Hodge sheaves underlying flat projective families

We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms for smooth maps and their functorial properties.


Introduction and main results
In a series of seminal works [Gri68a], [Gri68b], and [Gri70], Griffiths established that a degeneration of polarized Hodge structures (of fixed weight) in a smooth projective family f : X → B induces (i) a flat bundle (V, ∇) on B, equipped with a (ii) system of Hodge bundles (E , θ), and a (iii) natural analytic data defined by a harmonic metric.
Following this discovery, these fundamental results were later successfully developed further in two major new directions.Through nonabelian Hodge theory, Simpson [Sim92] and Mochizuki [Moc06] established topological characterizations of (i), (ii), and (iii), regardless of a geometric origin (a smooth projective family).In a different direction by replacing (i) and the Hodge filtration by filtered holonomic D-modules, Hodge modules were introduced by Saito [Sai90] as a generalization of variations of Hodge structures (VHS for short) for non-smooth families.None of these two general theories will be used in this paper.
For smooth projective families we know that the direct summands of E are represented by the cohomology of sheaves of relative Kähler forms; an algebro-geometric datum.From a geometric point of view the existence of (ii) for non-smooth families and one that is similarly of algebro-geometric origin is of special interest, 1 cf.1.B.
Our first goal in this paper is to establish that in fact any flat family of projective varieties gives rise to systems of Hodge sheaves (with no poles), as soon as the base of the family is smooth.Moreover, we will see that, similar to the smooth case, they arise from cohomology of objects-in the derived category-that play the role of relative Kähler forms for non-smooth families, cf.Section 4.
Theorem 1.1.Let f : X → B be a flat projective morphism of reduced complex schemes with connected fibers, where B is a smooth complex variety.Further let w ∈ N, 0 ≤ w ≤ dim(X/B).Then, there exists a functorial system of reflexive Hodge sheaves (E = w i=0 E i , θ), θ : B ⊗ E i+1 , of weight w on B. If in addition X has only rational singularities and w = dim(X/B), then E 0 ≃ (f * ω X/B ) * * .Remark 1.1.1.See Definition 4.3 and the subsequent paragraph for the definition of a system of reflexive Hodges sheaves and Subsection 1.A and Theorem 1.3 for the functoriality of such a system.Our next goal is to compare these Hodge sheaves to the logarithmic system (E 0 , θ 0 ) underlying the Deligne canonical extension [Del70, I.5.4]V 0 of integral variation of Hodge structures of weight w for a smooth model.Here we are following the standard parabolic notation for extensions of V.That is, for a tuple β = (β i ) i of real numbers β i , j : B \ D f → B is the inclusion map and D f = D i f is the discriminant locus of f (which is assumed to be normal crossing) is defined as follows.The sequence of holomorphic bundles V β is the decreasing filtration of j * V, defined by the lattice with respect to which res(∇)| D i f has eigenvalues in [β i , β i+1 ).Throughout this paper [β i , β i+1 ) is fixed to be equal for all i.More precisely, given a suitable resolution π : X → X and the resulting family f : X → B, we show that there is a nonnegative integer a f , that encodes how singular the family f is and measures the difference between E and E 0 , where E 0 is the Deligne extension of the integral VHS associated to the smooth locus of f .Theorem 1.2.In the setting of Theorem 1.1, let π : X → X be a good resolution with respect to f , and denote the resulting morphism by f : X → B. Further let D f denote the divisorial part of the discriminant locus of f , and assume that it is an snc divisor on B. (This can be achieved by base changing to an embedded resolution over B.) Then, there exists an integer for which we have an inclusion of systems of equal weights This isomorphism is in the category of smooth bundles.
Here, a good resolution with respect to f means a desingularization π for which f * D f has simple normal crossing (snc) support.For a more detailed and precise definition see Definition 2.3.In Theorem 1.2 and in the rest of this article (E 0 , θ 0 )(a f •D f ) denotes the naturally induced system of Hodge sheaves defined by The integer a f will be called the discrepancy of f with respect to π : X → X.Note that a f can be interpreted as a measure of degeneration in the family; the smaller this integer, the milder the singularity of the degeneration.In particular when f is smooth, we have a f = 0. 1.A. Functoriality.An important feature of the construction of (E , θ) in Theorem 1.1 is its functoriality.More precisely, one can consider a category Fam(n, d) of morphisms f : X → B as in Theorem 1.1, where dim(X) = n and dim(B) = d, and a category Hodge(d, w) of systems of Hodge sheaves of weight w (see 4.C for the precise definitions).The system (E , θ) in Theorem 1.1 then gives rise to a functor between these two categories.
Theorem 1.3.Let n, d, w ∈ N.There exists a functor χ w : Fam(n, d) → Hodge(d, w) defined by χ w (f : X → B) = (B, E , θ), where (E , θ) is the system in Theorem 1.1.Furthermore, for (f : X → B) ∈ Ob(Fam(n, d)) and any open subset V ⊆ B, we have 1.B.Singular families of varieties with base schemes of general type.Viehweg conjectured that for a projective morphism f : X → B of smooth projective varieties X and B with connected fibers and D denoting the (divisorial part of) the discriminant locus of f , if f has maximal variation, and its smooth fibers are canonically polarized, then (B, D) is of log general type, i.e., ω B (D) is big.
In higher dimensions, the minimal model program taught us that when positivity of canonical sheaves are involved, it is desirable to try to extend results to mildly singular cases.So, it is natural to ask whether Viehweg's conjecture extends to families of minimal models.The simple answer is that the desired positivity fails already, if one allows Gorenstein terminal singularities, arguably the mildest possible.In particular, the conjecture fails for Lefschetz pencils, cf.1.D.1.
This could be interpreted as a sign that there is no reasonable generalization of Viehweg's conjecture to singular families.However, here we offer a potential way to remedy the situation.Before we state that generalization, first recall that the initial step in the proof of essentially any result connected to Viehweg's conjecture has been a related result (which itself was a culmination of work of Fujita, Kawamata, Kollár, and Viehweg), which states that if a family of varieties of general type has maximal variation, then the line bundle det f * ω m X/B is big, i.e., has maximal Kodaira dimension.Reformulating Viehweg's conjecture in terms of the bigness of this line bundle has the advantage that it allows one to remove the condition that the fibers would be canonically polarized or even of general type.So, by including this initial step of the proof in the conjecture itself one may rephrase Viehweg's conjecture in terms of det f * ω m X/B being big, instead of requiring maximal variation and that the fibers be of general type.This formulation allows one to quantify (to some extent) the starting assumption for singular families and ask that not only det f * ω m X/B be big, but that it should be big compared to something else.
As an application of Theorem 1.2, we show that for Gorenstein families it is possible to obtain a result similar to Viehweg's conjecture along the lines outlined above.This requires that we take into account how singular the family is.More precisely, we show that if det f * ω m X/B is positive enough to balance the discrepancy of the family (discussed above), then the base of the family is indeed necessarily of (log) general type.
Theorem 1.4.Let X and B be projective varieties and f : X → B a flat family of geometrically integral varieties with only Gorenstein Du Bois singularities, such that B is smooth and the generic fiber of f has rational singularities.Further let D, D ′ ⊂ B be effective divisors such that Remark 1.5.Observe that this theorem includes Viehweg's conjecture: Assuming that the n-dimensional variety X is smooth and taking D = 0.This also shows that this statement is stronger than Viehweg's conjecture even in the original situation.Viehweg's conjecture predicted that maximal variation of the family implies that the base is of log general type with respect to the boundary divisor chosen to be the codimension one part of the discrepancy locus.Theorem 1.4 says that this can be improved: any divisor D that's part of the discrepancy locus and has the property that (det f * ω m X/B )(−mr m n • D) is big may be subtracted from the boundary divisor.In other words, if the pushforward of a pluricanonical sheaf is "bigger" than any part of the discrepancy locus, then one obtains that the base is of log general type with a smaller boundary divisor.In the extreme case that (det f * ω m X/B )(−mr m n • D f ) is big, this means that the base itself has to be of general type.This strengthening of Viehweg's conjecture is new even in the case when X is smooth, but in Theorem 1.4 we actually allow a singular X.This result could also be used in a reverse way to give a lower bound on discrepancy divisors of some families, or the discrepancy divisor of any of their resolutins.For instance, one obtains a bound for the notorious Lefschetz pencils.
Notice further, that in Theorem 1.4 there is no assumption on the Kodaira dimension of the fibers, which is another way this result is much more general than Viehweg's original conjecture.
On the other end of the spectrum, Theorem 1.4 implies that for every flat rational Gorenstein family we have the following implication: family, with X Gorenstein and B smooth, such that the general fiber of f has rational singularities, then f satisfies the assumption on the singularities in Theorem 1.4 by [KK10b] and hence Theorem 1.4 applies to such families.1.C.Hyperfiltered logarithmic forms in the derived category.Inspired by the works of Katz-Oda [KO68] our construction of (E , θ) in Theorem 1.1 fundamentally depends on the existence of a filtration, or more precisely the Koszul filtration, that is naturally available for Kähler forms of smooth families.In the absence of such objects with analogous properties for singular families, we pass on to the derived category D b (X), where an appropriate hyperfiltration F (in the derived sense, see Definition 2.2) was constructed in [Kov05] and applied to the complex of Deligne-Du Bois forms, which are objects in the bounded derived category of coherent sheaves of X.These objects have similar cohomological properties to the sheaves Ω p X in the smooth case (see Definition 3.1).For more details regarding the complexes of Deligne-Du Bois forms see Section 3, [DB81], [GNPP88], [Kov97, 3.1], [KS11,§4], and [PS08, 7.3.1].
For smooth projective families, through the Hodge-to-de Rham spectral sequence degeneration, one uses holomorphicity and transversality properties of ∇ to extract an underlying system of Hodge bundles.When f is singular, in the absence of such tools, including a filtered relative de Rham complex satisfying good degeneration properties, analogous results cannot be similarly established by the same methods.
To circumvent this difficulty we construct the complex of logarithmic Du Bois p-forms Ω p X (log ∆) ∈ Ob D b (X), which can be endowed with the structure of a Koszul-type hyperfiltration F f using the construction in [Kov05].Moreover, we show that for a morphism of snc pairs f : (X, ∆) , where F q K is the usual Koszul filtration.See Theorem 3.3 for details.In Section 3, we show that this hyperfiltration is functorial, and using this functoriality we establish a natural filtered pullback map from Ω p X to Ω p X (log ∆), twisted with a wellunderstood line bundle that encodes the singularity of the family f in terms of a f (the discrepancy of f with respect to π : X → X).On the other hand, (E 0 , θ 0 ) is determined by (Ω p X (log ∆), F q K ) by [Ste75] and [KO68].Now, the fact that, for each 0 ≤ p ≤ dim X/B, the two filtered objects Ω p X and Ω p X (log ∆) are functorially related then leads to the formation of (E , θ) compatible with (E 0 , θ 0 ) (in the sense of (1.2.2)), endowing the former with the structure of a system of Hodge sheaves.1.D. Singularities of Higgs fields underlying VHSs of geometric origin.The Gauss-Manin connection ∇ arising from a smooth projective family extends to V 0 with only logarithmic poles due to its integrability, as shown by Manin [Man65] and Deligne [Del70, I.5.4].However, in general such flat connections do not have trivial local monodromy at infinity and thus their singularities are often not removable.On the other hand, for a polarized VHS over a punctured polydisk with unipotent monodromy, the Hodge filtration extends to a holomorphic filtration of V 0 by Schmid's Nilpotent Orbit Theorem [Sch73] and the results of Cattani-Kaplan-Schmid [CKS86].It follows that the poles of (E 0 , θ 0 ), as a Higgs bundle, are at worst logarithmic.In fact, at least over a smooth quasi-projective variety, and for a suitable choice of extension, the same is true for all tame harmonic bundles [Moc07, 22.1].As a direct consequence of Theorem 1.1 and Theorem 1.3 we can show that there is always an extension of the Higgs bundle (E , θ) underlying (V, ∇) with zero residues2 .In other words, θ has removable singularities.We make this point more precise in the following remark.
Remark 1.6.In the setting of Theorem 1.1, further assume that X is smooth.Let D f denote the divisorial part of the discriminant locus of f and assume that D f and f * D f have simple normal crossing support.Then, for any fixed weight, the system (E , θ) in Theorem 1.1 is an extension of the Hodge bundle of the same weight underlying the VHS of the smooth locus of f , that is (E , θ)| B\D f ∼ = (E , θ).
In the context of Remark 1.6, we call (E , θ) a derived extension.We note that the inclusion (E , θ)(−a f • D f ) ⊆ (E 0 , θ 0 ) guarantees that there is always a subextension of (E 0 , θ 0 ) with vanishing residues.
Remark 1.6 can be interpreted as providing an analytic criterion for detecting when a VHS is not of geometric origin (and similarly for a complex VHS in the sense of [Sim90,p. 868]).
Corollary 1.7.Let B be a smooth complex variety, D ⊆ B a simple normal crossing divisor, and (V, ∇, E = E i , θ) an abstract real VHS on B \ D. If the given VHS is of geometric origin, then the singularity of θ is removable, i.e., there exists a reflexive Hodge sheaf (E ′ , θ ′ ) on B, with θ ′ : 1.D.1.Order of poles for Lefschetz pencils.We emphasize that Deligne extensions (or their underlying Hodge bundle) have logarithmic poles even in the case of very mild degenerations such as Lefschetz pencils of non-hyperelliptic curves (a particular instance of a stable family of curves).To see this, one may use the following observation.Note that f * ω X/P 1 is the first graded piece of the Hodge sheaves underlying the Deligne extension where f • denotes the smooth locus of f .By the weak positivity of f * ω X/P 1 (see for example [Vie95]) we know that every rank-one direct summand L j in the splitting f * ω X/P 1 ∼ = L j is nef.Over the smooth locus of f , the Higgs field θ 0 is locally equal to the derivative of the period map, so by the local Torelli theorem θ 0 = 0. Therefore, we have θ 0 (L j ) = 0, for some j.Now if θ 0 had no poles, by applying θ 0 to L j and using the weak negativity3 of the kernel of θ 0 , cf. [Gri84], [Sim90] or [Zuo00], we would get an induced injection It follows, again from the weak negativity of E 0 2 , that after taking determinants there is an injection 2 ) and (det E 0 2 ) −1 is nef.But this is absurd, showing that indeed θ 0 must have poles.
1.E.Acknowledgments.The second named author owes a special debt of gratitude to Gábor Székelyhidi for his support and encouragements.We are grateful to Erwan Rousseau for his help during our joint visit to CIRM.We would also like to thank Geordie Williamson and the referee for helpful comments and suggestions.

Preliminary definitions and notation
2.A. Families of pairs.The study of pairs or log varieties have led to many advances in birational geometry and moduli theory.For the questions investigated here a simple version of pairs will suffice, namely we will restrict to the case when the boundary divisor is reduced.
Definition 2.1.A reduced pair (X, ∆) consists of a normal scheme X and an effective reduced divisor ∆ ⊂ X.An snc pair is a reduced pair (X, ∆) such that X is smooth and ∆ is an snc divisor.A morphism of (reduced) pairs f : (X, ∆) → (B, D) is a morphism f : X → B of normal schemes such that supp ∆ ⊇ f −1 (supp D).Assuming that D is Q-Cartier, we will use the notation f −1 D : = (f * D) red to denote the reduced preimage of D. Using this notation the above criterion can be replaced by ∆ ≥ f −1 D. A morphism of snc pairs is a morphism of reduced pairs f : (X, ∆) → (B, D) such that both (X, ∆) and (B, D) are snc pairs.Consider a morphism of reduced pairs f : (X, ∆) → (B, D) and a decomposition ∆ = ∆ v + ∆ h into vertical and horizontal parts, i.e., such that codim B f (∆ v ) ≥ 1 and that f | ∆0 dominates B, for any irreducible component ∆ 0 ⊆ ∆ h .Using this decomposition, we call a morphism of snc pairs f : (X, ∆) The composition of two (snc) morphisms of pairs is also a (snc) morphism of pairs.Further note that the term "morphism of pairs" does not have a standard usage and it may be used to refer to a somewhat different situation by other authors.We added the extra word "reduced" to remind the reader of this potential difference.We are still not claiming that this definition is standard.We believe that an established standard usage of this phrase does not exist at this time.
Definition 2.2.Let f : (X, ∆) → (B, D) be an snc morphism.Then, after removing a codimension 2 subset of B, there exists a short exact sequence of locally free sheaves, For each 0 ≤ q ≤ dim(X/B) this induces a descending filtration, called the Koszul filtration and denoted by F q K Ω q X (log ∆) such that the associated graded quotients of the filtration satisfy that (2.2.1) The reader is referred to [GNPP88] for the definition of simplicial and cubic schemes.In this paper a hyperresolution will mean a cubic scheme, all of whose entries are smooth schemes of finite type over C. Definition 2.3.Let (X, ∆) be a reduced pair.A good resolution (or log resolution) of (X, ∆), is a proper birational morphism of pairs g : (Y, Γ) → (X, ∆) such that X is quasiprojective, the exceptional set E : = Ex(g) of g is a divisor, Γ = g −1 * ∆ + E and (Y, Γ) is an snc pair.
Let (X, ∆) be a reduced pair and f : X → B a morphism.A good resolution of (X, ∆) with respect to f is a good resolution g : (Y, Γ) → (X, ∆) such that in addition to the above, Γ + g −1 * D is also an snc divisor where D is the divisorial part of the discriminant locus of f • g.This can be constructed the following way: let g 0 : (Y 0 , Γ 0 ) → (X, ∆) be a (good) resolution of (X, ∆) and let D 0 ⊆ B denote the divisorial part of the discriminant locus of f 0 : = f • g 0 , i.e., the smallest effective reduced divisor Note that if ∆ = ∅, then we will often drop Γ from the notation and just say that g : Y → X is a good resolution (with respect to f ).
A good hyperresolution of (X, ∆), denoted by ε q : (X q, ∆ q) → (X, ∆) consists of a hyperresolution ε q : X q → X such that for each i ∈ N, dim X i ≤ dim X − i and for ∆ q : = X q \ (X q × X (X \ ∆)), either ∆ i is an snc divisor on X i , or ∆ i = X i .

2.B. Hyperfiltrations and spectral sequences.
Let A and B be abelian categories and D(A) and D(B) their derived categories respectively.Let Φ : A → B be a left exact additive functor and assume that RΦ : D(A) → D(B), the right derived functor of Φ exists.
) for j = l, . . ., k + 1, for some l, k ∈ Z and morphisms where F l K ≃ K and F k+1 K ≃ 0. F j K will be denoted by F j when no confusion is likely.
For convenience let F i K = K for i < l and F i K = 0 for i > k.The p-th associated graded complex of a hyperfiltration F q K is Gr p F q K := M (ϕ p ), the mapping cone of the morphism ϕ p .
Let F q A be a hyperfiltration of the object A and Ξ : D(A) → D(B) a functor.Then, there is a natural hyperfiltration of Ξ(A) given by for each j ∈ Z.We will always consider the object Ξ(A) with this natural hyperfiltration, unless otherwise specified.
Next let F q A and F q B be hyperfiltrations of the objects A and B of D(A) respectively.Then, a (hyper)filtered morphism4 between A and B is a collection of compatible morphisms F j α : F j A → F j B, i.e. for each j ∈ Z, the diagrams are commutative in D(A).Notice that in this case these morphisms induce a morphism α j : Gr j F q A → Gr j F q B , for each j ∈ Z.A filtered morphism α : A → B is a filtered quasi-isomorphism if the induced morphism α j : Gr j F q A ≃ −→ Gr j F q B is an isomorphism for each j ∈ Z.It is easy to see, and left to the reader, that a filtered quasi-isomorphism (of bounded complexes) is necessarily a quasiisomorphism.
Example 2.5.Let A ∈ C(A) be a complex of objects of the abelian category A and let defines a hyperfiltration of the object A.

Relative Du Bois complexes of p-forms
Our aim in this section is to construct, for all flat morphisms to regular base schemes, an analogue of relative logarithmic p-forms for morphisms of snc pairs.To do so, following the construction in [Kov05] (reviewed in Section 6), we will work in the derived category D b (X).We use the notation Ω p X/B (log ∆/D) to denote this object for a morphism of pairs f : (X, ∆) → (B, D).The "D" is included in the notation to emphasize the fact that the construction depends on D as well.
With hyperfiltrations playing a role here, similar to that of filtrations in an abelian category, our first goal is to use these objects to construct a functorial filtration of Ω p X (log ∆) (Theorem 3.3).Our next goal is to establish a connection between Ω p X and Ω p X (log ∆), as hyperfiltered objects (Theorem 3.7).This is where the notion of discrepancy (as was mentioned in the introduction) naturally appears.Our final goal in this section is to extend these relations to distinguished triangles arising from such hyperfiltrations (Corollary 3.11).The latter is of particular interest in the context of VHSs, as we will see in Section 4. We will use the terminology, notation and conventions developed in Section 2. Definition 3.1.Let (X, ∆) be a reduced pair (Definition 2.1) and ε q : (X q, ∆ q) → (X, ∆) a good hyperresolution (Definition 2.3).The logarithmic Deligne-Du Bois complex (or logarithmic DB complex for short) of (X, ∆) is defined as Ω q X (log ∆) : = R (ε q) * Ω q X q (log ∆ q).This is an object in the bounded filtered derived category of coherent sheaves on X, and the corresponding filtration (induced by the filtration bête on each component of X q) is denoted by F q DB : = F q DB Ω q X (log ∆).Both the obeject and this filtration is independent from the good hyperresolution used in the definition.The associated graded objects of this filtration give rise to the complexes of logarithmic DB p-forms: . The reader is referred to [GNPP88, IV.2.1] for details on this definition and basic properties of these complexes.
We will construct relative versions of these complexes, but first we need a notation.Definition 3.2.Let f : (X, ∆) → (B, D) be a morphism of snc pairs, Φ X : = f * Ω 1 B (log D), Ψ X : = Ω 1 X (log ∆), and θ X : Φ X → Ψ X the natural morphism induced by f .Using the notation of Section 6 (cf.[Kov05]), set F q f Ω q X/B (log ∆) : = F q p Ψ X and define Ω p X/B (log ∆/D) : = Q p θX , where Q p θX is the object constructed in Theorem 6.9 (cf.[Kov05, 2.7]).Next, let f : (X, ∆) → (B, D) be a morphism of pairs and assume that (B, D) is an snc pair.I.e., do not assume that (X, ∆) is snc.Let ε q : (X q, ∆ q) → (X, ∆) be a good hyperresolution.Then, as in Definition 3.1, the logarithmic Deligne-Du Bois complex of (X, ∆) is defined as Ω q X (log ∆) : = R (ε q) * Ω q X q (log ∆ q).Using this representative define a filtration as follows: Let n = dim X, d = dim B, and for each 0 ≤ p, q ≤ dim(X/B) = n − d, and 0 X) will be called the p th -relative logarithmic Deligne-Du Bois complex of f : (X, ∆) → (B, D) or simply the complex of relative logarithmic DB p-forms of f .
Next, we will prove that these objects are well-defined and satisfy a list of useful properties.
Theorem 3.3.Let f : (X, ∆) → (B, D) be a morphism of pairs and assume that (B, D) is an snc pair.Let n = dim X and d = dim B. Then, for each 0 ≤ p, q ≤ dim(X/B) = n − d, the objects Ω p X/B (log ∆/D) ∈ Ob D b (X) and a the hyperfiltration F q f Ω q X (log ∆) satisfy the following properties.
(i) The object Ω p X/B (log ∆/D) ∈ Ob D b (X) is independent from the good hyperresolution used in its definition.In other words, any two objects defined as in Definition 3.2 using possibly different good hyperresolutions are isomorphic in D b (X).
(ii) F 0 f Ω q X (log ∆) = Ω q X (log ∆), and F d+1 f Ω q X (log ∆) = 0. (iii) Let φ : ( X, ∆) → (X, ∆) be a log resolution.Then (v) The hyperfiltration F q f Ω q X (log ∆) is functorial in the following sense.Let g : (Y, Γ) → (X, ∆) be a morphism of pairs such that dim Y = dim X = n.Then, for each 0 ≤ q ≤ n − d, there exists a natural filtered morphism in D b (X) where F q K is the Koszul filtration (Definition 2.2).Remark 3.4.When ∆ and D are empty, we will suppress the "log" term from the notation.In particular, we will use the notation and Ω p X/B : = Ω p X/B (log ∅/∅).
Notation 3.5.To avoid cumbersome notation, as in Theorem 3.3(iv), we will use Gr p f to denote Gr p F q f , where F q f is the hyperfiltration F Proof of Theorem 3.3.First, assume in addition that (X, ∆) is also an snc pair.Then the statements (ii) and (iv) follow from Theorem 6.9, and (v) follows from [Kov05, 4.1].For (iii), first observe that both sides are independent of the choice of φ.This follows from [DB81, 6.3] for the left hand side and from [Kov11, 2.10] (cf.[Kol13, 10.34]) for the right hand side.In particular, in the snc case we may use φ = id, and in that case (iii) follows from Definition 6.8.Next, let (X, ∆) be arbitrary and let ε q : (X q, ∆ q) → (X, ∆) be a good hyperresolution.Using Definition 3.2, (ii), (iii), (iv), and (v) follow from the snc case above: (ii) follows directly, (iii) follows from the snc case, the definition of a good hyperresolution, Definition 2.3, and (3.2.2).Item (v) follows by the functoriality of the snc case.For (iv), further note that as (B, D) is an snc pair, f * Ω q−j B (log D) is locally free, so one can use the projection formula.
Statement (vi) follows by a descending induction on p.The induction can be started by (ii) and the inductive step follows from (iv) and (v).Indeed, choose a good hyperresolution µ q : (Y q, Γ q) → (Y, Γ), which is compatible with the chosen good hyperresolution of (X, ∆), i.e., there is a commutative diagram (Y q, Γ q) µ q g q / / (X q, ∆ q) Then, the following diagram is commutative by (v): (3.5.1) and hence induces a compatible natural morphism Gr This finishes the proof of (vi), and then (vii) follows from the construction of Q p θX carried out in this case (cf.Section 6, especially Definition 6.8, and [Kov05, §2]).The main point is that the cokernel of the morphism ) is locally free and hence its exterior powers satisfy the required properties, cf.Proposition 6.10.In fact, the construction outlined in Section 6 was modeled after this case.
Finally, to prove (i), observe that we have just proved that the other properties stated in the theorem hold for the corresponding object defined by any good hyperresolution of (X, ∆).For any two good hyperresolution there exists a third that maps to and is compatible with both of the others, so it is enough to prove (i) for two such good hyperresolutions.Then the proofs of (v) and (vi) show that there is a natural filtered morphism between the two objects defined by the two good hyperresolutions.It follows from (iii) that the induced morphism is an isomorphism for p = n − d and then descending induction using the commutative diagram (3.5.1) shows that (i) holds for all p.This finishes the proof of all the claims in the theorem.
Next we will compare these objects obtained with respect to different bases replacing (B, D).We will be using the standard ω B/B ′ : = ω B ⊗ τ * ω −1 B ′ notation.Theorem 3.6.Using the notation from Theorem 3.3, in addition let τ : (B, D) → (B ′ , D ′ ) be another morphism of pairs, such that (B ′ , D ′ ) is also an snc pair and τ is a dominant generically finite morphism.Let f ′ = τ • f and Γ : = D − τ * D ′ .Then for each 0 ≤ p, q ≤ dim(X/B) = n − d, and 0 ≤ j ≤ d = dim B = dim B ′ , (i) there exists a natural morphism (ii) there exists a natural morphism (iii) the natural morphisms in (i) and (ii) are compatible in the following sense.For each q and j there exists a commutative diagram of distinguished triangles, (iv) there exists a natural filtered morphism Proof.We will use the notation of Section 6.In particular, let Φ X , Φ ′ X and Ψ X be locally free sheaves on X of rank k, k ′ and n respectively, and let ̺ : Φ ′ X → Φ X and θ X : Φ X → Ψ X be two morphisms.Further let θ ′ Then ̺ induces a natural map between the filtration diagrams corresponding to the morphisms θ ′ X , θ X (the maps go from the ones associated to θ ′ X to those associated to θ X induced by the morphisms ∧ r ̺ : r Φ ′ X → r Φ X for various r).
In particular, let Φ X : = f * Ω 1 B (log D) and Φ ′ X : = (f ′ ) * Ω 1 B ′ (log D ′ ).Then for the objects F p i defined in 6.6 one obtains a natural morphism (3.6.1) ) , (3.6.3)(cf.Definition 6.8), and the fact that in this case we have ) we find that there exist natural morphisms This proves (i).The same argument, used for ) instead of the morphism in (3.6.1) and using the definition (3.6.4) (cf.Definition 6.8), gives which proves (ii).In fact, following this argument one is led to consider distinguished triangles of the form where the vertical arrows are induced by the morphism Finally, in order to prove (iv), first recall that F j+1 f Ω q X (log ∆) = 0 and F j+1 f ′ Ω q X (log ∆) = 0 by Theorem 3.3(ii), so we may assume that j < dim B = d.With that restriction, consider the morphism (3.6.5) This morphism allows us to add one more row to the above commutative diagram: Now, erasing the middle row gives us commutative diagrams, for each j, with the same line bundle multiplier in the last row.In other words this shows that there exists a natural filtered morphism as claimed in (iv) (cf.(3.6.5)).
Theorem 3.7.Using the notation from Theorem 3.3 and Theorem 3.6, we have that for each 0 ≤ p ≤ dim(X/B) there exists an integer 0 ≤ a p ≤ dim X − p, (i) a natural filtered morphism (ii) a natural morphism In addition, let g : (Y, Γ) → (X, ∆) be another morphism of pairs such that dim Y = dim X = n.Then there exist (iii) a natural filtered morphism (iv) a natural morphism Proof.Let D ′ : = ∅ and consider the morphism of pairs τ : (B, D) → (B, D ′ ).Observe that ω B/B ′ ≃ O B and Γ = D, hence (i) follows from Theorem 3.6(iv).Then (ii) follows from (i) and Theorem 3.3(iv).The required natural filtered morphisms in (iii) is simply the composition of the natural filtered morphisms in (i) and Theorem 3.3(v) (more precisely, the latter is twisted with the line bundle O X (a p •f * D)).Finally, (iv) follows from (ii) and Theorem 3.3(vi).
Definition 3.8.Let D ⊂ B be a reduced, effective divisor.The smallest non-negative integer a ∈ N for which a morphism as in Theorem 3.7(ii) for each 0 ≤ p ≤ dim(X/B) with the choice of a p = a ≤ dim X exists will be called the discrepancy of D with respect to f : X → B and will be denoted by a f (D).
3.9.Koszul triangles.Let f : (X, ∆) → (B, D) be a morphism of pairs and assume that (B, D) is an snc pair.Let n = dim X and d = dim B and 0 Let G 0,2 f denote the mapping cone of the morphism F 2 f Ω p X (log ∆) → F 0 f Ω p X (log ∆) and consider the commutative diagram of distinguished triangles, (3.9.1) Then the dotted arrows exist by [Nee91, Theorem 1.8] (cf.[Kov13, Theorem B.1]) and they maybe identified with the induced morphisms on the mapping cones.Therefore we obtain the distinguished triangle (3.9.2) in which each term is defined as the mapping cone of the vertical morphisms in (3.9.1) and the morphisms are the ones coming from the mapping cone construction.We will refer to this distinguished triangle in (3.9.2) as the p th -Koszul triangle of f : (X, ∆) → (B, D) and denote it by Kosz p f (log ∆).As before, in case ∆ = ∅ and D = ∅, then we will denote this by Kosz p X/B .Replacing Gr i f Ω p X (log ∆) for i = 0, 1 by isomorphic objects as in Theorem 3.3(iv) we obtain an alternative expression for Kosz p f (log ∆): (3.9.3) Remark 3.10.The nine lemma in triangulated categories is somewhat trickier than in abelian categories.It is not true that any morphism of triangles induce a distinguished triangle on their mapping cones.What [Nee91, Theorem 1.8] and [Kov13, Theorem B.1] state is that there exists a χ (see upper right side of (3.9.3)) such that the third row (of mapping cones) forms a distinguished triangle.In addition, it follows from [Kov13, Theorem B.1] that in the case of (3.9.1) the χ is in fact uniquely determined.This is, of course, not surprising given that the initial object of χ is 0, but one should remember that we are working in the derived category, so caution is warranted.We would also like to emphasize that we are not merely stating that a distinguished triangle exists with the given objects as in (3.9.2) and (3.9.3), but that the morphisms of the triangle are exactly the ones one would hope for, namely the morphisms induced by the mapping cone construction.In particular, this means that the Koszul triangles will inherit any natural property carried by the filtrations used in their definition.
Corollary 3.11.Using the above notation, let g : (Y, Γ) → (X, ∆) be another morphism of pairs such that dim Y = dim X = n.Then the morphisms κ p obtained in Theorem 3.7(iv) are compatible with Koszul triangles, that is, there exist natural compatible morphisms of the terms of the following Koszul triangles: (With a slight abuse of notation we will denote these morphisms of Koszul triangles by the same symbol).
Proof.Theorem 3.7 implies that there exist natural morphisms between the terms of the diagram (3.9.1) for f : (X, ∅) → (B, ∅) and for f g : (Y, Γ) → (B, D).It follows from Theorem 3.7(i) that these morphisms commute with the first two rows and all the columns.Then it follows that they also commute with the third row as well, which is exactly the desired statement.
The naturality of κ p follows from the naturality of the morphisms in Theorem 3.7 and the fact that the morphisms in Kosz p X/B are given by the mapping cone construction, as explained in Remark 3.10.Remark 3.12.One can slightly generalize Corollary 3.11 by considering The proof is the same as the one for Corollary 3.11.
Notation-Remark 3.13.For an snc morphism, f : (X, ∆) → (B, D), we use the notation Kosz p f (log ∆) to denote the triangle defined by the standard Koszul filtration F q K .Note that for such morphisms, using Theorem 3.3(vii), there is a natural isomorphism of triangles Kosz p f (log ∆) −→ Kosz p f (log ∆), defined explicitly by where the vertical quasi-isomorphims are the ones defined by Theorem 3.3(vii) (see also Remark 3.10).

Systems of Hodge sheaves and derived extensions
Our aim is now to use the complexes of relative logarithmic DB forms from Section 3 to construct systems of Hodge sheaves for arbitrary flat families.Since the proofs of Theorem 1.1 and Theorem 1.2 are interdependent, they will be presented together.First we need to introduce the notion of discrepancy for flat families that appears in the setting of Theorem 1.2.
Notation 4.1.Given a flat projective morphism g : X → B of regular schemes, we denote the reduced divisorial part of the discriminant locus disc(g) of g by D g .
Notation 4.2.Given a flat, projective family of schemes f : X → B with regular base B, let π : X → X be a log resolution of the pair and f : X → B the induced family.That is, assume that ∆ f := f −1 (D f ) is a divisor with normal crossings.By Theorem 3.7(ii) and Theorem 3.3(vi) (or Corollary 3.11), over the flat locus of f , there is a morphism (4.2.1) Following Definition 3.8, we use the notation a f,p to denote the smallest integer for which the morphism (4.2.1), with the choice of a p = a f ,p , exists.Furthermore, we use a f to denote the discrepancy a f (D f ) of D f with respect to f cf.Definition 3.8.
In the course of the proof of Theorem 1.1 and Theorem 1.2 it is helpful to differentiate between the various properties of systems of Hodge sheaves.To do so we introduce the following terminology.
Definition 4.3.Let W be an O B -module on a regular scheme B, w ∈ N. Then a W -valued system of weight w is a pair (E , τ ) where E is an O B -module and τ : E → W ⊗ E is sheaf homomorphism, such that there exists an O B -module splitting E = w i=0 E i with respect to which τ is Griffiths-transversal, that is, for every i = 0, . . ., w, τ : Using this terminology an Ω 1 B -valued system with an integrable and O B -linear map τ is a system of Hodge sheaves (of weight w).When E is reflexive, we call (E = E i , θ) a system of reflexive Hodge sheaves.

4.A.
Proof of Theorem 1.1 and Theorem 1.2.Let π : X → X be a good resolution with the induced map f : X → B. As introduced in Notation 4.2, a f denotes the discrepancy of the family with respect to f .Set m := n − d, where n = dim X and d = dim B. After removing a subscheme from B of codim B ≥ 2, defined by the complement of the flat locus of f , for every 0 ≤ i ≤ m, consider the map of distinguished triangles established in Corollary 3.11.By applying Rf * to (4.3.1)we find From the resulting cohomology sequence and the filtered quasi-isomorphism , and Notation-Remark 3.13 we find connecting homomorphisms τ i and Next, we define the two systems (F = By construction, the two systems (F , τ ) and (E 0 , θ 0 ) are Ω 1 B -valued and Ω 1 B (log It follows from (4.3.2) that there are sheaf morphisms ψ i : and let (G = G i , θ G ) denote its image.By extending the result of Katz-Oda [KO68] to the case of logarithmic relative de Rham complex and using [Ste75, 2.18] (cf.[Gri84, p.131]) one finds that (E 0 , θ 0 ) is the logarithmic system of Hodge bundles underlying the Deligne canonical extension of the flat bundle with ∇ denoting the Gauss-Manin connection.In particular θ a f is O B -linear and integrable.Consequently, so is θ G , that is (G , θ G ) is a system of Hodge sheaves.Furthermore, with (E a f , θ a f ) being locally free, the morphism ψ : The last part of Theorem 1.1 follows from the construction of (E , θ) and Theorem 3.3(iii).
More precisely, we have that . Furthermore, using the isomorphism π * ω X ≃ ω X , we find that As F 0 is torsion free, this implies that ψ 0 is injective.Therefore, F 0 can be identified with its image under ψ 0 , which we have denoted by G 0 .In particular we have G * * 0 ≃ F * * 0 ≃ (f * ω X/B ) * * .This completes the proof of Theorem 1.1 and Theorem 1.2.

4.B.
Explanation for Remark 1.6.Let f U : U → V be the smooth locus of f : X → B and i : U → X and j : V → B the natural inclusion maps.
Proof of Claim 4.4.This directly follows from flat base change and properties of complexes of relative DB forms.More precisely, we have , by the definitions of E l in (4.3.3) and ψ in (4.3.4) Thanks to [KO68] we know that the Higgs field underlying ∇ (the Gauss-Manin connection) is defined by the connecting homomorphisms of the long exact cohomology sequence associated to the right-hand-side of (4.4.1).On the other hand, by the construction of (E , θ), base change, and Claim 4.4, the left-hand-side of (4.4.1) similarly determines θ| V in (E , θ)| V .Therefore, we find that (E , θ) defines an extension of the Hodge bundle underlying ∇.
4.C.Functorial properties.The proof of Remark 1.6 already exhibits some of the functorial properties of the construction of (E , θ) in Theorem 1.1.This can be further formalized in the following way.Let Fam(n, d) be the category of projective surjective morphisms f : X → B with connected fibers between an n-dimensional reduced scheme X and a smooth quasi-projective scheme B of dimension d.A morphism (f ′ : 4.C.1.Proof of Theorem 1.3.This directly follows from the construction of Koszul triangles in 3.9 and the system (E , θ) in 4.A.More precisely, consider a log-resolution π : X → X as in Notation 4.2 .Similarly, let π ′ : X ′ → X ′ be a log-resolution factoring through the projection X × X X ′ → X ′ .By f ′ : X ′ → B ′ we denote the induced family.By construction, we have a commutative diagram of distinguished triangles The theorem now follows by applying Rf * to this diagram.
Remark 5.2.The above result remains valid if we replace nr by the discrepancy of D f with respect to f r .We opted to avoid a cumbersome notation, and instead use the upperbound rn, cf.Theorem 1.2.
Remark 5.3.One may also replace f * ω X/B in Proposition 5.1 by any of its subsheaves (and of course replace r with the corresponding rank).This is of interest for example in the setting of Fujita's Second Main Theorem (see [CD17] and references therein).
Before proving Proposition 5.1, we recall the following well-known fact regarding the functoriality of canonical extensions.
Let X r denote the rth fiber product X × B • • • × B X (r times) with the resulting morphism f r : X r → B. Now, as f is Gorenstein and flat we have Note, that X r has rational singularities by [KS16, Theorem E] (cf.[Elk78]).
Let n ′ := n−d.By slightly modifying F i in (4.3.3)we set , where ∆ ′ := f −1 D ′ .Since X r has only rational singularities, by Theorem 3.3(iii) we have * ω X r /B it follows that there is an injection (5.4.2) L −→ F 0 (−rnD).
To simplify our notation, we will replace f by f r in the sequel.Similar to (4.3.1)we have a morphism of triangles (Remark 3.12) whose weight is equal to the relative dimension.Denote the image of Proof of Claim 5.5.Aiming for a contradiction, assume that θ(L ) = 0. Let L denote the saturation of the image of the injection (5.4.2).Set C ⊆ B to be the smooth, complete intersection curve defined by H d−1 with the natural inclusion map γ : C → B. For a suitable choice of C we can ensure that C is in the locus of B over which E 0 /L is locally free.Let (E C , θ C ) be the logarithmic Hodge system defined in Fact 5.4.According to Fact 5.4 we have an injection (5.5.1) ).In particular we have an injection γ * L ֒−→ E 0 C,0 .From our initial assumption it follows that θ C,0 (γ * L ) = 0. On the other hand, since ker(θ 0 C | E 0 C,i ) is weakly negative by [Zuo00], this implies that deg(γ * L | C ) ≤ 0 and thus contradicting our initial assumption (5.4.1).This finishes the proof of the claim.Now, by applying θ to L we can find an integer k ≥ 1 such that where ⊗N , for some t ∈ N and pseudoeffective B := (det N k ) −1 .5.A.The general case.By using a cyclic covering construction (see also [VZ03], [Taj18] and [Taj20]), combined with the constructions in Section 4, we generalize Proposition 5.1 to the pluricanonical case.
Let Φ X and Ψ X be locally free sheaves on X of rank k and n respectively, and let θ X : Φ X → Ψ X be a morphism.
Let p, i ∈ N. We are going to define an object, ) .This will be done recursively, starting with i = 0 and then increasing i.Definition 6.3.The (p, 0)-filtration diagram of θ X is A 0-filtration morphism for some p, q, consists of locally free sheaves E , F and a morphism between Definition 6.4.The (p, 1)-filtration diagram of θ X consists of the predistinguished triangle, It is denoted by F p 1 .A 1-filtration morphism for some p, r, consists of locally free sheaves E , F and morphisms between the corresponding terms of F p 1 ⊗ E and F r 1 ⊗ F such that There exists a morphism, that makes the above diagram commutative.The diagram (6.4.1), combined with α gives a 1-filtration morphism Let Then there exists a distinguished triangle, Φ X +1 / / Definition 6.5.The (p, 2)-filtration diagram of θ X consists of the diagram, It is denoted by F p 2 .A 2-filtration morphism for some p, r, consists of locally free sheaves E , F and morphisms between the corresponding terms of F p 2 ⊗ E and F r 2 ⊗ F such that the resulting diagram is commutative.
More explicitly, the (p, 2)-filtration diagram of θ X is: Similarly, a 2-filtration morphism is: where the (p, 2)-filtration diagram, To define the (p, i)-filtration diagram of θ X and the i-filtration morphisms we will iterate this construction.6.6.Inductive Hypotheses.For a given i the following hold for each p, q, r ∈ N.
(i) The (p, i)-filtration diagram of θ X is defined and denoted by F p i .(ii) An i-filtration morphism, by definition, consists of locally free sheaves E , F and a morphism between the corresponding terms of F p i ⊗ E and F r i ⊗ F such that the resulting diagram is commutative.
(iii) F p i has a unique object, F p i , with only one adjacent arrow pointing out.(iv) F p i = 0 for p < i. (v) There exists an i-filtration morphism, θ,i q : F p i ⊗ det Φ X −→ F p−q i ⊗ k−q Φ X .
Proof.For the proof the reader is referred to [Kov05,2.5]. in Ob(D(X)).It follows that Q n−k θX = det Ψ X ⊗ (det Φ X ) −1 and that there is a distinguished triangle: Furthermore, for j ≥ p − n + k let F j p Ψ X be the class of The following theorem summarizes the above observations.Theorem 6.9.[Kov05, 2.7] Let Φ X and Ψ X be locally free sheaves on X of rank k and n respectively, and let θ X : Φ X → Ψ X be a morphism.Then there exists an object Q r θX ∈ Ob(D(X)) for each r ∈ Z, r ≥ −k with the following property.For each p ∈ N there exists a hyperfiltration F j p Ψ X of p Ψ X with j = 0, . . ., k + 1, such that Furthermore, for r > n − k, Q r θX ≃ 0.
Proposition 6.10.[Kov05, 2.9] Assume that θ X is injective.Then if Ξ X , the cokernel of θ X , is locally free, then Q p θX is isomorphic to the p-th exterior power of Ξ X .The filtration is given by p Ψ X = F 0 ⊃ F 1 ⊃ • • • ⊃ F p ⊃ F p+1 = 0, with quotients for each j.
Proof.By definition one has that Then the statement follows using descending induction, the filtration associated to the short exact sequence of locally free sheaves and the distinguished triangle,

Further
let Hodge(d, w) denote the category of triples (B, E , θ), where (E , θ) is a system of reflexive Hodge sheaves of weight w on the smooth quasi-projective scheme B of dimension d.A morphism Γ : (B ′ , E , θ ′ ) → (B, E , θ) in this category consists of a morphism γ : B ′ → B, such that the induced morphism E → Rγ * E ′ , fits into the commutative diagram

5.
Positivity of direct image sheaves and the discrepancy of the familyWe will continue using the notation introduced in Notation 4.2.First we show a somewhat weaker version of Theorem 1.4.Proposition 5.1.Let X and B be two projective varieties of dimension n and d, respectively, f : X → B a flat family of geometrically integral varieties with only Gorenstein Du Bois singularities, such that B is smooth and the generic fiber of f has rational singularities.Further let D ⊂ B be an effective divisor satisfying D ≤ D f , D ′ := D f − D, and let r := rank(f * ω X/B ).Then, one of the following holds.(i)Either c 1 (det(f * ω X/B )(−rnD − D ′ )) • H d−1 ≤ 0,for some ample divisor H B, or (ii) there exists a pseudo-effective line bundle B on B for which there is an injection

Fact 5. 4 .
Let f : X → B be a projective morphism of smooth quasi-projective varieties X and B. Assume that D f and ∆ := f −1 D f are simple normal crossing divisors.Let γ : C → B be a morphism of smooth quasi-projective varieties.Let X C be a strong resolution of X × B C, with f C : X C → C being the naturally induced family.Assume that the support of D fC and ∆ fC are simple normal crossing divisors.Let (E 0 C = E 0 C,i , θ 0 C ) be the associated system of Hodge sheaves underlying Deligne extension of the local system R j f * C XC \∆ f C of any given weight j.Then, as systems of Hodge sheaves, we have an inclusionγ * (E 0 , θ 0 ) ⊆ (E 0 C , θ 0 C ),which is an isomorphism over the flat locus of γ| C\D f C .Proof of Proposition 5.1.Let L := (det f * ω X/B )(−rnD − D ′ ) and assume that for some ample divisor H ⊂ B we have (5.4.1) c 1 Now we are ready to define Q p θX ∈ Ob(D(X)).Definition 6.8.Let p ∈ Z.For p > n − k let Q p θX = 0, and for−k ≤ p ≤ n − k let Q p θX be the class of F n−k−p n−k−p ⊗ (det Φ X ) −(n−k−p+1)
by the quasi-isomorphism in Theorem 3.3(vii).