The local Torelli theorem

X⁰

f⁰

S|V

b //S⁰

is a pullback square in the category of complex spaces. In particular, for alls∈V, the morphismiinduces an isomorphismXs ∼=X⁰_s, whenceXs ∈C.

3.4. The local Torelli theorem

First and foremost, we state and prove here our version of a local Torelli theorem for compact, connected, symplectic complex spacesXof Kähler type such thatΩ²_X(Xreg) is 1-dimensional and the codimension of the singular locus ofXis≥4 (cf. Theorem 3.4.4). Note that Y. Namikawa has proposed a local Torelli theorem for a slightly smaller class of spaces in [61, Theorem 8]. Our statement is more general in the following respects: Firstly, our spaces need neither be projective norQ-factorial (or similar, in the nonprojective case). Secondly, we do not require H¹(X,OX)to be trivial.

Moreover, we feel that we are after all the first to prove a local Torelli type statement in the context of singular symplectic complex spaces since, inloc. cit., Namikawa contents himself with referring to Beauville’s work [2] as a proof for the decisive point (3) of his theorem.

Beauville’s proof of the local Torelli theorem for irreducible symplectic manifolds certainly provides the basis for our line of reasoning below. Nonetheless, we would like to point out that in guise of Theorem 1.7.10 and Corollary 2.4.15, the upshots of our entire Chapters 1 and 2 enter the proof of Theorem 3.4.4.

Proposition 3.4.1. — Let X be a Cohen-Macaulay complex space and A a closed analytic subset of X such thatSing(X) ⊂ A andcodim(A,X) ≥ 3. Put Y := X\A. Then the evident restriction mapping

Ext¹(_Ω¹_X,OX)−→Ext¹(_Ω¹_Y,OY) is bijective.

Proof. — This follows from the analytic counterpart of [47, Lemma (12.5.6)].

Corollary 3.4.2. — Let f:X→S be a proper, flat morphism of complex spaces, A a closed analytic subset of X such thatSing(f) ⊂ A, and t ∈ S. Assume that S is smooth, f is semi-universal in t, Xtis Cohen-Macaulay, andcodim(A∩ |Xt|,Xt)≥3. Set Y:=X\A, and define g to be the composition of the inclusion Y→X and f . Then the Kodaira-Spencer map of g at t,

(3.4.2.1) KSg,t: T_S(t)−→H¹(Yt,ΘYt)

(cf. Notation 1.7.3), is an isomorphism of complex vector spaces.

Proof. — Denote T¹_Xtthe first (co-)tangent cohomology ofXtand (3.4.2.2) D_f,t: TS(t)−→T¹_Xt

the “Kodaira-Spencer map” for f int(cf. [66], [67], [39]). The construction of this generalized Kodaira-Spencer map is functorial so that the inclusionY→Xgives rise to a commutative diagram of complex vector spaces

T_S(t)

D_f,t

}}zzzzzzzz Dg,t

!!C

CC CC CC C

T¹_Xt //T¹_Yt

where the lower horizontal arrow is the morphism induced on (co-)tangent cohomol-ogy byYt→Xt.

Since the complex spacesXtandYtare reduced, the canonical maps Ext¹(Ω¹_Xt,OX_t)−→T¹_Xt,

Ext¹(_Ω¹_Y

t,OYt)−→T¹_Yt

are isomorphisms (cf. e.g., [68, (III.3.1), (iv) and (v)]). Moreover, the following diagram commutes:

Ext¹(_Ω¹_X

t,OX_t) ^//

T¹_Xt

Ext¹(Ω¹_Yt,OY_t) ^//T_Y¹_t

Since the morphism f: X →Sis semi-universal int, the Kodaira-Spencer map (3.4.2.2) is a bijection. Therefore, by Proposition 3.4.1 and the commutativity of the above diagrams we see that Dg,t is an isomorphism. However, through the composition of canonical maps

Ext¹(Ω¹_Yt,OY_t)−→Ext¹(OY_t,ΘY_t)−→H¹(Yt,ΘY_t)

the morphism Dg,t is isomorphic to the ordinary Kodaira-Spencer map (3.4.2.1), whence (3.4.2.1) is an isomorphism as claimed.

Lemma 3.4.3. — Let X be a symplectic complex manifold. Then the (adjoint) cup and contraction

γ:=γ^2,0_X : H¹(X,ΘX)−→Hom(_H^2,0(X),H^1,1(X))

(cf. Notation 1.7.4) is injective. Moreover, whenH^2,0(X)is1-dimensional over the field of complex numbers, thenγis an isomorphism.

Proof. — Let cbe an element of H¹(X,ΘX) such thatγ(c) = 0. Observe that, by definition,γarises from the “ordinary” cup and contraction

γ⁰: H¹(X,ΘX)⊗_CH⁰(X,Ω²_X)−→H¹(X,Ω¹_X)

through tensor-hom adjunction over C (cf. Notation 1.7.4). In particular, for all d∈H⁰(X,Ω²_X), we have

γ⁰(c⊗d) = (γ(c))(d) =0.

AsXis a symplectic complex manifold, there exists a symplectic structureσonX (cf. Definition 3.1.8). Denotesthe image ofσunder the canonical map

Ω²_X(X)−→H⁰(X,Ω²_X). Moreover, denoteφthe composition

ΘX −→ΘX⊗_XOX id⊗σ

−−−→ΘX⊗_XΩ²_X−→Ω¹_X

of morphisms of modules onX, where the first arrow stands for the inverse of the right tensor unit forΘX onX, theσ stands, by abuse of notation, for the unique morphismOX →Ω²_X of modules onXwhich sends the 1 ofOX(X)to the actualσ, and the last arrows stands for the sheaf-theoretic contraction morphism (cf. Notation 1.3.10)

γ²_X(Ω¹_X):ΘX⊗_XΩ²_X−→Ω¹_X. Then, sinceσis nondegenerate onX,

φ: ΘX−→Ω¹_X

is an isomorphism of modules onX(cf. Remarks 3.1.2 a)), whence H¹(X,φ): H¹(X,ΘX)−→H¹(X,Ω¹_X)

is an isomorphism of complex vector spaces by functoriality. By the definition of the cup product, we have

(H¹(X,φ))(c) =γ⁰(c⊗s).

Thus(H¹(X,φ))(c) =0, and, in consequence,c=0. This proves the injectivity ofγ.

Suppose thatH^2,0(X)is 1-dimensional and let f ∈Hom(_H^2,0(X)_,H^1,1(X))

be an arbitrary element. By the surjectivity of H¹(X,φ), there existsc∈H¹(X,ΘX) such that

(γ(c))(s) = (H¹(X,φ))(c) = f(s).

AsH^2,0(X)is nontrivial, we have dim(X)>_{0, whence}s6=0 inH^2,0(X)_. Accord-ingly, asH^2,0(X)is 1-dimensional,sgenerates H^2,0(X)over C. Thusγ(c) = f, which proves thatγis surjective.

Theorem 3.4.4(Local Torelli). — Let f:X → S be a proper, flat morphism of complex spaces such that S is simply connected and smooth and the fibers of f have rational singulari-ties, are of Kähler type, and have singular loci of codimension≥4. Furthermore, let t∈S such that f is semi-universal in t and Xtis symplectic and connected withΩ²_Xt((Xt)_reg)of dimension1over the field of complex numbers. Define g:Y→S to be the submersive share of f . Then the period mapping

P^2,2_t (g):S−→Gr(_H²(Yt)) (cf. Notation 1.7.2 b) is well-defined and the tangent map

(3.4.4.1) Tt(P^2,2_t (g)): TS(t)−→T_Gr(H2(Yt))(F²H²(Yt)) is an injection with1-dimensional cokernel.

Proof. — By Corollary 2.4.15, we know that the Frölicher spectral sequence of g degenerates in entries

I :={(ν,µ)∈Z×Z:ν+µ=2}

at sheet 1 in Mod(S); moreover, for all(p,q) ∈ I, the Hodge moduleH ^p,q(g)is locally finite free onSand compatible with base change in the sense that, for alls∈S, the Hodge base change map

β^p,q_g,s:(_H ^p,q(g))(s)−→H ^p,q(Ys)

is an isomorphism of complex vector spaces. Lets ∈ S. Then asXs has rational singularities,Xsis normal, whence locally pure dimensional according to Proposition 3.3.13. Thus, by Theorem 2.4.1, the Frölicher spectral sequence of Xs\Sing(Xs) degenerates in entries I at sheet 1 in Mod(C). Since the morphism f is flat, the inclusionY→Xinduces an isormophism of complex spaces

Ys−→Xs\Sing(Xs).

Therefore, the Frölicher spectral sequence ofYsdegenerates in entriesIat sheet 1 in Mod(C).

By Theorem 1.7.10 (applied to gin place of f andn = 2), the period mapping P^2,2_t (g)is well-defined (this is implicit in the theorem) and there exists a sequence ψe= (ψe^ν)_ν∈Zsuch that firstly, for allν∈Z,

ψe^ν: F^νH²(Yt)/F^ν+1H²(Yt)−→H^ν,2−ν(Yt) is an isomorphism in Mod(C)and secondly, setting

α:=ψe²◦coker(ι²_Yt(2, 3)) : F²H²(Yt)−→H^2,0(Yt),

β:= (ι²_Yt(₁)_/F²_H²(Yt))◦(ψe¹)⁻¹ _:H^1,1(Yt)−→H²(Yt)_/F²_H²(Yt)_,

the following diagram commutes in Mod(C):

(3.4.4.2) T_S(t) ^{KSg,t //}

Tt(P^2,2_t (g))

H¹(Yt,ΘY_t)

γ^2,0_Yt

Hom(H^2,0(Yt),H^1,1(Yt))

Hom(α,β)

T_Gr(H2(Y_t))(F²H²(Yt)) ^//

θ(H²(Yt),F²H²(Yt))

Hom(F²H²(Yt),H²(Yt)/F²H²(Yt)) commutes in Mod(C)(for the definition ofθ, see Notation 1.6.19).

SinceXthas rational singularities,Xtis Cohen-Macaulay. PutA:=Sing(f). Then clearly Ais a closed analytic subset of X and, due to the flatness of f, we have A∩ |Xt|=Sing(Xt), whence

3<4≤codim(Sing(Xt),Xt) =codim(A∩ |Xt|,Xt). Applying Corollary 3.4.2, we see that the Kodaira-Spencer map

KSg,t: TS(t)−→H¹(Yt,ΘY_t)

is an isomorphism in Mod(C). As Xt is a symplectic complex space, (Xt)_reg is a symplectic complex manifold. AsΩ²_Xt((Xt)_reg)is a 1-dimensional complex vector space,Ω²_(X

t)_reg((Xt)_reg)is a 1-dimensional complex vector space. SinceYt∼= (Xt)_reg (cf. above), the same assertions hold forYtinstead of(Xt)_reg. Therefore, by Lemma 3.4.3, the cup and contraction

γ^2,0_Y

t : H¹(Yt,ΘYt)−→Hom(_H^2,0(Yt),H^1,1(Yt))

is an isomorphism in Mod(C). Since F³H²(Yt) ∼=0, we know that the cokernel of the inclusion

ι²_Yt(2, 3): F³H²(Yt)−→F²H²(Yt)

is an isomorphism. Thus,αis an isomorphism. The morphismθin (3.4.4.2) is an isomorphism anyway. By the definition ofβ,βis certainly injective. So, Hom(α,β)is injective, and exploiting the commutativity of (3.4.4.2), we conclude that the tangent map (3.4.4.1) is injective.

Moreover, the dimension of the cokernel of the injection (3.4.4.1) equals the dimen-sion of the cokernel of the injection Hom(α,β), which in turn equals the dimension of the cokernel ofβsinceH^2,0(Yt), and likewise F²H²(Yt), is 1-dimensional. Now the cokernel ofβis obviously isomorphic toH²(Yt)_/F¹_H²(Yt)_{, and via}ψe⁰, i.e., by the degeneration of the Frölicher spectral sequence ofYtin entry(0, 2)at sheet 1, we have

H²(Yt)/F¹H²(Yt)∼=H^0,2(Yt).

According to Theorem 2.4.1, we have

H^0,2(Yt)∼=H^2,0(Yt).

Thus, we deduce that the cokernel of the tangent map (3.4.4.1) is 1-dimensional.

Reviewing the statement of Theorem 3.4.4, we would like to draw our reader’s attention to the fact that the period mappingP:=_P^2,2_t (g)depends exclusively on the submersive sharegof the original family f. Note that the submersive sharegof f is nothing but the family of regular loci of the fibers of f. Further on, note that the local system with respect to whichPis defined assigns to a pointsinSin principle the second complex cohomology H²((Xs)_reg,C)of(Xs)_reg. ThusPencompasses Hodge theoretic information of the(Xs)_reg, not however a priori of theXs themselves.

The upcoming series of results is aimed at sheding a little light on the relationship betweenXandXreg, forXa compact symplectic complex space of Kähler type, in terms of Hodge structures on second cohomologies. We also discuss ramifications of this relationship for the deformations theory ofX. To begin with, we introduce terminology capturing the variation of mixed Hodge structure in a family of spaces of Fujiki classC.

Construction 3.4.5. — Let f: X→Sbe a morphism of topological spaces,Aa ring, n an integer. Our aim is to construct, given that f satisfies certain conditions (cf.

below), anA-representation of the fundamental groupoid ofSwhich parametrizes theA-valued cohomology modules in degreenof the fibers of f.

Recall that we denote

f^A: (X,AX)−→(S,AS)

the canonical morphism of ringed spaces derived from f. Moreover, set Hⁿ(f,A)_:=_Rⁿ(f^A)_∗(A_X)_.

Thus Hⁿ(f,A)is a sheaf of AS-modules onS. Assume that Hⁿ(f,A)is a locally constant sheaf onS. Then by means of Remark 1.6.5, the sheaf Hⁿ(f,A)_{induces an} A-representation

ρ⁰:Π(S)−→Mod(A). Assume that, for alls∈S, the evident base change map

βs: (Hⁿ(_f,A))_s −→Hⁿ(_Xs,A) is a bijection. Then define

ρⁿ(f,A):Π(S)−→Mod(A) to be the unique functor such that, for alls∈S, we have (ρⁿ(f,A))₀(s) =Hⁿ(Xs,A)

and, for all(s,t)∈S×Sand all morphismsa: s→tinΠ(S), we have ((ρⁿ(f,A))₁(s,t)) (a) =βt◦((ρ⁰)₁(s,t))(a)◦(βs)⁻¹.

As a matter of fact, it would be more accurate to simply defineρ0andρ₁as indicated above, then setρⁿ(f,A):= (ρ₀,ρ₁)and assert that the so definedρis a functor from Π(S)to Mod(S).

When f: X → S is not a morphism of topological spaces but a morphism of complex spaces (resp. a morphism of ringed spaces), we agree on writingρⁿ(f,A)for ρⁿ(ftop,A)in case this makes sense.

Definition 3.4.6(Period mappings for MHS). — Let f: X → S be a proper mor-phism of complex spaces such thatXsis of Fujiki classC for alls∈S. Letnandpbe integers. We define F^pHⁿ(f)to be the unique function on|S|such that, for alls∈ |S|, we have

(F^pHⁿ(f))(s) =F^pHⁿ(Xs).

We call F^pHⁿ(f)thesystem of Hodge filtered pieces in degree p on n-th cohomology associ-ated to f.

Assume further thatStopis simply connected and Hⁿ(f,C)is a locally constant sheaf onStop. Then, for anyt∈S, we set (cf. Construction 1.6.3):

P_t^p,n(f)_MHS:=P^C_t(Stop,ρⁿ(f,C), F^pHⁿ(f)).

Note that this definition makes sense. In fact, ρⁿ(f,C)is a well-defined complex representation ofΠ(Stop)and F^pHⁿ(f)is a complex distribution inρ:=ρⁿ(f,C)as clearly,(F^pHⁿ(f))(s) =F^pHⁿ(Xs)is a complex vector subspace ofρ₀(s) =Hⁿ(Xs,C) for alls∈S.

Lemma 3.4.7. — Let X be a complex space of Fujiki classC having rational singularities and satisfying4≤codim(Sing(X),X).

a) The mapping

j^∗: H²(X,C)−→H²(Xreg,C) induced by the inclusion morphism j: Xreg→X is one-to-one.

b) The composition

H²(X,C)−→H²(Xreg,C)−→H²(Xreg) restricts to a bijection

F²H²(X)−→F²H²(Xreg).

Proof. — a). Let f:W→Xbe a resolution of singularities such that the exceptional locusEof f is a simple normal crossing divisor inWand f induces by restriction an isomorphism

f⁰:W\E−→Xreg

of complex spaces. Then, by Proposition B.2.4, sinceXhas rational singularities, we know that f induces a one-to-one map

f^∗: H²(X,C)−→H²(W,C)

on complex cohomology. Denote

i:W\E−→W,

i⁰: (W,∅)−→(W,W\E) the respective inclusion morphisms. Then the sequence

H²(W,W\E;C) ⁱ

0∗ //H²(W,C) ⁱ

∗ //H²(W\E,C)

is exact in Mod(C). Thus the kernel ofi^∗is precisely theC-linear span of the funda-mental cohomology classes[Eν]of the irreducible componentsEνofE. So, since

f^∗[H²(X,C)]∩Ch[Eν]i={0}, we see that

(f ◦i)^∗=i^∗◦f^∗: H²(X,C)−→H²(W\E,C)

is one-to-one. Therefore, j^∗ is one-to-one taking into account that f⁰ furnishes an isomorphism f◦i→jin the overcategoryAn/X.

b). By Proposition B.2.6, we know that

f^∗|F²H²(X): F²H²(X)−→F²H²(W)

is a bijection. By the functoriality of base change maps, we know that the following diagram, where the vertical arrows denote the respective inclusions, commutes in Mod(C):

(3.4.7.1) F²H²(W) ^//

F²H²(W\E)

H²(W) ^//H²(W\E)

Letp∈ {1, 2}andc∈_H^p,0(W\E). Then by Theorem 3.1.15, there exists one, and only one, elementb∈H ^p,0(W)such thatbis sent tocby the restriction mapping

H ^p,0(W)−→_H^p,0(W\E).

SinceWis a complex manifold of Fujiki classC, the Frölicher spectral sequene of Wdegenerates at sheet 1, whence, specifically,bcorresponds to a closed Kähler p-differential onW. In consequence,ccorresponds to a closed Kähler p-differential onW\E. Varyingcandp, we deduce that the Frölicher spectral sequence ofW\E degenerates in entries(2, 0)at sheet 1. Thus, there exist isomorphisms such that the diagram

F²H²(W) ^//

∼

F²H²(W\E)

∼

H^2,0(W) ^//_H^2,0(W\E) commutes in Mod(C).

As we have already noticed, the Hodge base change H^2,0(W)−→_H^2,0(W\E)

is an isomorphism. Thus from the commutativity of the diagram in (3.4.7.1) we deduce that the composition of morphisms

H²(W,C)−→_H²(W)−→_H²(W\E) restricts to an isomorphism

F²H²(W)−→F²H²(W\E);

note here that by the definition of the Hodge structure H²(W), the Hodge filtered piece F²H²(W)is the inverse image of F²H²(W)under the canonical map

H²(W,C)−→H²(W).

Finally, we observe that since f⁰:W\W →Xregis an isomorphism of complex spaces, f⁰induces an isomorphism

f^0∗: F²H²(Xreg)−→F²H²(W\E)

of complex vector spaces. Thus the commutativity in Mod(C)of the diagram H²(X,C) ^//

f^∗

H²(Xreg,C) ^//

f^0∗

H²(Xreg)

f^0∗

H²(W,C) ^//H²(W\E,C) ^//_H²(W\W) yields our claim.

Proposition 3.4.8. — Let f:X→S be a proper, flat morphism of complex spaces such that S is smooth and simply connected,H²(f,C)is a locally constant sheaf on Stop, and the fibers of f have rational singularities, are of Kähler type, and have singular loci of codimension

≥ 4. Furthermore, let t ∈ S. Define f⁰: X⁰ → S to be the submersive share of f , set P:=P^2,2_t (f)_MHSandP⁰:=P^2,2_t (f⁰), and denote

φt: H²(Xt,C)−→H²(X_t⁰,C)−→_H²(X⁰_t) the composition of canonical mappings. Then, for all s∈S, we have

P⁰(s) =φt[P(s)].

Proof. — Set ρ := ρ²(f,C)(cf. Construction 3.4.5); note that this makes sense as H²(f,C)is a locally constant sheaf onStopandf is proper. By Proposition 2.4.15 a) and c), the algebraic de Rahm moduleH²(f⁰)is locally finite free onS. LetH⁰be the module of horizontal sections of

∇²_GM(f⁰):H²(f⁰)−→_Ω¹_S⊗_SH²(f⁰)

onS (cf. Notations 1.5.7 and 1.6.7). Then by Proposition 1.7.1 b) and Proposition 1.6.10 a),H⁰is a locally constant sheaf onS. Defineρ⁰⁰to beC-representation ofΠ(S)

associatedH⁰(cf. Construction 1.6.4 and Remark 1.6.5). For anys∈S, defineβ⁰_sto be the composition of the following morphisms

(H⁰)_s−→(_H²(f⁰))_s−→(_H²(f⁰))(s)−→_H²(X⁰_s)

in Mod(C), where the first arrow stands for the stalk-at-smorphism onStopassociated to the inclusion morphismH⁰→_H²(f⁰), the second arrow is the evident “quotient morphism”, and the third arrow stands for the de Rham base change in degree 2 for

f⁰ats. Define(ρ⁰)₀to be the function on|S|given by the assignment s7−→_H²(X⁰_s).

Define(ρ⁰)₁to be the unique function on|S| × |S|such that, for all(r,s)∈ |S| × |S|, (ρ⁰)₁(r,s)is the unique function on(Π(S))₁(r,s)satisfying, for alla∈(Π(S))₁(r,s):

((ρ⁰)₁(r,s))(a) =β⁰_s◦((ρ⁰⁰)₁(r,s))(a)◦(β⁰_r)⁻¹. Setρ⁰:= ((ρ⁰)₀,(ρ⁰)₁). Thenρ⁰is functor fromΠ(S)to Mod(_C).

Defineψto be the composition

H²(f,C)−→H²(f⁰,C)−→H⁰

of morphisms of sheaves ofCS-modules onS, where the first arrow signifies the evident base change map and the second arrow denotes the unique morphism from H²(f⁰,C)toH⁰which factors the canonical morphism H²(f⁰,C)→_H²(f⁰)through the inclusionH⁰→H²(f⁰). Letφ= (φs)s∈Sbe the family of morphisms

φ_s: H²(Xs,C)−→H²(X_s⁰,C)−→H²(X_s⁰);

note that this fits with the notation “φt” introduced in the statement of the proposition.

Then by the functoriality of base change maps, the following diagram commutes in Mod(_C)_{for all}s∈S:

(H²(f,C))_s ^{ψs //}

βs

(H⁰)s β⁰_s

H²(Xs,C)

φs

//H²((X⁰)_s)

In consequence,

φ:ρ−→ρ⁰

is a natural transformation of functors from Π(S) to Mod(C). Thus employing Lemma 3.4.7, we see that, for alls ∈ S, letting astand for the unique element of (Π(S))₁(s,t), we have

φ_t[P(s)] =φ_t[ρ₁(s,t)(a)[F²H²(Xs)]] = (φ_t◦ρ₁(s,t)(a))[F²H²(Xs)]

= (ρ⁰₁(s,t)(a)◦φs)[F²H²(Xs)] =ρ⁰₁(s,t)(a)[φs[F²H²(Xs)]]

=ρ⁰₁(s,t)(a)[F²H²(X_s⁰)] =_P⁰(s), which is what had to be proven.

Corollary 3.4.9. — Let f and t be as in Proposition 3.4.8. Then there exists a unique morphism of complex spaces

P⁺:S−→G:=Gr(H²(Xt,C))

such that the underlying function ofP⁺is preciselyP:=_P^2,2_t (f)_MHS; in particular,Pis a holomorphic mapping from S to G. Moreover, the diagram

S

P⁺

yysssssssssss

P⁰

%%J

JJ JJ JJ JJ J Gr(H²(Xt,C))

(φ_t)∗

//Gr(_H²(X⁰_t))

where f⁰,P⁰ andφthave the same meaning as in Proposition 3.4.8 and(φt)∗ := Gr(φt), commutes in the category of complex spaces.

Proof. — By Proposition 3.4.8, we know that |P⁰| = |(φt)_∗| ◦P(in the plain set-theoretic sense) since, for alll ∈ Gr(H²(Xt,C)), we have|(φt)∗|(l) = φt[l]by the definition of(φ_t)∗. Therefore,P⁰(S)⊂(φ_t)∗[G]. Asφ_tis a monomorphism of complex vector spaces,(φt)∗is a closed embedding of complex spaces. Hence, given that the complex spaceSis reduced (for it is smooth by assumption), there exists a morphism of complex spacesP⁺: S→Gsuch that(φt)∗◦_P⁺=_P⁰. From this we obtain

|(φt)∗| ◦ |_P⁺|=|_P⁰|=|(φt)∗| ◦_P,

which implies|P⁺|=Pas|(φ_t)∗|is a one-to-one function. This proves the existence ofP⁺.

WhenP⁺₁ is another morphism of complex spaces fromStoGsuch that|_P1⁺|=_P, we have|_P⁺₁| =|_P⁺|and thusP⁺₁ =_P⁺ by the reducedness ofS. This shows the uniqueness ofP⁺.

Proposition 3.4.10. — Let f:X →S be a proper, flat morphism of complex spaces and t∈S such that Xtis connected and symplectic,Ω²_Xt((Xt)reg)is1-dimensional, S is simply connected,H²(f,C)is a locally constant sheaf on S, and f is fiberwise of Fujiki classC.

a) WhenP:=_P^2,2_t (f)_MHSis a continuous mapping from S to G1:=Gr(1, H²(Xt,C)), there exists a neighborhood V of t in S such thatP(V)⊂QX_t.

b) When S is smooth andPis a holomorphic mapping from S to G₁, we haveP(S)⊂Q_Xt. Proof. — a). Let r be half the dimension of Xt. As Xt is nonempty, connected, symplectic, and of Fujiki classC, there exists a normed symplectic classwonXt(cf.

Proposition 3.1.25 and Definition 3.2.13). We know thatwis a nonzero element of F²H²(Xt). SincePhas image lying in the Grassmannian of 1-dimensional subspaces of H²(Xt,C)andP(t) =F²H²(Xt), we see that F²H²(Xt)is 1-dimensional, whence generated byw. By Proposition 3.1.21, we have:

H²(Xt,C) =H^0,2(Xt)⊕H^1,1(Xt)⊕H^2,0(Xt).

DefineEto be the hyperplane inG₁which is spanned by H^0,2(Xt)⊕H^1,1(Xt). More-over, putV:=P⁻¹(G1\E).

Lets ∈ Vbe arbitrary. Since F²H²(Xs)is 1-dimensional, there exists a nonzero elementv ∈F²H²(Xs). AsH :=H²(f,C)is a locally constant sheaf onSandSis simply connected, the stalk mapH(S)→Hsis one-to-one and onto. Asf is proper, the base change mapHs →H²(Xs,C)is one-to-one and onto, too. Thus there exists a uniqueγ∈H(S)which is sent tovby the composition of the latter two functions.

Writeafor the image ofγin H²(Xt,C). Then, by the definition ofP, we have P(s) =Ca⊂H²(Xt,C).

Thus, by the definition ofQX_t(cf. Definition 3.2.23), we haveP(s)∈QX_tif and only ifqX_t(a) =0.

According to the Hodge decomposition on H²(Xt,C), there exist complex numbers λandλ⁰as well as an elementb∈H^1,1(Xt)such that

a=λw+b+λ⁰w.

By Proposition 3.2.18, the following identity holds:

(3.4.10.1)

Z

Xt

(a^r+1w^r−1) = (r+1)λ^r−1q_Xt(a).

Denoteδthe unique lift ofwwith respect to the functionH(S)→H²(Xt,C). Denote cthe image ofδunderH(S) →H²(Xs,C). As dim(Xs) =2r(due to the flatness of f) andv ∈F²H²(Xs), we see thatv^r+1= 0 in H^∗(Xs,C). In consequence, we have v^r+1c^r−1=0 in H^∗(Xs,C). As the mapping

(H^∗(f,C))(S)−→H^∗(Xs,C)

is a morphism of rings, it sendsγ^r+1δ^r−1tov^r+1c^r−1. Therefore, as (H^4r(f,C))(S)−→H^4r(Xs,C)

is one-to-one, we see thatγ^r+1δ^r−1=0 in(H^∗(f,C))(S). But then,a^r+1w^r−1=0 in H^∗(Xt,C)as this is the image ofγ^r+1δ^r−1under

(H^∗(f,C))(S)−→H^∗(Xt,C).

Thus the left hand side of equation (3.4.10.1) equals zero. AsP(s) =Ca∈/E, we have λ6=0. So, we obtainqX_t(a) =0, whenceP(s)∈QX_t. Asswas an arbitrary element ofV, we deduceP(V)⊂QX_t.

b). We apply a) and conclude by means of the Identitätssatz for holomorphic functions.

Theorem 3.4.11(Local Torelli, II). — Let f: X→ S be a proper, flat morphism of com-plex spaces such that S is smooth and simply connected and the fibers of f have rational singularities, are of Kähler type, and have singular loci of codimension≥4. Furthermore, let t ∈S and suppose that Xtis connected and symplectic withΩ²_Xt((Xt)_reg)of dimension1

overC. Define g:Y→S to be the submersive share of f , setP⁰ :=_P^2,2_t (g), and assume that the tangent map

Tt(P⁰): TS(t)−→T_Gr(H2(Yt))(F²H²(Yt))

is an injection with1-dimensional cokernel. Moreover, assume thatH²(f,C)is a locally constant sheaf on Stop, and setP:=_P^2,2_t (f)_MHS.

a) There exists one, and only one, morphism of complex spaces P⁺:S−→G:=Gr(H²(Xt,C)) such that|_P⁺|=_P.

b) There exists one, and only one, morphism of complex spaces P:S−→Q_Xt

such that j◦P=P⁺, where j:QX_t →G denotes the inclusion morphism.

c) Pis locally biholomorphic at t.

d) The tangent map

Tt(_P⁺): T_S(t)−→T_G(F²H²(Xt)) is an injection with1-dimensional cokernel.

e) The mapping

H²(Xt,C)−→H²((Xt)_reg,C) induced by the inclusion(Xt)_reg→Xtis a bijection.

Proof. — Assertion a) is an immediate consequence of Corollary 3.4.9.

Let us write φt for the composition of the following canonical morphisms in Mod(_C):

H²(Xt,C)−→H²(Yt,C)−→^∼ H²(Yt).

Then by Lemma 3.4.7 a), noting that, due to the flatness off, the morphismit:Yt→Xt

induces an isomorphismYt→(Xt)_reg, we infer thatφtis a monomorphism of complex vector spaces. Denote

(φt)_∗:G=Gr(H²(Xt,C))−→Gr(_H²(Yt)) =:G⁰.

the induced morphism of Grassmannians. By Corollary 3.4.9, we haveP⁰= (φt)∗◦P⁺. Since by assumptionΩ²_Xt((Xt)_reg)is of dimension 1 overC, we see that F²H²(Yt) is of dimension 1 overC. Thus the range ofP⁰ is a subset of the Grassmannian of lines inH²(Yt). In consequence, the range ofP⁺is a subset of the GrassmannianG₁ of lines in H²(Xt,C), whenceP=|_P⁺|is a holomorphic map fromStoG₁. As the fibers of f are compact, reduced, and of Kähler type, the fibers of f are of Fujiki class C. Therefore, by Proposition 3.4.10, we have assertion b).

From b), we deduce thatP⁰ = (φt)∗◦j◦P. Thus by the functoriality of tangent maps, we obtain:

Tt(_P⁰) =T_P(t)((φt)∗)◦T_P(t)(j)◦Tt(_P).

By assumption, Tt(_P⁰)is a monomorphism with cokernel of dimension 1. Therefore, Tt(P)is certainly a monomorphism. Besides, T_P(t)(j)and T_P(t)((φ_t)∗)are monomor-phisms since jand(φt)∗are closed immersions. Since the quadric QX_t is smooth at P(t) = P(t), the cokernel of T_P(t)(j) has dimension 1. Thus both Tt(P) and T_P(t)((φt)∗)are isomorphisms. SinceSis smooth att, we deduce c).

As

Tt(_P⁺) =_TP(t)(j)◦Tt(_P)_,

we see that Tt(P)is an injection with 1-dimensional cokernel, which proves d).

Furthermore, T_P(t)((φt)_∗)is an isomorphism (if and) only ifφtis an isomorphism.

Therefore, as H²(Yt,C)→H²(Yt)is an isomorphism anyway, we see that i^∗_t: H²(Xt,C)−→H²(Yt,C)

is an isomorphism. As we have already pointed out, the morphismitis isomorphic to the inclusion morphism(Xt)_reg →Xtin the overcategoryAn/X_t, whence we deduce e) by means of the functoriality of H²(−,C).

Im Dokument Irreducible symplectic complex spaces (Seite 167-180)