Period mappings of Hodge-de Rham type

Hom(F(t)_,E)

Hom(id_F(t),π|E)

T_Gr(H(t))(F(t))

θ(H(t),F(t))

//Hom(F(t),H(t)/F(t))

Hence, the image ofvunder the composition θ(_H(t),F(t))◦Tt(_P)

is represented by the matrix (1.6.20.2) with respect to the bases(f0(t), . . . ,fd−1(t)) and(π(e0(t)), . . . ,π(ec−1(t))). On the other hand, we have for allj∈d:

∇_U(α_j) =

∑

i<c

(d_S)_U(λ_ij)⊗(e_i|U), whence

(∇_ι)_U(α_j) =

∑

i<c

(dS)_U(λ_ij)⊗ei,

whereeidenotes the image ofei|Uunder the mappingH(U)→(H/F)(U). Put A:=AS,t(F,H/F)(∇_ι). Then by Contruction 1.6.15, we have:

Av(α_j(t)) =

∑

i<c

(vy(d_S)_U(λ_ij))·e_i(t).

Evidently, for allj∈ d, the mappingι(t):F(t)→H(t)sendsα_j(t)(evaluation in F) to

α_j(t) = f_j(t) +

∑

i<c

λ_ij(t)·e_i(t) = f_j(t)

(evaluation inH); thus σ(α_j(_t)) = _fj(_t). Likewise, for alli ∈ c, the mapping (cokerι)(t):H(t) →(_H/F)(t)sendse_i(t)toe_i(t); thusτ(π(e_i(t))) = e_i(t). This proves the commutativity of (1.6.20.1).

1.7. Period mappings of Hodge-de Rham type

After the ground laying work of the previous § , we are now in the position to analyze period mappings of “Hodge-de Rham type”; the concept will be made precise in the realm of Notation 1.7.2 b) below. As a preparation we need the following result.

Proposition 1.7.1. — Let n be an integer.

a) Let (f,g)be a composable pair of submersive morphisms of complex spaces. Then

∇_GMⁿ (f,g)is a flat g-connection onH ⁿ(f).

b) Let f:X→S be a submersive morphism of complex spaces such that S is a complex manifold. Then∇_GMⁿ (f)is a flat S-connection onHⁿ(f).

Proof. — Clearly, assertion b) follows from assertion a) letting g = a_S. For the verification of Leibniz’s rule and the flatness condition for connections we refer our reader to [44, Section 2].

Notation 1.7.2. — Let f:X→Sbe a submersive morphism of complex spaces such thatSis a simply connected complex manifold. Letnandpbe integers andt∈ S.

Assume that Hⁿ(f)is a locally finite free module onS and F^pHⁿ(f)is a vector subbundle ofHⁿ(f)onS.

a) We put:

(1.7.2.1) P^0p,n_t (f):=Pt(S,(Hⁿ(f),∇ⁿ_GM(f)), F^pH ⁿ(f)),

where the right hand side is to be interpreted in the sense of Construction 1.6.11. Note that (1.7.2.1) makes sense in particular because by Proposition 1.7.1 b)∇ⁿ_GM(f)is a flatS-connection onHⁿ(f), whence(Hⁿ(f),∇ⁿ_GM(f))is a flat vector bundle onS.

Note that by means of Proposition 1.6.13 we may regardP^0p,n_t (_f)as a morphism of complex spaces:

P^0p,n_t (f)_:S−→Gr((_Hⁿ(f))(t))_. b) Assume that for alls∈Sthe base change maps

φⁿ_f,s:(Hⁿ(f))(s)−→Hⁿ(Xs) φ^p,n_f,s :(F^pHⁿ(f))(s)−→F^pHⁿ(Xs) are isomorphisms in Mod(C). Write

ρ⁰:Π(S)−→Mod(_C)

for the C-representation of the fundamental groupoid of S which is defined for (H,∇):= (Hⁿ(f),∇ⁿ_GM(f))in Construction 1.6.11. Let

ρ:Π(S)−→Mod(_C)

be the functor which is obtained by “composing”ρ⁰with the family of isomorphisms φ := (φⁿ_f,s)_s∈S. DefineFto be the unique function onSsuch that, for alls∈ S, we have:

F(s) =F^pHⁿ(Xs). Then clearlyFis aC-distribution inρ. We set:

P_t^p,n(f):=_P^C_t(S,ρ,F),

cf. Construction 1.6.3. Note thatφis an isomorphism of functors fromΠ(S)to Mod(C) fromρ⁰toρ. Moreover, whenF⁰denotes the unique function onSsuch that, for all s∈S, we have

F⁰(s) =im((ιⁿ_f(p))(s):(F^pHⁿ(f))(s)−→(_Hⁿ(f))(s)), then

φⁿ_f,s[F⁰(s)] =F(s)

for alls∈S. Therefore the following diagram commutes inSet:

S C-vector spaces), we may viewP_t^p,n(f)as a morphism of complex spaces fromSto Gr(Hⁿ(Xt)). We callP^p,n_t (f)the(Hodge-de Rham) period mappingin bidegree(p,n)of

f with basepointt.

Next, we introduce the classical concept of Kodaira-Spencer maps. Our definition shows how to construct these maps out of the Kodaira-Spencer class given by Notation 1.5.2 and Notation 1.3.17. As an auxiliary means, we also introduce “Kodaira-Spencer maps without base change”.

Notation 1.7.3(Kodaira-Spencer maps). — Let f: X → S be a submersive mor-phism of complex spaces such thatS is a complex manifold. Then, by means of Notation 1.5.2, we may speak of the Kodaira-Spencer class of f, writtenξ_KS(f), which is a morphism

ξ_KS(f):OS −→_Ω¹_S⊗R¹f∗(_Θf)

of modules onS. We write KS_f for the composition of the following morphisms in Mod(S):

to be the composition of the inverse of the canonical isomorphismΘS(t) →T_S(t) with KS_f(t):ΘS(t)→(R¹f∗(Θ_f))(t). We call KS⁰_f,ttheKodaira-Spencer map without base changeof f att. Furthermore, define

KSf,t: TS(t)−→H¹(Xt,ΘX_t)

to be the composition of KS⁰_f,twith the evident base change morphism:

β¹_f,t: (R¹f∗(Θ_f))(t)−→H¹(Xt,ΘX_t). We call KS_f,ttheKodaira-Spencer mapof f att.

Notation 1.7.4(Cup and contraction, II). — Letf:X→Sbe an arbitrary morphism of complex spaces. Letpandqbe integers. We define

γ^p,q_f : R¹f∗(Θf)⊗H^p,q(f)−→H^p−1,q+1(f) to be the composition in Mod(S)of the cup product morphism

^^1,q_f (Θf,Ω^p_f): R¹f∗(Θf)⊗R^qf∗(Ω^p_f)−→R^q+1f∗(Θf⊗Ω^p_f) and the R^q+1f∗(−)of the contraction morphism:

γ_X^p(Ω¹_f): Θf ⊗Ω^p_f −→Ω^p−1_f ,

cf. Notation 1.3.10.γ^p,q_f is called thecup and contractionin bidegree(p,q)for f. As a shorthand, we writeγ_X^p,qforγ^p,q_aX.

By means of tensor-hom adjunction onS(with respect to the modules R¹f∗(Θf), H ^p,q(f), andH^p−1,q+1(f)), the morphismγ^p,q_f corresponds to a morphism

R¹f∗(Θf)−→HomS(H ^p,q(f),H^p−1,q+1(f))

in Mod(_S)_{. Lett}∈S. Then evaluating the latter morphism attand composing the result in Mod(C)with the canonical morphism

(_Hom_S(_H ^p,q(f),H^p−1,q+1(f)))(t)−→Hom

(_H^p,q(f))(t),(_H ^p−1,q+1(f))(t), yields:

γ^0p,q_f,t :(R¹f∗(Θf))(t)−→Hom

(H ^p,q(f))(t),(H ^p−1,q+1(f))(t).

We refer toγ^0p,q_f,t as thecup and contraction without base changein bidegree(p,q)of f at t.

The following two easy lemmata pave the way for the first essential statement of

§ 1.7, which is Proposition 1.7.7.

Lemma 1.7.5. — Let f: X→S be a submersive morphism of complex spaces such that S is a complex manifold. Let p and q be integers and t∈S. Then the following identity holds in Mod(C):

(1.7.5.1) A_S,t(_H ^p,q(f),H^p−1,q+1(f))(γ_KS^p,q(f)) =γ^0p,q_f,t ◦KS⁰_f,t.

Proof. — We argue in several steps. To begin with, observe that the following diagram commutes in Mod(S):

Thus, by the naturality of tensor-hom adjunction, the next diagram commutes in Mod(S), too: From this we deduce by means of evaluation attthat the diagram

ΘS(t)

commutes in Mod(C). Plugging in the inverse of the canonical isomorphism ΘS(t)−→T_S(t)

and taking into account the definitions of A_S,tand KS⁰_f,t, we deduce the validity of (1.7.5.1).

Then the following diagram commutes inMod(S):

Proof. — By Proposition 1.5.10, there exists an ordered pair(ζ,ζ)of morphisms in Mod(S/C)such that the following two identities hold in Mod(S/C)_:

Proposition 1.7.7. — Let f: X → S be a submersive morphism of complex spaces such that S is a simply connected complex manifold. Let n and p be integers and t∈S. In addition, letψ^pandψ^p−1be such that the following diagram commutes inMod(S)forν=p,p−1:

Moreover, set

F⁰(t):=im((ιⁿ_f(p))(t):(F^pHⁿ(f))(t)−→(Hⁿ(f))(t)) and write

σ:(_F^p_Hⁿ(f))(t)−→ F⁰(t)_,

τ:(Hⁿ(f))(t)/F⁰(t)−→(Hⁿ(f)/F^pH ⁿ(f))(t)

for the evident morphisms. Thenσandτare isomorphisms inMod(C)and the following diagram commutes inMod(_C):

(1.7.7.2) TS(t)

KS⁰_f,t

//

Tt(P⁰_t^p,n(f))

(_R¹_f∗(_Θf))(_t)

γ⁰_f^p,n_,t ^−p

Hom((_H ^p,n−p(f))(t),(_H^{p−1,n−p+1}(f))(t))

Hom(α⁰(t)◦σ⁻¹,τ⁻¹◦β⁰(t))

T_Gr((Hn(f))(t))(F⁰(t)) //

θ((Hⁿ(f))(t),F⁰(t))

Hom(F⁰(t),(_Hⁿ(f))(t)/F⁰(t)) Proof. — Introduce the following notational shorthands:

H :=_Hⁿ(f), θ:=θ(_H(t),F⁰(t)), F^∗:=F^∗Hⁿ(f).

Furthermore, set:

∇_ι:= (id_Ω1

S⊗coker(ιⁿ_f(p)))◦ ∇ⁿ_GM(f)◦ιⁿ_f(p).

Then by Lemma 1.6.20,σ and τare isomorphisms in Mod(C)and the following diagram commutes in Mod(C):

T_S(t) ^AS,t(F

p,H/F^p)(∇_ι)//

Tt(P⁰_t^p,n(f))

Hom(_F^p(t),(_H/F^p)(t))

Hom(σ⁻¹,τ⁻¹)

T_Gr(H(t))(F⁰(t))

θ

//Hom(F⁰(t),H (t)/F⁰(t))

By Lemma 1.7.6, setting

ι:=ιⁿ_f(p−1)_/F^p:F^p−1/F^p−→_H/F^p, the diagram in (1.7.6.1) commutes in Mod(S). Therefore, with

ι:=coker(ιⁿ_f(p,p+1)):F^p−→F^p/F^p+1 we have according to Remark 1.6.17:

A_S,t(_F^p,H/F^p)(∇_ι) =Hom(ι(t),ι(t))◦A_S,t(_F^p_/F^p+1,F^p−1/F^p)(∇_GM^p,n(f)).

Hence the following diagram commutes in Mod(C):

By Theorem 1.5.13, the following diagram commutes in Mod(S): F^p/F^{p+1 ∇}

Thus making use of Remark 1.6.17 again, we obtain:

A_S,t(_F^p_/F^p+1,F^p−1/F^p)(∇_GM^p,n(f)) Hence, this next diagram commutes in Mod(C):

TS(t) ^//

Employing Lemma 1.7.5, we infer the commutativity of (1.7.7.2).

The next result is a variant of the previous Proposition 1.7.7 incorporating base changes.

Theorem 1.7.8. — Let f: X→S be a submersive morphism of complex spaces such that S is a simply connected complex manifold. Let n and p be integers and t∈S. Letψ_X^p

Letω_X^p−1

t be a left inverse ofψ^p−1_X

t inMod(_C). Assume thatHⁿ(f)is a locally finite free module on S,F^pHⁿ(f)is a vector subbundle ofH ⁿ(f)on S, and the base change morphisms

φⁿ_f,s:(_Hⁿ(f))(s)−→_Hⁿ(Xs), φ^p,n_f,s :(_F^p_Hⁿ(f))(s)−→F^pHⁿ(Xs)

are isomorphisms inMod(C)for all s ∈ S. Assume that there existψ^p,ψ^p−1, andω^p−1 such that firstly, the diagram in(1.7.7.1)commutes inMod(S)forν=p,p−1and secondly, ω^p−1is a left inverse ofψ^p−1inMod(S). Moreover, assume that the Hodge base change map

β^p,n−p_f,t : (_H^p,n−p(f))(t)−→H ^p,n−p(Xt) is an isomorphism inMod(C). Then, setting

α:=κⁿ_Xt(σ^=pΩq

Xt)◦ψ_X^p_t◦coker(ιⁿ_Xt(p,p+1)), β:= (ιⁿ_Xt(p−1)_/F^p_Hⁿ(Xt))◦ω_X^p−1

t ◦(κⁿ_Xt(σ^=p−1Ωq

X_t))⁻¹_, the following diagram commutes inMod(_C):

(1.7.8.2) TS(t) ^{KSf,t //}

T_t(P_t^p,n(f))

H¹(Xt,ΘX_t)

γ_Xt^p,n−p

Hom(H ^p,n−p(Xt),H ^{p−1,n−p+1}(Xt))

Hom(α,β)

T_Gr(Hn(Xt))(F^pHⁿ(Xt)) //

θ(Hⁿ(X_t),F^pHⁿ(X_t))

Hom(F^pHⁿ(Xt),Hⁿ(Xt)/F^pHⁿ(Xt))

Proof. — We set

φ:=φⁿ_f,t:(Hⁿ(f))(t)−→Hⁿ(Xt).

Then by Notation 1.7.2, the following diagram commutes in the category of complex spaces:

S

P⁰_t^p,n(f)

xxqqqqqqqqqqqq _P^p,n

t (f)

%%K

KK KK KK KK KK Gr((_Hⁿ(f))(t))

Gr(φ) //Gr(_Hⁿ(Xt)) In consequence, letting

F⁰(t):=im((ιⁿ_f(p))(t):(F^pHⁿ(f))(t)−→(_Hⁿ(f))(t)),

the following diagram commutes in Mod(C):

the morphism which is induced byφthe obvious way, the following diagram com-mutes in Mod(C)by means of the the naturality ofθ(−,−), cf. Notation 1.6.19:

Then by Proposition 1.7.7,σandτare isomorphisms in Mod(C)and the diagram in (1.7.7.2) commutes in Mod(C). Let

φ₁:(_Hⁿ(f)/F^pH ⁿ(f))(t)−→_H ⁿ(Xt)/F^pHⁿ(Xt)

be the morphism which is naturally induced byφ—similar toφabove—using the fact that

coker(ιⁿ_f(p))(t):(_Hⁿ(f))(t)−→(_Hⁿ(f)/F^pHⁿ(f))(t)

is a cokernel for(ιⁿ_f(p))(t)in Mod(C)as the evaluation functor “−(t)” is a right exact functor from Mod(S)to Mod(C). Then it is a straightforward matter to verify these identities:

Similarly, comparingβ⁰andβ, we see that (1.7.8.7) (H ^p−1,q+1(f))(t)

for the evident base change morphism. Then, since the cup product morphisms

^^1,qas well as the contraction morphismsγ^pare compatible with base change, the following diagram commutes in Mod(C):

(R¹f∗(_Θf)⊗_SH ^p,q(f))(t)

Therefore, given thatβ^p,q_f,t is an isomorphism by assumption, the following diagram

According to the definition of the Kodaira-Spencer map KS_f,tin Notation 1.7.3, the following diagram commutes in Mod(C):

(1.7.8.9) Tt(S)

Taking all our previous considerations into account, we obtain:

θ(_Hⁿ(Xt), F^pHⁿ(Xt))◦Tt(_Pt^p,n(f))

When it comes to applying Theorem 1.7.8, one is faced with the problem of deciding whether there exist morphismsψ^ν(resp.ψ^ν_Xt) rendering commutative in Mod(S)(resp.

Mod(C)) the diagram in (1.7.7.1) (resp. (1.7.8.1)). Let us formulate two (hopefully) tangible criteria.

Proposition 1.7.9. — Let n andν be integers and f: X → S an arbitrary morphism of complex spaces. Denote by E the Frölicher spectral sequence of f .

a) The following are equivalent:

(i) E degenerates from behind in(ν,n−ν)at sheet1inMod(S);

(ii) there existsψ^νrendering commutative inMod(S)the diagram in(1.7.7.1).

b) The following are equivalent:

(i) E degenerates in(_ν,n−ν)at sheet1inMod(S);

(ii) there exists an isomorphismψ^νrendering commutative inMod(S)the diagram in (1.7.7.1).

Proof. — The above statements are special cases of standard interpretations of the degeneration of a spectral sequence associated to a filtered complex. We refer our readers to Deligne’s treatment [12, § 1].

Theorem 1.7.10. — Let n be an integer and f:X→S a submersive morphism of complex spaces such that S is a simply connected complex manifold. Assume that:

(i) the Frölicher spectral sequence of f degenerates in entries I:={(p,q)∈Z×Z:p+q=n} at sheet1inMod(_S)_;

(ii) for all(p,q)∈ I,H ^p,q(f)is a locally finite free module on S;

(iii) for all s∈S, the Frölicher spectral sequence of Xsdegenerates in entries I at sheet1 inMod(C);

(iv) for all s∈S and all(p,q)∈I, the Hodge base change map β^p,q_f,s:(H^p,q(f))(s)−→H ^p,q(Xs) is an isomorphism inMod(C).

Let t∈S. Then there exists a sequence(ψe^ν)_ν∈Zof isomorphisms inMod(C), ψe^ν: F^νHⁿ(Xt)/F^ν+1Hⁿ(Xt)−→_H^ν,n−ν(Xt),

such that, for all p∈Z, the diagram in(1.7.8.2)commutes inMod(_C), where we set:

α:=ψe^p◦coker(ιⁿ_Xt(p,p+1)),

β:= (ιⁿ_Xt(p−1)/F^pHⁿ(Xt))◦(ψe^p−1)⁻¹. (1.7.10.1)

Proof. — By assumption (i) we know using Proposition 1.7.9 that, for all integers ν, there exists one, and only one,ψ^νsuch that the diagram in (1.7.7.1) commutes in Mod(S)(note that the uniqueness ofψ^νfollows from the fact that bothλⁿ_f(ν)and coker(ιⁿ_f(ν,ν+1)), and whence their composition, are epimorphisms in Mod(S));

moreover,ψ^νis an isomorphism. Further on, for all integersν, κⁿ_f(σ^=νΩq

f): Rⁿf_∗(σ^=νΩq

f)−→Rⁿf∗(σ^=νΩq

f) =R^n−pf∗(_Ω^p_f) is an isomorphism. Thus for allν∈Z, there exists an isomorphism

F^νHⁿ(f)/F^ν+1Hⁿ(f)−→_H^ν,n−ν(f)

in Mod(S). Now since, for allν ∈ Z≥n+1, F^νHⁿ(f)is a zero module onSand in particular locally finite free, we conclude by descending induction onνstarting at ν=n+1 that, for allν∈Z, F^νHⁿ(f)is a locally finite free module onS; along the way we make use of (ii). Specifically, since F⁰Hⁿ(f) =_Hⁿ(f), we see thatHⁿ(f)is a locally finite free module, i.e., in the terminology of Definition 1.6.9, a vector bundle, onS. Furthermore, for all integersµandνsuch thatµ≤ν, there exists a short exact sequence

0−→F^µ/F^ν−→F^µ−1/F^ν−→F^µ−1/F^µ−→0,

where we write F^∗as a shorthand for F^∗Hⁿ(f). Therefore, we see, using descending induction onµ, that for all integersνand all integersµsuch thatµ≤νthe quotient F^µ/F^νis a locally finite free module onS. Specifically, we see that for all integersν, the quotientHⁿ(f)/F^νH ⁿ(f)is a locally finite free module onS. Thus we conclude that, for all integersν, F^νHⁿ(f)is a vector subbundle ofHⁿ(f)onS.

For the time being, fix an arbitrary elementsofS. Then by assumption (iii) and Proposition 1.7.9 we deduce that, for all integersν, there exists a (unique) isomorphism ψ^ν_Xs such that the diagram in (1.7.8.1), where we replacetbys, commutes in Mod(C). As base change commutes with taking stupid filtrations, the following diagram has exact rows and commutes in Mod(C)for all integersν:

0 //F^ν(s)

(ιⁿ_f(ν−1,ν))(s)

//

φ^ν,n_f,s

F^ν−1(s) ^//

φ^ν_f⁻_,s^1,n

H^ν,n−ν(s) ^//

β^ν,n_f,s^−p

0

0 //F^νHⁿ(Xs)

ιⁿ_Xs(ν−1,ν)//F^ν−1Hⁿ(Xs) ^//H^ν,n−ν(Xs) ^//0 Therefore, using a descending induction onνstarting at ν = n+1 together with assumption (iv) and the “short five lemma”, we infer that, for allν ∈ Z, the base change mapφ^ν,n_f,s is an isomorphism in Mod(C). Specifically, sinceφ^0,n_f,s =φⁿ_f,s, we see that the de Rham base change mapφⁿ_f,sis an isomorphism in Mod(C).

Abandon the fixation ofsand define aZ-sequenceψeby putting, for anyν∈Z:

ψe^ν:=κ_Xⁿ_t(σ^=νΩq

X_t)◦ψ_X^ν_t.

Letpbe an integer. Then definingαandβaccording to (1.7.10.1), the commutativity of the diagram in (1.7.8.2) is implied by Theorem 1.7.8.

DEGENERATION OF THE FRÖLICHER SPECTRAL SEQUENCE

Later, in Chpater 3, we would like to make use of Theorem 1.7.10 in order to establish a local Torelli theorem for certain compact, symplectic complex spaces of Kähler type, cf. Theorem 3.4.4. The basic idea in the application of Theorem 1.7.10 thereby is the following. Consider a family of compact complex spaces over a smooth base, i.e., a proper, flat morphism of complex spaces f: X → S such that S is a complex manifold. Wanting to talk about period mappings, we assume additionally that the spaceSbe simply connected, although this assumption is not essential for the problems of Chapter 2. Now defineg:Y→Sto be the “submersive share” of f, by which we mean thatYis the open complex subspace ofXinduced on set of points ofXin which the morphism f is submersive andgis the composition of the inclusion Y→Xand f. Thengis certainly a submersive morphism of complex spaces with smooth and simply connected base, so that we might think of applying Theorem 1.7.10 tog(in place of f, as in the formulation of the theorem) and an integern. This leads us to the task of determining circumstances, in terms of f andn, under which the following assertions hold—observe that these correspond to conditions (i)–(iv) of Theorem 1.7.10:

a) the Frölicher spectral sequence ofgdegenerates in entries I:={(p,q)∈Z×Z:p+q=n} at sheet 1 in Mod(S);

b) for all(p,q)∈ I, the Hodge moduleH ^p,q(g)is a locally finite free module onS;

c) for alls∈ S, the Frölicher spectral sequence ofYsdegenerates in entries Iat sheet 1 in Mod(C);

d) for alls∈Sand all(p,q)∈I, the Hodge base change map β^p,qg,s:(_H ^p,q(g))(s)−→_H ^p,q(Ys) is an isomorphism in Mod(C).

In view of assertion c), we remind the reader of a result of T. Ohsawa (cf. [65, Theorem 1]):

Theorem 2.0.1. — Let X be a compact, pure dimensional complex space of Kähler type and A a closed analytic subset of X such thatSing(X)⊂A. Then the Frölicher spectral sequence of X\A degenerates in entries

{(p,q)∈Z×Z:p+q+2≤codim(A,X)}

at sheet1inMod(C).

So, suppose that the fibers of f are altogether pure dimensional (e.g., normal and connected) and of Kähler type. Then Theorem 2.0.1 guarantees the validity of assertion c) when we have

(∗) n+₂≤codim(_Sing(Xs)_,Xs) _{for all}s∈S;

note that due to the flatness of f, we know that, for alls∈S, the complex spacesYs

and(Xs)_reg =Xs\Sing(Xs)are isomorphic. Inspired by this observation, we pose the following

Question 2.0.2. — Let f andnbe as above; in particular, we assume that (∗) holds.

Definegto be the submersive share of f. Which of the assertions b), d), and a) are then fulfilled?

We put forward another question, which is wider in scope.

Question 2.0.3. — Let f:X→Sbe a proper (and flat) morphism of complex spaces (such thatSis a complex manifold),Aa closed analytic subset ofXsuch that the restrictiong:Y:=X\A→Sof f is submersive,nan integer such that

n+2≤codim(A∩Xs,Xs) for alls∈S.

Assume that f is

(i) locally equidimensional, i.e., the functionx 7→dimx(X_f(x))is locally constant onX, and

(ii) weakly Kähler (cf. [5, (5.1)]).

Do assertions b), d), and a) hold then?

Our goal in this chapter is to give several positive answers in the direction of Question 2.0.3 and Question 2.0.2—unfortunately we do not manage to answer either of the proposed questions in its entirety.

In § 2.1, we investigate the coherence of the Hodge modulesH ^p,q(g)_{by means} of standard techniques of local cohomology as a first step towards the local finite freeness stated in b). In § 2.2, we study the degeneration behavior of the Frölicher spectral sequence when passing from one infinitesimal neighborhood of a fiber of g:Y→Sto the next. In § 2.3, we invoke a comparison theorem between formal and ordinary higher direct image sheaves due to C. B˘anic˘a and O. St˘an˘a¸sil˘a (cf. [9]) in

order to establish b) and d). Finally, in § 2.4, we draw conclusions for the degeneration of the the Frölicher spectral sequence ofg.

2.1. Coherence of direct image sheaves

Notation 2.1.1(Depth). — LetAbe a commutative ring,Ian ideal ofA, andMan A-module. Then we define:

(2.1.1.1) prof_A(I,M):=sup{n∈_N:(∀N∈ T)(∀i∈n)Extⁱ_A(N,M) =0}, whereTdenotes the class of all finite typeA-modules for which there exists a natural numbermsuch thatI^mN=0. Note that the set in (2.1.1.1) over which the supremum is taken certainly contains 0, whence is nonempty. prof_A(I,M)is called theI-depthof MoverA.

WhenAis a commutative local ring, we define:

prof_A(M)_:=_profA(m(A)_,M)_.

LetXbe an analytic space (or else a commutative locally ringed space). Letxbe an element of the set underlyingXandFa module onX. Then we set:

prof_X,x(F)_:=_profO

X,x(Fx)_.

Proposition 2.1.2. — Let A be a commutative ring and I an ideal of A.

a) For all A-modules M and M⁰, we have:

prof_A(I,M⊕M⁰) =min(prof_A(I,M), prof_A(I,M⁰)).

b) For all A-modules M and all r∈N, we haveprof_A(I,M^⊕r) =prof_A(I,M). Proof. — Follows from the fact that, for allA-modulesNand alli∈N(resp.i∈Z), Extⁱ_A(N,−)is an additive functor from Mod(A)to Mod(A)hence commutes with the formation of finite sums.

Notation 2.1.3. — LetXbe a complex space (or else a commutative locally ringed space),Fa module onX, andman integer. Then we define:

Sm(X,F):={x∈X: prof_X,x(F)≤m}. Sm(X,F)is called them-thsingular setofFonX.

Notation 2.1.4(Sheaves of local cohomology). — LetXbe a topological space (re-spectively a ringed space or complex space),Aa closed subset ofX. We denote

Γ_A(X,−): Ab(X)−→Ab(X) (resp. Mod(X)−→Mod(X))

thesheaf of sections on X with supports in A functor. That is, for any abelian sheaf (resp. sheaf of modules) F on X, we define Γ_A(X,F) to be the abelian subsheaf (resp. subsheaf of modules) ofF on Xsuch that, for all open subsetsU ofX, the set(_ΓA(X,F))(U)comprises precisely those elements ofF(U)which are sent to the

zero ofF(U\A)by the restriction mappingF(U)→F(U\A). Note that the functor Γ_A(X,−)is additive as well as left-exact. For any integernwe write

Hⁿ_A(X,−): Ab(X)−→Ab(X) (resp. Mod(X)−→Mod(X)) for then-th right derived functor ofΓ_A(X,−), cf. § A.6.

Proposition 2.1.5. — Let X be a topological space (resp. a ringed space or complex space), A a closed subset of X. Write i:X\A→X for the inclusion morphism. Then, for all abelian sheaves (resp. sheaves of modules) F on X, there exists an exact sequence inAb(X)(resp.

Mod(X)),

0−→H⁰_A(X,F)−→ F−→R⁰i∗(i^∗(F))−→H¹_A(X,F)−→0, and, for all integers q≥1, we have

R^qi∗(_i^∗(_F))∼=H^q+1_A (_X,F)_.

Proof. — The topological space case is [9, Chapter II, Corollary 1.10]; the ringed space case is proven along the very same lines.

Lemma 2.1.6. — Let X be a commutative ringed space.

a) For all morphismsφ: F→G inMod(X)such that F and G are coherent on X, both ker(φ)andcoker(φ)are coherent on X.

b) For all short exact sequences

0−→F−→G−→H−→0

inMod(X), when F and H are coherent on X, then G is coherent on X.

Proof. — See [69, I, § 2, Théorème 1 and Théorème 2].

Corollary 2.1.7. — Let X be a ringed space, A a closed subset of X, and F a coherent module on X. Then, for all n∈Z, the following are equivalent:

(i) For all q∈Zsuch that q≤n, the moduleR^qi∗(i^∗(F))is coherent on X.

(ii) For all q∈Zsuch that q≤n+1, the moduleH^q_A(X,F)is coherent on X.

Proof. — This is clear from Proposition 2.1.5 and Lemma 2.1.6.

Theorem 2.1.8. — Let X be a complex space, A a closed analytic subset of X, and F a coherent module on X. Denote i:X\A→X the inclusion morphism. Then, for all natural numbers n, the following are equivalent:

(i) For all integers k, we have

(2.1.8.1) dim(A∩Sk+n+1(X\A,i^∗(F)))≤k,

where the bar refers to taking the closure in Xtop. Note that imposing(2.1.8.1)hold for all integers k<0is equivalent to requiring that

A∩Sn(X\A,i^∗(F)) =_∅.

(ii) For all q∈N(or q∈Z) such that q≤n, the moduleH^q_A(X,F)is coherent on X.

Proof. — This is [9, Chapter II, Theorem 4.1].

Proposition 2.1.9. — Let X be a locally pure dimensional complex space, A a closed analytic subset of X, and F a coherent module on X. Assume that,

(2.1.9.1) for all x∈X\A, prof_X,x(F) =dimx(X).

Denote by i the inclusion morphism of complex spaces from X\A to X. Then, for all q∈Z such that q+2≤codim(A,X), the moduleR^qi∗(i^∗(F))is coherent on X.

Proof. — Assume thatXis pure dimensional. WhenA=∅, the morphismiis the identity onX, hence we have R⁰i∗(i^∗(F)) ∼= Fand R^qi∗(i^∗(F)) ∼=0 for all integers q6=0 in Mod(X). Thus, for all integersq, the module R^qi∗(i^∗(F))is coherent onX.

Now assume thatA6=_∅and putn:=codim(A,X)−1. WriteSmas a shorthand for Sm(X\A,i^∗(F)). For allx∈X\A, we have:

prof_X\A,x(i^∗(F)) =prof_X,x(F) =dimx(X) =dim(X).

Therefore,Sm = ∅for all integersmsuch thatm< dim(X). Letkbe an arbitrary integer. When dim(A)≤k, then

dim(A∩Sk+n+1)≤dim(A)≤k.

For allx∈A, we have

dimx(A) =dimx(X)−codimx(A,X)≤dim(X)−codim(A,X), which implies that

dim(A)≤dim(X)−codim(A,X). In turn, whenk<dim(A), we have

k+n+1=k+codim(A,X)<dim(X), whenceS_k+n+1=_{∅. Thus,}A∩S_k+n+1=_∅and so again,

dim(A∩Sk+n+1)≤k.

We see that assertion (i) of Theorem 2.1.8 holds. Hence by Theorem 2.1.8, assertion (ii) holds too, so that, for all integersqwithq≤n, the module H^q_A(X,F)is a coherent module onX. Corollary 2.1.7 implies that, for all integersqwithq+₂≤codim(A,X)_, i.e.,q≤n−1, the module R^qi∗(i^∗(F))is coherent onX.

Abandon the assumption thatXis pure dimensional. Letqbe an integer such that q+2 ≤ codim(A,X). Let x ∈ Xbe any point. Then, sinceXis locally pure dimensional, there exists an open neighborhood Uof x in X such that the open complex subspace ofXinduced onUis pure dimensional. PutY:=X|U,B:=A∩U, andG:=F|U. By what we have already proven, and the fact that

q+2≤codim(A,X)≤codim(B,Y),

we infer that R^qj∗(j^∗(G))is coherent onY, wherejstands for the canonical morphism fromY\BtoY. Since R^qi∗(i^∗(F))|Uis isomorphic to R^qj∗(j^∗(G))in Mod(Y), we see that R^qi∗(i^∗(F))is coherent onXinx. Asxwas an arbitrary point ofX, the module R^qi∗(i^∗(F))is coherent onX.

Corollary 2.1.10. — Let X be a locally pure dimensional complex space, A a closed analytic subset of X, and F a coherent module on X. Assume that X is Cohen-Macaulay in X\A and F is locally finite free on X in X\A. ThenR^qi∗(i^∗(F))is a coherent module on X for all integers q satisfying q+2≤codim(A,X), where i denotes the canonical immersion from X\A to X.

Proof. — Letx ∈ X\A. AsFis locally finite free onXinx, there exists a natural numberrsuch that Fx is isomorphic to(_OX,x)^⊕r in the category ofOX,x-modules, hence using Proposition 2.1.2, we obtain:

prof_X,x(F) =prof_OX,x(Fx) =prof_OX,x((_OX,x)^⊕r) =prof_OX,x(_OX,x)

=dim(_OX,x) =dimx(X).

Asxwas an arbitrary element ofX\A, we see that (2.1.9.1) holds. Thus our claim follows readily from Proposition 2.1.9.

Theorem 2.1.11(Grauert’s direct image theorem). — Let f: X → S be a proper mor-phism of complex spaces. Then, for all coherent modules F on X and all integers q, the module R^qf∗(F)is coherent on S.

Proof. — This is [24, “Hauptsatz I”, p. 59]. See also [9, Chapter III, Theorem 2.1] as well as [26, Chapter III, § 4] and references there.

Proposition 2.1.12. — Let f:X→S and g:Y→X be morphisms of complex spaces. Let G be a module on Y and n an integer. Put h := f ◦g. Suppose that f is proper and, for all integers q≤n, the moduleR^qg∗(G)is coherent on X. Then, for all integers k≤n, the moduleR^kh∗(G)is coherent on S.

Proof. — We employ the following fact: There exists a spectral sequence E with values in Mod(S)such that:

(i) for allp,q∈Z,E₂^p,q ∼=R^pf∗(R^qg∗(G));

(ii) for allk ∈ Z, there exists a filtrationFon R^kh∗(G)such thatF⁰ = R^kh∗(G), F^k+1∼=0, and for allp∈Zthere existsr∈N≥2such thatF^p/F^p+1∼=Er^p,k−p.

GivenE, we claim: For allr∈N≥2and allp,q∈Z,Er^p,qis coherent onSwhen any of the following conditions is satisfied:

a) p+q≤n;

b) there exists∆∈N≥1such thatp+q=n+_{∆, and}

(2.1.12.1) q≤n−

min(∆,r−1) ν=2

∑

(r−ν)

! ,

where the sum appearing on the right hand side in (2.1.12.1) is defined to equal 0 in case min(∆,r−1)<2.

The claim is proven by means of induction. By Theorem 2.1.11 and property (i) we know that, for allp,q∈_Zsuch thatq≤norp<0,E₂^p,qis coherent onS; note that in

The claim is proven by means of induction. By Theorem 2.1.11 and property (i) we know that, for allp,q∈_Zsuch thatq≤norp<0,E₂^p,qis coherent onS; note that in

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