduli space of smooth hyperelliptic curves, H ctg ⊂ M g denote the moduli space of stable hyperelliptic curves with compact Jacobians and j ( H g ) ⊂ j ( H ctg ) ⊂ T g denote the images under the Torelli map. Note that the Zariski closure of j ( H g ) in A g is j ( H ctg ).A Kuga or Shimura curve U ⊂ A g is said to be contained generically in the Torelli locus j ( H ctg ) of hyperelliptic curves , if U ⊆ j ( H ctg ) and U ∩ j ( H g ) = ∅ . It is clear that if f is a Kuga (resp. Shimura) family of hyperelliptic curves, then theimage of U under the Torelli map is a Kuga (resp. Shimura) curve lying generically in j ( H ctg ) . —3— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
Conversely, given a Kuga (resp. Shimura) curve U lies generically in j ( H ctg ) . We like toshow U ⊂ j ( H ctg ) is induced by a Kuga (resp. Shimura) family of hyperelliptic curves.By taking an n -level structure, we may assume that U ⊆ T g,n ⊆ A g,n , hence, U carriesa universal family h : V → U of abelian varieties, which is the Kuga (resp. Shimura) familyof abelian varieties over U classifying the level n -structure. We consider now the Torellimap j : M g,n → A g,n By Oort-Steenbrink (cf. [31]), j is a 2-to-1 morphism exactly outside the hyperelliptic locus H g,n ⊂ M g,n . Furthermore, the restriction of j to the hyperelliptic locus j : H g,n → j ( H g,n ) ⊂ A g,n is injective and immersion. So, we may regard j : H ctg,n → j ( H ctg,n )as the blowing up along the subvariety j ( H ctg,n ) \ j ( H g,n ) . Since U is a smooth and closedcurve in j ( H ctg,n ) and U ∩ j ( H g,n ) = ∅ , the proper transformation ˆ U ⊂ H ctg,n of U under theblowing up j is isomorphic to U j ˆ U : ˆ U ≃ U. Hence, the pullback of the Kuga (resp. Shimura) family of abelian varieties h : V → U under the isomorphism j ˆ U is again a Kuga (resp. Shimura) family j ∗ ˆ U ( h ) : j ∗ ˆ U ( V ) → ˆ U By the definition of the Torelli map j , it is the Jacobian of the pullback to ˆ U → H ctg,n ofthe universal family of stable hyperelliptic curves of compact type to ˆ U → H ctg,n , saying f : S → ˆ U .
Theorem 1.3.
For g ≥ , there does not exist a Kuga family f : S → B of hyperellipticcurves. In particular, for g ≥ there does not exist a Shimura family f : S → B ofhyperelliptic curves. By the above discussion, as a consequence of Theorem 1.3, we obtain
Theorem 1.3 ′ . For g ≥ , there does not exist a Kuga curve, which lies generically in theTorelli locus j ( H ctg ) of hyperelliptic curves. In particular, for g ≥ there does not exist aShimura curve, which lies generically in j ( H ctg ) . Let f : S → B be a semi-stable family of curves of genus g ≥
2. Let Υ → ∆ denote thesemi-stable singular fibres, Υ c → ∆ c denote the singular fibres with compact Jacobians,Υ nc , Υ \ Υ c → ∆ nc , ∆ \ ∆ c correspond to singular fibres with non-compact Jacobians, U := B \ ∆ nc , and S := f − ( U ). Then the logarithmic Higgs bundle associated to theVHS of the relative Jacobian jac ( f ) : J ac ( S ) −→ U —4— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo is decomposed as Higgs bundles( E , ⊕ E , , θ ) = ( A , ⊕ A , , θ | A , ) ⊕ ( F , ⊕ F , , , where θ | A , : A , −→ A , ⊗ Ω B (log ∆ nc )is described on Page 2. Since the family f : S → B is semi-stable, it is well known that (cid:0) E , ⊕ E , , θ (cid:1) ∼ = (cid:16) f ∗ Ω S/B (log Υ) ⊕ R f ∗ O S , θ (cid:17) . Theorem 1.2 is then a consequence of the following facts: (i).
Given a semi-stable family f : S → B of curves with strictly maximal Higgs field, theArakelov equality for the characterization of the relative Jacobian jac ( f ) : J ac ( S ) −→ U to be a Shimura family becomes (cf. [37]):deg f ∗ Ω S/B (log Υ) = deg A , = rank A , · deg Ω B (log ∆ nc ) = g · deg Ω B (log ∆ nc ) . (1-1) (ii). The following two theorems on a non-isotrivial semi-stable family of curves.Let f : S → B be as above. For every singular fibre F , let δ i ( F ) be the number of nodesof F of type i (1 ≤ i ≤ [ g/ q of F is said to be of type i (1 ≤ i ≤ [ g/ F at q consists of two connected components of arithmeticgenera i and g − i . Let δ i (Υ) = X F ∈ Υ δ i ( F ) , δ h (Υ) = [ g/ X i =2 δ i (Υ); δ i (Υ c ) = X F ∈ Υ c δ i ( F ) , δ h (Υ c ) = [ g/ X i =2 δ i (Υ c ) . (1-2) Theorem 1.4.
Let f : S → B be a non-isotrivial semi-stable family of curves of genus g ≥ as above, and ω S/B = ω S ⊗ f ∗ ω ∨ B the relative canonical sheaf. Then ω S/B ≥ g − g · deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) + 3 g − g δ (Υ) + 7 g − g δ h (Υ) . (1-3) Theorem 1.5.
Let f : S → B be the same as in Theorem . Then ω S/B ≤ (2 g − · deg (cid:0) Ω B (log ∆ nc ) (cid:1) + 2 δ (Υ c ) + 3 δ h (Υ c ) . (1-4) Moreover, if ∆ nc = ∅ or ∆ = ∅ , then the above inequality is strict. Theorem 1.4 is a direct consequence of Moriwaki’s sharp slope inequality (cf. [27]). WhileTheorem 1.5 is based on Miyaoka’s theorem (cf. [21]) for the bound on the number ofquotient singularities in a surface. The base change technique is also a key point.The proof of Theorem 1.3 is much more complicated, but the idea is the same as that ofTheorem 1.2. Without the assumption on the strictly maximality of Higgs field, we have tobound the rank of the flat part of the Higgs bundle and to give an analogous lower boundof ω S/B for a semi-stable family f : S → B of hyperelliptic curves with positive relativeirregularity q f := q ( S ) − g ( B ). —5— n Shimura curves in the Torelli locus of curves X. Lu & K. ZuoTheorem 1.6.
Let f : S → B be as in Theorem . If f is a hyperelliptic family, thenafter passing to a finite ´etale cover of B , the local subsystems F , and F , become to triviallocal systems. So, by Theorem 1.6 together with Deligne’s global invariant cycle theorem (cf. [4, § q f is equal to rank F , after passing a finite ´etale cover of B , i.e., q f = rank F , = g − rank A , . Thus the Arakelov equality (cf. [24]) for the characterization of the relative Jacobian of f : S → B to be a Kuga family becomesdeg f ∗ Ω S/B (log Υ) = g − q f · deg Ω B (log ∆ nc ) . (1-5) Theorem 1.7.
Let f : S → B be as in Theorem and q f = q ( S ) − g ( B ) the relativeirregularity. Assume that f is a hyperelliptic family. Then ω S/B ≥ g − g − q f · deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) + (1-6) g − (8 q f + 1) g + 10 q f − g + 1)( g − q f ) δ (Υ) + 7 g − (16 q f + 9) g + 34 q f − g + 1)( g − q f ) δ h (Υ) , if ∆ nc = ∅ ; [ g/ X i =1 (cid:18) g + 1 − q f ) i ( g − i )(2 g + 1)( g − q f ) − (cid:19) δ i (Υ) , if ∆ nc = ∅ . Moreover, if ∆ nc = ∅ and q f ≥ , then [ g/ X i = q f (2 i + 1)(2 g + 1 − i ) g + 1 · δ i (Υ) ≥ q f − X i =1 i (2 i + 1) · δ i (Υ) . (1-7)There are two ingredients in the proof of Theorem 1.6. The first one is Lemma 5.2 on theglobal invariant cycle with unitary locally constant coefficient. The second one is Bogo-molov’s lemma on Kodaira dimension of an invertible subsheaf of the sheaf of logarithmicdifferential forms (cf. [34]). The proof of Theorem 1.7 is based on formulas given by Cor-nalba and Harris (cf. [3]). When q f >
0, the observation that the smooth double coverinduced by the hyperelliptic involution is fibred (cf. Proposition 4.4) plays a crucial role.In order to illustrate the idea how Theorem 1.2 (and Theorem 1.3) follows from theabove ingredients, we just consider here the simplest case: U is non-compact (i.e., ∆ nc = ∅ )and the logarithmic Higgs bundle associated to the family has strictly maximal Higgs field.By (1-1) together with (1-3), one has ω S/B ≥ (2 g − · deg (cid:0) Ω B (log ∆ nc ) (cid:1) + 3 g − g δ (Υ) + 7 g − g δ h (Υ) . Note that (1-4) in Theorem 1.5 is strict if ∆ nc = ∅ . Hence by Theorem 1.5, we have ω S/B < (2 g − · deg (cid:0) Ω B (log ∆ nc ) (cid:1) + 2 δ (Υ c ) + 3 δ h (Υ c ) . —6— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
By the definition, 0 ≤ δ (Υ c ) ≤ δ (Υ) , and 0 ≤ δ h (Υ c ) ≤ δ h (Υ) . Combining all these together, one obtains0 > g − g · (cid:0) δ (Υ) + 4 δ h (Υ) (cid:1) . Because both δ (Υ) and δ h (Υ) are non-negative, it follows that g <
4, i.e., there does notexist a Kuga family f : S → B of curves of genus g ≥ nc = ∅ and strictly maximalHiggs field.The paper is organized as follows: In Section 2, we introduce some notations and ter-minology.In Section 3, we prove Theorems 1.4 and 1.5 for arbitrary semi-stable families. Theorem1.4 is derived from Moriwaki’s sharp slope inequality (cf. [27, Theorem D]) together withNoether’s formula. The proof of Theorem 1.5 is based on a generalized Miyaoka-Yau’sinequality (cf. [21, Theorem 1.1]) and the base change technique.In Section 4, we prove Theorem 1.7. When q f = 0, (1-6) is a direct consequence of Cornalba-Harris’ formula (cf. [3, Proposition 4.7]) together with Noether’s formula. When q f > π : S → S/ h σ i induced bythe hyperelliptic involution is fibred. As a consequence, the branched divisor of π is veryspecial. And the proof of Theorem 1.7 is completed in Section 4.3 by combining this withCornalba-Harris’ formula.Theorem 1.6 is proved in Section 5, which is based on two ingredients. The first one isLemma 5.2 on the global invariant cycle with unitary locally constant coefficient, whichgeneralizes Deligne’s original theorem with the constant coefficient. The second one comesfrom Bogomolov’s lemma on Kodaira dimension of an invertible subsheaf in the sheaf oflogarithmic differential forms on a smooth projective surface (cf. [34, Lemma 7.5]).In Section 6, we are in the position to prove Theorems 1.2 and 1.3 with the idea demon-strated on Page 6. Assume f : S → B is a Kuga family of curves of genus g . Then theArakelov inequality for f becomes to an equalitydeg f ∗ Ω S/B (log Υ) = rank A , · deg Ω B (log ∆ nc ) . Theorems 1.4 and 1.5 together with this equality for rank A , = g give rise to the requiredbound g ≤
4, which proves Theorem 1.2. If f is hyperelliptic, according to Theorems 1.5and 1.7 together with the above equality for rank A , = g − q f by Theorem 1.6, we obtaininequalities of g as a rational function of the variable g in the different subcases. We proveTheorem 1.3 by showing that g ≤ g = 3 and 4 respectively. In particular, theHiggs field in the example of genus g = 4 is strictly maximal, which shows that the boundin Theorem 1.2 is optimal. —7— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
2. Preliminaries
In this section, we introduce notations and terminology that will be used in the paper.A curve F is called semi-stable (resp. stable ) if it is a reduced nodal curve, and everysmooth rational component intersects the rest part of F at least two (resp. three) points.A morphism f : S → B is called a semi-stable family (resp. stable family ) of curves ofgenus g , if f is a morphism from a projective surface S to a smooth projective curve B with connected fibres, the general fibre is a connected nonsingular complex projective curveof genus g , and all the singular fibres of f are semi-stable (resp. stable). In this paper,when we talk about a semi-stable family f : S → B as above, we always assume the totalsurface S is smooth. If the general fibre of f is a hyperelliptic curve, then we call f a hyperelliptic family . f is called smooth if all its fibres are smooth, isotrivial if all its smoothfibres are isomorphic to each other. f is called relatively minimal , if there is no ( − f . Here a curve C is called a ( − k ) -curve if it is a smooth rationalcurve with self-intersection C = − k . Note that by definition, f is relatively minimal if f is semi-stable.Let ω S (resp. ω B ) be the canonical sheaf of S (resp. B ). Denote by ω S/B = ω S ⊗ f ∗ ω ∨ B the relative canonical sheaf of f . let b = g ( B ), p g = h ( S, ω S ), q = h ( S, Ω S ), χ ( O S ) = p g − q + 1, and χ top ( X ) the topological Euler characteristic of a variety X , where Ω S is thedifferential sheaf of S . For a semi-stable family f : S → B of genus g ≥ / ∆, we consider the following relative invariants: ω S/B = ω S − g − b − ,δ f = χ top ( S ) − g − b −
1) = X F ∈ Υ δ ( F ) , deg f ∗ Ω S/B (log Υ) = χ ( O S ) − ( g − b − , (2-1)where δ ( F ) = χ top ( F ) + (2 g − . They satisfy the Noether’s formula:12 deg f ∗ Ω S/B (log Υ) = ω S/B + δ f . (2-2)These invariants are nonnegative. And deg f ∗ Ω S/B (log Υ) = 0 (equivalently, ω S/B = 0) ifand only if f is isotrivial. Note that for a singular fibre F , δ ( F ) is also equal to the numberof nodes contained F . Hence δ f = 0 iff f is smooth, in which case, f is called a Kodairafamily if moreover f is non-isotrivial.By contracting all ( − f : S → B ,one gets a stable family f : S → B , S / / f (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ S f ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ B In this case, of course, S is not necessarily smooth. For any singular point q of S , ( S , q )is a rational double point of type A λ q , here λ q is the number of ( − S over q .—8— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
For each singular fibre F of a semi-stable family f : S → B of genus g ≥
2, we define δ i ( F ) for 0 ≤ i ≤ [ g/
2] in the following way. A singular point q of F is said to be of type i (0 ≤ i ≤ [ g/ q consists of two connected componentsof arithmetic genera i and g − i for i >
0, and is connected for i = 0. Then δ i ( F ) is thenumber of singular points in F of type i . We call also define δ i ( F ) according to its stablemodel F ⊆ S . To do this, first we similarly define singular points of type i as before.Around a singular point q ∈ F in S , locally S is of the form xy = t m q , where t is alocal coordinate of B around f ( q ). We call m q is the multiplicity of q . Then δ i ( F ) is thenumber of singular points of F of type i counting multiplicity. We remark that ( S , q ) isa rational double point of type A m q − , if m q > q .Let Υ → ∆ denote the singular fibres, Υ c → ∆ c denote those singular fibres withcompact Jacobians, and Υ nc , Υ \ Υ c → ∆ nc , ∆ \ ∆ c correspond to singular fibres withnon-compact Jacobians. Define δ h ( F ) = [ g/ P i =2 δ i ( F ), and δ i (Υ) = X F ∈ Υ δ i ( F ) , δ i (Υ c ) = X F ∈ Υ c δ i ( F ) , δ i (Υ nc ) = X F ∈ Υ nc δ i ( F ) .δ h (Υ) = [ g/ X i =2 δ i (Υ) , δ h (Υ c ) = [ g/ X i =2 δ i (Υ c ) . (2-3)Then δ ( F ) = [ g/ X i =0 δ i ( F ) = δ ( F ) + δ ( F ) + δ h ( F ) ,δ f = [ g/ X i =0 δ i (Υ) = δ (Υ) + δ (Υ) + δ h (Υ) . (2-4)Let M g (resp. M g ) be the moduli space of smooth (resp. stable) complex curvesof genus g . By [5], the boundary M g \ M g is of codimensional one and has [ g/
2] + 1irreducible components, saying, ∆ , ∆ , · · · , ∆ [ g/ . The geometrical meaning of the indexis as follows. A general point of ∆ i ( i >
0) corresponds to a stable curve consisting of acurve of genus i and a curve of genus g − i joint at one point, and a general point of ∆ represents an irreducible stable curve with one node. These boundaries define divisor classes (cid:8) ∆ , ∆ , · · · , ∆ [ g/ (cid:9) ∈ Pic ( M g ) ⊗ Q . There is also a natural class λ ∈ Pic ( M g ) ⊗ Q ,called the Hodge class. A non-isotrivial semi-stable family f : S → B of curves of genus g ≥ ϕ : B → M g . Then one has (cf. [5])deg f ∗ Ω S/B (log Υ) = deg ϕ ∗ ( λ ) , δ i (Υ) = deg ϕ ∗ (∆ i ) . (2-5)For a singular points in a stable hyperelliptic curve F , we have a more detail descriptionby using the induced double cover. To see this, first note that F has a semi-stable model F which is an admissible double cover (cf. [3] or [10]) of a stable (2 g + 2)-pointed nodedcurve Γ of arithmetic genus zero. Let ψ : F → Γ be the covering map, and let p be asingular point of Γ. The complement of p has two connected components Γ ′ and Γ ′′ , so theset of marked points of Γ breaks up into two subsets: those lying on Γ ′ and those lying onΓ ′′ ; let α and 2 g + 2 − α ≥ α be the orders of these two subsets. Following [3, P ], α is—9— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo called the index of the point p . Note that α ≥
2. If p has odd index α = 2 k + 1, then ψ mustbe branched at p , and the unique singular point q lying above p is a singular point of type k . Suppose that the index α = 2 k + 2 is even, then ψ is unbranched at p , so ψ − ( p ) consistsof two points q ′ and q ′′ , and ψ − (Γ ′ ) and ψ − (Γ ′′ ) are semi-stable hyperelliptic curves ofgenera k and g − k −
1, joint at q ′ and q ′′ . In particular, both q ′ and q ′′ are singular pointsof type 0.Let f : S → B be a semi-stable family of hyperelliptic curves, and F a singular fibre.Let e B → B be a base change of degree d and totally branched over f ( F ), ˜ f : e S → e B thecorresponding semi-stable family, and e F the pre-image of F . If d is sufficiently large, thenwe see that e F is an admissible double cover of a stable (2 g + 2)-pointed noded curve e Γof arithmetic genus zero. Let ξ ( e F ) equal to two times the number of singular points of e Γ of index 2, and ξ j ( e F ) equal to the number of singular points of e Γ of index 2 j + 2 for1 ≤ j ≤ [( g − / ξ j ( F ) = ξ j ( e F ) d , ∀ ≤ j ≤ [( g − / . It is clear that ξ j ( F )’s are independent on the choices of the base change, and hence well-defined invariants. And δ ( F ) = ξ ( F ) + 2 [( g − / X j =1 ξ j ( F ) . We also define ξ j (Υ) = X F ∈ Υ ξ j ( F ) , ∀ ≤ j ≤ [( g − / . Let H g ⊆ M g (resp. H g ⊆ M g ) be the moduli space of smooth (resp. stable) hyper-elliptic complex curves of genus g . The above discussion shows that the intersection of ∆ i and H g ( i >
0) is still an irreducible divisor (see [3] for more details). By abuse of notations,we still denote it by ∆ i . The intersection of ∆ and H g , however, is reducible. Let Ξ j bethe locus of all curves in H g such that the corresponding marked pointed noded curve Γ ofarithmetic genus zero has a singular point of index 2 j + 2. Then (cf. [3])∆ ∩ H g = Ξ ∪ Ξ ∪ · · · ∪ Ξ [( g − / . A general point in Ξ corresponds to an irreducible stable hyperelliptic curve with onenode. A general point of Ξ j (1 ≤ j ≤ [( g − / j and a hyperelliptic curve of genus g − j − h ∗ (∆ ) = Ξ + 2 [( g − / X j =1 Ξ j , where h : H g ֒ → M g is the embedding.Hence if f : S → B is a non-isotrivial semi-stable family of hyperelliptic curves of genus g with singular locus Υ / ∆, and ϕ : B → H g the induced map, then ξ j (Υ) = deg ϕ ∗ (Ξ j ) , ∀ ≤ j ≤ [( g − / δ (Υ) = deg ϕ ∗ (Ξ ) + 2 [( g − / X j =1 deg ϕ ∗ (Ξ j ); δ i (Υ) = deg ϕ ∗ (∆ i ) , ∀ ≤ i ≤ [ g/ . (2-6)—10— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
3. Bounds of ω S/B for arbitrary families ω S/B
The subsection aims to prove Theorem 1.4. It follows directly from Moriwaki’s sharpslope inequality (cf. [27, Theorem D]) together with Noether’s formula (2-2).
Proof of Theorem . From [27, Theorem D], (2-2) and (2-4), it follows that(8 g + 4) deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) ≥ gδ (Υ) + [ g/ X i =1 i ( g − i ) δ i (Υ)= gδ f + [ g/ X i =1 (cid:0) i ( g − i ) − g (cid:1) δ i (Υ)= g (cid:16)
12 deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) − ω S/B (cid:17) + [ g/ X i =1 (cid:0) i ( g − i ) − g (cid:1) δ i (Υ) . Hence ω S/B ≥ g − g · deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) + [ g/ X i =1 i ( g − i ) − gg δ i (Υ) ≥ g − g · deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) + 3 g − g δ (Υ) + 7 g − g δ h (Υ) . ω S/B
The purpose of this subsection is to prove Theorem 1.5. The proof is based on ageneralized Miyaoka-Yau’s inequality (see Theorem 3.1 below). What we do is to take asuitable base change and to choose suitable components contained in singular fibres (butnot the entire singular fibres).First we recall the generalized Miyaoka-Yao’s theorem (cf. [21]).Let X x the germ of a quotient singularity of ( C /G x ) (in the analytic sense), where G x is a finite subgroup of GL(2 , C ) with the origin 0 being its unique fixed point. Let X E be the minimal resolution of X x and E the exceptional divisor (= the inverse image of x ).Let v ( x ) , χ top ( E ) − | G x | . (3-1) Theorem 3.1 (Miyaoka [21, Theorem 1.1]) . Let X be a projective surface with only quo-tient singularities, and Λ the singular locus of X . Let D be a reduced normal crossingcurve which lies on the smooth part of X . Let X be the minimal resolution of X and E ⊆ X the inverse image of Λ (with reduced structure). Assume the negative part in theZariski decomposition of ω X + D + E has the form N + N ′ such that supp N is disjointwith E and supp N ′ ⊆ E . Then X x ∈ Λ v ( x ) ≤ χ top ( X ) − χ top ( D ) −
13 ( ω X + E + D ) + 13 ( N ′ ) + 112 N . —11— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo If X contains at most rational singularities of type A (cf. [2, § III-3]), X is minimal andof general type, and D is composed of some disjoint smooth elliptic curves, then χ top ( D ) = 0and the negative part in the Zariski decomposition of ω X + D + E is just E , which is somechains of ( − (cid:0) ω X + D + E (cid:1) = ( ω X + D ) + E . Note that for a singularity x of type A k , the invariant v ( x ) defined in (3-1) is equal to( k + 1) − k +1 . Therefore we get Theorem 3.2.
Let conditions be the same as that of Theorem . . Assume that each point x ∈ Λ is a quotient singularities of type A k x , X is minimal and of general type, and D iscomposed of some disjoint smooth elliptic curves. Then X x ∈ Λ (cid:18) ( k x + 1) − k x + 1 (cid:19) ≤ χ top ( X ) −
13 ( ω X + D ) . We are now able to prove Theorem 1.5.
Proof of Theorem . We use the notations introduced in Sections 1 and 2.Consider first the case g = 2. In this case, f is a semi-stable hyperelliptic family, whenceΥ = ∅ ; otherwise, from [3, Proposition 4.7], it follows that deg f ∗ (cid:0) Ω S/B (log Υ) (cid:1) = 0, whichis impossible, since f is assumed to be non-isotrivial. Note that for a singular fibre F ∈ Υ c , δ ( F ) ≥
1. Hence δ (Υ c ) ≥ c ). Therefore when g = 2, our theorem is a directconsequence of the strict canonical class inequality (cf. [35]): K f < (2 g − · deg (cid:0) Ω B (log ∆) (cid:1) = 2 deg (cid:0) Ω B (log ∆ nc ) (cid:1) + 2 · c ) ≤ (cid:0) Ω B (log ∆ nc ) (cid:1) + 2 δ (Υ c ) . In the rest part of the proof, we assume g ≥ s c = c , s nc = nc . For any p ∈ ∆, let F p = f − ( p ), and E p = X j E p,j (cid:12)(cid:12)(cid:12) E p,j ⊆ F p is a ( − , ∀ p ∈ ∆; (3-2) D p = X j D p,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D p,j ⊆ F p is a smooth elliptic curve such that D p,j · E p = 0 , and D p,j = − , ∀ p ∈ ∆ c . (3-3)Let S → S be the contraction of P p ∈ ∆ E p ⊆ S , and f : S → B the induced morphism.Then f : S → B is nothing but the stable model of f . Note also that, for any p ∈ ∆ c ,the image of D p on S lies on the smooth part of S , which we still denote by D p . Forany singular point q of S , ( S , q ) is a rational double point of type A λ q , here λ q is thenumber of ( − S over q . For convenience, we also denote by q the singular pointof the fibres on the smooth part of S , in which case, λ q = 0. So a singular point ( S , q )—12— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo of type A is understood as a node of the fibres but a smooth point of S . Let F p be theimage of F p on S for p ∈ ∆, andΥ = X p ∈ ∆ F p , Υ c = X p ∈ ∆ c F p , Υ nc = X p ∈ ∆ nc F p . Then δ ( F p ) = [ g/ X i =0 δ i ( F p ) = X q ∈ F p ( λ q + 1) , ∀ p ∈ ∆ . (3-4)We claim Claim 3.2.1.
For each p ∈ ∆ c , D p is smooth (not necessary irreducible), and X q ∈ F pλq> ( λ q + 1) + | D p | ≥ δ ( F p ) , (3-5)where | D p | is the number of irreducible components of D p .We leave the proof of the claim at the end of the section.Let φ : B → B be a cover of B , such that deg φ = de and φ ramifies uniformly over∆ nc with ramification index equaling to e . By Kodaira-Parshin construction, such a coverexists for all e if b = g ( B ) > e ≥ b = 0 (cf. [38] or [35]). Let ¯ b = g ( B ) bethe genus of B . According to Hurwitz formula, we get2(¯ b −
1) = de · (cid:18) b −
1) + e − e · s nc (cid:19) . (3-6)Let S = B × B S be the fibre-product, S → S the minimal resolution of singularities.We have the following commutative diagram: S ¯ f (cid:27) (cid:27) (cid:15) (cid:15) Φ / / S (cid:15) (cid:15) f (cid:2) (cid:2) S (cid:15) (cid:15) Φ / / S (cid:15) (cid:15) B φ / / B For p ∈ ∆ c , the inverse image of a singular point ( S , q ) of type A λ q with q ∈ F p is de singular points of the same type A λ q in S . For p ∈ ∆ nc , the inverse image of a singularpoint ( S , q ) of type A λ q with q ∈ F p is d singular points of type A e ( λ q +1) − in S . Let D = X p ∈ ∆ c (cid:0) Φ (cid:1) − ( D p ) . Since φ is unbranched over ∆ c , D is smooth and lies on the smooth part of S , and thenumber of irreducible components in D is | D | = de · X p ∈ ∆ c | D p | . —13— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
Hence 2 ω S · D + D = | D | = de · X p ∈ ∆ c | D p | . (3-7)Because f is semi-stable, ¯ f : S → B is also semi-stable, and δ ¯ f = de · δ f , ω S/B = de · ω S/B . (3-8)It is not difficult to see that S is minimal and of general type if ¯ b = g ( B ) ≥
1, whichis satisfied when de is large enough. Hence applying Theorem 3.2 to the case by setting X = S , X = S , and D as above, we get de · X q ∈ Υ c (cid:18) ( λ q + 1) − λ q + 1 (cid:19) + d · X q ∈ Υ nc (cid:18) e ( λ q + 1) − e ( λ q + 1) (cid:19) ≤ χ top ( S ) − (cid:0) ω S + D (cid:1) = de · δ f − ω S/B − X p ∈ ∆ c | D p | + 13 de · (2 g − (cid:18) b − e − e · s nc (cid:19) . (3-9)We use (2-1), (3-6), (3-7) and (3-8) in the last step above. Note that δ f = X p ∈ ∆ δ ( F p ) = X q ∈ Υ c ( λ q + 1) + X q ∈ Υ nc ( λ q + 1) , (3-10) X q ∈ Υ c λ q + 1 = X q ∈ Υ cλq =0 λ q + 1 + X q ∈ Υ cλq> λ q + 1 ≤ X q ∈ Υ cλq =0 λ q + 1) + X q ∈ Υ cλq> λ q + 1)4= X q ∈ Υ c λ q + 1) − X q ∈ Υ cλq> λ q + 1)4 ≤ X q ∈ Υ c λ q + 1) − X q ∈ Υ cλq> ( λ q + 1) . (3-11)Combining (3-11) with (3-4) and (3-5), one gets X q ∈ Υ c λ q + 1 − X p ∈ ∆ c | D p | ≤ X q ∈ Υ c λ q + 1) − X p ∈ ∆ c X q ∈ F pλq> ( λ q + 1) + | D p | ≤ X q ∈ Υ c λ q + 1) − X p ∈ ∆ c δ ( F p ) = 2 δ (Υ c ) + 2 δ h (Υ c ) . (3-12)—14— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
Therefore by (3-9), (3-10) and (3-12), we get ω S/B ≤ (2 g − b − s nc ) + X q ∈ Υ c λ q + 1 − X p ∈ ∆ c | D p | + 1 e · X q ∈ Υ nc λ q + 1 − (2 g − s nc e ≤ (2 g − b − s nc ) + 2 δ (Υ c ) + 2 δ h (Υ c ) + 1 e · X q ∈ Υ nc λ q + 1 − (2 g − s nc e . Letting e tend to infinity, we get the required inequality (1-4). If s nc >
0, then letting e belarge enough, one has 1 e · X q ∈ Υ nc λ q + 1 − (2 g − s nc e < . Hence if ∆ nc = ∅ , then the inequality (1-4) is strict. Finally, if ∆ = ∅ , then f is a Kodairafamily, and deg (cid:0) Ω B (log ∆ nc ) (cid:1) = 2 b − , δ (Υ c ) = δ h (Υ c ) = 0 , so it follows from [19, Corollary 0.6] that (1-4) is also strict. Remark 3.3.
If (1-4) is indeed an equality, i.e., ω S/B = (2 g − · deg (cid:0) Ω B (log ∆ nc ) (cid:1) + 2 δ (Υ c ) + 3 δ h (Υ c ) , (3-13)then ∆ nc = ∅ ; E p = ∅ for p ∈ ∆ c by (3-11); and δ (Υ c ) = P p ∈ ∆ c | D p | by (3-5). Hence (3-13)is equivalent to c Ω S (cid:18) log (cid:16) X p ∈ ∆ c D p (cid:17)(cid:19) = 3 c Ω S (cid:18) log (cid:16) X p ∈ ∆ c D p (cid:17)(cid:19) . It follows that S \ S p ∈ ∆ c D p ! is a ball quotient by [16] or [22]. Proof of Claim . First we prove that D p is smooth. Assume that D p is not smooth.As each irreducible component of D p is smooth by definition, there are two irreduciblecomponents D p, , D p, contained in D p such that D p, ∩ D p, = ∅ . Since D p,j = −
1, eachirreducible component D p,j of D p intersects F p − D p,j in exactly one point. So F p = D p = D p, + D p, with ω S · D p, = ω S · D p, = 1, and hence 2 g − ω S · F p = 2. It is impossible,since g ≥ F p of F p . As F p has compactJacobian, F p is tree of smooth curves. Hence a singular point in F p of type 1 is a point q such that the partial normalization at q consists of a smooth elliptic curve D q and a curveof arithmetic genera g −
1. And δ ( F p ) is by definition is the number of singular points in F p of type 1 counting multiplicity.Let m q be the multiplicity of a singular point q ∈ F p of type 1. If m q = 1, then( S , q ) is smooth at q , hence the inverse image of D q in S is a smooth elliptic curve withself-intersection equal to − E p . It means that the inverse image of D q is contained in D p . If m q >
1, then ( S , q ) is a rational double point of type A m q − ,hence λ q = m q − >
0. Therefore, (3-5) follows immediately.—15— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
4. Lower bound of ω S/B for hyperelliptic families
The section aims to prove Theorem 1.7. So we always assume that f : S → B is a non-isotrivial semi-stable family of hyperelliptic curves of genus g ≥ q f = q ( S ) − g ( B ).When q f = 0, it is a direct consequence of the Noether’s formula and the followingformula given in [3, Proposition 4.7]:deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) = g g + 1) ξ (Υ)+ [ g/ X i =1 i ( g − i )2 g + 1 δ i (Υ) + [( g − / X j =1 ( j + 1)( g − j )2(2 g + 1) ξ j (Υ) . (4-1)When q f >
0, the proof starts from the observation that the double cover π : S → S/ h σ i isfibred, where σ is the involution on S induced by the hyperelliptic involution on fibres of f .From this it follows a restriction on those invariants δ i (Υ)’s and ξ j (Υ)’s (cf. Proposition 4.1).And the proof of Theorem 1.7 is completed in Section 4.3 by combining this with (4-1). (1-6) for q f = 0 By (2-4) and (2-6), one has δ f = ξ (Υ) + [ g/ X i =1 δ i (Υ) + 2 [( g − / X j =1 ξ j (Υ) . (4-2)From the above equation together with Noether’s formula and (4-1), it follows that ω S/B = g − g + 1 ξ (Υ)+ [ g/ X i =1 (cid:18) i ( g − i )2 g + 1 − (cid:19) δ i (Υ) + [( g − / X j =1 (cid:18) j + 1)( g − j )2 g + 1 − (cid:19) ξ j (Υ) . (4-3)Hence ω S/B − g − g · deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) = [ g/ X i =1 i ( g − i ) − gg δ i (Υ) + [( g − / X j =1 j + 1)( g − j ) − gg ξ j (Υ) ≥ g − g δ (Υ) + 7 g − g δ h (Υ) , if ∆ nc = ∅ ; [ g/ X i =1 i ( g − i ) − gg δ i (Υ) , if ∆ nc = ∅ . Hence (1-6) is proved for q f = 0. —16— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
The main purpose of this subsection is to prove the following proposition for a semi-stable hyperelliptic family with positive relative irregularity.
Proposition 4.1.
Let f : S → B be a non-isotrivial semi-stable family of hyperellipticcurves of genus g with singular locus Υ / ∆ . Let δ i (Υ) ’s and ξ j (Υ) ’s be defined in Section 2.Assume that the relative irregularity q f = q ( S ) − g ( B ) > . Then [ g/ X i = q f (2 i + 1)(2 g + 1 − i ) g + 1 δ i (Υ) + [( g − / X j = q f j + 1)( g − j ) g + 1 ξ j (Υ) ≥ ξ (Υ) + q f − X i =1 i (2 i + 1) δ i (Υ) + q f − X j =1 j + 1)(2 j + 1) ξ j (Υ) . (4-4)As said before, the key point is the observation that the induced double cover π : S → S/ h σ i is fibred. To be more precise, let f : S → B be as in the above proposition,and f : S → B the stable model. The hyperelliptic involution induces a double cover π : S → Y . By resolving the singular points, one gets a double cover ˜ π : e S → e Y between smooth surfaces with smooth branched divisor e R ⊆ e Y . Let ˜ f : e S → B and˜ h : e Y → B be the induced morphism. e S ˜ π / / (cid:15) (cid:15) e Y (cid:15) (cid:15) S π / / f ❇❇❇❇❇❇❇❇ Y h } } ⑤⑤⑤⑤⑤⑤⑤⑤ B Figure 4.2-1: Hyperelliptic involution.We would like to show that ˜ π : e S → e Y is fibred. First we recall the following definition. Definition 4.2 ([15]) . A double cover π : X → Z of smooth projective surfaces withbranched divisor R ⊆ Z is called fibred if there exist a double cover π ′ : C → D of smoothprojective curves and morphisms ̟ : X → C and ǫ : Z → D with connected fibres, suchthat the diagram X π / / ̟ (cid:15) (cid:15) Z ǫ (cid:15) (cid:15) C π ′ / / D is commutative, R is contained in the fibres of ǫ , and q ( X ) − q ( Z ) = g ( C ) − g ( D ), where g ( C ) (resp. g ( D )) is the genus of C (resp. D ), q ( X ) = dim H ( X, Ω X ), and q ( Z ) =dim H ( Z, Ω Z ).The next theorem is proved in [15]. For readers’ convenience, we reprove it here.—17— n Shimura curves in the Torelli locus of curves X. Lu & K. ZuoTheorem 4.3 ([15, Theorem 1]) . If the geometrical genus p g ( Z ) = dim H ( Z, Ω Z ) = 0 .Then any double cover π : X → Z with smooth branched divisor R ⊆ Z and with q ( X ) >q ( Z ) is fibred.Proof. Note that the Galois group Gal(
X/Z ) ∼ = Z has a natural action on H ( X, Ω X ). Let H ( X, Ω X ) = H ( X, Ω X ) ⊕ H ( X, Ω X ) − be the eigenspace decomposition. Then H ( X, Ω X ) = π ∗ H ( Z, Ω Z ) , and k , dim H ( X, Ω X ) − = q ( X ) − q ( Z ) . Let ω , · · · , ω k be a basis of H ( X, Ω X ) − . First we prove that there exists a morphism ̟ : X → C to a curve C with connected fibres, such that there exist α , · · · , α k ∈ H ( C, Ω C )satisfying ω i = ̟ ∗ ( α i ) , ∀ ≤ i ≤ k. (4-5)If k ≥
2, then ω i ∧ ω j ∈ ∧ H ( X, Ω X ) − ⊆ H ( X, Ω X )is invariant under the action of the Galois group for any i = j . Hence it belongs to H ( X, Ω X ) = π ∗ (cid:0) H ( Z, Ω Z ) (cid:1) , which is zero by our assumption p g ( Z ) = dim H ( Z, Ω Z ) = 0. By [2, § IV-5], there exists amorphism ̟ : X → C with connected fibres such that (4-5) holds.For the case k = 1, note that π : X → Z induces a surjective morphism Alb( π ) :Alb( X ) → Alb( Z ) between abelian varieties as follows. X π / / Alb X (cid:15) (cid:15) Z Alb Z (cid:15) (cid:15) Alb( X ) Alb( π ) / / Alb( Z )By assumption dim Alb( X ) − dim Alb( Z ) = q ( X ) − q ( Z ) = k = 1 . According to the theory on abelian varieties (cf. [29]), there exists a one-dimensional abelianvariety (i.e., an elliptic curve) C such that Alb( X ) is isogenous to Alb( Z ) × C , i.e., thereexists a morphism ϕ : Alb( X ) → Alb( Z ) × C with finite kernel. Let pr : Alb( Z ) × C → C be the projection, and ̟ , pr ◦ ϕ ◦ Alb X : X −→ C be the composition. As Alb X ( X ) generates Alb( X ), ̟ is surjective. By Stein factorization(cf. [11, § III-11]), we get a morphism ̟ : X → C with connected fibres. And (4-5) isclearly satisfied.Note that the property (4-5) implies that the morphism ̟ : X → C is unique. Inparticular, the Galois group Gal( X/Z ) ∼ = Z induces an automorphism group G on C . Let D = C/G , and π ′ : C → D be the natural morphism. Then by construction, there exists amorphism ǫ : Z → D such that ǫ ◦ π = π ′ ◦ ̟ .—18— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
Let σ be the non-identity element of Gal( X/Z ). Then the fixed locus Fix( σ ) of σ isclearly contained in the fibres of ̟ . So R = π (cid:0) Fix( σ ) (cid:1) is contained in the fibres of ǫ . By(4-5), one sees that the eigenspace decomposition of H ( C, Ω C ) with respect to the actionof G is H ( C, Ω C ) = ( π ′ ) ∗ H ( D, Ω D ) ⊕ H ( C, Ω C ) − , where H ( C, Ω C ) − is generated by α , · · · , α k . So q ( X ) − q ( Z ) = dim H ( X, Ω X ) − = dim H ( C, Ω C ) − = g ( C ) − g ( D ) . Coming back to our case. Note that q ( e S ) = q ( S ) and q ( e Y ) = g ( B ). If q f = q ( S ) − g ( B ) >
0, it follows that q ( e S ) > q ( e Y ). As e Y is a ruled surface, the geometric genus p g ( e Y ) = 0.Hence by Theorem 4.3 above, we get Proposition 4.4.
The double cover ˜ π : e S → e Y is fibred, i.e., there exist a double cover π ′ : B ′ → D of smooth projective curves and morphisms ˜ f ′ : e S → B ′ and ˜ h ′ : e Y → D withconnected fibres, such that the diagram e S π / / ˜ f ′ (cid:15) (cid:15) e Y ˜ h ′ (cid:15) (cid:15) B ′ π ′ / / D is commutative, e R is contained in the fibres of ˜ h ′ and q f = q ( e S ) − q ( e Y ) = g ( B ′ ) − g ( D ) . (4-6)Our purpose is to prove Proposition 4.1. Before going to the detailed proof, we showthat the curve D in the above proposition is actually isomorphic to P , whence g ( B ′ ) = q f by (4-6).Let e F and e F ′ be any fibres of ˜ f : e S → B and ˜ f ′ : e S → B ′ respectively. By restrictions,we get the following two morphisms: ( ˜ f ′ | e F : e F −→ B ′ , ˜ f | e F ′ : e F ′ −→ B. (4-7)It is clear that ˜ f ′ | e F and ˜ f | e F ′ are surjective and have the same degree d = deg (cid:16) ˜ f ′ | e F (cid:17) = deg (cid:16) ˜ f | e F ′ (cid:17) = e F · e F ′ . (4-8) Proposition 4.5.
Let f : S → B be a non-isotrivial semi-stable family of hyperellipticcurves of genus g ≥ , and d be defined in (4-8) . Then d ≥ , D ∼ = P , and q f = g ( B ′ ) ≤ g − d + 1 . (4-9)—19— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
Proof. If d = 1, then it follows that e S is birational to e F × B ′ . It is a contraction, since f is non-isotrivial. So d ≥ h : e Y → B is the ruling of the ruled surface e Y , a general fibre e Γ of ˜ h is isomorphicto P . By the discussion above, e Γ is mapped surjective to D by ˜ h ′ . Hence D ∼ = P , i.e., g ( D ) = 0. By (4-6), g ( B ′ ) = q f . According to Hurwitz formula for algebraic curves, we get2 g − g ( e F ) − ≥ d · (2 g ( B ′ ) −
2) = 2 d · ( q f − . So (4-9) is proved.
Remark 4.6.
Let f : S → B be as in the above proposition with b = g ( B ) ≥
1. It isnot difficult to show that d = deg (cid:16) ˜ f ′ | e F (cid:17) is nothing but the degree of the Albanese map S → Alb ( S ). Xiao ([39]) proved that if q f = g − d + 1 , then f is isotrivial. Proof of Proposition . In order to prove (4-4), we may limit ourselves to the familywhose stable model f : S → B comes from an admissible double cover (cf. [3] or [10]);that is, a double cover of a family h : Y → B of stable (2 g + 2)-pointed noded curvesof arithmetic genus zero, branched along the 2 g + 2 disjoint sections σ i of h and possiblyat some of the nodes of fibres of h . Actually, for any semi-stable family of hyperellipticcurves over a curve, we may get a family of admissible covers by base change and blowing-ups of singular points in the fibres. These operations have the effect of multiplying allthe invariants δ i (Υ)’s and ξ j (Υ)’s by the same constant, and the relative irregularity q f isnon-decreasing under these operation.Let Λ = { p i } be the set of points of Y which are nodes of their fibres. Following [3,P ], if the local equation of Y at p i is xy = t m i , then we say that p i has multiplicity m i . We also denote by α i the index of p i , i.e., the two connected components of thepartial normalization of the fibre Γ through p i intersect those 2 g + 2 sections in α i and2 g + 2 − α i ≥ α i points respectively.Let e Y → Y be the resolution of singularities on Y , and ˜ π : e S → e Y the smoothdouble cover with branched divisor e R as in Figure 4.2-1. The pullbacks of those 2 g + 2disjoint sections σ i ’s are still disjoint sections of ˜ h : e Y → B , and by abuse of notation, westill denote them by σ i ’s. Let e R h = g +2 P i =1 σ i . Then e R is the union of e R h and some disjoint( − h : e Y → B .Let e Λ = { q l } be the set of points of e Y which are nodes of fibres of ˜ h . For a node q l ∈ e Γin a fibre e Γ of ˜ h , we also define the index of q l to be β l if the two connected componentsof the partial normalization of the fibre e Γ at q l intersect those 2 g + 2 sections in β l and2 g + 2 − β l ≥ β l points respectively. Then a node p i in a fibre of h of index α i withmultiplicity m i would introduce m i nodes in the corresponding fibre of ˜ h with the sameindices α i .Let ˜ h ′ : e Y → D ∼ = P be the morphism given in Proposition 4.4. Let ˜ ρ : e Y → ˆ Y be thelargest contraction of ‘vertical’ ( − h ′ : ˆ Y → D ,here ‘vertical’ means such a curve is mapped to a point on B .—20— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo e Y ˜ ρ / / ˜ h ′ ❋❋❋❋❋❋❋❋❋ ˆ Y ˆ h ′ { { ①①①①①①①①① D ∼ = P This means that any ‘vertical’ ( − Y is mapped surjectively onto D by ˆ h ′ . Since e R is contained in fibres of ˜ h ′ by Proposition 4.4, ˆ R = ˜ ρ ( e R ) is contained in fibres of ˆ h ′ . Soin particular any ‘vertical’ ( − Y is not contained in ˆ R ⊆ ˆ Y . Let ˆ R h = ˜ ρ ( e R h ). Claim 4.6.1.
For any ‘vertical’ ( − C in ˆ Y , C · ˆ R ≥ q f + 2. In particular, C · ˆ R h ≥ q f + 1 . Proof of the claim.
Note that ˆ R is the union of ˆ R h and some curves in fibres of ˆ h , whereˆ h : ˆ Y → B is the induced morphism from ˜ h : e Y → B . Hence for any ‘vertical’ ( − C , let ˆΓ be the fibre of ˆ h containing C . Then C · ( ˆ R − ˆ R h ) ≤ C · ˆΓ = − C = 1 . Therefore it suffices to prove C · ˆ R ≥ q f + 2.Let C ′ ⊆ e S and e C ⊆ e Y be the strict inverse image of C on e S and e Y respectively. Thenby construction, C ′ is mapped surjectively onto B ′ by ˜ f ′ , and C · ˆ R ≥ e C · e R. Applying Hurwitz formula to the double cover C ′ → e C ∼ = P , whose branched locus is atmost e C ∩ e R , one gets 2 g ( C ′ ) − ≤ − e C ∩ e R ) . As C ′ is mapped surjectively onto B ′ , g ( C ′ ) ≥ g ( B ′ ) = q f . Hence C · ˆ R ≥ e C · e R ≥ e C ∩ e R ) ≥ g ( C ′ ) + 2 ≥ q f + 2 . Now we contract ˆ ρ : ˆ Y → Y to be a P -bundle h : Y → B in such a way that the orderof any singularity of R h = ˆ ρ ( ˆ R h ) is at most g + 1. It is easy to see that such a contractionexists. e Y ˜ ρ / / ˜ h & & ▼▼▼▼▼▼▼▼▼▼▼▼▼ ˆ Y ˆ ρ / / ˆ h (cid:15) (cid:15) Y h x x qqqqqqqqqqqqq B Let ρ = ˆ ρ ◦ ˜ ρ . Then ρ can be viewed as a sequence of blowing-ups ρ l : Y l → Y l − centeredat y l − ∈ Y l − with Y t + s = e Y , Y s = ˆ Y , and Y = Y . Let R h,l ⊆ Y l be the image of e R h , y l − a singularity of R h,l − of order n l − . Then one sees that each blowing-up ρ l creates a node q ∈ e Λ with index β = n l − . Hence e R h = R h − X q l ∈ e Λ β l . (4-10)—21— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
By Claim 4.6.1, for 1 ≤ l ≤ s , any blowing-up ρ l : Y l → Y l − is centered at a point y l − with n l − ≥ q f + 1. In other words, for 1 ≤ l ≤ s , each ρ l creates a node q ∈ e Λ withindex at least 2 q f + 1. We divide the nodes e Λ of fibres of ˜ h into two parts: one, denotedby e Λ ˆ ρ , is created by blowing-ups contained in ˆ ρ ; the other one, denoted by e Λ ˜ ρ , is createdby blowing-ups contained in ˜ ρ . Then β l ≥ q f + 1 , ∀ q l ∈ e Λ ˆ ρ ; (4-11)ˆ R h = R h − X q l ∈ e Λ ˆ ρ β l . (4-12)Note that e R h consists of 2 g + 2 disjoint sections σ i ’s. According to [3, Lemma 4.8], it followsthat e R h = g +2 X i =1 σ i · σ i = − X p i ∈ Λ m i α i (2 g + 2 − α i )2 g + 1 = − X q l ∈ e Λ β l (2 g + 2 − β l )2 g + 1 . (4-13)Combining (4-10), (4-12) and (4-13), one getsˆ R h = e R h + X q l ∈ e Λ ˜ ρ β l = X q l ∈ e Λ ˜ ρ (2 g + 2) β l ( β l − g + 1 − X q l ∈ e Λ ˆ ρ β l (2 g + 2 − β l )2 g + 1 . (4-14)Now according to Proposition 4.4, e R h ⊆ e R is contained in the fibres of ˜ h ′ . By our construc-tion, ˆ R h = ˜ ρ ( e R h ) is contained in the fibres of ˆ h ′ . In particular, ˆ R h ≤
0. Hence by (4-14),we obtain X q l ∈ e Λ ˜ ρ β l ( β l − ≤ X q l ∈ e Λ ˆ ρ β l (2 g + 2 − β l )2 g + 2 . (4-15)Let ǫ k (resp. ν k ) be the number of points q l ∈ e Λ of index 2 k + 1 (resp. 2 k + 2). Then itis clear that ǫ k (resp. ν k ) is also the number of points p i ∈ Λ of index 2 k + 1 (resp. 2 k + 2),counted according to their multiplicity. Hence (cf. [3, (4.10)]) ξ (Υ) = 2 ν ; δ i (Υ) = ǫ i / , ∀ ≤ i ≤ [ g/ ξ j (Υ) = ν j , ∀ ≤ j ≤ [( g − / . (4-16)—22— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
Combining all together, one gets [ g/ X i = q f (2 i + 1)(2 g + 1 − i ) g + 1 δ i (Υ) + [( g − / X j = q f j + 1)( g − j ) g + 1 ξ j (Υ)= [ g/ X i = q f (2 i + 1) (cid:0) (2 g + 2) − (2 i + 1) (cid:1) g + 2 ǫ i + [( g − / X j = q f (2 j + 2) (cid:0) (2 g + 2) − (2 j + 2) (cid:1) g + 2 ν j = X β l (2 g + 2 − β l )2 g + 2 , the sum is taken over all q l ∈ e Λ with index β l ≥ q f + 1, ≥ X q l ∈ e Λ ˆ ρ β l (2 g + 2 − β l )2 g + 2 , since any point q l ∈ e Λ ˆ ρ is of index β l ≥ q f + 1 by (4-11), ≥ X q l ∈ e Λ ˜ ρ β l ( β l − , by (4-15) , ≥ X β l ( β l − , the sum is taken over all q l ∈ e Λ with index β l < q f + 1,and such points are all contained in e Λ ˜ ρ by (4-11),= 2 ν + q f − X i =1 i (2 i + 1) ǫ i + q f − X j =1 j + 1)(2 j + 1) ν j = ξ (Υ) + q f − X i =1 i (2 i + 1) δ i (Υ) + q f − X j =1 j + 1)(2 j + 1) ξ j (Υ) . This completes the proof.The next lemma will be used in Section 6.
Lemma 4.7.
Let f : S → B be a non-isotrivial semi-stable family of hyperelliptic curvesof genus g with singular locus Υ / ∆ , and d be defined in (4-8) . If d = 2 , then δ i (Υ) = ξ i (Υ) = 0 , ∀ ≤ i ≤ q f − . Proof.
As what we did in the proof of Proposition 4.1, we may assume that the stablemodel f : S → B comes from an admissible cover. We also use the same symbols andnotations introduced there.According to (4-16), it suffices to prove that ǫ i = µ i = 0 , ∀ ≤ i ≤ q f − . Since any point q l ∈ e Λ ˆ ρ is of index β l ≥ q f + 1, it is enough to prove that for any point q l ∈ e Λ ˜ ρ is of index β l = 2.Let k = β l . Assume that q l ∈ e Λ ˜ ρ is created by a blowing-up ρ l : Y l → Y l − centered at y l − ∈ R h,l − ⊆ Y l − . Then y l − is a singularity of R h,l − of order k . Let { τ , · · · , τ k } ⊆ R h,l − be those sections passing through y l − , and h l − : Y l − → B , h ′ l − : Y l − → D the induced morphisms. Since R h,l − is contained in fibres of h ′ l − by construction, and { τ , · · · , τ k } have a common point y l − , it follows that { τ , · · · , τ k } must be contained inone fibre of h ′ l − . —23— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo e S (cid:15) (cid:15) ˜ f ′ / / ˜ f (cid:28) (cid:28) B ′ (cid:15) (cid:15) Y l − h l − (cid:15) (cid:15) ˜ h ′ l − / / DB Denote by ˆΓ the fibre of h ′ l − containing { τ , · · · , τ k } , and by e F ′ the corresponding fibreof ˜ f ′ : e S → B ′ . Then it is not difficult to see that2 = d = deg (cid:16) ˜ f | e F ′ (cid:17) = deg (cid:0) h l − | ˆΓ (cid:1) ≥ k X i =1 deg ( h l − | τ i ) = k, where ˜ f | e F ′ : e F ′ → B (resp. h l − | ˆΓ : ˆΓ → B , resp. h l − | τ i : τ i → B ) is the restricted mapof e F ′ (resp. ˆΓ, resp τ i ) to B . This completes the proof. q f > The subsection aims to prove Theorem 1.7 for the case q f >
0. It is based on theformula (4-1) given by Cornalba-Harris and (4-4) obtained in Proposition 4.1.We consider first the case ∆ nc = ∅ . By (4-1) and (4-3), one gets ω S/B − g − g − q f deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) = − ( g − q f (2 g + 1)( g − q f ) ξ (Υ)+ [ g/ X i =1 (cid:18) g − q f + 1) i ( g − i )(2 g + 1)( g − q f ) − (cid:19) δ i (Υ)+ [( g − / X j =1 (cid:18) g − q f + 1)( j + 1)( g − j )(2 g + 1)( g − q f ) − (cid:19) ξ j (Υ) . Combining this with (4-4), one gets ω S/B − g − g − q f deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) ≥ q f − X i =1 a i δ i (Υ) + [ g/ X i = q f b i δ i (Υ) + q f − X j =1 c j ξ j (Υ) + [( g − / X j = q f d j ξ j (Υ) , where a i = (cid:18) g − q f + 1) i ( g − i )(2 g + 1)( g − q f ) − (cid:19) + ( g − q f (2 g + 1)( g − q f ) · i (2 i + 1) ,b i = (cid:18) g − q f + 1) i ( g − i )(2 g + 1)( g − q f ) − (cid:19) − ( g − q f (2 g + 1)( g − q f ) · (2 i + 1)(2 g + 1 − i ) g + 1 ,c j = (cid:18) g − q f + 1)( j + 1)( g − j )(2 g + 1)( g − q f ) − (cid:19) + ( g − q f (2 g + 1)( g − q f ) · j + 1)(2 j + 1) ,d j = (cid:18) g − q f + 1)( j + 1)( g − j )(2 g + 1)( g − q f ) − (cid:19) − ( g − q f (2 g + 1)( g − q f ) · j + 1)( g − j ) g + 1 . —24— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo If q f = 1, then b = 3 g − g + 1 ; b i = 4 i ( g − i ) − g − g + 1 ≥ g − g + 1 , ∀ ≤ i ≤ [ g/ d j = 2 (cid:0) ( j + 1)( g − j ) − ( g + 1) (cid:1) g + 1 ≥ , ∀ ≤ j ≤ [( g − / . If q f ≥
2, then a ≥ g − (8 q f + 1) g + 10 q f − g + 1)( g − q f ) ; a i ≥ g − (16 q f + 9) g + 34 q f − g + 1)( g − q f ) , ∀ ≤ i ≤ q f − b i ≥ g − (16 q f + 9) g + 34 q f − g + 1)( g − q f ) , ∀ q f ≤ i ≤ [ g/ c j ≥ , ∀ ≤ j ≤ q f − d j ≥ , ∀ q f ≤ j ≤ [( g − / . Hence (1-6) holds for ∆ nc = ∅ .Now we consider the case that ∆ nc = ∅ . Note that in this case, ξ j (Υ) = 0 , ∀ ≤ j ≤ [( g − / . (4-17)Hence by (4-1) and (4-3), we get ω S/B − g − g − q f deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) = [ g/ X i =1 (cid:18) g + 1 − q f ) i ( g − i )(2 g + 1)( g − q f ) − (cid:19) δ i (Υ) . Hence (1-6) holds too for ∆ nc = ∅ . If moreover q f ≥
2, then according to (4-4) and (4-17),we get [ g/ X i = q f (2 i + 1)(2 g + 1 − i ) g + 1 · δ i (Υ) ≥ q f − X i =1 i (2 i + 1) · δ i (Υ) . So (1-7) is proved.
5. Flat part of R f ∗ C for hyperelliptic families The purpose of the section is to prove Theorem 1.6. It is based on two lemmas. The firstone is Lemma 5.2, coming from a discussion with Chris Peters, on the global invariant cyclewith unitary locally constant coefficient, which generalizes Deligne’s original theorem withthe constant coefficient. The second one is Bogomolov’s lemma on Kodaira dimension ofan invertible subsheaf of the sheaf of logarithmic differential forms on a smooth projectivesurface (cf. [34, Lemma 7.5]). —25— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
Let f : S → B be a non-isotrivial semi-stable family of hyperelliptic curves of genus g ,Υ → ∆ the singular fibres of f , and S → B \ ∆ the smooth part of f . The direct imagesheaf R f ∗ C S is a local system on B \ ∆, which underlies a variation of Hodge structuresof weight one. Let (cid:16) E ∼ = f ∗ Ω S/B (log Υ) ⊕ R f ∗ O S , θ (cid:17) be the Higgs bundle by taking the graded bundle of the Deligne extension of R f ∗ C S ⊗O B \ ∆ . According to [8] or [17], (cid:0) E, θ (cid:1) = (cid:0) A, θ | A (cid:1) ⊕ (cid:0) F, (cid:1) where A , is an ample vector bundle over B and F , is a flat vector bundle coming from arepresentation of the fundamental group ˜ ρ F : π (cid:0) B \ ∆ (cid:1) −→ U ( r ) into a unitary group ofrank r = rank F , . Note that the monodromy around ∆ is unipotent, since f is semi-stable.Hence ˜ ρ F actually factors through π ( B ): π (cid:0) B \ ∆ (cid:1) ˜ ρ F / / i ∗ % % ❑❑❑❑❑❑❑❑❑❑ U ( r ) π ( B ) ρ F ; ; ✇✇✇✇✇✇✇✇✇ Theorem 5.1.
Let f : S → B be a semi-stable family of hyperelliptic curves over B asabove. Then after a suitable base change which is unbranched over B \ ∆ , F , is a trivialbundle, i.e., F , = r M i =1 O B , where r = rank F , . Note that F , and F , are dual to each other. Hence Theorem 1.6 follows fromTheorem 5.1. Indeed, from Theorem 5.1 it follows that the image of ˜ ρ F is finite. Because˜ ρ F factors through π ( B ) and i ∗ is surjective, one gets that ρ F has also finite image. Itimplies that after a suitable base change which is unbranched over B , F , (hence also itsdual F , ) becomes a trivial bundle. So it remains to prove Theorem 5.1.The proof of Theorem 5.1 depends on the following general statement on the globalinvariant cycle with unitary locally constant coefficient, The proof stated below comes froma discussion with Chris Peters. We thank him very much. Lemma 5.2.
Let f : X → B be a smooth proper morphism. Take X ⊃ X to be a smoothcompactification of X , and let U be a locally constant sheaf U over X , which comes froma representation of π ( X, ∗ ) into the unitary group U ( n ) . Then the canonical morphism: H k ( X, U ) −→ H ( B , R k f ∗ U ) is surjective.Proof. The proof is, in fact, along the same line as what Deligne did for the original casewhen U = Q (cf. [4, § U on X carries in a natural way a polarized variationof Hodge structure, say, of pure type (0 , n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo is an induced pure Hodge structure of weight k on H k ( X, U ) as well as on H k ( X b , U | X b )where X b is any (smooth projective) fibre of f : X → B .We first show the ”edge-homomorphism” H k ( X , U ) −→ H ( B , R k f ∗ U )is surjective by the following argument from the proof of [33, Proposition 1.38].It is not difficult to see that we only need to find a class h ∈ H ( X , Q ) such thatcup-products satisfy the hard Lefschetz property, i.e., the following homomorphism is anisomorphism for any 0 ≤ k ≤ m , where m is the dimension of a general fibre of f [ ∪ h ] k : R m − k f ∗ U −→ R m + k f ∗ U . As for the class h ∈ H ( X , Q ), we just take an embedding X ֒ → P N , and let h bethe restriction of the hyperplane class. Then the hard Lefschetz property can be verifiedfiber-by-fiber. On each fiber the natural locally constant metric on U induces a Hodgedecomposition of the cohomology with coefficients in U , hence the hard Lefschetz propertyholds. So, we show that the above ”edge-homomorphism” is surjective.Hence, it also induces surjective morphisms between the weight-filtrations of the bothcohomologies, in particular, on the lowest weight kW k ( H k ( X , U )) ։ W k ( H ( B , R k f ∗ U )) . Since the restriction morphism (as monodromy invariant) H ( B , R k f ∗ U ) → H k ( X b , U | X b )is injective and H k ( X b , U | X b ) carries a pure Hodge structure of weigh- k , H ( B , R k f ∗ U )carries a pure Hodge structure of weight- k. So the above surjective morphism becomes W k ( H k ( X , U )) ։ W k ( H ( B , R k f ∗ U )) = H ( B , R k f ∗ U ) . Finally, according to [32], W k ( H k ( X , U )) is nothing but the image of the restrictionhomomorphism H k ( X, U ) → H k ( X , U ) . Put the above two surjective morphisms together, we obtain a surjective morphism: H k ( X, U ) ։ W k ( H k ( X , U )) ։ H ( B , R k f ∗ U ) . The proof is finished.
Corollary 5.3.
Let f : S → B be a semi-stable family of projective curves (not necessarilyhyperelliptic) over a smooth projective curve B, with semi-stable singular fibres f : Υ → ∆ . Let S = S \ Υ . Given a vector subbundle
U ⊆ f ∗ Ω S/B (log Υ) , which underlies a unitarylocally constant subsheaf U ⊆ V C , R f ∗ C S , then it lifts to a morphism f ∗ U → Ω S , such that the induced canonical morphism U → f ∗ Ω S → f ∗ Ω S (log Υ) → f ∗ Ω S/B (log Υ) coincides with the subbundle
U ⊆ f ∗ Ω S/B (log Υ) . —27— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
Proof.
Since the local monodromy of V around ∆ is unipotent and the local monodromyof the subsheaf U around ∆ is semisimple, so U extends on B as a locally constant sheaf.The morphism U ⊂ V C corresponds to a section η ∈ H ( B \ ∆ , V C ⊗ U ∨ ) = H ( B \ ∆ , R f ∗ ( C S ⊗ f ∗ U ∨ )) . Applying Lemma 5.2, η lifts to a class ˜ η ∈ H ( S, f ∗ U ∨ ) under the canonical morphism H ( S, f ∗ U ∨ ) → H ( B \ ∆ , R f ∗ ( C S ⊗ f ∗ U ∨ )) . Note that this canonical morphism is a morphism between pure Hodge structures of weight-1, and by the construction η is of type (1,0), so ˜ η is of type (1,0), i.e.,˜ η ∈ H ( S, Ω S ⊗ f ∗ U ∨ ) , which corresponds to a morphism f ∗ U → Ω S , such that under the canonical morphism it goes back to U ⊂ f ∗ Ω S/B (log Υ) . In the rest part of this section, we prove Theorem 5.1. It follows from Corollary 5.3and Bogomolov’s lemma on Kodaira dimension of an invertible subsheaf in the sheaf oflogarithmic differential forms on a smooth projective surface (cf. [34, Lemma 7.5]).According to Corollary 5.3, for any flat vector subbundle
U ⊆ F , , there is a sheafmorphism f ∗ U → Ω S , such that the induced canonical morphism U → f ∗ Ω S → f ∗ Ω S (log Υ) → f ∗ Ω S/B (log Υ)coincides with the subbundle
U ⊆ f ∗ Ω S/B (log Υ) . Let ˜ π : e S → e Y be the smooth double cover described in Figure 4.2-1, and ϑ : e S → S be the blowing-ups. By pulling back, we obtain a sheaf morphism˜ f ∗ U = ϑ ∗ f ∗ U −→ Ω e S , where ˜ f = f ◦ ϑ, which corresponds to an element ˜ η ∈ H ( e S, Ω e S ⊗ ˜ f ∗ U ∨ ) . By pushing-out, we also obtain an element (where ˜ h : e Y → B is the induced morphism)˜ π ∗ (˜ η ) ∈ H (cid:16) e Y , ˜ π ∗ (cid:16) Ω e S ⊗ ˜ f ∗ U ∨ (cid:17) (cid:17) = H (cid:16) e Y , ˜ π ∗ (cid:16) Ω e S (cid:17) ⊗ ˜ h ∗ U ∨ (cid:17) . So one gets a sheaf morphism ˜ h ∗ U → ˜ π ∗ (cid:16) Ω e S (cid:17) . The Galois group Gal( e S/ e Y ) ∼ = Z acts on˜ π ∗ (cid:16) Ω e S (cid:17) . One obtains the eigenspace decomposition˜ h ∗ ( U ) → ˜ π ∗ (cid:16) Ω e S (cid:17) , ˜ h ∗ ( U ) → ˜ π ∗ (cid:16) Ω e S (cid:17) − . —28— n Shimura curves in the Torelli locus of curves X. Lu & K. ZuoLemma 5.4.
The image of the map ̺ : ˜ h ∗ ( U ) → ˜ π ∗ (cid:16) Ω e S (cid:17) − . is an invertible subsheaf M such that M is numerically effective (nef ) and M = 0 . Let e Γ be a general fibre of ˜ h , and D be any component of the branch divisor e R ⊆ e Y of the doublecover ˜ π : e S → e Y . Then M · D = 0 , rank U = dim H ( e Γ , O e Γ ( M )) . Proof.
First of all, we want to show that ̺ = 0. It is known that˜ π ∗ (cid:16) Ω e S (cid:17) = Ω e Y , ˜ π ∗ (cid:16) Ω e S (cid:17) − = Ω e Y (cid:16) log( e R ) (cid:17) ( − e L ) , (5-1)where e R ≡ e L ( ≡ stands for linearly equivalent) is the defining data of the double cover˜ π : e S → e Y . Note that the induced map U = ˜ h ∗ ˜ h ∗ U −→ ˜ h ∗ (cid:18) ˜ π ∗ (cid:16) Ω e S (cid:17) ⊕ ˜ π ∗ (cid:16) Ω e S (cid:17) − (cid:19) = ˜ h ∗ (cid:18) Ω e Y (cid:16) log( e R ) (cid:17) ( − e L ) (cid:19) ⊆ E , (5-2)is just the inclusion U ⊆ E , . Hence in particular, ̺ = 0.We claim that the image of ̺ is a subsheaf of rank one. Otherwise, it is of rank two,and so the second wedge product ∧ ˜ h ∗ U ∧ ̺ −→ ∧ (cid:16) ˜ π ∗ (Ω e S ) − (cid:17) = ω e Y is a non-zero map. Note that the image of that map is a quotient sheaf of ∧ g ∗ U comingfrom a unitary local system, so the image sheaf is semi-positive. But, it is impossible, since ω e Y can not contain any non-zero semi-positive subsheaf.So the image of ̺ is a rank one subsheaf M ⊗ I Z , where M is an invertible subsheafand dim Z = 0. Actually, Z = ∅ ; otherwise by a suitable blowing-up ρ : X → e Y , we mayassume the image of ρ ∗ ˜ h ∗ U is ρ ∗ ( M ) ⊗ ( − E ), where E is a combination of the exceptionalcurves. As U comes from a unitary local system, we get ρ ∗ ( M ) ⊗ ( − E ) is semi-positive and0 ≤ ( ρ ∗ ( M ) − E ) = M + E . So M is semi-positive and M ≥ − E >
0, which implies that the Kodaira dimension of M is 2. On the other hand, by (5-1), we get the following inclusion of sheaves, O e Y ( e L ) ⊗ M ⊆ Ω e Y (cid:16) log( e R ) (cid:17) . (5-3)As 2 e L ≡ e R is effective, the Kodaira dimension of e L ⊗ M is also 2, which is impossible byBogomolov’s lemma (cf. [34, Lemma 7.5]).Hence the image of ̺ is an invertible subsheaf M , which is semi-positive since it is aquotient sheaf of a vector bundle coming from a unitary local system. Note that we stillhave the inclusion (5-3). So again by Bogomolov’s lemma (cf. [34, Lemma 7.5]), we get M = 0 , and M · D = 0 . —29— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
Finally, according to (5-2), we haverank U = dim H (cid:0)e Γ , O e Γ ( M ) (cid:1) . Proof of Theorem . Similarly as the proof of Proposition 4.1, we may restrict ourselvesto the situation that the induced double cover π : S → Y in Figure 4.2-1 comes froman admissible double cover (cf. [3] or [10]). In fact, for any semi-stable family f : S → B of hyperelliptic curves, we may get a family of admissible covers by a base change which isunramified over B \ ∆ and blowing-ups of singular points in the fibres.We first prove that in such a situation, F , is a direct sum of line bundles F i on B , i.e., F , = r M i =1 F i , where r = rank F , . (5-4)By assumption, the branched divisor e R ⊆ e Y of the induced smooth double cover ˜ π : e S → e Y is a union of 2 g + 2 sections and some curves contained in fibres of ˜ h : e Y → B . Let D besuch a section, and F , = t M i =1 U i , be a decomposition into irreducible components.If rank U i ≥ i , then by setting U = U i in Lemma 5.4, we obtain M · D = 0.Equivalently if we write F = O D ( M ), thendeg F = 0 . Note that M is a quotient of ˜ h ∗ U . As D is a section, D ∼ = B . Hence we may view F is aninvertible subsheaf on B , which is a quotient of U . As U comes from a unitary local system, U is poly-stable. Thus U = F ⊕ U ′ , which is a contradiction, since U = U i is irreducible byour assumption. Hence we obtain the required decomposition (5-4).Now applying [4, § F i is torsion in Pic ( B ). Henceafter a suitable base change which is unbranched over B , F i ∼ = O B . Therefore, the proof isfinished.
6. Conclusions
The purpose of the section is to prove our main results, Theorems 1.2 and 1.3. Asillustrated in Section 1, the proof follows from the Arakelov equality for the characterizationfor f being a Kuga family together with those bounds on ω S/B given in Theorems 1.4, 1.5and 1.7.Let f : S → B be a Kuga family of curves of genus g ≥
2. Let Υ / ∆ denote semi-stablesingular fibres, Υ c / ∆ c denote those singular fibres with compact Jacobians, and Υ nc / ∆ nc correspond to singular fibres with non-compact Jacobians. Then the logarithmic Higgsbundle associated to the VHS of f is decomposed as Higgs subbundles (cid:16) f ∗ Ω S/B (log Υ) ⊕ R f ∗ O S , θ (cid:17) = (cid:0) A , ⊕ A , , θ | A , (cid:1) ⊕ (cid:0) F , ⊕ F , , (cid:1) , —30— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo where θ | A , : A , → A , ⊗ Ω B (log ∆ nc )is described on Page 2. As f is a Kuga family, θ | A , is an isomorphism by [24] or [37]. Inother words, one hasdeg (cid:16) f ∗ Ω S/B (Υ) (cid:17) = deg A , = rank A , · deg (cid:0) Ω B (log ∆ nc ) (cid:1) . (6-1) The case ∆ nc = ∅ is already proved in Section 1. We consider here only the case∆ nc = ∅ . In this case, ∆ = ∆ c and Υ = Υ c .Since the logarithmic Higgs bundle associated to the family has strictly maximal Higgsfield, by (6-1), we have deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) = g · deg Ω B . Combining this with (1-3), one has ω S/B ≥ (2 g − · deg Ω B + 3 g − g δ (Υ) + 7 g − g δ h (Υ) . (6-2)Together with (1-4), we get 0 ≥ g − g · (cid:0) δ (Υ) + 4 δ h (Υ) (cid:1) . (6-3)Note that both δ (Υ) and δ h (Υ) are non-negative. If one of them is positive, then g ≤ δ (Υ) = δ h (Υ) = 0, then Υ = Υ c = ∅ , i.e., ∆ = ∆ c = ∅ . Hence the inequality(1-4) in Theorem 1.5 is strict. So (6-3) is also strict, which is impossible. Remarks 6.1. (i). If g = 4, then (1-4) must be an equality according the proof above.Hence S \ S p ∈ ∆ c D p ! is a ball quotient by Remark 3.3, where D p is defined in (3-3). Werefer to Example 7.2 for such an example.(ii). There is also another way to show that δ (Υ) and δ h (Υ) cannot be zero simul-taneously if f is a Kuga family with Υ = Υ c and strictly maximal Higgs field. Assume δ (Υ) = δ h (Υ) = 0, then (6-3) is an equality, which implies that (6-2) is also an equality.So we have ω S/B = (2 g − · deg Ω B = 4( g − g · deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) . This implies that f must be a hyperelliptic family by [3, Theorem (4.12)]. However, fora hyperelliptic family with no singular fibres, deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) = 0 by (4-1), which isimpossible. —31— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
The subsection is aimed to prove Theorem 1.3. The idea is similar to that of provingTheorem 1.2. It is based on the Arakelov equality for the characterization for f being aKuga family together with those bounds on ω S/B given in Theorems 1.5 and 1.7. We alsoneed the fact that the rank of flat part of the logarithmic Higgs bundle associated to f isexactly the relative irregularity q f up to some unbranched base change.We assume in this subsection that f : S → B is a Kuga family of hyperelliptic curves.According to Theorem 1.6 together with Deligne’s global invariant cycle theorem (cf. [4, § B by somesuitable unbranched cover, one has rank A , = g − q f . Note that the property that the Higgs field θ is maximal remains true under any unbranchedbase change. Hence the Arakelov equality for the characterization for f being a Kuga familyreads as deg (cid:16) f ∗ Ω S/B (Υ) (cid:17) = g − q f · deg Ω B (log ∆ nc ) . (6-4)By the definition, 0 ≤ δ i (Υ c ) ≤ δ i (Υ) , ∀ ≤ i ≤ [ g/ . (6-5)Combining this with (6-4), (1-4) and (1-6), one obtains that if ∆ nc = ∅ , then0 > g − (6 q f + 3) g + 12 q f − g + 1)( g − q f ) δ (Υ) + 4 g − (13 q f + 12) g + 37 q f − g + 1)( g − q f ) δ h (Υ) , (6-6)and if ∆ nc = ∅ , then0 ≥ (cid:18) g + 1 − q f )( g − g + 1)( g − q f ) − (cid:19) δ (Υ) + [ g/ X i =2 (cid:18) g + 1 − q f ) i ( g − i )(2 g + 1)( g − q f ) − (cid:19) δ i (Υ) . (6-7)One might imagine that it is impossible if g is large enough, since δ i (Υ)’s are non-negative;and one of δ i (Υ)’s must be positive by (4-1) if ∆ nc = ∅ . In other words, there should notexist a Kuga family of hyperelliptic curves of genus g when g is sufficiently large. The detailcomputation is complicated and occupies the rest of the section. Proof of Theorem . We divide the proof into two cases: ∆ nc = ∅ and ∆ nc = ∅ . Case I. ∆ nc = ∅ .In this case, we prove that q f ≤ g ≤ ( , if q f = 0;7 , if q f = 1 . (6-8)First we prove q f ≤
1. Assume that q f ≥
2. Then by Propositions 4.4 and 4.5, weget a morphism ˜ f ′ : e S → B ′ with g ( B ′ ) = q f ≥
2, where e S → S is the blowing-up of S centered at those points fixed by the hyperelliptic involution. Clearly ˜ f ′ factors through e S → S , hence we obtain a morphism f ′ : S → B ′ . It is easy to see that the restricted—32— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo map f ′ | F : F → B ′ is surjective, where F is any fibre of f : S → B . Let F be a singularfibre of f over ∆ nc . Then F has a non-compact Jacobian by assumption. As f reaches theArakelov equality, by [20, Corollary 1.5] and its proof, one gets that the geometric genusof F is g ( F ) = q f . As f ′ | F is surjective, there is at least one irreducible component of F , saying C , mapped surjectively onto B ′ . Hence g ( C ) ≥ g ( B ′ ) = q f by Hurwitz formula.Thus g ( F ) = g ( C ) = g ( B ′ ) = q f , (6-9)and C is a section of f ′ : S → B ′ since q f ≥
2. By (6-9), we see that any componentof F other than C is rational and hence contracted by f ′ . This implies deg ( f ′ | F ) =deg ( f ′ | F ) = 1 for any general fibre F of f . Hence f ′ | F is an isomorphism between F and B ′ for a general fibre F of f . It follows that q f = g ( B ′ ) = g ( F ) = g , which is a contradictionto (4-9). Therefore q f ≤ q f = 0, then by (6-4), the Higgs field associated to f is strictlymaximal. So (6-8) follows from Theorem 1.2, in which we prove that g ≤ nc = ∅ . It remains to consider the case q f = 1. According to (6-6), one gets0 > g − g + 1 δ (Υ) + 4 g − g + 1 δ h (Υ) . This implies that g <
8, i.e., g ≤ Case II. ∆ nc = ∅ .In this case, we prove that q f ≤ g ≤ , if q f = 0;5 , if q f = 1 or 2;6 , if q f = 3 . (6-10)We divide the proof into three subcases: Subcase A: q f ≤ . By (6-7), we get0 ≥ (cid:18) g + 1 − q f )( g − g + 1)( g − q f ) − (cid:19) δ (Υ) + [ g/ X i =2 (cid:18) g + 1 − q f ) i ( g − i )(2 g + 1)( g − q f ) − (cid:19) δ i (Υ) . = g − g · (cid:0) δ (Υ) + 4 δ h (Υ) (cid:1) , if q f = 0;2 g − g + 1 · δ (Υ) + 8 g − g + 1 · δ h (Υ) , if q f = 1 . Note that δ (Υ) ≥ δ h (Υ) ≥
0, and they cannot be zero simultaneously by (4-1). (6-11)Hence g ≤ , if q f = 0; g ≤ , if q f = 1 . Subcase B: q f = 2 . —33— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
In this subcase, (6-7) reads as0 ≥ g − g + 26(2 g + 1)( g − · δ (Υ) + [ g/ X i =2 (cid:18) g − i ( g − i )(2 g + 1)( g − − (cid:19) δ i (Υ) . (6-12)When g ≥
8, it is easy to show that4(2 g − i ( g − i )(2 g + 1)( g − − > g − g + 26(2 g + 1)( g − > , ∀ ≤ i ≤ [ g/ . But this is impossible by (6-11) and (6-12). Hence we may assume g ≤
7. So2 g − g + 26(2 g + 1)( g − < . As q f = 2, by (1-7), one has δ (Υ) ≤ [ g/ X i =2 (2 i + 1)(2 g + 1 − i )12( g + 1) δ i (Υ) . (6-13)Combining this with (6-12), we obtain0 ≥ [ g/ X i =2 (cid:18) g − i ( g − i )(2 g + 1)( g − − g − g + 26(2 g + 1)( g − · (2 i + 1)(2 g + 1 − i )12( g + 1) (cid:19) δ i (Υ)= [ g/ X i =2 (cid:18) g − g − g + 1)(2 g + 1)( g − · i ( g − i ) − g − g + 2612( g + 1)( g − (cid:19) δ i (Υ) ≥ (cid:18) g − g − g + 1)(2 g + 1)( g − · g − − g − g + 2612( g + 1)( g − (cid:19) δ h (Υ)= (cid:18) g − g − g + 1)(2 g + 1) + 2 g − g + 2612( g + 1)( g − (cid:19) δ h (Υ) . Thus if g ≥
6, then it follows that 0 ≥ δ h (Υ), so δ h (Υ) = 0. According to (6-13), we get δ (Υ) = 0 too. However, it is a contradiction by (6-11). Therefore, we have proved that g ≤ q f = 2. Subcase C: q f ≥ . In this subcase, it suffices to prove g ≤
6, from which it follows that q f = 3 by Propo-sition 4.5. To our purpose, we assume g ≥ nc = ∅ , b = g ( B ) ≥ d be the degree of the Albanese map S → Alb ( S ). According to Proposition 4.5 and Remark 4.6, it is known that d ≥ d = 2, then by Lemma 4.7, δ i (Υ) = 0 , ∀ ≤ i ≤ q f − . (6-14)Note that by a remarkable result of Xiao (cf. [39] or Remark 4.6), q f ≤ g since f isnon-isotrivial. So4(2 g + 1 − q f ) q f g + 1 ≥ min (cid:26) g + 1 − · · g + 1 , g + 1 − · g ) · g g + 1 (cid:27) ≥ , since we assume that g ≥ . —34— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
Hence according to (6-7), we get0 ≥ [ g/ X i = q f (cid:18) g + 1 − q f ) i ( g − i )(2 g + 1)( g − q f ) − (cid:19) δ i (Υ) ≥ [ g/ X i = q f (cid:18) g + 1 − q f ) q f ( g − q f )(2 g + 1)( g − q f ) − (cid:19) δ i (Υ) ≥ [ g/ X i = q f δ i (Υ) . By (6-11) and (6-14), we see that it is impossible.Finally we assume d ≥
3. By (4-9) and Remark 4.6, q f ≤ g +13 . It follows that g ≥ q f − ≥
8. According to (6-7), we get [ g/ X i =2 (cid:16) (2 g + 1 − q f ) i ( g − i ) − (2 g + 1)( g − q f ) (cid:17) · δ i (Υ) ≤ (cid:0) − g + (7 + 6 q f ) g + 4 − q f (cid:1) · δ (Υ) . (6-15)Note that for 2 ≤ i ≤ [ g/ g + 1 − q f ) i ( g − i ) − (2 g + 1)( g − q f ) ≥ (2 g + 1 − q f ) · · ( g − − (2 g + 1)( g − q f ) ≥ (2 g + 1)( g − − (4 g − q f ≥ (2 g + 1)( g − − (4 g − · g + 13= 2 g − g + 13 > , since g ≥ , − g + (7 + 6 q f ) g + 4 − q f ≥ . Combining (6-15) with (1-7), we obtain [ g/ X i =2 (cid:16) (2 g + 1 − q f ) i ( g − i ) − (2 g + 1)( g − q f ) (cid:17) · δ i (Υ) ≤ Θ · · [ g/ X i = q f (2 i + 1)(2 g + 1 − i ) g + 1 δ i (Υ) − · q f − X i =2 i (2 i + 1) δ i (Υ) i.e., 0 ≥ q f − X i =2 a i δ i (Υ) + [ g/ X i = q f b i δ i (Υ) , (6-16)—35— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo where a i = 4 (cid:0) (2 g + 1 − q f ) i ( g − i ) − (2 g + 1)( g − q f ) (cid:1) + Θ3 · i (2 i + 1) ,b i = 4 (cid:0) (2 g + 1 − q f ) i ( g − i ) − (2 g + 1)( g − q f ) (cid:1) − Θ12 · (2 i + 1)(2 g + 1 − i ) g + 1 . It is not difficult to check that a i > ≤ i ≤ q f −
1. For b i with q f ≤ i ≤ [ g/ b i = 4 (cid:18) (2 g + 1 − q f ) − Θ12( g + 1) (cid:19) · i ( g − i ) − g + 1)( g − q f ) − Θ · (2 g + 1)12( g + 1) ≥ (cid:18) (2 g + 1 − q f ) − Θ12( g + 1) (cid:19) · q f · ( g − q f ) − g + 1)( g − q f ) − Θ · (2 g + 1)12( g + 1)= (2 g + 1)( g − q f )12( g + 1) q f (13 g − q f + 8) − g −
51+ 4 (cid:0) ( q f − g + ( g + 1 − q f )( g − (cid:1) g − q f ! ≥ (2 g + 1)( g − q f )12( g + 1) · (cid:16) q f (13 g − q f + 8) − g − (cid:17) . Let f ( g, q f ) = 4 q f (13 g − q f + 8) − g −
51. Since g ≥
8, one gets f ( g,
3) = 106 g − > ,f (cid:18) g, g + 13 (cid:19) = 13 · (24 g − g − > . Hence for any 3 ≤ q f ≤ g +13 , we have f ( g, q f ) ≥ min (cid:26) f ( g, , f (cid:18) g, g + 13 (cid:19)(cid:27) > . Thus for any 3 ≤ q f ≤ g +13 , b i > . Now from (6-16) it follows that δ i (Υ) = 0 for all 2 ≤ i ≤ [ g/ δ (Υ) = 0 too.This is a contradiction by (6-11). Therefore the proof is complete.
7. Examples
In this section, we construct two Shimura curves contained generically in the Torellilocus of hyperelliptic curves of genus 3 and 4 respectively.The idea is to construct first non-isotrivial semi-stable families of hyperelliptic curvesof genus g = 3 and 4 respectively by taking double covers of ruled surfaces branched oversuitable branched locus. Then we show that their corresponding Jacobian families reach theArakelov bound. By [37] (or [24]), a semi-stable one-dimensional family of g -dimensionalabelian variety reaching the Arakelov bound gives a Kuga curve in A g . Hence we obtain—36— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo two Kuga curves contained generically in the Torelli locus of hyperelliptic curves of genus g = 3 and 4 respectively.To show that those two Kuga curves are indeed Shimura curves, first we note that theHiggs field associated to the family is actually strictly maximal for g = 4, hence by virtueof [37], it is a Shimura curve. For g = 3, we present two ways to prove that such a Kugacurve is also Shimura. Example 7.1.
Shimura curve contained generically in the Torelli locus of hyperellipticcurves of genus g = 3.Let C , H x ⊆ X = P × P be defined respectively by1 + (4 t − x + x = 0 , and x = x , where t and x are the coordinates of the first and second factor of X respectively. Theprojection of C to the first factor P of X branches exactly over three points, i.e., { , , ∞} .Locally, it looks like the following. rr t = 0 − rr t = 1 −√− √− rr t = ∞ ∞ Let ϕ : P → P be the cyclic cover of degree 4 defined by t = ( t ′ ) , totally ramifiedover { , ∞} . Let X be the normalization of the fibre-product X × P P and R the inverseimage of C ∪ H ∪ H − ∪ H ∪ H ∞ . Then R is a double divisor, i.e., we can construct a double cover S → X branched exactlyover R . Let S ′ → X r be the canonical resolution, and f : S → P the relatively minimalsmooth model as follows. S ′ (cid:15) (cid:15) (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ / / X r (cid:15) (cid:15) S f (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ S / / (cid:15) (cid:15) X / / τ ~ ~ ⑤⑤⑤⑤⑤⑤⑤ X τ (cid:15) (cid:15) P ϕ / / P By the theory of double covers (cf. [2, § III.22]), it is not difficult to show that f : S → P is a semi-stable hyperelliptic family of genus g = 3. In fact, there are exactly 6 singularfibres in the family f , i.e., those fibres Υ over ∆ := ϕ − (0 ∪ ∪ ∞ ). More precisely, forany fibre F over ϕ − (1), F is an irreducible singular elliptic curve with exactly two nodes,hence ξ ( F ) = 2 , and δ ( F ) = ξ ( F ) = 0;—37— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo for any fibre F over ϕ − (0 ∪ ∞ ), F is a chain of three smooth elliptic curves, hence δ ( F ) = 2 , and ξ ( F ) = ξ ( F ) = 0 . So ξ (Υ) = 8 , δ (Υ) = 4 , and ξ (Υ) = 0 . Therefore by (2-2), (4-1) and (2-4), δ f = 12 , deg (cid:0) f ∗ Ω S/ P (log Υ) (cid:1) = 2 , ω S/ P = 12 . By definition, those fibres over ∆ c := ϕ − (0 ∪ ∞ ) have compact Jacobians, while thoseover ∆ nc := ϕ − (1) have non-compact Jacobians. Hence the Jacobian of f admits exactly nc ) = 4 singular fibres over P . By [37, § P with exactly 4 singular fibres must be maximal. i.e., it reachesthe Arakelov bound. Hence f is a Kuga family.Let F , ⊕ F , be the flat part of the logarithmic Higgs bundle associated to the VHSof the Jacobian of f as on Page 2. Since the base of the family is P , q f = rank F , . Hencedeg (cid:0) f ∗ Ω S/ P (log Υ) (cid:1) = g − q f · deg (cid:0) Ω P (log ∆ nc ) (cid:1) = 3 − q f , from which it follows that q f = 1.It remains to show that f is in fact a Shimura family. We present here two ways.(i). Since the base P of f is simply connected, by [37, Theorem 0.2], the Jacobian of f is isogenous over P to a product E × P E × P E , (7-1)where E is a constant elliptic curve, and E → P is a family of semi-stable elliptic curvesreaching the Arakelov bound. To show that f is a Shimura family, it suffices to prove thatthe constant part E has a complex multiplication.It is not difficult to see that our family is actually defined by y = (cid:0) t ′ ) − x + x (cid:1) · ( x − · x. (7-2)Let E be a constant elliptic curve defined by u = v · ( v + 1) . Then it is clear that E has complex multiplication by Z (cid:2) √− (cid:3) . Define a morphism fromthe family f to the constant family E by( u, v ) = ψ ( x, y ) = √ · t ′ y ( x − , t ′ ) x ( x − ! . It can be checked easily that ψ is well-defined. Hence the Jacobian of f contains a constantpart E . Note that the constant part E in the decomposition (7-1) is unique up to isogenous,and the property with a complex multiplication is invariant under isogenous. Therefore,the constant part E ∼ E has a complex multiplication, and so f is a Shimura family.—38— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo (ii). We prove that f is a Shimura family by showing that our family f is actuallyisomorphic to a known Shimura family constructed by Moonen and Oort [26].Let u = 1 + x − x , v = 2 y (1 − x ) , w = (cid:18) x − x (cid:19) . Then by virtue of (7-2), we see that our family is isomorphic to U t ′ : u = w,v = (cid:16) t ′ ) w − (cid:0) ( t ′ ) − (cid:1)(cid:17) · ( w − . Such a family can be viewed as a family of abelian covers of P branched exactly over4 points with Galois group Z × Z and local monodromy of the branched points being (cid:0) (1 , , (1 , , (0 , , (0 , (cid:1) . And it is just the family (22) given in [26, § Table 2 ], whichis Shimura. So is f .We remark that by [26], we do not know whether the corresponding Shimura curve iscomplete or not (i.e., whether ∆ nc = ∅ or not). Our concrete description shows that sucha Shimura curve is a non-complete rational Shimura curve. Example 7.2.
Shimura curve with strictly maximal Higgs field contained generically inthe Torelli locus of hyperelliptic curves of genus g = 4.The construction is similar to Example 7.1.Let C , H x and X be the same as those in Example 7.1. Let π : B → P be a cover ofdegree 8, ramified uniformly over { , , ∞} with ramification indices equal to 4. It is easyto see that such a cover exists, and g ( B ) = 2 , , where ∆ = ϕ − (0 ∪ ∪ ∞ ). Let X be the normalization of X × P B and R the inverseimage of C ∪ H ∪ H − ∪ H √− ∪ H −√− ∪ H ∪ H ∞ . Then R is a double divisor, i.e., we can construct a double cover S → X branched exactlyover R . Let f : S → B the relatively minimal smooth model as follows. S f (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ o o / / ❴❴❴ S / / (cid:15) (cid:15) X / / τ ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ X τ (cid:15) (cid:15) B ϕ / / P By [2, § III.22], one can show that f : S → B is a semi-stable hyperelliptic family ofgenus g = 4 with 6 singular fibres, i.e., those fibres Υ over ∆. More precisely, for any fibre F ∈ Υ, F consists of two smooth elliptic curves D , D , and a smooth curve e D of genus 2,such that D does not intersect D , and e D intersects each D i in one point for i = 1 , δ ( F ) = 2 , and δ ( F ) = ξ ( F ) = ξ ( F ) = 0 . So δ (Υ) = 12 , and δ (Υ) = ξ (Υ) = ξ (Υ) = 0 . —39— n Shimura curves in the Torelli locus of curves X. Lu & K. Zuo
Therefore by (2-2), (4-1) and (2-4), δ f = 12 , deg (cid:0) f ∗ Ω S/B (log Υ) (cid:1) = 4 , ω S/B = 36 . By definition, any singular fibre of f has a compact Jacobian, so the Jacobian of f is asmooth family of abelian varieties of dimension 4. Let A , ⊆ f ∗ Ω S/B (log Υ) be the amplepart described on Page 2. Then according to Arakelov inequality, we have4 = deg (cid:0) f ∗ Ω S/B (log Υ) (cid:1) ≤ rank A , · deg Ω B = rank A , ≤ rank f ∗ Ω S/B (log Υ) = g = 4 . Hence the Jacobian of f reaches the Arakelov bound with rank A , = g , i.e., the Higgsfield associated to f is strictly maximal. Therefore f is a Shimura family.We remark that in this example, c (cid:0) Ω S (log D ) (cid:1) = 3 c (cid:0) Ω S (log D ) (cid:1) = 72 , where D is the union of those 12 smooth disjoint elliptic curves contained in Υ. Hence S \ D is a ball quotient by [16] or [22]. Acknowledgements.
We would like to thank Chris Peters, Guitang Lan and JinxingXu for discussions on the topic related to global invariant cycles with locally constantcoefficients. Especially, the proof of Lemma 5.2 comes from a discussion with Chris Peters.We would also like to thank Shengli Tan and Hao Sun for discussing with us on Miyaoka-Yau’s inequality and the slope inequality. We are grateful to Yanhong Yang for her interests,careful reading and valuable suggestions.
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