On the Negativity of Moduli Spaces for Polarized Manifolds

Given a log base space (Y, S), parameterizing a smooth family of complex projective varieties with semi-ample canonical line bundle, we briefly recall the construction of the deformation Higgs sheaf and the comparison map on (Y, S) made in the work by Viehweg–Zuo. While almost all hyperbolicities in the sense of complex analysis such as Brody, Kobayashi, big Picard and Viehweg hyperbolicities of the base U = Y ∖ S (under some technical assumptions) follow from the negativity of the kernel of the deformation Higgs bundle we pose a conjecture on the topological hyperbolicity on U. In order to study the rigidity problem we then introduce the notions of the length and characteristic varieties of a family f : X → Y, which provide an infinitesimal characterization of products of sub log pairs in (Y, S) and an upper bound for the number of subvarieties appearing as factors in such a product. We formulate a conjecture on a characterization of non-rigid families of canonically polarized varieties.


Introduction
Let U be a complex quasi-projective manifold andȲ = U ∪S a smooth compactification. Hodge theory plays a fundamental role in studying the geometry of U . We are particularly interested on U as a base space f : V → U parameterizing smooth projective varieties. The powerful theory on variation of Hodge structures developed by P. Griffiths [11][12][13] suggests certain negativity of the sheaf of logarithmic holomorphic vector fields of the log pair (Ȳ ,S). We detail here the resulting program of the complex hyperbolicity of U in its various aspects as well as the program motivated by the original Shafarevich conjecture.
In Section 2, we summarize all these relevant properties for a base admitting a locally injective Torelli map. C. Simpson [25] has introduced a basic notion in nonabelian Hodge theory, the so-called graded Higgs bundle or the system of Hodge bundles (E, θ) as the grading of a polarized variation of Hodge structures (PVHS). These O Y -linear objects are better understandable in algebraic geometry and enjoy a crucial property in complex geometry: The kernel of the Kodaira-Spencer map (the Higgs field) ker(θ) is semi-negative.
In Section 3, we discuss the work by Viehweg-Zuo on families without the assumption of local injective Torelli maps. We emphasize the following crucial new ingredients in Viehweg-Zuo's construction: -The deformation Higgs bundle arising from the Kodaira-Spencer theory. One notes that if the family has maximal variation then the Kodaira-Spencer map is generically injective. -Constructing a comparison map between the deformation Higgs bundle and a Higgs bundle of geometric origin.
With the help of this comparison map, the negativity of the kernel of the Higgs field mentioned in Section 2 is transformed into that of the kernel of Kodaira-Spencer map on the deformation Higgs bundle. However we do not achieve all known negativity properties which are enjoyed by families admitting locally injective Torelli map. We pose several questions and conjectures. In particular, we raise a conjecture on the topological hyperbolicity of the base space, which is motivated by a question asked by Jürgen Jost some years ago on the negativity of Riemannian sectional curvature of a base space admitting a locally injective Torelli map.
In Section 4 we consider the rigidity problem of families, or equivalently products of subvarieties of the base space. The original Shafarevich program asked for the so-called finiteness of the set H of isomorphic classes of families over a fixed log base. The finiteness is decomposed into two basic problems: -The boundedness of H.
-The rigidity of points in H.
The boundedness has been proven by various people. For example, for semi-stable families of abelian varieties over a base curve Faltings [9] has shown the boundedness. Jost and Yau [16] have used Yau's form of Schwarz inequality and given another proof of the boundedness for families of abelian varieties over curves. In general, Viehweg and Zuo [29] have shown the boundedness for families of varieties with semi-ample canonical line bundles over a base curve without requiring the existence of a locally injective Torelli map. Finally, Kovács, Lieblich [18] have shown the boundedness for families over a higher dimensional base.
The rigidity problem is subtle. In fact, there exists non-rigid families of higher dimensional varieties. So characterizing non-rigid families will be important for Shafarevich program for families of higher dimension varieties. In Section 4.1 we introduce the notion of the length of a family, which generalizes the rank of a bounded symmetric domain. In Section 4.2 we define the characteristic varieties of a family, which gives an infinitesimal characterization of products of subvarieties of the base space. The characteristic varieties has been introduced by N. Mok originally for bounded symmetric domain for studying the metric rigidity problems. Motivated by Mok's work, M. Sheng and the author have constructed characteristic varieties for PVHS of Calabi-Yau type. With help of such new invariants we hope to get a better understanding of the Arakelov rigidity problem for families of higher dimensional varieties.
Very recently, Javanpeykar, Sun and Zuo [15] proved a new version of the Shafarevich conjecture, the so-called finiteness of pointed families f : X → Y of h-dimensional polarized varieties. A log map from a log curve φ : (C, S C ) → (Y, S) has only the trivial deformation φ t if φ t fixes no less than 1 2 (h − 1)deg Ω 1 C (log S C ) number of points in C \ S C . We remark that the slop of the ample vector bundle f * ω ν X/C satisfies the following Arakelov inequality Based on the Griffiths curvature formula for the Hodge metric on E and the asymptotic behavior of the Hodge metric on the quasi-canonical extension onS due to Cattani-Kaplan-Schmid [3], one shows that the curvature current of the Hodge metric on K is semi negative. In the same way, using the polarized variation of mixed Hodge structures along S, one shows (in [32]) that K is also semi-negative alongS.

Semi-negativity of TȲ (− logS )
The most important negativity is that of the logarithmic tangent bundle TȲ (− logS). Assuming injectivity of the local Torelli, i.e., the injectivity of the derivative of the period map φ dφ : TȲ (− logS) → End(E) on U (in general dφ is required to be injective at at least one point in U ), since the original Higgs map satisfies the integrability condition: θ ∧ θ = 0, this is equivalent to where the Higgs map θ end comes from the tensor algebra and gives also the grading of the Gauss-Manin connection on End(V , ∇, Fil * , Ψ ). In particular, TȲ (− logS) is seminegative in the sense of the curvature current of the Hodge metric. Consequently, it implies that is semi-positive in the usual sense of Algebraic Geometry, i.e., for any projective curveC ⊂Ȳ and any quotient bundle Ω 1 Y (logS)|C → Q → 0 one has det Q ≥ 0.

Bigness of ωȲ (S )
The theorem due to Griffiths-Schmid [14] tells us more! It asserts that the holomorphic sectional curvature is strictly negative. This property together with semi-negativity of TȲ (− logS) shows that ωȲ (S) is big. Griffiths [10,13] constructed the so-called augmented line bundle L by taking the product of determinants of the Hodge bundles of suitable powers. L is nef on Y and ample on U . Alternatively, one uses the iteration of the Kodaira-Spencer map and shows directly L = rωȲ (S) − P + N , where r is a positive rational number, P is an effective divisor and N is a rational semi-negative divisor. This shows ω Y (S) is big ([32, Lemma 2.2]).

Bigness of Ω 1 Y (logS )
The bigness of Ω 1 Y (logS) should have a close relation to the strict negativity of holomorphic sectional curvature due to Griffiths-Schmid. There are several different notions of ampleness and bigness for torsion free sheaves on Y . The first one has been introduced by Viehweg and mainly used in Viehweg-Zuo's papers. Here we just concentrate on vector bundles.

Proposition 1 1. E is big in Definition 2 if and only if some symmetric power S n (E)
contains a generically ample subbundle.

A generically ample vector bundle is big.
Conversely a big vector bundle is not necessarily generically ample. For example, the cotangent bundle on a locally bounded symmetric domain of rank > 1 is big but not generically ample. However, the cotangent bundle on a complex ball quotient is ample.
In general, the sheaf of log differential forms on a log base (Ȳ ,S) admitting a locally injective Torelli map is big. This is due to Brunebarbe, Klingler and Totaro ([2, Theorem 1.1]). The crucial point in Lemma 1.4 in their paper is to use the strict negativity of holomorphic sectional curvature to produce a generically ample subsheaf in a symmetric power S n (Ω 1 Y (logS)). In fact, the existence of an ample subsheaf in the symmetric power of sheaf of log differential forms can be constructed directly via Kodaira-Spencer map as follows. If the first Hodge bundle E n,0 is generically ample as in Definition 1, for example in the case of variation of middle cohomology of Calabi-Yau n-folds or abelian varieties, then one obtains a generically ample subbundle E n,0 ⊗ ker(θ) ∨ → S m (Ω 1 Y (logS)) by applying the maximal non-zero iteration of Kodaira-Spencer maps. In general we consider the Griffiths augmented line bundle L. Then L is generically ample. Deng [6] observed that L can be realized as a sub-line bundle in a suitable type of tensor product of the original Higgs bundle. By running the maximal non-zero iteration of Kodaira-Spencer map starting from L as the initial line bundle one obtains non-zero map L ⊗ ker(θ ⊗( * ) ) ∨ → S m Ω 1 Y (logS). As the sheaf on the left-hand side is a tensor product of a generically ample line bundle with a non-negative sheaf and the image is generically ample and Ω 1 Y (logS) is semi-positive, the bigness of ω Y (S) follows.

Y (logS) for a Log Base (Ȳ,S) Parametrizing Varieties with Semi-ample Canonical Line Bundles
The following construction has been introduced in [29,30]. Let f : V → U be a smooth family of polarized manifolds of dimension n with semi-ample canonical line bundle and with maximal variation. We take a good partial compactification f : X → Y of the original family f : V → U , which satisfies: f : X → Y is a log smooth projective morphism between the log pairs (X, Δ) and (Y, S). -Y has a smooth projective compactificationȲ such thatȲ \ U =:S is a normal crossing divisor and codim(Ȳ \ Y ) ≥ 2.

Deformation Higgs Bundle (Sheaf) (F 0 , τ 0 ) Arising from Kodaira-Spencer Map
We start with the classical Kodaira-Spencer map on the log smooth family f : X → Y : Tensoring the Kodaira-Spencer map τ n,0 0 with R q f * T q X/Y (− log Δ) and then composing with the wedge product ∧, we define the extended Kodaira-Spencer maps τ p,q 0 as the composition: Putting all terms together we obtain the so-called deformation Higgs bundle (sheaf) attached to f : X → Y : One checks that the sum of the extended Kodaira-Spencer maps τ 0 = ⊕τ p,q 0 satisfies the integrability condition τ 0 ∧ τ 0 = 0 using the associativity and anti-commutivity of the cup product on Dolbeault cohomology of the tangent sheaf. By taking the reflexive hull F p,q of F p,q 0 onȲ , the Higgs field So we obtain a Higgs bundle (sheaf) (F, τ ) onȲ as an extension of the Kodaira-Spencer map of f : X → Y . We note that for a family of Calabi-Yau n-folds the Higgs bundle (F, τ ) is nothing but the graded Higgs bundle of the VHS associated to the middle cohomology of f after tensoring with the Hodge line bundle f * Ω n X/Y (log Δ) ∨ . Motivated by the negativity of the kernel of the Higgs map arising from PVHS, we would like to compare (F, τ ) with a Higgs bundle arising from the geometric origin and show the negativity of the kernel of Kodaira-Spencer map τ .
Via linear algebra , θ arising from PVHS of the middle cohomology of f . Indeed, if L has a non-zero section, then it induces a natural Higgs sheaf map from (F, τ ) into (E, θ). In Section 3.5 we will see that this idea can be realized once some positive power of L has non-zero sections.

Comparing (F , τ ) with Higgs Bundle Arising from the Geometry Origin
Given an ample line bundle A onȲ there are two versions of the comparison map.
Version 1. Replacing f : X → Y by taking a self-fibre product to a higher power By taking a self fibre product f (n) :X (n) →Ȳ to a higher power and a cyclic cover π :Z → X (n) we obtain a Higgs sheaf map Version 2. Replacing Y by a Kawamata cover Y → Y to raise the power of A By taking a Kawamata cover ψ : Y → Y and a cyclic cover π : Z → X of the fibre product f : X → Y induced by the base change, we obtain a Higgs sheaf map where (E, θ) is the logarithmic graded Higgs bundle of the quasi-canonical extension of PVHS of the middle cohomology of the induced family g : We emphasize the following crucial property of the comparison map. Although the Higgs field θ on E has in general bigger singularityS +T , its restriction to ρ(F (n) ) has only singularity on the original degeneration locusS. We will briefly describe the construction of (E, θ) in Section 3.5.

Negativity of the Kernel of Kodaira-Spencer Map
The Higgs sheaf map ρ is a sort of Torelli map, which relates the negativity of the kernel of θ to the negativity of the kernel of τ . First we discuss the injectivity of ρ.

Proposition 2
We take (F, τ ) either as the deformation Higgs bundle associated to a higher power self fibre product of the original family or as the pulled back of the deformation Higgs bundle associated to the original family via a Kawamata base change. Writing ρ = ρ p,q , The injectivity in (1) is tautological. The injectivity in (2) respectively in (3) follows from Kodaira type respectively Bogomolov-Sommese vanishing theorem [29]. The comparison map in (4) is similar (in every way) to the Kodaira-Spencer map on a family of Calabi-Yau n-folds over U . Viehweg and Zuo show the map is non-zero and Deng [5] shows it is injective at generic point. Both arguments are global and crucially rely on the seminegativity of ker(θ n,0 ) and the strictly negativity of A −1 . The injectivity of θ n,0 • ρ n,0 in (4) is weaker than the injectivity in (3), but strong enough to show all hyperbolicities on U except for Kobayashi hyperbolicity.
As a consequence of the the non-triviality of the comparison map, we get the negativity of the kernel of Kodaira-Spencer map. To illustrate the idea we assume that the canonical line bundle along the fibres are ample (in general one works with the image ρ(F, τ ) ⊂ (E, θ) ⊗ A −1 and pays the attention to the property that the restriction of θ to ρ(F ) has only singularities inS and is injective at the generic point). We have an embedding of Higgs bundles Since (E, θ) is the quasi-canonical extension of a polarized VHS on U ⊂Ȳ = U ∪S ∪T , Griffiths curvature formula for ker(θ ) yields the semi-negativity on ker(θ); more precisely, the Hodge metric on ker(θ) outside S + T extends to a degenerated metric onȲ with seminegative curvature in the sense of currents. Hence, we see that ker(τ ) is negative onȲ with the following consequences.
its dual gives rise to a generically ample subsheaf Proof 3. Take the second version of the comparison map where ρ is injective. Let r := rank G, then the sub-Higgs line bundle lies in the kernel of the Higgs field on the right-hand side, as a graded Higgs line bundle has vanishing Higgs field. This shows that φ * det G ∨ is big onȲ and hence det G ∨ is big onȲ .

Remark 1 It would be interesting to compare the bigness of the line bundle
with the bigness of the Griffiths augmented line bundle from PVHS.

Pseudo-effectivity of Ω 1 Y (logS ) ⊗N and Various Hyperbolicities of U
Campana and Paun [4] proved that Ω 1 Y (logS) is pseudo-effective. Namely, for any quotient sheaf Q of Ω 1 Y (logS) ⊗N and for any movable curveC ⊂Ȳ one has c 1 (Q)C ≥ 0. The proof relies on the existence of a generically ample subsheaf A ⊂ S l Ω 1 Y (logS). Again combining with this ample subsheaf A they showed the log pair (Ȳ ,S) is of log general type, the so-called conjecture on Viehweg hyperbolicity.
The maximal non-zero iteration of Kodaira-Spencer map on the image Higgs bundle ρ((F, τ ) ⊗ A) defines a complex Finsler pseudometric (a further modification of the Fubini-Study metric on A is eventually needed [30] and [21]) such that the holomorphic sectional curvature is strictly negative. All hyperbolicities on U except the Kobayashi hyperbolicity follows from this metric. In particular, very recently Deng et al. [7] have shown the big Picard theorem holds true in U . As for Kobayashi hyperbolicity, a further modification by taking the Finsler metric as the sum of iterations of all lengths by To and Yeung [26] (also see [5]) is needed for constructing a non-degenerated and strictly negatively curved Finsler metric.
The following proposition shall give a Hodge theoretical counterpart and interpretation of the strict negativity of the holomorphic sectional curvature of the modified Finisler metric defined by taking the sum of iterations of all lengths.

Proposition 4 The determinant of the image Higgs sheaf by taking iterations of Kodaira-Spencer map of all lengths
is non-positive.

Comments 1
The pseudo-effectivity of Ω 1 Y (log S) together with the big subsheaf A in the symmetric power of Ω 1 Y (log S) makes the log pair (Ȳ ,S) similar to a base space admitting a locally injective Torelli map. For more applications we would like to ask for the positivity of log differential forms pulled back to subvarieties. More precisely, regarding the second component of the deformation Higgs bundle F n−1,1 = R 1 f * T X/Y (− log S) as the pulled back of the log tangent sheaf on the moduli space via the moduli map one would like to know if any type non-positivity of F n−1,1 restricted to any subvariety holds true? In a recent paper [15] Javanpeykar, Sun and Zuo used the argument by taking the sum of iterations of Kodaira-Spencer map specifically on an invertible subsheaf in the pulled back φ * F n−1,1 via a log map φ into (Y, S) and showed that Proposition 5 [15] Let φ : (C, S C ) → (Y, S) be a non constant map from a log curve (C, S), then any invertible subsheaf L ⊂ φ * F n−1,1 one has Consequently, Javanpeykar, Sun and Zuo proved a new version of the Shafarevich conjecture, the so-called finiteness of pointed families of polarized varieties. Conjecture 1 should have more stronger consequences on the geometry of (Y, S). For example, it implies the one pointed Shafarevich Conjecture asked in [15].
So far we have discussed various notions of hyperbolicity in the sense of complex analysis, to end this section we like to pose a conjecture on the bigness of the fundamental group of U .
One notes that a punctured Riemann surface U is hyperbolic if and only if π 1 (U ) is infinite and nonabelian.
Definition 3 (Milnor [19]) A growth function associated to a finitely generated group G is defined as follows: For each positive integer s let (s) be the number of distinct group elements which can be expressed as words of length ≤ s with a fixed choice of generators and their inverses.

Theorem 2 (Milnor [19]) The fundamental group of a compact Riemannian manifold M with all Riemannian sectional curvatures less than zero has exponential growth, i.e.
(s) ≥ a s for some a > 1.
The proof relies on Günther's volume comparison theorem on the exponential growth of the volume of the geodesic ball on the universal coverM. We make Conjecture 2 Let U be a base parameterizing polarized manifolds with semi-ample line bundle and of maximal variation. Then π 1 (U ) grows at least exponentially.
The idea supporting Conjecture 2 goes back to a question asked by Jürgen Jost many years ago. A horizontal subvariety in a period domain is locally embedded into the associated real symmetric space under the natural projection. As the real symmetric space has non-positive Riemannian sectional curvature, Jost asked if the Hodge metric, as a Kähler metric on the horizontal subvariety enjoys the same type of property for the Riemannian sectional curvature. However, Conjecture 1 asks more, via the comparison map and the maximal non-zero iteration of the Kodaira-Spencer map one hopes that the negativity of the Riemannian sectional curvature on the real symmetric space can be further "transformed" into the degenerated Finsler metric on U discussed here. Similar to the approach in proving the complex hyperbolicity on U we hope to construct a Riemann-Finsler metric on U via the iteration of Kodaira-Spencer map, whose curvature has certain negativity. In our situation the complex Finsler metric naturally induces a Riemann-Finsler metric ds 2 RFin on U . We are aware that in general the Riemannian curvature decreasing principle does not hold true for real sub-manifolds. Very recently together with Steven Lu and Ruiran Sun we observed the fact that pluriharmonicity of the composition of the horizontal period map with the projection to the symmetric space of non-compact type implies decreasing Riemannian curvature still holds true in a weak form. We expect this weak form of the negative sectional curvatures can be used to show Günter's volume comparison inequality and get the solution of Conjecture 2 by Milnor's original argument.

Constructing the Comparison Map in Section 3.2, a Hodge Theoretic Interpretation
The main issue in the comparison map is that we would like to give the relativeČech cohomology of log tangent sheaf T p X/Y (− log Δ) a Hodge theoretical interpretation. Hence, the negativity (or positivity) results from Hodge theory can be applied. The construction is the gluing of the fiberwise construction, which goes back to the works of Esnault-Viehweg and Kawamata.

Cyclic (Kummer) Cover in the Absolute Case
Let X be a smooth projective variety. Cyclic covers of X are classical and were invented mainly to prove Kodaira type vanishing theorems for theČech cohomology of differential forms (twisted with an anti big line bundle) of type p + q < dim X, see [8] for details. Note that for the comparison map we focus on the cohomology of type p + q = dim X.
Let L be a line bundle on X with H 0 (X, L ν ) = 0 for some positive integer ν. Then any non-zero section s ∈ H 0 (X, L ν ) will induce naturally a comparison map where Z stands for the ν-th cyclic (Kummer) cover of s, see [8,Section 3]. Assume that the zero divisor div(s) =: D is smooth. The ν-th root out of s defines a smooth cyclic cover π : Z → X ramified on D ⊂ X. The Galois group of the cover (Z/νZ) acts on Z and on π * (Ω p Z ). The eigenspace decomposition reads as:

which induces an isomorphism
In general, D could be singular and have some components with multiplicities. We first take a blowing up δ : W → X to make δ * (D) =: B = j α j B j being normal crossing. Then we choose a suitable Kummer cover, the so-called Kawamata cover, π : Z → W by adding the ν-th root of π * s and the ν-th roots of some sections from additional line bundles such that Z is smooth and the ramification locus is a normal crossing divisor containing the divisor j : where (δ * L) (−1) is slightly large than δ * L −1 , see [8] for details.
• Logarithmic rank one local system (L −1 , ∇) on X arising from the cyclic cover π : Z → X. In fact, the eigenspace associated to 1 which has Hodge-to-de Rham E 1 -degeneration w.r.t. the truncated de Rham subcomplexes using twisted harmonic forms defined by a singular locally constant metric below. One checks that the Hodge cohomology of this de Rham complex is

Cyclic (Kummer) Cover for a Family f : X → Y in the Relative Case and the Comparison Map
Choose a generic non-zero section s of L ν ⊗ f * (A −ν ) such that the zero divisor D := div(s) intersects a general fibre of f smoothly. Let T ⊂ Y denote the closure of the discriminant of the map H ∩ V → U , which is the locus where the intersection D y := D ∩ f −1 (y) becomes singular. Leaving out some codimension two subschemes, we may assume that S + T is a smooth divisor. Let Σ := f * (T ) and we keep the notation Δ = f * (S). We take a blowing up δ : W → X with centers in D + Δ + Σ and such that δ * (D + Δ + Σ) is normal crossing and the composition map is log smooth as a morphism between log pairs h : W, δ * (D + Δ + Σ) → (Y, (S + T )).
We write M := δ * (L ⊗ f * A −1 ) and B := δ * (D), then M ν = O W (B). One takes the ν-th cyclic cover of the divisor B and choose Z to be a desingularization of this covering and obtains the induced new family T ) is a smooth family. We may assume thatS +T is normal crossing inȲ . The quasi-canonical extension of the filtered de Rham bundle of the locally constant system R n f * (C Z 0 ) gives rise to a locally free filtered logarithmic de Rham bundle ) and the graded Higgs bundle The Galois group of g acts on the local system hence on the Higgs bundle. Let (E, θ) 1 denote 1st eigen sub Higgs bundle. We now make the comparison between the deformation Higgs bundle (F, τ ) for f : X → Y with (E, θ) 1 ⊗ A −1 . Just remember that the cyclic cover π : Z → W is defined by taking the ν-root out of the divisor δ * D ∈ |(δ * L ⊗ h * A −1 ) ν |. Therefore (E, θ) 1 ⊗ A −1 can be computed by taking higher direct image R • h * of the tautological exact sequence of the log smooth map On the other hand, the Higgs bundle can be computed by taking higher direct images R • f * of the tautological exact sequence of the log smooth map f : (X, Δ) → (Y, S) twisted with L −1 . The Hurwitz formula for the pullback of logarithmic differential forms via δ : W → X induces an inclusion of the short exact sequence δ * (I I ) → (I ), which induces a map between higher direct images ρ p,q : F p,q 0 → E p,q 0 and commuting with the edge morphisms.
• Comparison map by taking Kawamata cover. The construction here is similar to the previous case. Fixing an ample line bundle then L ν ⊗ A −1 is big for ν 0. Instead of taking self fibre product to higher power we take a Kawamata cover ψ : Y → Y so that ψ * A = A ν for an ample line bundle on Y . Let f : X → Y denote the fibre product of the base change and L = f * Ω n X /Y (log Δ ) then (L ⊗ A −1 ) ν ψ * (L ν ⊗ A −1 ) is big. By taking the ν-th cyclic cover π : Z → X of a section s of (Ω n X /Y (log Δ ) ⊗ f * A −1 ) ν we get the comparison map by the same type of the construction as before.

Introducing the Notion of the Length of a Family and a Criterion for the Rigidity
Given a family f : X → Y of varieties of dimension n with of semi-ample canonical line bundle and such that the moduli map into the moduli space is quasi finite. Recall the i-th iteration of the Kodaira-Spencer map For i = n (dimension of the fibres) one obtains a coupling If f is a family of Calabi-Yau manifolds or in general replacing the deformation Higgs bundle by the graded Higgs bundle of the VHS of the middle cohomology of the family then τ n coincides with the Yukawa coupling or in general Griffiths-Yukawa coupling.

Definition 4
We introduce the length of the iteration of Kodaira-Spencer map of the family f : X → Y to be is called the length of the iteration under a comparison map ρ of the second version. It is clear that ζ(ρ) ≤ ζ(f ) and rank A ρ ≤ h ζ(ρ) (X y , T ζ(ρ) X y ). Proposition 6 (Viehweg-Zuo [29]) If there exists a generically finite map from a product

Non-rigid Families and Products Base
This proposition suggests that ζ(f ) looks like the rank of a locally bounded symmetric space. Indeed, applying Proposition 6 to the universal family of polarized abelian g-folds one shows that the maximal number of subvarieties in a product inĀ g is equal to g. It is well-known that there exists products of g copies of modular curves as the so-called polydisc embedding inĀ g parameterizing abelian varieties isogeny to products of g-copies of elliptic curves. Here g is defined as the rank of the group Sp(2g, R).

Proposition 7
1. If there exists a generically finite morphism from a product Z ζ(f ) (log S ζ(f ) ) (see the proof for Corollary 6.4 in [29] and to be more precise we shall work on the Kawamata cover ψ ρ : Y → Y ). Restricting the above map, for example, to the factor Z 1 passing through the generic point, then A Z 1 is generically ample and from the inclusion map We obtain a non-zero map α : Claim 1 α is generically surjective.
Proof of Claim 1 Consider the non-zero sub sheaf α( 2. For such a family f : X → Y of surfaces we have then the rank ζ(f ) = 1 or 2.

Claim 2
The length ζ(f W ) of the family f W is equal to 2.

Proof of Claim 2
If ζ(f W ) = 1, then the maximal non-zero iteration of Kodaria-Spncer map of the family f W has length 1 and induces a non-zero map Since ζ(f W ) = 2 we obtain a non-zero map Since A Z×T is generically ample, the restriction of S 2 Ω 1 Z (log S Z ) to the T -factor and the restriction of S 2 Ω 1 T (log S T ) to the Z-factor are trivial sheaves, therefor the map τ 2 factors through τ 2 : A Z×T → Ω 1 Z (log S Z ) Ω 1 T (log S T ). Applying Bogomolov lemma to the restriction of τ 2 to a generic Z-factor Assume f : X → Y in Proposition 7 is a family of Calabi-Yau manifolds or hypersurfaces in P N of degree d ≥ N + 1 [31] shows a much more stronger result: Proposition 8 Let f : V → U = U 1 × · · · × U l be a smooth family of Calabi-Yau m-folds or a normalized family of hypersurfaces in P N of degree d ≥ N + 1 (admitting a particle good compactification f : X → Y with Y = Y 1 × · · · × Y l removed a codimension-2 sub scheme) and let the corresponding moduli map be generically finite. Then the natural map induced by k-th iteration of the Kodaira-Spencer map on the deformation Higgs bundle is injective. In particular, Propositions 6, 7 and 8 support the following conjecture.

Characteristic Varieties Attached to a Family
We would like to introduce a finer infinitesimal invariant to characterize products in (Y, S).
In the study of the metric rigidity problem on a locally bounded symmetric domain D, N. Mok [20] has introduced the so-called characteristic bundles {S i } in the projective tangent bundle of D using a maximal set of strongly orthogonal positive non-compact roots in a Cartan subalgebra h of the Lie algebra of the Lie group of Hermitian type. Note that S 1 consists of those tangents with minimal holomorphic sectional curvature. Motivated by Mok's work Sheng-Zuo [23] have introduced the notion of characteristic varieties for PVHS of Calabi-Yau type using iterations of Kodaira-Spencer map on the graded Higgs bundle. For the case of universal families of abelian varieties over Shimura varieties, the characteristic varieties coincides with Mok's characteristic bundles over Shimura varieties. Sheng, Xu and Zuo [24] and Robles [22] have found further applications in characterization of Gross's canonical variations of Hodge structure of Calabi-Yau type over bounded symmetric tube domains. Motivated by the above construction, we introduce characteristic varieties attached to a family f : X → Y of n-dim varieties with semi-ample canonical line bundle along the fibres and with the maximal variation as follows.
Assume the canonical line bundle along the fibres is ample: by taking k-th iterated Kodaira-Spencer map on the deformation Higgs bundle we define the k-th characteristic variety S k ⊂ P(T Y (− log S)) fibrewise by S k y = v ∈ T Y (− log S)| y : τ k+1 (v ⊗(k+1) ) = 0 .
For the case the canonical line bundle is semi-ample along the fibres we perform this construction via a comparison map over a Kawamata cover. In this way we obtain an increasing filtration of subvarieties in the projective bundle of log tangent bundle over Y ∅ = S 0 ⊂ S 1 ⊂ · · · ⊂ S k ⊂ S k+1 ⊂ · · · ⊂ S ζ(f )−1 ⊂ S ζ(f ) = P(T Y (− log S)).
The characteristic varieties provide an infinitesimal characterization of products of subvarieties in (Y, S) in the following sense.

Proposition 9
If there is a generically finite map φ from the log pair l i=1 (Z i , S i ) into (Y, S) then each factor Z i lies in leaves of the distribution defined by S ζ(f )−l+1 , i.e.
Proof We take the map restricted to the factor the l−1 i=1 (Z i , S i ), then the family pulled back to l−1 i=1 (Z i , S i ) is not rigid. By Corollary 6.5 in [29] P for some d 1 ≥ 1. Repeating the argument for further (l − 2) times we finish the proof.
Motivated by the above proposition we introduce the notion on special sub varieties in (Y, S).

Conjecture 4
Given a family f : X → Y 1 × · · · × Y l of canonically polarized varieties over a product base. Then the family is dominated by a finite map from the product of l families f i : X i → Y i , 1 ≤ i ≤ l.
Faltings [9] found examples of families of abelian varieties such that Conjecture 4 does not hold true. However, the VHS attached to those families does decompose over a number field. If the canonical line bundle of the fibres is ample one hopes the decomposition of the base induces a decomposition on the fibers.