Finite dimensional reduction

Proof. Let (X,MX) be a fine log Banach analytic space proper and anaflat over a Banach analytic space S. Let s₀ ∈ S and i : (X(s₀),M_X(s0)) → (X₀,M_X0) a log isomorphism. Sincei^∗q^†₀is a triangularly privileged log cuirasse (see Definition 5.14) on (X(s₀),M_X(s0)), we have thatQ_S(X,M_X) is smooth overSin a neighborhood ofi^∗q^†₀ (Proposition 5.19). Therefore there exists a local relative log cuirasseq^† on (X,MX) defined in a neighborhood S⁰ of s₀ in S. Hence, taking ϕ_q^† : S → Z^log (Definition 5.21) and the S⁰-isomorphismα_q^†|_S⁰ : (X_ϕ

q†,M_Xϕ

q†)→(X,M_X) (Proposition 5.23), the statement follows.

5.2. VERSAL DEF. OF COMPACT FINE LOG COMPLEX SPACES 91 For the sake of simplicity, if we identify (X,MX)(s) with (X_log,MX_log)(ϕ_σ^†(s)) and hence σ^†(s) with α_σ^†(s)^∗σ^†(s), then we can rewrite (5.10) as

ψ_σ^† :S →Q_Z^log(X_log,M_Xlog)

s7→(z_σ^†_(s), σ^†(s)) . (5.11) By construction, ψ_σ^† factors through Z^log (5.9). Let qand ψ_q:S →Z given by (4.4) and (4.8) respectively. If we denote with π_Z :Z^log →Z the forgetful morphism, then, clearly, π_Z◦ψ_q^† =ψ_q. Let q₀^† be a triangularly privileged cuirasse on (X₀,M_X0) (see Definition 5.14) and z_q^†

0 ∈Z^log the associated log puzzle (see Definition 5.20), hence (z_q^†

0, q^†₀)∈Z^log. Let

α_q^†

0

: (X_log,MX_log)(z_q^†

0

)→(X₀,MX0)

be the log isomorphism given by Proposition 5.23. Let π : Q_Zlog(X_log,M_Xlog) → Z^log be the projection and i : Z^log ,→ Q_Z^log(X_log,M_Xlog) the canonical injection. Set (X_Z^log,M_X

Zlog) :=i^∗π^∗(X_log,M_Xlog) and consider the log isomorphism i^∗π^∗α_q^†

0 : (X_Z^log,M_X

Zlog)(z_q^†

0, q^†₀)→(X₀,M_X0).

Proposition 5.26. The morphism ((X_Z^log,M_X

Zlog) → (Z^log,(z_q^†

0, q₀^†)), i^∗π^∗α_q^†

0) is a complete deformation of (X₀,M_X0).

Proof. Let (X,M_X) be a fine log Banach analytic space proper and anaflat over a Banach analytic space S. Let s₀ ∈ S and i : (X(s₀),M_X(s0)) → (X₀,M_X0) a log isomorphism. Since i^∗q^†₀ is a triangularly privileged log cuirasse (see Definition 5.14) on (X(s₀),M_X(s0)), we have that Q_S(X,M_X) is smooth over S in a neighborhood of i^∗q₀^† (Proposition 5.19). Therefore, there exists a local relative log cuirasse q^† on (X,M_X) defined in a neighborhoodS⁰ ofs₀ inS. Letϕ_q^† :S →Z^log be the morphism given by Definition 5.21 and ψ_q^† : S → Z^log given by Definition 5.10. Let π ◦ i : Z^log → Q_Z^log(X_log,MX_log) → Z^log be the composition of the canonical projection π with the canonical injection i. Since π◦i◦ψ_q^† =ϕ_q^†, we have ψ_q^∗†(X_Z^log,M_X

Zlog) = ϕ^∗_q†(X_log,M_Xlog).

In what follows, in the same way as done in Section 4.2, we are going to decompose Z^loginto a product Σ^log×R^log, where Σ^log is a Banach manifold (see Definition 3.4) and R^log is a finite dimensional complex analytic space satisfying the versality property in a neighborhood of (z_q^†

0, q₀^†). To do that, we start by noticing that the restriction of q^† (5.7) to Z^log gives us a canonical relative log cuirasse on (X_Z^log,M_X

Zlog) over Z^log σ_Zlog :Z^log → Q_Zlog(X_Zlog,M_X

Zlog). (5.12)

Via the projection π : Q_Z^log(X_Zlog,MX_Zlog) → Z^log, we get a canonical relative log cuirasse π^∗σ_Zlog on π^∗(X_Zlog,M_X

Zlog) over Q_Zlog(X_Zlog,M_X

Zlog), namely π^∗σ_Z^log :Q_Z^log(X_Z^log,M_X

Zlog)→ QQ

Zlog(X_Zlog,M_X

Zlog)(π^∗(X_Z^log,M_X

Zlog))

(z_q^†, q^†, p^†)7→(z_q^†, q^†, p^†, p^†) . (5.13) By Proposition 5.23, we get a log Q_Z^log(X_Z^log,M_X

Zlog)-isomorphism α_π^∗_σ

Zlog :ϕ^∗_π∗σ_Zlog(X_log,M_Xlog)→π^∗(X_Z^log,M_X

Zlog). (5.14) Let

ψ_Z^log :Q_Z^log(X_Z^log,M_X

Zlog)→Z^log s^† := (z_q^†, q^†, p^†)7→(z_(απ∗σ

Zlog(s^†))^∗p^†,(α_π^∗_σ

Zlog(s^†))^∗p^†) (5.15) be the versal morphism associated to the relative log cuirasseπ^∗σ_Z^logonπ^∗(X_Z^log,M_X

Zlog) over Q_Z^log(X_Z^log,M_X

Zlog) (see (5.10)). In simpler terms, by (5.11), we can write ψ_Z^log :Q_Z^log(X_Z^log,M_X

Zlog)→Z^log

(z_q^†, q^†, p^†)7→(z_p^†, p^†). (5.16) The proof of Proposition 5.27 is identical to the one of Proposition 4.35.

Proposition 5.27. Let S be a Banach analytic space and f₁, f₂ : S → Z^log mor-phisms. If f₁^∗(X_Z^log,M_X

Zlog) ' f₂^∗(X_Z^log,M_X

Zlog), then there exists a morphism k^† : S → Q_Z^log(X_Z^log,M_X

Zlog), such that the following diagram commutes Z^log

S Q_Z^log(X_Z^log,M_X

Zlog)

Z^log

f1

f2

k^†

ψ_Zlog

π

. (5.17)

Proof. Let

f₁ :S →Z^log s7→(z_q^†

1(s), q₁^†(s)) (5.18)

and

f₂ :S →Z^log s7→(z_q^†

2(s), q^†₂(s)). (5.19)

5.2. VERSAL DEF. OF COMPACT FINE LOG COMPLEX SPACES 93 Let α : f₁^∗(X_Zlog,MX_Zlog) → f₂^∗(X_Zlog,MX_Zlog) be a log S-isomorphism. For each s ∈S, it induces an isomorphism of spaces of absolute log cuirasses

α_Q,s :Q((X_Z^log,M_X

Zlog)(f₁(s)))→ Q((X_Z^log,M_X

Zlog)(f₂(s)))

q₁^†7→α(s)∗q₁^† . (5.20) Hence, we have (f₂(s), α(s)_∗q₁^†(s))∈ Q_Zlog(X_Zlog,M_X

Zlog). Let απ^∗σ_Zlog :ϕ^∗_π^∗_σ

Zlog(X_log,MX_log)→π^∗(X_Zlog,MX_Zlog) be the log Q_Z^log(X_Z^log,M_X

Zlog)-isomorphism given by (5.14). By the definition of the fibre product (Corollary 3.26 and Corollary 2.31), we can canonically write

π^∗(X_Z^log,M_X

Zlog)(f₂(s), α(s)∗q^†₁(s)) = (X_Z^log,M_X

Zlog)(f₂(s)).

On the other hand, we can canonically write

(ϕ^∗_π∗σ_Zlog(X_log,M_Xlog))(f₂(s), α(s)∗q₁^†(s)) = (X_log,M_Xlog)(z_α(s)

∗q^†₁(s))

see Definition 5.21. Since α(s) is a log isomorphism, by the construction of log puz-zle associated to a log cuirasse (Definition 5.20), we have z_α(s)

∗q₁^†(s) = z_q^†

1(s). Hence, (X_log,M_Xlog)(z_α(s)

∗q^†₁(s)) = (X_log,M_Xlog)(z_q^†

1(s)). Moreover, again by the definition of the fibre product, we can canonically write

(X_log,M_Xlog)(z_q^†

1(s)) = (X_Zlog,M_X

Zlog)(z_q^†

1(s), q₁^†(s)).

Therefore, (ϕ^∗_π^∗_σ

Zlog(X_log,M_Xlog))(f₂(s), α(s)∗q₁^†(s)) = (X_Z^log,M_X

Zlog)(f₁(s)).

Thus, for each s∈S, α_π^∗_σ

Zlog naturally induces a log isomorphism α⁰(s) : (X_Z^log,M_X

Zlog)(f₁(s))→(X_Z^log,M_X

Zlog)(f₂(s)).

Therefore, the map

k^†:S → Q_Zlog(X_Zlog,M_X

Zlog)

s7→(f₂(s), α⁰(s)_∗q^†₁(s)) (5.21) is well defined. Then, by (5.15) and by the construction of the isomorphism α⁰(s), the statement follows.

Now, by Proposition 5.19, we have that the canonical projection π:Q_Zlog(X_Zlog,M_X

Zlog)→Z^log

is smooth in a neighborhood of (z_q^†

0, q₀^†, q₀^†). Therefore, setting Q^log₀ := Q(X0,MX0), the space of log cuirasses on (X₀,M_X0) (see Definition 5.10), we can choose a local trivialization

γ^†:Z^log ×Q^log₀ → Q_Z^log(X_Z^log,M_X

Zlog). (5.22)

Let σ_Z^log given by (5.12) and α_q^†

0 : (X_Z^log,M_X

Zlog)(z_q^†

0, q^†₀)→(X₀,M_X0)

the log isomorphism given by Proposition 5.23¹. We naturally get an induced isomor-phism α_Qlog

0 :Q^log₀ → Q((X_Zlog,M_X

Zlog)(z_q^†

0, q^†₀)). We choose γ^† such that γ^†|_Zlog×{q₀^†} =σ_Z^log and γ^†|_{(z

q† 0

,q₀^†)}×Q^log₀ =α_Q^log

0 . (5.23)

Let ψ_Z^log given by (5.15). We can construct a morphism ω^†:Z^log ×Q^log₀ ^γ

†

→ Q_Zlog(X_Zlog,M_X

Zlog)^ψ→^Zlog Z^log (5.24) via ω^†:=ψ_Z^log ◦γ^†. Since γ^†|_Zlog×{q^†₀} =σ_Z^log (5.23), we get that

ω^†|_Zlog×{q^†₀} =ψ_Z^log◦σ_Z^log = Id_Z^log. (5.25) The proof of Proposition 5.28 is identical to the proof of Proposition 4.36.

Proposition 5.28. LetS be a Banach analytic space andf, g:S →Z^log morphisms.

Then f^∗(X_Z^log,M_X

Zlog)'g^∗(X_Z^log,M_X

Zlog), if and only if there exists h^† :S → Q^log₀ such that the following diagram commutes

Z^log ×Q^log₀

S

Z^log

(f, h^†)

g

ω^†

. (5.26)

Proof. Letf, g:S →Z^log be morphisms, withf^∗(X_Z^log,MX_Zlog)'g^∗(X_Z^log,MX_Zlog).

Using Proposition 5.27, with f₁ := g and f₂ := f, we get a morphism k^† : S → Q_Z^log(X_Z^log,M_X

Zlog), such that ψ_Z^log ◦k^† =g and π◦k^† =f. Let γ^† :Z^log×Q^log₀ →

1Letπ◦i:Z^log → Q_Zlog(Xlog,MX_log)→Z^log be the canonical projection, we identifyα_q† 0

with i^∗π^∗α_q†

0

.

5.2. VERSAL DEF. OF COMPACT FINE LOG COMPLEX SPACES 95 Q_Z^log(X_Zlog,MX_Zlog) be the local trivialization chosen in (5.22). Then, setting h^† :=

π₂◦(γ^†)⁻¹◦k^† and using (5.24), we get

ω^†◦(f, h^†) =ψ_Z^log ◦γ^†◦(f, π₂◦(γ^†)⁻¹◦k^†), that is, using π◦k^†=f

ω^†◦(f, h^†) =ψ_Z^log◦γ^†◦(π◦k^†, π₂◦(γ^†)⁻¹◦k^†), and equivalently

ω^†◦(f, h^†) =ψ_Z^log ◦k^†.

Usingψ_Z^log◦k^† =g, the statement follows. Conversely, let us assume the existence of a morphism h^† :S → Q^log₀ such that ω^†◦(f, h^†) =g. Letπ : Q_Z^log(X_Z^log,M_X

Zlog)→ Z^log be the projection and ψ_Z^log : Q_Z^log(X_Z^log,M_X

Zlog) → Z^log given by (5.15). By Definition 5.15 and Proposition 5.26,

π^∗(X_Zlog,M_X

Zlog)'ψ_Z^∗log(X_Zlog,M_X

Zlog).

Set ρ := γ^†◦(f, h^†). Since π◦ρ = f and ψ_Zlog ◦ρ = g (by (5.24)), the statement follows.

Let us denote with Ex¹(X₀,M_X0) the set of equivalence classes of infinitesimal deformations of (X₀,MX0). Using (5.24), we define

δ^†:=ω^†|_{(z

q† 0

,q^†₀)}×Q^log₀ . (5.27) LetD= ({·},C[]/²) be the double point. Using Proposition 5.28 forS =D, we get that

Ex¹(X₀,M_X0) = T_(z

q† 0

,q₀^†)Z^log/ImT_q^†

0δ^†.

As in the classical case (see Section 4.2), we show in Proposition 5.31 that T_q^†

0δ^† is a Fredholm operator of index 0. This fact, together with Proposition 5.28, allows us to get a finite dimensional versal deformation space (see Theorem 5.32).

Definition 5.29. Let (X,M_X) be a fine log Banach analytic space proper and anaflat over a Banach analytic space S. Let q^† : S → QS(I; (X,MX)) be a relative log cuirasse on (X,M_X) overS. We recall from Definition 5.10 thatq^†(s) = (q(s),(θ_i(s) : P_i → M_Xϕq(s)|_Y^◦

i (s)),(η_jⁱ(s) : P_i → O^×_Y◦

j(s))), s ∈ S. We say that q^† is extendable if there exists a type of cuirasse ˆI (see Definition 4.8) and a relative log cuirasse

ˆ

q^† : S → Q_S(ˆI; (X,M_X)) of type ˆI on (X,M_X) over S such that the relative cuirasse ˆq underlying ˆq^† extends the relative cuirasse q underlying q^† (see Definition 4.38) and, for each i∈I,

1. θi = ˆθi|Yi; 2. η_jⁱ = ˆηⁱ_j|Yj.

Exactly as in the classical case (see Definition 4.38), extendable relative log cuirasses exist on a fine log Banach analytic spaces (X,M_X) proper and anaflat over a Banach analytic spaceS. Indeed, let ˆI:= ( ˆI•,( ˆK_i)_i∈Iˆ,(Kˆ˜_i)_i∈Iˆ,( ˆK⁰_i)_i∈Iˆ0∪Iˆ1) be a type of cuirasse and ˆqa relative log cuirasse of type ˆIon (X,M_X) overS. Then, by slightly shrinking each polycylinder ˆK_i,Kˆ˜_i and ˆK_i⁰ in ˆI, we can get polycylinders K_i, ˜K_i and K_i⁰ respectively and hence a type of cuirasse I := ( ˆI•,(K_i)_i∈Iˆ,( ˜K_i)_i∈Iˆ,(K_i⁰)_i∈Iˆ0∪Iˆ1), such that I b I. Then,ˆ q := ˆq|_I is an extendable relative log cuirasse on (X,M_X) over S. By Proposition 5.19, the projection π : Q_Z^log(X_log,M_Xlog) → Z^log is smooth in a neighborhood of (z_q^†

0, q₀^†). Therefore, we can find an extendable local relative log cuirasse

σ^†:Z^log → Q_Z^log(X_log,M_Xlog) (5.28) on (X_log,M_Xlog) overZ^log, such that σ^†(z_q^†

0) = (z_q^†

0, q₀^†). Moreover, we can find a local trivialization

ρ^†:Q_Z^log(X_log,MX_log)→Z^log×Q^log₀ (5.29) of the form (π, p^†), where

p^†:Q_Z(X_log,M_Xlog)→Q^log₀ (5.30) is a morphism such that

p^†◦σ^†={q^†₀}. (5.31)

Furthermore, let α_q^†

0 : (X_log,M_Xlog)(z_q^†

0)→ (X₀,M_X0) be the log isomorphism given by Proposition 5.23. We naturally get an induced isomorphism

α_Q^log

0 :Q((X_log,M_Xlog)(z_q^†

0))→Q^log₀ . We choose p^† so that

p^†|Q((X_log,M_X

log)(z

q† 0

))=α_q^†

0. (5.32)

The proof of Proposition 5.30 is identical to the proof of Proposition 4.39.

Proposition 5.30. The morphism p^† :Z^log →Q^log₀ is of relative finite dimension in a neighborhood of (z_q^†

0, q^†₀).

Proof. Let ϕ_q^† be as in (5.8), π :Q_Z^log(X_log,M_Xlog) →Z^log the canonical projection, σ^† the extendable relative log cuirasse on (X_log,M_Xlog) overZ^log chosen in (5.28) and p^† :Q_Zlog(X_log,M_Xlog)→Q^log₀ the morphism chosen in (5.30). We can construct two morphisms

(p^†, π)◦σ^†:Z^log →Z^log× {q^†₀}

z^†7→(z^†, q₀^†) (5.33)

5.2. VERSAL DEF. OF COMPACT FINE LOG COMPLEX SPACES 97 and

(p^†, ϕ_q^†)◦σ^†:Z^log →Z^log× {q^†₀}

z^†7→(z_σ^†_(z^†₎, q₀^†). (5.34) On one hand we have that (p^†, π)◦σ^† is nothing but the identity map Id_Zlog on Z^log. On the other hand, sinceσ^†is extendable (see Definition 5.29), the morphismϕ_q^†◦σ^† factors through the restriction morphism

j^†:Z^ˆI,log →Z^log

(( ˆY_i),(ˆg^j_i),( ˆβ_i),(ˆηⁱ_j))7→(( ˆY_i∩K_i),(ˆg_i^j|_Yˆ

j∩K_j),( ˆβ_i|_Yˆ

i∩K_i),(ˆη_jⁱ|_Yˆ

j∩K_j)), (5.35) with I bˆI. Since the morphism j^† is a compact morphism (see [Dou74], p. 595 and [Sti88], p. 17), we get that ϕ_q^† ◦σ^† is compact. By Proposition 3.33, we get that

T^log := ker((p^†, ϕ_q^†)◦σ^†,Id_Zlog)⊂Z^log

is of finite dimension. From (5.31), we getσ^†(T^log) = (p^†, π)⁻¹({q₀^†}×Z^log) = (p^†)⁻¹(q^†₀).

SinceZ^log = ker((p^†◦ϕ_q^†),(p^†◦π)), we have thatσ^†(T^log) = (p^†|_Zlog)⁻¹(q₀^†). Hence, we have that the fibre (p^†|_Zlog)⁻¹(q₀^†) ofp^†|_Zlog :Z^log →Q^log₀ overq^†₀ is of finite dimension.

Using Proposition 3.37, the statement follows.

We can draw the following diagram

Q^log₀ Z^log Q^log₀ ×C^m

Q^log₀

ι^†

p^† δ^†

π1

. (5.36)

Analogously, the proof of the Proposition 5.31 is identical to the proof of Proposition 4.40.

Proposition 5.31. The linear tangent map:

T_q^†

0(p^†◦δ^†) :T_q^†

0Q^log₀ →T_q^†

0Q^log₀ is of the form 1−v^†, with v^† compact.

Proof. Let Q^log₀ ×(X₀,M_X0) → Q^log₀ be the trivial deformation of (X₀,M_X0) over Q^log₀ . The triangularly privileged log cuirasse q^†₀ on (X₀,M_X0) induces a relative log cuirasseq^†₀ :Q^log₀ → Q_Q^log

0 (Q^log₀ ×(X₀,M_X0)) =Q^log₀ ×Q^log₀ onQ^log₀ ×(X₀,M_X0) over Q^log₀ via q^† 7→ (q^†, q^†₀). Let β^† : Q^log₀ ×(X₀,M_X0) → (X_log,M_Xlog) be the morphism

induced by the morphism ϕ_q^†

0

:Q^log₀ →Z^log associated toq^†₀ (see Definition 5.21). We have the following commutative diagram

Q^log₀ ×(X₀,M_X0) (X_log,M_Xlog)

Q^log₀ Z^log

β^†

π1

ϕq^†₀

. (5.37)

Let ρ^† := (π, p^†) :Q_Z^log(X_log,MX_log) → Z^log ×Q^log₀ be the local trivialization chosen in (5.29). From (5.37), we get the following induced commutative diagram

Z^log×Q^log₀

Q^log₀ ×Q^log₀ Q_Z^log(X_log,M_Xlog)

Q^log₀ Z^log

(ϕ_q† 0

,p˜^†) (π, p^†)

β^†

π1 π

ϕ_q† 0

, (5.38) where ˜p^† := p^†◦β^†. Let σ^† : Z^log → Q_Z^log(X_log,M_X„log) be the extendable relative log cuirasse chosen in (5.28). Set τ^† := ϕ^∗

q^†₀σ^† : Q^log₀ → Q^log₀ ×Q^log₀ . Since σ^† is a section of the projection π : Q_Zlog(X_log,M_Xlog) → Z^log, we get that τ^† is of the form (Id_Q^log

0 , θ^†). Moreover, since σ^† is extendable (see Definition 5.29), T_q^†

0θ^† is a compact morphism (see Definition 3.32). Indeed, θ^† is of the form j^† ◦θˆ^†, where θˆ^† : Q^log₀ → Q(ˆI; (X₀,M_X0)), with I b I, andˆ j^† : Q(ˆI; (X₀,M_X0)) → Q^log₀ is the restriction morphism, which is a compact morphism (see [Dou74], p. 595 and [Sti88], p. 17). Because τ^†=ϕ^∗

q^†₀σ^†, we get that p^†◦σ^†◦ϕ_q^†

0 = ˜p^†◦τ^†. Hence, using (5.31), we get that

T_q^†

0(˜p^†◦τ^†) = 0. (5.39)

Let i₁ : Q^log₀ → Q^log₀ ×Q^log₀ and i₂ : Q^log₀ → Q^log₀ ×Q^log₀ be the injections defined by q^†7→(q^†, q₀^†) and q^†7→(q^†₀, q^†) respectively. Then, since τ^†= (Id_Q^log

0 , θ^†), we get T_q^†

0

(˜p^†◦τ^†) =T_q^†

0

(˜p^†◦(Id_Qlog

0 , θ^†)) =T_q^†

0

(˜p^†◦i₁) +T_q^†

0

(˜p^†◦i₂)◦T_q^†

0

θ^†. (5.40)

5.2. VERSAL DEF. OF COMPACT FINE LOG COMPLEX SPACES 99 From (5.32), we obtain

˜

p^†◦i₂ =p^†◦β^†◦i₂ =Id_Qlog

0 . (5.41)

Thus, since T_q^†

0(˜p^†◦τ^†) = 0 by (5.39), from (5.40), using (5.41), we get T_q^†

0

(˜p^†◦i₁) = −T_q^†

0

θ^†. (5.42)

Let ψ_q^†

0

: Q^log₀ → Z^log be the morphism associated to q^†₀ (see (5.10)). From the definition of δ^† :Q^log₀ → Q_Z^log(X_log,MX_log) (see (5.27)), we get that δ^†=ψ_q^†

0

. Let 4: Q^log₀ → Q^log₀ ×Q^log₀ ,q^†7→(q^†, q^†), be the diagonal morphism. From the commutative diagram (5.38) and from π◦ψ_q^†

0

=ϕ_q^†

0

(5.10), we getβ^†◦ 4=δ^†. Hence,

p^†◦δ^† =p^†◦β^†◦ 4= ˜p^†◦ 4. (5.43) Thus, using (5.43), (5.42) and (5.41), we get

T_q^†

0(p^†◦δ^†) = T_q^†

0(˜p^†◦ 4) =T_q^†

0(˜p^†◦i₁) +T_q^†

0(˜p^†◦i₂) =−T_q^†

0θ^†+ Id_Q^log

0 .

From Proposition 5.31, we get that kerT_q^†

0(p^† ◦δ^†) is finite dimensional and a direct subspace of T_q^†

0Q^log₀ . By Proposition 5.30 kerT_q^†

0(p^†◦δ^†)⊃kerT_q^†

0δ^† = kerT_q^†

0(ι^†◦δ^†), we get that kerT_q^†

0(ι^†◦δ^†) is of finite dimension. Moreover, from Proposition 5.31 we have that ImT_q^†

0(p^† ◦δ^†) has finite codimension in T_(z

q† 0

,q^†₀)Z^log. Therefore, using again Proposition 5.30, we get that ImT_q^†

0(ι^†◦δ^†) is of finite codimension. Thus, let us identify Z^log with its image inQ^log₀ ×C^m underι^†(5.36). Let Σ^log be the subspace of Q^log₀ such that

T_q^†

0Σ^log⊕kerT_q^†

0δ^† =T_q^†

0Q^log₀ .

We choose a retractionr^†:Q^log₀ ×C^m →δ^†(Σ^log) and we set R^log := (r^†)⁻¹(q₀^†)∩Z^log. Let i^†:R^log ,→Z^log be the inclusion and set

(X_R^log,M_X

Rlog) := (i^†)^∗(X_Z^log,M_X

Zlog).

Moreover, we setr₀^†:= (z_q^†

0, q₀^†). Let α_q^†

0

: (X_Rlog,MX_Rlog)(r^†₀)→(X₀,MX0) be the log isomorphism given by Proposition 5.23. The proof of the next Theorem is identical to the one of Theorem 4.43.

Theorem 5.32. The morphism (X_R^log,M_X

Rlog) → (R^log, r₀^†), endowed with the iso-morphism α_q^†

0

is a versal deformation of (X₀,M_X0).

Proof. Let ((X,MX) →(S, s₀), i) be a deformation of (X₀,MX0). Since, by Propo-sition 5.26, we have that ((X_Zlog,M_X

Zlog) → Z^log,(z_q^†

0, q^†₀)) is versal, there exists a morphism ψ_† : S → Z^log such that (X,M_X) is isomorphic to ψ_†^∗(X_Zlog,M_X

Zlog).

Let π_R^log : R^log × Σ^log → R^log and π_Σ^log : R^log × Σ^log → Σ^log be the projec-tions. Exactly as in Proposition 4.42, using Lemma 4.41, we get that the mor-phism ω^†|_Rlog×Σ^log : R^log × Σ^log → Z^log (5.24) is an isomorphism. Thus, setting g :=π_R^log ◦(ω^†|_R^log_×Σ^log)⁻¹◦ψ† and h^†:=π_Σ^log◦(ω^†|_R^log_×Σ^log)⁻¹◦ψ†, we obtain

ω^†◦(g, h^†) = ψ†. Hence, by Proposition 5.28, we get

g^∗(X_R^log,M_X

Rlog)'ψ_†^∗(X_Z^log,M_X

Zlog)'(X,M_X).

Hence, we get completeness. Moreover, by construction, T_q^†

0

R^log = Ex¹(X₀,M_X0).

Hence, our deformation is effective. Now, let ((S, s₀),(X,M_X), i) be a log deforma-tion of (X₀,M_X0) and (S⁰, s₀) a subgerm of (S, s₀). Let h⁰ : (S⁰, s₀) → (R^log, r₀) such that (X,MX)|S⁰ ' h^0∗(X_Rlog,MX_Rlog). Let q^† be the canonical relative log cuirasse on (X_Z^log,M_X

Zlog) over Z^log (see (5.7)). Then, h^0∗q^† is a relative log cuirasse on (X,M_X)|_S⁰ over S⁰, whose associated morphism (see (5.10)) coincides with h⁰. Since, by Proposition 5.19, QS(X,MX) is smooth over S in a neighborhood of q₀^† ∈ Q((X(s₀),M_X(s0)), there exists a relative cuirasse q^† on (X,M_X) over S, such that q^†|S⁰ = h^0∗q^†. Let ˜h : S → Z^log be the morphism associated to q^† (see (5.10)) and π_R^log : Z^log → R^log the projection. Then, h := π_R^log ◦˜h satisfies (X,M_X) ' h^∗(X_R^log,M_X

Rlog) and h|_S⁰ =h⁰.

Im Dokument Existence of a versal deformation for compact complex analytic spaces endowed with logarithmic structure (Seite 90-100)