1 Introduction

The main aim of this work is to extend the following classical results in analytic deformation theory, Theorems 1.1 and 1.2, to the category of compact fine log complex spaces.

Theorem 1.1

([4, 9, 11, 27]) Every compact complex analytic space admits a semi-universal deformation.

Theorem 1.2

([7, p. 130]) Every morphism between compact complex analytic spaces admits a semi-universal deformation.

We start by briefly reviewing some results in analytic deformation theory and by fixing some notation. For background material on complex analytic geometry, we recommend [6], whereas, as references for log geometry, we recommend [12, 26] and [21]. The latter, in particular, explicitly deals with log structures on complex analytic spaces.

Definition 1.3

Let $$X_0$$ be a compact complex analytic space. A deformation of $$X_0$$ is a triple ((S, 0), Xi) consisting of a flat and proper morphism of complex spaces $$\pi :X\rightarrow S$$ and an isomorphism $$i:X_0\rightarrow X(0)$$, where $$X(0):=\pi ^{-1}(0)$$.

A deformation ((S, 0), Xi) of a compact complex space $$X_0$$ is called complete, if it contains, in a small neighborhood of the base point $$0\in S$$, all possible deformations of $$X_0$$. Technically, this means that if ((T, 0), Yj) is another deformation of $$X_0$$, then there exists a morphism of germs $$\varphi :(T,0)\rightarrow (S,0)$$ and an isomorphism $$\alpha :Y\rightarrow \varphi ^{*}X$$, such that $$\alpha \circ j=\varphi ^{*}i$$.

Let $$D:=(\{\cdot \},{\mathbb {C}}[\epsilon ]/\epsilon ^2)$$ be the double point and (S, 0) a germ of complex spaces. Denote with $${\text {Hom}}(D,(S,0))$$ the set of morphisms of germs $$D\rightarrow (S,0)$$. We have a bijection

\begin{aligned}{\text {Hom}}(D,(S,0))\rightarrow \text {T}_0S\end{aligned}

sending $$u:D\rightarrow (S,0)$$ to $$du(v)\in \text {T}_0S$$, where $$v\in \text {T}D$$ is a basis element. If we denote with $${\text {Ex}}^{1}(0)$$ the set of isomorphism classes of deformations of $$X_0$$ over D, we get a natural morphism

\begin{aligned} \text {ks}:\text {T}_0S\rightarrow {\text {Ex}}^{1}(0), \end{aligned}
(1)

via $$u\mapsto u^*\pi$$. This morphism is called the Kodaira–Spencer map. If ((S, 0), Xi) is a complete deformation of $$X_0$$, then $$\text {ks}$$ is an epimorphism. If $$\text {ks}$$ is an isomorphism, the deformation is called effective (see, for instance, the discussion in [28, pp. 130–134]). In 1958, Kodaira, Nirenberg and Spencer ([23]) proved that if $$X_0$$ is a compact complex manifold with $$H^2(X_0;{\mathcal {T}}_{X_0})=~0$$, then $$X_0$$ admits a complete and effective deformation with smooth base space. In 1962, Kuranishi ([24]) proved the existence of a complete and effective deformation without the condition $$H^2(X_0;{\mathcal {T}}_{X_0})=0$$. In this case, the base space is a germ of complex spaces, in general singular. In 1964, A. Douady ([2]), using his theory of Banach analytic spaces, succeeded in giving a very elegant exposition of the results of Kuranishi.

Definition 1.4

([4, p. 601, Proposition 1], [34, p. 5, Definition 0.8]) Let $$X_0$$ be a compact complex space. A deformation ((S, 0), Xi) is called versal if given any other deformation ((T, 0), Yj) of $$X_0$$, a subgerm $$(T',0)$$ of (T, 0) and a morphism $$h':(T',0)\rightarrow (S,0)$$ such that $$Y|_{T'}\simeq h^{'*}X$$, there exists a morphism $$h:(T,0)\rightarrow (S,0)$$ such that $$Y\simeq h^{*}X$$ and $$h|_{T'}=h'$$.

In literature, a versal and effective deformation is called semi-universal or miniversal. By a general result of Flenner ([8, Satz 5.2]), every versal deformation gives a semi-universal deformation.

We outline the key ideas in Douady’s construction of a semi-universal deformation of a compact complex space. We start by noticing that we can cover a compact complex space $$X_0$$ with finitely many open subsets $$(U_i)_{i\in I_0}$$, such that, for each $$i\in I_0$$, there exists a closed subset $$Z_i\subset W_i$$, for some $$W_i$$ open in $${\mathbb {C}}^{n_i}$$, and an isomorphism

\begin{aligned} f_i:Z_i\rightarrow U_i. \end{aligned}
(2)

Moreover, we can find an isomorphism of the form (2) for any double $$U_{ij}:=U_i\cap U_j$$ and triple $$U_{ijk}:=U_i\cap U_j\cap U_k$$ intersection. The collection of closed subspaces $$((Z_i),(Z_{ij}),(Z_{ijk}))$$ is a disassembly of $$X_0$$, where the assembly instructions are encoded into the isomorphisms $$((f_i),(f_{ij}),(f_{ijk}))$$ via the transition maps $$(f_i^{-1}\circ f_j)$$. A deformation of $$X_0$$ is obtained by deforming each closed subspace $$Z_i$$, together with the gluing morphisms $$f_i$$, and by assembling together the obtained deformed subspaces.

Douady’s key insight was to choose special (“privileged”) subspaces $$(Y_i)$$ of given polycylinders $$(K_i\subset {\mathbb {C}}^{n_i})$$ for the closed subspaces $$(Z_i)$$, and to show that the collection of all privileged subspaces of a given polycylinder can be endowed with an analytic structure. More precisely, given a polycylinder $$K_i\subset {\mathbb {C}}^{n_i}$$, we can consider the Banach algebra

\begin{aligned} B(K):=\{h:K_i\rightarrow {\mathbb {C}}| h\text { is continuous on }K_i\text { and analytic on its interior}\}. \end{aligned}

An ideal $$I\subset B(K)$$ is called direct if there exists a $${\mathbb {C}}$$-vector subspace J of B(K), such that $$B(K)=I\oplus J$$ as $${\mathbb {C}}$$-vector spaces. Douady showed in [3, p. 34], that the set

\begin{aligned} {\mathcal {G}}(B(K)):=\{I\subset B(K)| I \text { is direct}\} \end{aligned}

can be endowed with the structure of a Banach manifold (see [3, p. 16]; [1, p. 38, Example 3.15]). The space $${\mathcal {G}}(B(K))$$ is called the Grassmannian of B(K). Furthermore, if we consider B(K) as a module over itself, the set

\begin{aligned} {\mathcal {G}}_{B(K)}(B(K)):=\{I\in {\mathcal {G}}(B(K))| I \text { is a }B(K)\text {-submodule of } B(K)\} \end{aligned}

can be endowed with the structure of a Banach analytic space (see [3, pp. 29–30]; [1, p. 39, Example 3.21]) and the subset

\begin{aligned} {\mathcal {G}}(K):=\{I\in {\mathcal {G}}_{B(K)}(B(K))|I \text { admits a finite free resolution}\} \end{aligned}
(3)

is open in $${\mathcal {G}}_{B(K)}(B(K))$$. The privileged subspaces of a given polycylinder $$K_i$$ are precisely those subspaces corresponding to the direct ideals of B(K) admitting a finite free resolution (see [4, p. 577] and [25, p. 256]). In [3, p. 62, Theorem 1], Douady showed that every compact complex space can be covered with finitely many privileged subspaces of polycylinders.

Now, given a covering of a compact complex space $$X_0$$ with privileged charts $$(f_i:Y_i\rightarrow X_0)$$, since intersections of privileged polycylinders are not in general privileged, one needs to cover the intersections too. In order to have the transition maps well-defined, one needs to work with two polycylinders

\begin{aligned} {\tilde{K}}_i\subset \mathring{K}_i \end{aligned}
(4)

for double intersections and three polycylinders

\begin{aligned} K'_i\subset \mathring{{\tilde{K}}}_i, {\tilde{K}}_i\subset \mathring{K}_i \end{aligned}
(5)

for triple intersections. We rewrite (4) and (5) using the following notation

\begin{aligned} {\tilde{K}}_i\Subset K_i \end{aligned}
(6)

and

\begin{aligned} K'_i\Subset {\tilde{K}}_i\Subset K_i \end{aligned}
(7)

respectively. Let

\begin{aligned} {\mathfrak {I}}:=(I_{\bullet }, (K_{i})_{i\in I},({\tilde{K}}_{i})_{i\in I},(K'_{i})_{i\in I_{0}\cup I_{1}}), \end{aligned}
(8)

where $$I_{\bullet }$$ is a finite simplicial set of dimension 2 (see, for instance, [4, p. 587]) and the collections of polycylinders satisfy (6) and (7). A cuirasse q of type $${\mathfrak {I}}$$ on a compact complex space $$X_0$$ is a disassembly of $$X_0$$ given by a collection of pairs $$q:=\{(Y_i,f_i)\}_{i\in I}$$, where $$Y_i\subset K_i$$ is privileged, $$f_i:Y_i\rightarrow X_0$$ is a morphism, and they satisfy gluing relations on double and triple intersections (see [4, p. 587]).

In [4, p. 588], Douady showed that the set of all cuirasses of a fixed type $${\mathfrak {I}}$$ on a compact complex space $$X_0$$

\begin{aligned} {\mathcal {Q}}(X_0):=\{q \text { is a cuirasse on } X_0\} \end{aligned}
(9)

can be endowed with the structure of a Banach analytic space. Moreover, if $$X\rightarrow S$$ is a deformation of $$X_0$$, a choice of a cuirasse $$q_s$$ on each fibre $$X_s$$ is called a relative cuirasse on X over S. More precisely, in [4, p. 588], Douady showed that the set

\begin{aligned} {\mathcal {Q}}_{S}(X):=\{(s,(Y_{i},f_{i})_{i\in I})|s\in S, (Y_{i},f_{i})_{i\in I}\in {\mathcal {Q}}(X(s))\}, \end{aligned}
(10)

that is

\begin{aligned} {\mathcal {Q}}_{S}(X)=\bigsqcup _{s\in S}{\mathcal {Q}}(X(s)), \end{aligned}

can be endowed with the structure of a Banach analytic space. Then, a (local) relative cuirasse on X over S is defined as a (local) section

\begin{aligned} q:S\rightarrow {\mathcal {Q}}_{S}(X) \end{aligned}
(11)

of the natural projection $$\pi : {\mathcal {Q}}_{S}(X)\rightarrow S$$.

On the other side stands the notion of puzzle. Informally speaking, a puzzle is a compact complex space delivered in pieces, together with the assembly manual. Technically, a puzzle z is given by a collection $$z:=\{(Y_i,g^j_i)\}_{i\in I, j\in \partial i}$$, where $$Y_i\subset K_i$$ is a privileged subspace and $$g^j_i:Y_j\rightarrow Y_i$$ is a morphism. This collection of data satisfies gluing axioms ([4, p. 589]). The collection of puzzles

\begin{aligned} {\mathfrak {Z}}:=\{(Y_i,g^j_i)_{i\in I, j\in \partial i}\} \end{aligned}
(12)

form a Banach analytic space, each puzzle z glues to a compact complex space $${\mathfrak {X}}_z$$ and the collection of compact complex spaces $$({\mathfrak {X}}_z)_{z\in {\mathfrak {Z}}}$$ glues to a proper Banach analytic family $${\mathfrak {X}}$$ over $${\mathfrak {Z}}$$ (see [4, p. 591]), which is anaflat (see [3, p. 66, Definition and Proposition 1]).

Now, let $$X\rightarrow S$$ be a deformation of $$X_0$$. The aim is to produce a map $$\varphi :S\rightarrow {\mathfrak {Z}}$$, such that, in a neighborhood of some base point $$z_0\in {\mathfrak {Z}}$$, with $${\mathfrak {X}}_{z_0}\simeq X_0$$, we have $$\varphi ^{*}{\mathfrak {X}}\simeq X$$. To achieve this end, a special role is played by triangularly privileged cuirasses on $$X_0$$ (see [4, p. 588]). Informally speaking, these are cuirasses on $$X_0$$ that extend to cuirasses on the nearby fibres $$X_s$$. Douady showed that every compact complex space $$X_0$$ admits a triangularly privileged cuirasse ([4, p. 588])

\begin{aligned} q_0\in {\mathcal {Q}}(X_0). \end{aligned}
(13)

This means that if $$X\rightarrow S$$ is a deformation of $$X_0$$, with base point $$0\in S$$, and $$q_0$$ is a triangularly privileged cuirasse on $$X_0$$, then we get the existence of a continuous family of cuirasses $$\{q_s\}_{s\in S}$$, where $$q_s$$ is a cuirasse on the fibre $$X_s$$, for s in a small neighborhood of 0. Namely, we can find a (local) relative cuirasse $$q:S\rightarrow {\mathcal {Q}}_S(X)$$ on X over S, such that $$q(0)=q_0$$. Now, since every cuirasse $$q_s=\{(Y_i,f_i)\}$$ naturally produces an associated puzzle ([4, p. 590]) via

\begin{aligned} z_{q_s}:=(Y_i,g_{i}^{j}:=f_{i}^{-1}\circ f_{j})_{i\in I,j\in \partial i}, \end{aligned}
(14)

we get a morphism ([4, p. 591])

\begin{aligned} \begin{aligned} \varphi _q:S&\rightarrow {\mathfrak {Z}}\\ s&\mapsto z_{q_s}.\\ \end{aligned} \end{aligned}
(15)

Because a cuirasse $$q_s$$ is a disassembly of a compact complex space $$X_s$$ and the associated puzzle $$z_{q_s}$$ glues to a compact complex space $${\mathfrak {X}}_{z_{q_s}}$$, it is reasonable to expect that $${\mathfrak {X}}_{z_{q_s}}$$ is isomorphic to $$X_s$$. In fact, we have an S-isomorphism ([4, p. 592])

\begin{aligned} \alpha _q:\varphi ^{*}_q{\mathfrak {X}}\rightarrow X. \end{aligned}
(16)

In other words, the Banach analytic family $${\mathfrak {X} \rightarrow \mathfrak {Z}}$$ contains all possible deformations of $$X_0$$ in a neighborhood of $$z_{q_0}$$. That is, the family is complete.

An involved finite-dimensional reduction procedure (“a cure d’amaigrissement”) is used to obtain a finite-dimensional semi-universal deformation of $$X_0$$ out of the complete infinite-dimensional family $${\mathfrak {X}}\rightarrow {\mathfrak {Z}}$$ (see [4, pp. 593–599], [34, pp. 20–46] and Sect. 2.3). This ends our survey about Douady’s construction of a semi-universal deformation of a compact complex space.

Now, we assume that $$X_0$$ comes endowed with a fine log structure $${\mathcal {M}}_{X_0}$$. We view $$X_0$$ as a log space over the point $${\text {Spec}}{\mathbb {C}}$$ with trivial log structure.

Definition 1.5

A deformation of a compact fine log complex space $$(X_0,{\mathcal {M}}_{X_0})$$ is a triple $$((S,s_0), ({\mathfrak {X}},{\mathcal {M}}_{\mathfrak {X}}), i)$$, where S is a complex space endowed with trivial log structure, $$s_0\in S$$, $$p:({\mathfrak {X}},{\mathcal {M}}_{\mathfrak {X}})\rightarrow (S,{\mathcal {O}}_S^{\times })$$ is a log morphism between fine log complex spaces with underlying map of complex spaces $${\mathfrak {X}}\rightarrow S$$ proper and flat, and $$i:(X_0,{\mathcal {M}}_{X_0})\rightarrow ({\mathfrak {X}},{\mathcal {M}}_{\mathfrak {X}})(s_0):=p^{-1}(s_0)$$ is a log isomorphism.

A deformation is complete if for any other deformation $$((T,t_0),(X,{\mathcal {M}}_X),j)$$ of $$(X_0,{\mathcal {M}}_{X_0})$$, there exists a morphism $$\psi :(T,{\mathcal {O}}_T^{\times })\rightarrow (S,{\mathcal {O}}_S^{\times })$$, sending $$t_0$$ to $$s_0$$, and a log T-isomorphism

\begin{aligned} \alpha :(X,{\mathcal {M}}_X)\rightarrow ({\mathfrak {X}},{\mathcal {M}}_{\mathfrak {X}})\times _{(S,{\mathcal {O}}_S^{\times })}(T,{\mathcal {O}}_T^{\times }), \end{aligned}

such that $$\alpha \circ j=i$$. For the sake of readability, in what follows, we shall mostly denote a complex space endowed with trivial log structure $$(S,{\mathcal {O}}^{\times }_S)$$ just by S.

One of the key in points, in the construction of deformations of log spaces, is to find a proper way to deform the log structure $${\mathcal {M}}_{X_0}$$ coherently with the deformation of the underlying analytic space $$X_0$$. We show, in Sect. 2.1, that we can disassemble $${\mathcal {M}}_{X_0}$$ using log charts satisfying gluing conditions on double and triple intersections (Proposition A.5). That is, the log structures associated to the log charts glue to a global log structure $${\mathcal {M}}^a_{X_0}$$ on $$X_0$$ isomorphic to $${\mathcal {M}}_{X_0}$$. We call this collection of log charts a set of directed log charts (Definition 2.1). This insight leads to the notion of log cuirasse (Definition 2.10) and log puzzle (Definition 2.5).

In Sect. 2.2, we construct an infinite-dimensional log family $$({\mathfrak {X}},{\mathcal {M}}_{{\mathfrak {X}}})\rightarrow {\mathfrak {Z}}^{\log }$$ (Proposition 2.8). Given a log deformation $$({\mathcal {Y}},{\mathcal {M}}_{{\mathcal {Y}}})\rightarrow T$$ of $$(X_0,{\mathcal {M}}_{X_0})$$, with base point $$t_0$$, an essential point is to show that a triangularly privileged log cuirasse $$q_0^{\dagger }$$ exists on $$(X_0,{\mathcal {M}}_{X_0})\simeq ({\mathcal {Y}},{\mathcal {M}}_{{\mathcal {Y}}})(t_0)$$ and it extends to a log cuirasse $$q^{\dagger }_t$$ on the fibre $$({\mathcal {Y}}_t,{\mathcal {M}}_{{\mathcal {Y}}_t})$$, for t in a neighborhood of $$t_0$$ (Propositions 2.18 and 2.19). This allows us to show the completeness of the log family $$({\mathfrak {X}},{\mathcal {M}}_{{\mathfrak {X}}})\rightarrow {\mathfrak {Z}}^{\log }$$.

In Sect. 2.3, we proceed with a finite-dimensional reduction procedure, which produces a semi-universal deformation of $$(X_0,{\mathcal {M}}_{X_0})$$ out of the complete log family $$({\mathfrak {X}},{\mathcal {M}}_{{\mathfrak {X}}})\rightarrow {\mathfrak {Z}}^{\log }$$. The finite-dimensionality is achieved with the exact same procedure used by Douady in the classical case. This is because the space $${\mathfrak {Z}}^{\log }$$ of log puzzles does not come endowed with a non-trivial log structure. We prove

Theorem 1.6

(Theorem 2.32) Every compact fine log complex space $$(X_0,{\mathcal {M}}_{X_0})$$ admits a semi-universal deformation $$((S,s_0), ({\mathfrak {X}},{\mathcal {M}}_{\mathfrak {X}}), i)$$.

For a construction of a semi-universal deformation in the non-fine log context see, for instance, [32] where a semi-universal family is obtained by means of Artin approximation (see, also, [31]).

The existence of semi-universal deformations of morphisms between compact complex analytic spaces follows naturally from Douady’s results (see [7, p. 130]). Analogously, we take a further step in our work studying semi-universal deformations of log morphisms.

Given a morphism of log complex spaces, we have the notion of log smoothness (see, for instance, [12, p. 107]) and log flatness (see [17]). These notions generalize and extend the classical notions of smoothness and flatness, which are retrieved if we consider complex spaces endowed with trivial log structures. In [19], K. Kato writes that a log structure is “magic by which a degenerate scheme begins to behave as being non-degenerate”.

For example, the affine toric variety $${\text {Spec}}_{\textrm{an}}{\mathbb {C}}[P]$$, with its canonical divisorial log structure, is log smooth over $${\text {Spec}}{\mathbb {C}}$$ (equipped with the trivial log structure), despite almost always not being smooth in the usual sense. In what follows, we denote the analytic spectrum $$\text {Spec}_{\text {an}}{\mathbb {C}}[P]$$ of a monoid ring simply by $${\text {Spec}}{\mathbb {C}}[P]$$.

In Sect. 3, we prove the following

Theorem 1.7

(Theorem 3.4 and Proposition 3.12) Every morphism of compact fine log complex spaces $$f_0:(X_0,{\mathcal {M}}_{X_0})\rightarrow (Y_0,{\mathcal {M}}_{Y_0})$$ admits a semi-universal deformation f over a germ of complex spaces $$(S,s_0)$$. Moreover, if $$f_0$$ is log flat (or log smooth), then f is log flat (or log smooth) in an open neighborhood of $$s_0$$.

As a corallary result (Corollary 3.6), we obtain a relative semi–universal deformation of a compact fine log complex space $$(X_0,{\mathcal {M}}_{X_0})$$ over a fine log complex space $$(Y_0,{\mathcal {M}}_{Y_0})$$ (Definition 3.5). Notice that, in this case, $$Y_0$$ needs not to be compact. If $$(X_0,{\mathcal {M}}_{X_0})$$ is a log subspace of $$(Y_0,{\mathcal {M}}_{Y_0})$$, we get a semi–universal deformation of a log subspace in a fixed ambient log space (Remark 3.7).

The focus of this work is the construction of analytic deformations via Douady’s patching method rather than a comprehensive treatment of deformations of analytic log spaces. In particular, we do not discuss infinitesimal or formal deformations. The classical treatment of these topics in the algebraic geometric setup (see [20] and [18]) readily carry over to the analytic setup treated here. See also [5], for a more recent treatment of log smooth deformations from the point of view of differential graded algebras.

2 Semi-universal deformations of compact fine log complex spaces

In what follows, we construct a semi-universal deformation in the general case of a compact complex space $$X_{0}$$ endowed with a fine log structure $${\mathcal {M}}_{X_{0}}$$.

2.1 Gluing log charts

Let $$(X_0,{\mathcal {M}}_{X_0})$$ be a compact fine log complex space. Denote by $$\alpha :{\mathcal {M}}_{X_0}\rightarrow {\mathcal {O}}_{X_0}$$ the structure map. The sheaf of monoids

\begin{aligned} \overline{{\mathcal {M}}}_{X_0}:={\mathcal {M}}_{X_0}/{{\,\textrm{Im}\,}}\alpha ^{-1}={\mathcal {M}}_{X_0}/{\mathcal {O}}^{\times }_{X_0}, \end{aligned}

written additively, is called the ghost sheaf of $${\mathcal {M}}_{X_0}$$. We assume that $$\overline{{\mathcal {M}}}^{\text {gp}}_{X_0}$$ is torsion free.

We want to find a universal setup for constructing log structures from gluing of log charts. This is quite analogous to the case of sheaves, see for example [16, Exercise II.1.22]. Assume we have a covering of $$X_0$$ by open sets $$U_i$$ for an ordered index set $$J_0$$, and for each $$U_i$$ a log chart

\begin{aligned} \theta _i:P_i\longrightarrow \Gamma (U_i,{\mathcal {M}}_{X_0}). \end{aligned}

We identify $$\theta _i$$ with the corresponding map of monoid sheaves $$\underline{P}_i\rightarrow {\mathcal {M}}_{X_0}|_{U_i}$$. For $$l=1,2$$, set

\begin{aligned} J_l:=\{(i_0,\ldots ,i_l)\in J_0^{l+1}: U_{i_0}\cap \ldots \cap U_{i_l}\ne \emptyset \}. \end{aligned}

We get maps $$d_m:J_l\rightarrow J_{l-1}$$, for $$0\le m\le l$$ and $$1\le l\le 2$$, sending $$(i_0,\ldots ,i_m,\ldots ,i_l)$$ to $$(i_0,\ldots ,i_{m-1},i_{m+1},\ldots ,i_l)$$. We set

\begin{aligned} J&:=\bigcup ^2_{l=0}J_l,\\ \partial {i}&:=\left\{ d_{0}i, \ldots , d_{l}i \right\}{} & {} \text {, if } i\in J_{l}. \end{aligned}

The set J, together with the maps $$(d_m)$$, is called a simplicial set of order 2.

For each $$j:=(i_0,i_1)\in J_1$$, assume that there is a log chart

\begin{aligned} \theta _{j}:P_{j}\longrightarrow \Gamma (U_{j},{\mathcal {M}}_{X_0}) \end{aligned}

and comparison maps

\begin{aligned} \varphi ^{i}_{j}: P_{i}\longrightarrow & {} P_{j}\oplus \Gamma (U_{j},{\mathcal {O}}^{\times }_{X_0}), \end{aligned}

for $$i\in \partial j$$, with the property

\begin{aligned} \big (\theta _{j}\cdot {{\,\textrm{Id}\,}}_{{\mathcal {O}}^{\times }_{X_0}|_{U_j}}\big )\circ \varphi ^{i}_{j}= \theta _i|_{U_{j}}. \end{aligned}
(17)

Each $$\theta _i$$ defines an isomorphism of $${\mathcal {M}}_{U_i}$$ with the log structure $${\mathcal {M}}_i$$ associated to the pre-log structure $$\beta _i:=\alpha \circ \theta _i$$. Similarly, the pre-log structure $$\beta _{j}:=\alpha \circ \theta _{j}$$ defines a log structure $${\mathcal {M}}_{j}$$ and $$\theta _j$$ defines an isomorphism of log structures $${\mathcal {M}}_{U_{j}}\simeq {\mathcal {M}}_{j}$$. From this point of view, equation (17) means that $$\varphi ^i_{j}$$ provides an isomorphism between $${\mathcal {M}}_i|_{U_{j}}$$ and $${\mathcal {M}}_{j}$$, and this isomorphism is compatible with the isomorphisms $${\mathcal {M}}_i|_{U_{j}}\simeq {\mathcal {M}}_{U_{j}}$$ and $${\mathcal {M}}_{j}\simeq {\mathcal {M}}_{U_{j}}$$.

Now, if we have $$\theta _i$$, $$\theta _{j}$$, $$\varphi ^i_{j}$$, fulfilling (17), we need compatibility on triple intersections for the patching of the $${\mathcal {M}}_i$$ to be consistent. To formulate this cocycle condition in terms of log charts, assume, for each $$k:=(i_0,i_1,i_2)\in J_2$$, a third system of charts

\begin{aligned} \theta _{k}: P_{k}\longrightarrow \Gamma (U_{k},{\mathcal {M}}_{X_0}) \end{aligned}

and comparison maps

\begin{aligned} \varphi ^{j}_{k}: P_{j}\longrightarrow & {} P_{k}\oplus \Gamma (U_{k},{\mathcal {O}}^{\times }_{X_0}), \end{aligned}

for $$j\in \partial k$$. The analogue of the compatibility condition (17) is

\begin{aligned} \big (\theta _{k}\cdot {{\,\textrm{Id}\,}}_{{\mathcal {O}}^{\times }_{X_0}|_{U_{k}}}\big )\circ \varphi ^{j}_{k}= \theta _{j}|_{U_{k}}. \end{aligned}
(18)

Again, the $$\varphi ^{j}_{k}$$ define an isomorphism between the log structure $${\mathcal {M}}_{j}|_{U_{k}}$$ on $$U_{k}$$ and the log structure $${\mathcal {M}}_{k}$$ associated to the pre-log structure $$\beta _{k}:=\alpha \circ \theta _k$$. In particular, all the isomorphisms of log structures are compatible and the $$({\mathcal {M}}_i)_{i\in J_0}$$ glue in a well-defined fashion, as do their structure maps, to a log structure on $$X_0$$ isomorphic to $${\mathcal {M}}_{X_0}$$. This is just standard sheaf theory, for sheaves of monoids.

Definition 2.1

A set of directed log charts is a set of log charts $$(\theta _i:P_i\rightarrow {\mathcal {M}}_{U_i})_{i\in J}$$ covering $$(X_{0},{\mathcal {M}}_{X_0})$$, together with a morphism

\begin{aligned} \varphi ^{i}_j:P_i\rightarrow P_j\oplus {\mathcal {O}}^{\times }_{U_j}, \end{aligned}

for each $$j\in J_1\cup J_2$$ and $$i\in \partial j$$, such that

\begin{aligned} (\theta _j\cdot {{\,\textrm{Id}\,}}_{{\mathcal {O}}^{\times }_{U_j}})\circ \varphi ^i_j=\theta _i|_{U_j}. \end{aligned}

In Proposition A.5, we show that every compact fine log complex space can be covered with a finite set of directed log charts.

Now, let us forget that the $$(\theta _i)_{i\in J_0}$$, $$(\theta _{j})_{j\in J_1}$$ and $$(\theta _{k})_{k\in J_2}$$ are charts for the given log structure. Let $$(U_i)_{i\in J_0}$$ be an open cover of $$X_0$$ and J as above. Assume we have pre-log structures $$(\beta _i)_{i\in J}$$ and comparison maps $$(\varphi ^i_{j})_{j\in J_1\cup J_2,i\in \partial j}$$ satisfying equations (17) and (18). Then the log structures $$({\mathcal {M}}_i)_{i\in J_0}$$ glue to a log structure $${\mathcal {M}}_{X_0}$$ on $$X_0$$ in such a way that the gluing data $$(\beta _{j})_{j\in J_1}$$ and compatibility $$(\beta _{k})_{k\in J_2}$$ arise from identifying $${\mathcal {M}}_{j}$$ and $${\mathcal {M}}_{k}$$ with restrictions of $${\mathcal {M}}_{X_0}$$ to $$U_{j}$$ and $$U_{k}$$ respectively.

Definition 2.2

Let $$X_0$$ be a compact complex space. With the above notation, we call a pre-log atlas on $$X_0$$ a collection of data

\begin{aligned}\{(\beta _i:P_i\rightarrow {\mathcal {O}}_{U_i})_{i\in J},( \varphi ^{i}_{j}: P_{i}\rightarrow P_{j}\oplus {\mathcal {O}}^{\times }_{U_j})_{j\in J_1\cup J_2,i\in \partial j}\}\end{aligned}

satisfying

\begin{aligned} \big (\beta _{j}\cdot {{\,\textrm{Id}\,}}_{{\mathcal {O}}^{\times }_{U_j}}\big )\circ \varphi ^{i}_{j}= \beta _i|_{U_{j}}. \end{aligned}
(19)

2.2 Infinite dimensional construction

The notion of log structure can be naturally extended to the category of Banach analytic spaces. Indeed, let $$(X,\Phi )$$ be a Banach analytic space (see [3, pp. 22–25]; [1, p. 38, Definition 3.16]). Setting $${\mathcal {O}}_X:=\Phi ({\mathbb {C}})$$, we get a ringed space $$(X,{\mathcal {O}}_X)$$.

Definition 2.3

A pre-log structure on a Banach analytic space $$(X,\Phi )$$ is a sheaf of monoids $${\mathcal {M}}_{X}$$ on X together with a homomorphism of sheaves of monoids:

\begin{aligned} \alpha _{X}:{\mathcal {M}}_{X}\rightarrow {\mathcal {O}}_{X}, \end{aligned}

where the monoid structure on $${\mathcal {O}}_X$$ is given by multiplication. A pre-log structure is a called a log structure if

\begin{aligned} \alpha _{X}:\alpha _{X}^{-1}({\mathcal {O}}_{X}^{\times })\rightarrow {\mathcal {O}}_{X}^{\times } \end{aligned}

is an isomorphism.

The notion of fine log structure extends naturally to the Banach analytic setting. In what follows, we shall mostly denote a log Banach analytic space endowed with the trivial log structure $$(S,{\mathcal {O}}^{\times }_S)$$ just by S. Moreover, for the sake of readability, we shall often write Banach analytic morphisms just set-theoretically.

Let $$(X_0, {\mathcal {M}}_{X_{0}})$$ be a compact fine log complex space.

Definition 2.4

Let $$(\theta _{i,0}:P_i\rightarrow {\mathcal {M}}_{U_i})_{i\in J}$$, with comparison morphisms

\begin{aligned}(\varphi ^i_{j,0}:=(\phi ^{i}_{j,0},\eta ^i_{j,0}):P_i\rightarrow P_j\oplus {\mathcal {O}}^{\times }_{U_j})_{j\in J_1\cup J_2, i\in \partial j},\end{aligned}

be a finite set of directed log charts covering $$(X_0,{\mathcal {M}}_{X_0})$$ (Definition 2.1) given by Proposition A.5.

Let

\begin{aligned} {\mathfrak {I}}:=(I_{\bullet }, (K_{i})_{i\in I},({\tilde{K}}_{i})_{i\in I},(K'_{i})_{i\in I_{0}\cup I_{1}}) \end{aligned}

be as in (8) and $${\mathfrak {Z}}$$ the space of puzzles (12). Without loss of generality, we assume that the index sets I and J (Definition 2.4) coincide. We can define the notion of log puzzle, which, informally speaking, is a compact fine log complex space delivered in pieces with the instructions to glue them together.

Definition 2.5

A log puzzle is a pair (zl), where $$z:=(Y_{i},g^{i}_j)\in {\mathfrak {Z}}$$ is a puzzle and l is a collection of data

\begin{aligned}((\beta _i:P_i\rightarrow {\mathcal {O}}_{Y^{\circ }_i})_{i\in I},(\eta ^i_{j}:P_{i}\rightarrow {\mathcal {O}}^{\times }_{Y^{\circ }_j})_{j\in I_1\cup I_2,i\in \partial j}),\end{aligned}

satisfying

\begin{aligned} \big ((\beta _{j}\cdot {{\,\textrm{Id}\,}}_{{\mathcal {O}}^{\times }_{Y^{\circ }_{j}}})\circ \varphi ^{i}_{j}= \beta _i|_{Y^{\circ }_{j}}\big )_{j\in I_1\cup I_2, i\in \partial j}, \end{aligned}
(20)

where $$\varphi ^{i}_{j}:=(\phi ^{i}_{j,0},\eta ^i_{j})$$, with $$\phi ^{i}_{j,0}:P_i\rightarrow P_j$$ given by Definition 2.4.

Definition 2.6

We denote the set of log puzzles by $${\mathfrak {Z}}^{\log }$$.

The set of log puzzles $${\mathfrak {Z}}^{\log }$$ can be endowed with a Banach analytic structure. Indeed, for each polycylinder $$K_i$$, let us consider the Grassmannian $${\mathcal {G}}(K_i)$$ (3) and let $${{\,\textrm{Id}\,}}:{\mathcal {G}}(K_i)\rightarrow {\mathcal {G}}(K_i)$$ be the identity map. Identifying $${{\,\textrm{Id}\,}}$$ with its graph, we get a universal $${\mathcal {G}}(K_i)$$-anaflat subspace $$\underline{Y}_i\subset {\mathcal {G}}(K_i)\times {\mathcal {G}}(K_i)\subset {\mathcal {G}}(K_i)\times K_i$$ (see [4, p. 579], [25, pp. 258–259] and [30, p. 183, Theorem 4.13]). Let us consider the Banach analytic space

\begin{aligned}{} & {} {\mathfrak {M}}:={{\,\mathrm{\mathop {{\mathop {\prod }\nolimits _{{\mathfrak {Z}}}}}\limits }\,}}_{i\in I} \mathfrak {Mor}_{{\mathfrak {Z}}}(\underline{Y}_i,{\text {Spec}}{\mathbb {C}}[P_{i}]\times {\mathfrak {Z}})\times _{{\mathfrak {Z}}}{{\,\mathrm{\mathop {{\mathop {\prod }\nolimits _{{\mathfrak {Z}}}}}\limits }\,}}_{j\in I_1\cup I_2}\\{} & {} \quad {{\,\mathrm{\mathop {{\mathop {\prod }\nolimits _{{\mathfrak {Z}}}}}\limits }\,}}_{i\in \partial j}\mathfrak {Mor}_{{\mathfrak {Z}}}(\underline{Y}_j,{\text {Spec}}{\mathbb {C}}[P^{{{\text {gp}}}}_{i}]\times {\mathfrak {Z}}). \end{aligned}

Each morphism

\begin{aligned} \eta ^{i}_j(z)^{{{\text {gp}}}}:P^{{{\text {gp}}}}_i\rightarrow {\mathcal {O}}^{\times }_{\underline{Y}^{\circ }_{j}(z)} \end{aligned}

induces a morphism $$\eta ^i_j(z):P_i\rightarrow {\mathcal {O}}^{\times }_{\underline{Y}^{\circ }_{j}(z)}$$. Hence, a point in $${\mathfrak {M}}$$ can be written as

\begin{aligned} (z,(\beta _{i}(z):P_i\rightarrow {\mathcal {O}}_{\underline{Y}^{\circ }_{i}(z)})_{i\in I},(\eta ^i_j(z):P_i\rightarrow {\mathcal {O}}^{\times }_{\underline{Y}^{\circ }_{j}(z)})_{j\in I_1\cup I_2,i\in \partial j}). \end{aligned}

Thus, we naturally get an injective map

\begin{aligned}{} & {} \rho :{\mathfrak {Z}}^{\log }\hookrightarrow {{\,\mathrm{\mathop {{\mathop {\prod }\nolimits _{{\mathfrak {Z}}}}}\limits }\,}}_{i\in I} \mathfrak {Mor}_{{\mathfrak {Z}}}(\underline{Y}_i,{\text {Spec}}{\mathbb {C}}[P_{i}]\times {\mathfrak {Z}})\times _{{\mathfrak {Z}}}\nonumber \\{} & {} \quad {{\,\mathrm{\mathop {{\mathop {\prod }\nolimits _{{\mathfrak {Z}}}}}\limits }\,}}_{j\in I_1\cup I_2}{{\,\mathrm{\mathop {{\mathop {\prod }\nolimits _{{\mathfrak {Z}}}}}\limits }\,}}_{i\in \partial j}\mathfrak {Mor}_{{\mathfrak {Z}}}(\underline{Y}_j,{\text {Spec}}{\mathbb {C}}[P^{\text {gp}}_{i}]\times {\mathfrak {Z}}). \end{aligned}
(21)

Proposition 2.7

The universal space of log puzzles $${\mathfrak {Z}}^{\log }$$ is Banach analytic.

Proof

Let $$(\phi _{j,0}^i:P_i\rightarrow P_j)_{j\in I_1\cup I_2,i\in \partial j}$$ given by Definition 2.4, we set

\begin{aligned}\varphi ^{i}_{j}(z):=(\phi _{j,0}^i,\eta ^i_j(z)):P_i\rightarrow P_j\oplus {\mathcal {O}}^{\times }_{\underline{Y}^{\circ }_{j}(z)}.\end{aligned}

The subset $$\rho ({\mathfrak {Z}}^{\log })$$ of $${\mathfrak {M}}$$ (21) is defined by the equations

\begin{aligned} \big ((\beta _{j}(z)\cdot {{\,\textrm{Id}\,}}_{{\mathcal {O}}^{\times }_{\underline{Y}^{\circ }_{j}(z)}})\circ \varphi ^{i}_{j}(z)= \beta _i(z)|_{\underline{Y}^{\circ }_{j}(z)}\big )_{j\in I_1\cup I_2, i\in \partial j}. \end{aligned}
(22)

Thus, we can define a double arrow

\begin{aligned}(\rho _1,\rho _2):{\mathfrak {M}} \rightrightarrows {{\,\mathrm{\mathop {{\mathop {\prod }\nolimits _{{\mathfrak {Z}}}}}\limits }\,}}_{j\in I_1\cup I_2}\mathfrak {Mor}_{{\mathfrak {Z}}}(\underline{Y}_{j},{\text {Spec}}{\mathbb {C}}[P_{j}]\times {\mathfrak {Z}})\end{aligned}

by

\begin{aligned}\rho _1:(z,(\beta _i(z)),(\eta ^{i}_j(z)))\mapsto (z,(\beta _i(z)|_{\underline{Y}^{\circ }_{j}(z)}))\end{aligned}

and

\begin{aligned}\rho _2:(z,(\beta _i(z)),(\eta ^{i}_j(z)))\mapsto (z,((\beta _{j}(z)\cdot {{\,\textrm{Id}\,}}_{{\mathcal {O}}^{\times }_{\underline{Y}^{\circ }_{j}(z)}})\circ \varphi ^{i}_{j}(z))).\end{aligned}

Then $${\mathfrak {Z}}^{\log }$$ is given by the kernel of the double arrow defined by $$\rho _1$$ and $$\rho _2$$:

\begin{aligned} {\mathfrak {Z}}^{\log }={\text {ker}}(\rho _1,\rho _2). \end{aligned}

$$\square$$

Let $$p:{\mathfrak {Z}}^{\log } \rightarrow {\mathfrak {Z}}$$ be the canonical projection and consider the Banach analytic space $${\mathfrak {X}}_{\log }:=p^*{\mathfrak {X}}$$ over $${\mathfrak {Z}}^{\log }$$.

Proposition 2.8

The Banach analytic space $${\mathfrak {X}}_{\log }$$ comes naturally endowed with a fine log structure $${\mathcal {M}}_{{\mathfrak {X}}_{\log }}$$.

Proof

By [30, p. 192, Theorem 5.13] (see, also, [4, p. 579]), we have universal morphisms

\begin{aligned} \begin{aligned} ({\underline{\beta }}_{i}&: P_i \rightarrow {\mathcal {O}}_{p^{*}\underline{Y}^{\circ }_{i}})_{i\in I}\\ ({\underline{\varphi }}^{i}_{j}:=(\phi ^{i}_{j,0},{\underline{\eta }}^{i}_j)&: P_{i}\rightarrow P_{j}\oplus {\mathcal {O}}^{\times }_{p^{*}\underline{Y}^{\circ }_{j}})_{j\in I_1\cup I_2,i\in \partial j} \end{aligned}. \end{aligned}
(23)

By construction, they satisfy

\begin{aligned} \big (({\underline{\beta }}_{j}\cdot {{\,\textrm{Id}\,}}_{{\mathcal {O}}^{\times }_{\underline{Y}^{\circ }_{j}}})\circ {\underline{\varphi }}^{i}_{j}= {\underline{\beta }}_i|_{\underline{Y}^{\circ }_{j}}\big )_{j\in I_1\cup I_2, i\in \partial j}. \end{aligned}

On the other hand, we have that the space $${\mathfrak {X}}$$ is canonically isomorphic to

\begin{aligned} \coprod _{i\in I_0}\underline{Y}^{'}_{i}/{\mathcal {R}}, \end{aligned}

where $${\mathcal {R}}(x,x^{'})$$ if $$x\in \underline{Y}'_{i}$$ and $$x'\in \underline{Y}'_{i'}$$ are such that there exists $$j\in I_1$$ and $$y\in \underline{Y}'_{j}$$ with $$dj=(i,i')$$, $$\underline{g}^{j}_{i}(y)=x$$ and $$\underline{g}^{j}_{i'}(y)=x'$$ (see [4, p. 592]). Therefore, $${\mathfrak {X}}_{\log }$$ is canonically isomorphic to

\begin{aligned} \coprod _{i\in I_0}p^{*}\underline{Y}^{'}_{i}/{\mathcal {R}}. \end{aligned}

Hence, the collection of universal morphisms $$(({\underline{\beta }}_{i}),({\underline{\varphi }}^{i}_{j}))$$ defines a pre-log atlas (see Definition 2.2) on $${\mathfrak {X}}_{\log }$$, which glues to a fine log structure $${\mathcal {M}}_{{\mathfrak {X}}_{\log }}$$ on $${\mathfrak {X}}_{\log }$$ (see Sect. 2.1).$$\square$$

We show that the universal family of log puzzles $$({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})\rightarrow {\mathfrak {Z}}^{\log }$$ gives a complete deformation of $$(X_0,{\mathcal {M}}_{X_0})$$. To do that, we introduce the notion of log cuirasse. We recall that if S is a Banach analytic space, X a Banach analytic space proper and anaflat over S and q a relative cuirasse on X, then we get a morphism $$\varphi _q:S\rightarrow {\mathfrak {Z}}$$ (15), a Banach analytic space $$X_{\varphi _q}$$ over S obtained by gluing the pieces of the puzzle $$z_q$$ associated to q (14), and an S-isomporphism $$\alpha _q:X_{\varphi _q}\rightarrow X$$ (16). Now, let $$(X_0,{\mathcal {M}}_{X_0})$$ be a compact fine log complex space admitting a collection of directed log charts $$((\theta _i:P_i\rightarrow {\mathcal {M}}_{U_i})_{i\in I},(\varphi ^{i}_j:=(\phi ^{i}_j,\eta ^i_j):P_i\rightarrow P_j\oplus {\mathcal {O}}^{\times }_{U_j})_{j\in I_1\cup I_2, i\in \partial j})$$ (see Definition 2.1). We assume that $$\phi ^{i}_j$$ coincide with the $$\phi ^{i}_{j,0}$$ given by Definition 2.4. Let $$q_0\in {\mathcal {Q}}(X_0)$$ be a cuirasse on $$X_0$$. We have an isomorphism (16)

\begin{aligned} \alpha _{q_0}:X_{\varphi _{q_0}}\rightarrow X_0. \end{aligned}

Definition 2.9

We naturally get a fine log structure on $$X_{\varphi _{q_0}}$$ via

\begin{aligned} {\mathcal {M}}_{X_{\varphi _{q_0}}}:=\alpha _{q_0}^{*}{\mathcal {M}}_{X_0}. \end{aligned}

Definition 2.10

A log cuirasse $$q_0^{\dagger }$$ on $$(X_0, {\mathcal {M}}_{X_0})$$ is a pair given by a cuirasse $$q_0=(Y_i,f_i)_{i\in I}$$ on $$X_0$$ and a collection of directed log charts $$((\theta _i:P_i\rightarrow {\mathcal {M}}_{X_{\varphi _{q_0}}}|_{Y^{\circ }_i}), (\eta ^{i}_j:P_i\rightarrow O^{\times }_{Y^{\circ }_j}))$$ on $$(X_{\varphi _{q_0}},{\mathcal {M}}_{X_{\varphi _{q_0}}})$$ (Definition 2.1). We denote the set of log cuirasses on $$(X_0, {\mathcal {M}}_{X_0})$$ by $${\mathcal {Q}}(X_0,{\mathcal {M}}_{X_0})$$.

Remark 2.11

In Definition 2.10 we need to give the set of comparison morphisms $$(\eta ^{i}_{j})$$ in order to define, in Definition 2.20, the log puzzle associated to a log cuirasse.

Analogously to the classical case (11), we can define the notion of relative log cuirasse. Let S be a Banach analytic space and $$(X,{\mathcal {M}}_{X})$$ a fine log Banach analytic space proper and anaflat over S. Given the local nature of the problem, we can assume that $$(X,{\mathcal {M}}_{X})$$ can be covered by finitely many log charts $$(\theta _{i}:P_i\rightarrow {\mathcal {M}}_{U_i})_{i\in I}$$ such that $$U_{i}\cap X(s)\ne \emptyset$$, for each $$i\in I$$ and $$s\in S$$.

Definition 2.12

Let S be a Banach analytic space and $$(X,{\mathcal {M}}_{X})$$ a fine log Banach analytic space proper and anaflat over S. We define the set of relative log cuirasses on $$(X,{\mathcal {M}}_{X})$$ over S by

\begin{aligned} {\mathcal {Q}}_{S}(X,{\mathcal {M}}_{X}):=\{(s,q)|s\in S, q\in {\mathcal {Q}}(X(s),{\mathcal {M}}_{X}\arrowvert _{X(s)})\}= \coprod _{s\in S} {\mathcal {Q}}(X(s),{\mathcal {M}}_{X}\arrowvert _{X(s)}). \end{aligned}

Definition 2.13

We call a section $$q^{\dagger }:S\rightarrow {\mathcal {Q}}_{S}(X,{\mathcal {M}}_{X})$$, of the canonical projection $$\pi :{\mathcal {Q}}_{S}(X,{\mathcal {M}}_{X})\rightarrow S$$, a relative log cuirasse on $$(X,{\mathcal {M}}_{X})$$ over S.

Definition 2.14

Let $$(X_0,{\mathcal {M}}_{X_0})$$ be a compact fine log complex space. A log cuirasse $$q_0^{\dagger }$$ on $$(X_0,{\mathcal {M}}_{X_0})$$ is called triangularly privileged if the underlying cuirasse $$q_0\in {\mathcal {Q}}({\mathfrak {I}};X_0)$$ on $$X_0$$ is triangularly privileged (13).

Since every compact complex space $$X_0$$ admits a triangularly privileged cuirasse, every compact fine log complex space $$(X_0,{\mathcal {M}}_{X_0})$$ admits a triangularly privileged log cuirasse. The set of log cuirasses can be endowed with the structure of a Banach analytic space in a neighborhood of a triangularly privileged log cuirasse. To prove it, we need the following three Lemmas.

Lemma 2.15

Let $$(X,{\mathcal {M}}_X)$$ be a fine log Banach analytic space over a Banach analytic space S. Let $$q_0=(Y_{i,0},f_{i,0})$$ be a triangularly privileged cuirasse on the central fibre $$(X_0,{\mathcal {M}}_{X_0})$$ over $$s_0\in S$$. Assume that $$\Gamma (Y_{i,0},f^{-1}_{i,0}\overline{{\mathcal {M}}}_{X_0})$$ is globally generated, for each $$i\in I$$. Then there exists a local relative cuirasse $$q=(Y_i,f_i)$$ on X defined on a neighborhood $$S'$$ of $$s_0$$ in S, such that $$\Gamma (Y_i,f^{-1}_i\overline{{\mathcal {M}}}_{X})$$ is globally generated, for each $$i\in I$$.

Proof

Since $$q_0$$ is triangularly privileged, there exists a local relative cuirasse $$q=(Y_i,f_i)$$ on X defined in a neighborhood $$S'$$ of $$s_0$$ in S ([4, p. 585, Proposition 2]). Now, up to shrinking $$Y_{i,0}$$, for each $$i\in I$$ and for each $$y\in Y_{i,0}$$ there exists an open set $$U_y=V_y\times W_y\subset S\times K_i$$, $$y\in U_y$$, such that the canonical map $$\Gamma (U_y,f_i^{-1}\overline{{\mathcal {M}}}_X)\rightarrow \overline{{\mathcal {M}}}_{X,f_i(y)}$$ is an isomorphism. Since each $$Y_i$$ is compact, we can find a finite set J and finitely many subsets $$U_j$$ such that $$\Gamma (\bigcup _{j\in J} U_j,f_i^{-1}\overline{{\mathcal {M}}}_X)$$ is isomorphic to $$\Gamma (Y_i,f_i^{-1}\overline{{\mathcal {M}}}_X)$$. Hence, possibly after shrinking $$S'$$, we can assume that for each $$j\in J$$, $$V_j=S'$$ and $$f^{-1}_i(X)\subset \bigcup _{j\in J}U_j$$. Thus, we get that for each $$i\in I$$, $$\Gamma (Y_i,f^{-1}_i\overline{{\mathcal {M}}}_{X})$$ is isomorphic to $$\Gamma (Y_{i,0},f^{-1}_{i,0}\overline{{\mathcal {M}}}_{X})$$.$$\square$$

A holomorphic line bundle with $$c_1=0$$ is topologically trivial, hence analytically isomorphic to the trivial line bundle by the following Lemma 2.16.

Lemma 2.16

([10, p. 268]) Let X be a Stein space and $${\mathcal {L}}$$ a $${\mathcal {O}}^{\times }_X$$-torsor. If $$c_1({\mathcal {L}})=0$$, then $${\mathcal {L}}$$ is trivial.

Lemma 2.17

Let $$(X,{\mathcal {M}}_X)$$ be a fine log Banach analytic space. Assume that $$P:=\Gamma (X,\overline{{\mathcal {M}}}_X)$$ is globally generated and torsion free. Assume that for each $${\overline{m}}\in \Gamma (X,\overline{{\mathcal {M}}}_X)$$, the torsor $${\mathcal {L}}_{{\overline{m}}}=\kappa ^{-1}({\overline{m}})$$, with $$\kappa :{\mathcal {M}}_X\rightarrow \overline{{\mathcal {M}}}_X$$ the canonical map, is trivial. Then there exists a chart $$P\rightarrow \Gamma (X,{\mathcal {M}}_X)$$.

Proof

Let $$p_1,\ldots ,p_r \in P$$ be generators, that is we have a surjective map $${\mathbb {N}}^r\rightarrow \Gamma (X,\overline{{\mathcal {M}}}_X)$$ sending $$e_i$$ to $$p_i$$. For each $$i\in \{1,\ldots ,r\}$$, choose a section $$m_i\in {\mathcal {L}}_{{\overline{m}}}$$. We obtain a chart $$\phi :{\mathbb {N}}^r\rightarrow \Gamma (X,{\mathcal {M}}_X)$$. Now, we want to modify $$\phi$$ so that it factors through P. Let $$K:={\text {ker}}({\mathbb {Z}}^r\rightarrow P^{gp})$$, we have $$P={\mathbb {N}}^r/K$$. We get the following exact sequence

\begin{aligned} 0\rightarrow K\rightarrow {\mathbb {Z}}^r\rightarrow P^{gp}\rightarrow 0. \end{aligned}

Since, by assumption, P is torsion free, we can find a section $$\pi :{\mathbb {Z}}^r\rightarrow K$$. Set $$h_i:=\phi ^{gp}(\pi (p_i))$$. Clearly, if $$\sum a_ip_i=\sum b_jp_j$$, for $$a_i, b_j \ge 0$$, then it holds $$\prod h^{a_i}_i=\prod h^{b_j}_j$$ in $$\Gamma (X,{\mathcal {M}}^{gp}_X)$$. We get a chart by

\begin{aligned} \begin{aligned} {\tilde{\psi }}:{\mathbb {N}}^r&\rightarrow \Gamma (X,{\mathcal {M}}_X)\\ e_i&\mapsto h_i^{-1}m_i\\ \end{aligned}. \end{aligned}
(24)

$$\square$$

Now, let $${\tilde{\psi }}^{gp}:{\mathbb {Z}}^r\rightarrow \Gamma (X,{\mathcal {M}}^{gp}_X)$$ and $$\sum a_ie_i\in K$$. If $${\tilde{\psi }}^{gp}(\sum a_ie_i)=1$$, we get that $${\tilde{\psi }}^{gp}$$ induces a chart $$\psi :P\rightarrow \Gamma (X,{\mathcal {M}}_X)$$. Hence, assume $$\sum a_ie_i\in K$$, $$a_i\in {\mathbb {Z}}$$. Then $${\tilde{\psi }}^{gp}(\sum a_ie_i)=\prod {\tilde{\psi }}^{gp}(e_i)^{a_i}=\prod h^{-a_i}_im_i^{a_i}=\prod \phi ^{gp}(\pi (p_i))^{-a_i}\phi ^{gp}(e_i)^{a_i}=\phi ^{gp}(\pi (-\sum a_ip_i)+\sum a_ie_i))=1$$.

Proposition 2.18

Let $$(X,{\mathcal {M}}_X)$$ be a fine log Banach analytic space over a Banach analytic space S. Let $$s_0\in S$$ and $$q^{\dagger }_0$$ a triangularly privileged log cuirasse on $$(X(s_0),{\mathcal {M}}_{X(s_0)})$$. Then the set of log cuirasses $${\mathcal {Q}}_S(X,{\mathcal {M}}_X)$$ on $$(X,{\mathcal {M}}_X)$$ over S can be endowed with the structure of a Banach analytic space in a neighborhood of $$(s_0,q_0^{\dagger })$$.

Proof

Let us consider the projection $$\pi :{\mathcal {Q}}_S(X,{\mathcal {M}}_X)\rightarrow Q_S(X)$$. By Lemmas 2.15 and 2.16, we can use Lemma 2.17 and get the existence around $$(s_0,q_0)$$ of a local section $$\rho :Q_S(X)\rightarrow {\mathcal {Q}}_S(X,{\mathcal {M}}_X)$$, such that $$\rho (s_0,q_0)=q^{\dagger }_0$$. Now, let $$(s,q)\in Q_S(X)$$, in a small neighborhood of $$(s_0,q_0)$$, and consider $$\rho (s,q)\in {\mathcal {Q}}_S(X,{\mathcal {M}}_X)$$. We have that $$\rho (s,q)=(s,q=(Y_i,f_i),(\theta _i),(\eta ^{i}_j))$$, where $$(\theta _i),(\eta ^{i}_j)$$ is a directed collection of log charts on $$(X_{\varphi _q},{\mathcal {M}}_{X_{\varphi _q}})$$ (see Definition 2.10). Any other directed set of log charts $$((\theta '_i),(\eta ^{'i}_j))$$ on $$(X_{\varphi _q},{\mathcal {M}}_{X_{\varphi _q}})$$ is obtained by $$\theta '_i=\chi _i\cdot \theta _i$$ and $$\eta ^{'i}_j=\chi _i^{-1}\cdot \chi _j\cdot \eta ^i_j$$, for morphisms $$\chi _i:P_i\rightarrow {\mathcal {O}}^{\times }_{Y^{\circ }_i}$$, for $$i\in I$$. Therefore, let $$\underline{Y}_i$$ be the universal $${\mathcal {G}}(K_i)$$-anaflat subspace of $${\mathcal {G}}(K_i)\times K_i$$, for $$i\in I$$ ([4, p. 579], [25, pp. 258–259] and [30, p. 183, Theorem 4.13]). We can define a map

\begin{aligned}{} & {} \gamma :{\mathcal {Q}}_S(X)\times _{\prod {\mathcal {G}}(K_i)\times S}{{\,\mathrm{\mathop {{\mathop {\prod }\nolimits _S}}\limits }\,}}_{i\in I} \mathfrak {Mor}_{{\mathcal {G}}(K_i)\times S}(\underline{Y}_i\times S, {\text {Spec}}{\mathbb {C}}[P^{gp}_{i}]\times \\ {}{} & {} {\mathcal {G}}(K_i)\times S)\rightarrow {\mathcal {Q}}_S(X,{\mathcal {M}}_X) \end{aligned}

via

\begin{aligned} (s,q,(\chi _i))\mapsto (s,q,(\chi _i\cdot \theta _i), (\chi _i^{-1}\cdot \chi _j\cdot \eta ^i_j)), \end{aligned}

which defines a structure of Banach analytic space on $${\mathcal {Q}}_S(X,{\mathcal {M}}_X)$$ in a neighborhood of $$(s_0,q^{\dagger }_0)$$.$$\square$$

Proposition 2.19

Let $$(X,{\mathcal {M}}_X)$$ be a fine log Banach analytic space proper and anaflat over a Banach analytic space S. Let $$s\in S$$ and $$q^{\dagger }(s)$$ a triangularly privileged log cuirasse on $$(X(s),{\mathcal {M}}_{X(s)})$$. Then

\begin{aligned}\pi :{\mathcal {Q}}_S(X,{\mathcal {M}}_X)\rightarrow S\end{aligned}

is smooth in a neighborhood of $$q^{\dagger }(s)$$.

Proof

Let $$q^{\dagger }(s)$$ be a triangularly privileged log cuirasse on $$(X(s),{\mathcal {M}}_{X(s)})$$. By Proposition 2.18, we have that in a neighborhood of $$(s,q^{\dagger }(s))$$, the space $${\mathcal {Q}}_{S}(X,{\mathcal {M}}_{X})$$ is isomorphic to

\begin{aligned} {\mathcal {Q}}_S(X)\times _{\prod {\mathcal {G}}(K_i)\times S}{{\,\mathrm{\mathop {{\mathop {\prod }\nolimits _S}}\limits }\,}}_{i\in I} \mathfrak {Mor}_{{\mathcal {G}}(K_i)\times S}(\underline{Y}_i\times S, {\text {Spec}}{\mathbb {C}}[P^{gp}_{i}]\times {\mathcal {G}}(K_i)\times S). \end{aligned}

Let $$q(s)=(Y_i,f_i)$$ be the triangularly privileged cuirasse on X(s) underlying $$q^{\dagger }(s)$$. By [4, p. 589, Corollary 2], $$\pi :{\mathcal {Q}}_{S}(X)\rightarrow S$$ is smooth in a neighborhood of (sq(s)). Furthermore, by [4, p. 585, Proposition 2], we have that

\begin{aligned} \mathfrak {Mor}_{{\mathcal {G}}(K_i)\times S}(\underline{Y}_i\times S, {\text {Spec}}{\mathbb {C}}[P^{gp}_{i}]\times {\mathcal {G}}(K_i)\times S)\rightarrow S \end{aligned}

is smooth in a neighborhood of $$(s,Y_i,\chi _i)$$. Hence, the statement follows.$$\square$$

Analogously to the classical case (14), we define the notion of log puzzle associated to a log cuirasse. Let $$(X_0,{\mathcal {M}}_{X_0})$$ be a compact fine log complex space. Let $$q_0$$ be a cuirasse on $$X_0$$. By Definition 2.9, we get a compact fine log complex space $$(X_{\varphi _{q_0}},{\mathcal {M}}_{X_{\varphi _{q_0}}})$$, which is isomorphic to $$(X_0,{\mathcal {M}}_{X_0})$$. Let $$q^{\dagger }_0=(q_0,(\theta _i),(\eta ^{i}_j))$$ be a log cuirasse on $$(X_0,{\mathcal {M}}_{X_0})$$ (see Definition 2.10). Let $$\alpha _{X_{\varphi _{q_0}}}:{\mathcal {M}}_{X_{\varphi _{q_0}}}\rightarrow {\mathcal {O}}_{X_{\varphi _{q_0}}}$$ be the structure log morphism and $$z_{q_0}\in {\mathfrak {Z}}$$ the puzzle associated to $$q_0$$.

Definition 2.20

We call

\begin{aligned} z_{q^{\dagger }_0}:=(z_{q_0},(\alpha _{X_{\varphi _{q_0}}}\circ \theta _i),(\eta ^{i}_j ))\end{aligned}

the log puzzle associated to $$q^{\dagger }_0$$.

Clearly, $$z_{q^{\dagger }_0}\in {\mathfrak {Z}}^{\log }$$ (Definition 2.5). Let $$(X,{\mathcal {M}}_{X})$$ be a fine log Banach analytic space proper and anaflat over a Banach analytic space S. Let $$q^{\dagger }$$ be a relative log cuirasse on $$(X,{\mathcal {M}}_{X})$$ over S.

Definition 2.21

We can define a morphism

\begin{aligned} \begin{aligned} \varphi _{q^{\dagger }}:S&\rightarrow {\mathfrak {Z}}^{\log }\\ s&\mapsto z_{q^{\dagger }(s)}. \\ \end{aligned} \end{aligned}
(25)

Let $$q^{\dagger }=(q,(\theta _i),(\eta ^i_j))$$ be a log cuirasse on $$(X,{\mathcal {M}}_{X})$$ over S. Let $$(X_{\varphi _q},{\mathcal {M}}_{X_{\varphi _q}})$$ given by Definition 2.9 and $$\alpha _{X_{\varphi _q}}:{\mathcal {M}}_{X_{\varphi _q}}\rightarrow {\mathcal {O}}_{X_{\varphi _q}}$$ the structure log morphism. For each $$i\in I$$, let $${\mathcal {M}}^{a}_{X_{\varphi _q},i}$$ be the log structure associated to the pre-log structure $$\alpha _{X_{\varphi _q}}\circ \theta _i$$. The collection of log structures $$({\mathcal {M}}_{X_{\varphi _q},i}^{a})$$ glues to a log structure $${\mathcal {M}}_{X_{\varphi _q}}^{a}$$ on $$X_{\varphi _q}$$ (Sect. 2.1).

Definition 2.22

We set

\begin{aligned} (X_{\varphi _{q^{\dagger }}},{\mathcal {M}}_{X_{\varphi _{q^{\dagger }}}}):=(X_{\varphi _{q}},{\mathcal {M}}^{a}_{X_{\varphi _q}}). \end{aligned}

The fine log Banach analytic space $$(X_{\varphi _{q^{\dagger }}},{\mathcal {M}}_{X_{\varphi _{q^{\dagger }}}})$$ is obtained by gluing the pieces of the log puzzle $$z_{q^{\dagger }}$$ associated to the cuirasse $$q^{\dagger }$$.

Proposition 2.23

Let $$(X,{\mathcal {M}}_X)$$ be a fine log Banach analytic space proper and anaflat over a Banach analytic space S. Let $$q^{\dagger }$$ be a relative log cuirasse on $$(X,{\mathcal {M}}_X)$$ over S. Then, there exists a log S-isomorphism

Proof

By Proposition 16, we have an S-isomorphism $$\alpha _q:X_{\varphi _q}\rightarrow X$$. Moreover, we have $${\mathcal {M}}_{X_{\varphi _q}}:=\alpha _q^{*}{\mathcal {M}}_{X}$$ (see Definition 2.10). Hence, $$\alpha _q$$ induces a S-log isomorphism $$\alpha _q:(X_{\varphi _q},{\mathcal {M}}_{X_{\varphi _q}})\rightarrow (X,{\mathcal {M}}_X)$$. Now, the log cuirasse $$q^{\dagger }$$ gives us a collection of directed log charts $$((\theta _i),(\eta ^i_j))$$ for $${\mathcal {M}}_{X_{\varphi _q}}$$. Let $$\alpha _{X_{\varphi _q}}:{\mathcal {M}}_{X_{\varphi _q}}\rightarrow {\mathcal {O}}_{X_{\varphi _q}}$$ be the structure log morphism and $${\mathcal {M}}^{a}_{X_{\varphi _q},i}$$ the log structure associated to the pre-log structure $$\alpha _{X_{\varphi _q}}\circ \theta _i$$, for each $$i\in I$$. By the definition of log chart ([26, p. 249]), we have an isomorphism $$\alpha ^{\flat }_i:{\mathcal {M}}_{X_{\varphi _q},i}^{a}\rightarrow {\mathcal {M}}_{X_{\varphi _q}}$$. Then the collection of log structures $$({\mathcal {M}}_{X_{\varphi _q},i}^{a})$$, together with the isomorphisms $$(\alpha ^{\flat }_i)$$, glues to a log structure $${\mathcal {M}}_{X_{\varphi _q}}^{a}$$ on $$X_{\varphi _q}$$, together with an isomorphism $$\alpha ^{\flat }:{\mathcal {M}}_{X_{\varphi _q}}^{a}\rightarrow {\mathcal {M}}_{X_{\varphi _q}}$$ (see Sect. 2.1). Hence, set $$(X_{\varphi _{q^{\dagger }}},{\mathcal {M}}_{X_{\varphi _{q^{\dagger }}}}):=(X_{\varphi _{q}},{\mathcal {M}}^{a}_{X_{\varphi _q}})$$ and $$\alpha :=({{\,\textrm{Id}\,}}, \alpha ^{\flat })$$, we get an isomorphism $$\alpha :(X_{\varphi _{q^{\dagger }}},{\mathcal {M}}_{X_{\varphi _{q^{\dagger }}}})\rightarrow (X_{\varphi _{q}},{\mathcal {M}}_{X_{\varphi _q}})$$. Set $$\alpha _{q^{\dagger }}:=\alpha _q\circ \alpha$$.$$\square$$

Remark 2.24

Clearly,

\begin{aligned} (X_{\varphi _{q^{\dagger }}},{\mathcal {M}}_{X_{\varphi _{q^{\dagger }}}})=\varphi _{q^{\dagger }}^{*}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }}). \end{aligned}

We are ready to prove the existence of an infinite-dimensional complete deformation of a fine compact log complex space $$(X_0,{\mathcal {M}}_{X_0})$$. With the due modifications, the proof of Theorem 2.25 is identical to the proof of Theorem 1.1 ([4, p. 592]). Let $$q_0=(Y_{i,0},f_{i,0})$$ be a triangularly privileged cuirasse on $$X_0$$ and $$((\theta _{i,0}),(\eta ^{i}_{j,0}))$$ the collection of directed log charts on $$(X_{0},{\mathcal {M}}_{X_0})$$ as in Definition 2.4. Then,

\begin{aligned} q^{\dagger }_{0}:=(q_0,(f^{*}_{i,0}\theta _i),(f^{*}_{j,0}\eta ^{i}_j)) \end{aligned}

is a triangularly privileged log cuirasse on $$(X_{0},{\mathcal {M}}_{X_0})$$ (see Definition 2.14). Let $$z_{q^{\dagger }_{0}}$$ be the log puzzle associated to $$q^{\dagger }_{0}$$ (see Definition 2.20). Let $$({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})\rightarrow {\mathfrak {Z}}^{\log }$$ be the universal space of log puzzles (see Proposition 2.8) and

\begin{aligned} \alpha _{q^{\dagger }_{0}}:({\mathfrak {X}}_{\log }(z_{q^{\dagger }_{0}}),{\mathcal {M}}_{{\mathfrak {X}}_{\log }(z_{q^{\dagger }_{0}})})\rightarrow (X_{0},{\mathcal {M}}_{X_0}) \end{aligned}

the log isomorphism given by Proposition 2.23.

Theorem 2.25

The triple $$(({\mathfrak {Z}}^{\log }, z_{q^{\dagger }_0}), ({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }}), \alpha _{q^{\dagger }_{0}})$$ is a complete deformation of $$(X_{0},{\mathcal {M}}_{X_0})$$.

Proof

Let $$(X,{\mathcal {M}}_{X})$$ be a fine log Banach analytic space proper and anaflat over a Banach analytic space S. Let $$s_0\in S$$ and $$i:(X(s_0),{\mathcal {M}}_{X(s_0)})\rightarrow (X_0,{\mathcal {M}}_{X_0})$$ a log isomorphism. Since $$i^{*}q^{\dagger }_{0}$$ is a triangularly privileged log cuirasse (see Definition 2.14) on $$(X(s_{0}),{\mathcal {M}}_{X(s_0)})$$, we have that $${\mathcal {Q}}_S(X,{\mathcal {M}}_X)$$ is smooth over S in a neighborhood of $$i^{*}q^{\dagger }_{0}$$ (Proposition 2.19). Therefore there exists a local relative log cuirasse $$q^{\dagger }$$ on $$(X,{\mathcal {M}}_X)$$ defined in a neighborhood $$S'$$ of $$s_{0}$$ in S. Hence, taking $$\varphi _{q^{\dagger }}:S\rightarrow {\mathfrak {Z}}^{\log }$$ (Definition 2.21) and the $$S'$$-isomorphism

\begin{aligned} \alpha _{q^{\dagger }}|_{S'}:(X_{\varphi _{q^{\dagger }}},{\mathcal {M}}_{X_{\varphi _{q^{\dagger }}}})\rightarrow (X,{\mathcal {M}}_X) \end{aligned}

(Proposition 2.23), the statement follows.$$\square$$

2.3 Finite dimensional reduction

Let $$({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})\rightarrow {\mathfrak {Z}}^{\log }$$ be the complete deformation of $$(X_0,{\mathcal {M}}_{X_0})$$ given by Theorem 2.25 and $${\mathcal {Q}}_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})$$ the space of relative log cuirasses on $$({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})$$ over $${\mathfrak {Z}}^{\log }$$ (Definition 2.10). Since the finite-dimensional reduction is performed on the Banach analytic space $${\mathcal {Q}}_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})$$, which does not come endowed with a non-trivial log structure, the finite-dimensional reduction in the log setting is identical to the one in the classical setting (see [4, pp. 593–599] and [34, pp. 20–46]). In what follows, we give an account of the main steps of the finite-dimensional reduction procedure (in the log setting). For more details, see [1, pp. 90–100].

We recall from subection 2.2 that the space $${\mathfrak {Z}}^{\log }$$ (Definition 2.6) parametrizes all log puzzles $$z^{\dagger }$$ of type $${\mathfrak {I}}$$ (Definition 2.5). Each fibre $$({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})(z^{\dagger })$$ of the map $$({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})\rightarrow {\mathfrak {Z}}^{\log }$$ is obtained by gluing the “pieces”, $$(Y_i)_{i\in I}$$ and $$(\beta _i:P_i\rightarrow {\mathcal {O}}_{Y^{\circ }_i})_{i\in I}$$, of the log puzzle $$z^{\dagger }$$.

Each point in $${\mathcal {Q}}_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})$$ is a pair $$(z^{\dagger },q^{\dagger })$$, where $$z^{\dagger }\in {\mathfrak {Z}}^{\log }$$ is a log puzzle and $$q^{\dagger }$$ is a log cuirasse on the fibre $$({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})(z^{\dagger })$$. To the log cuirasse $$q^{\dagger }$$ we can naturally associate another log puzzle $$z_{q^{\dagger }}\in {\mathfrak {Z}}^{\dagger }$$ (Definition 2.20). In principle, $$z^{\dagger }\ne z_{q^{\dagger }}$$ although

\begin{aligned} ({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})(z^{\dagger })\simeq ({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})(z_{q^{\dagger }}), \end{aligned}

(Proposition 2.23). However, we can consider the subspace $$Z^{\log }\subset {\mathcal {Q}}_{\mathfrak {Z^{\log }}}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})$$ defined by selecting, in each fibre $${\mathcal {Q}}(({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})(z^{\dagger }))$$ of the canonical projection $$\pi :{\mathcal {Q}}_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})\rightarrow {\mathfrak {Z}}^{\log }$$, all log cuirasses $$q^{\dagger }$$ on $$({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})(z^{\dagger })$$ whose associated log puzzle $$z_{q^{\dagger }}$$ coincides exactly with $$z^{\dagger }$$. More precisely, there exists a canonical relative log cuirasse $${\mathfrak {q}}^{\dagger }$$ on

\begin{aligned} \pi ^{*}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})\rightarrow {\mathcal {Q}}_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }}), \end{aligned}
(26)

see [4, p. 593], [25, p.267] and [1, p. 90]. By Definition 2.21, we get an associated morphism

\begin{aligned} \begin{aligned} \varphi _{{\mathfrak {q}}^{\dagger }}:{\mathcal {Q}}_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})&\rightarrow {\mathfrak {Z}}^{\log }\\ (z^{\dagger },q^{\dagger })&\mapsto z_{q^{\dagger }}\\ \end{aligned}. \end{aligned}
(27)

Then, the subspace $$Z^{\log }$$ is obtained as the kernel of the double arrow $$(\pi ,\varphi _{{\mathfrak {q}}^{\dagger }})$$:

\begin{aligned} Z^{\log }:={\text {ker}}(\pi ,\varphi _{{\mathfrak {q}}^{\dagger }})\subset {\mathcal {Q}}_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }}). \end{aligned}
(28)

The space $$Z^{\log }$$ parametrizes all log cuirasses on compact fine log complex spaces “close” to $$(X_0,{\mathcal {M}}_{X_0})$$. This space is not as pathological as $$\mathfrak {Z^{\log }}$$ (see [4, p. 590, Remark]) and it still gives a complete deformation of $$(X_0,{\mathcal {M}}_{X_0})$$. Indeed, given any log Banach analytic space $$(X,{\mathcal {M}}_X)$$ proper and anaflat over a Banach analytic space S, we get a map from the space of relative log cuirasses $${\mathcal {Q}}_S(X,{\mathcal {M}}_X)$$ into the space of log puzzles $${\mathfrak {Z}}^{\log }$$ by

\begin{aligned} \begin{aligned} \varphi _{(X,{\mathcal {M}}_X)/S}:{\mathcal {Q}}_S(X,{\mathcal {M}}_X)&\rightarrow {\mathfrak {Z}}^{\log }\\ (s,q^{\dagger })&\mapsto z_{q^{\dagger }}.\\ \end{aligned} \end{aligned}
(29)

If $$\sigma ^{\dagger }:S\rightarrow {\mathcal {Q}}_S(X,{\mathcal {M}}_X)$$ is a relative log cuirasse on $$(X,{\mathcal {M}}_X)$$ over S, that is a section of the projection $$\pi :{\mathcal {Q}}_S(X,{\mathcal {M}}_X)\rightarrow S$$, the composition

\begin{aligned} \varphi _{\sigma ^{\dagger }}:=\varphi _{(X,\mathcal {M_X})/S}\circ \sigma ^{\dagger }:S\rightarrow {\mathfrak {Z}}^{\log } \end{aligned}

is a morphism satisfying the completeness property (see Definition 2.21 and Theorem 2.25). Indeed, for each $$s\in S$$, the fibre $$({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})(\varphi _{\sigma ^{\dagger }}(s))$$ is isomorphic to the fibre $$(X,{\mathcal {M}}_X)(s)$$ via an isomorphism $$\alpha _{\sigma ^{\dagger }}$$ (Proposition 2.23). Identifying these two isomorphic fibres, we get a map

\begin{aligned} \begin{aligned} \psi _{\sigma ^{\dagger }}:S&\rightarrow Q_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})\\ s&\mapsto (z_{\sigma ^{\dagger }(s)},\sigma ^{\dagger }(s))\\ \end{aligned}. \end{aligned}
(30)

For more details, see [1, pp. 70–71 and pp. 90–91]. We can draw the following commutative diagram:

In fact, $$\psi _{\sigma ^{\dagger }}$$ is the unique morphism from S to $$Q_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})$$ making the above diagram commutative (see [4, p. 593]). By construction, $$\psi _{\sigma ^{\dagger }}$$ factors through $$Z^{\log }\subset Q_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})$$ and it is used to prove Proposition 2.26. Let

\begin{aligned} i:Z^{\log }\hookrightarrow {\mathcal {Q}}_{{\mathfrak {Z}}^{\log }} ({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})\end{aligned}

be the canonical injection, we set

\begin{aligned}({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}} _{{\mathfrak {X}}_{Z^{\log }}}):=i^{*}\pi ^{*}({\mathfrak {X}}_{\log }, {\mathcal {M}}_{{\mathfrak {X}}_{\log }}). \end{aligned}

Let $$q^{\dagger }_0$$ be a triangularly privileged log cuirasse on $$(X_0,{\mathcal {M}}_{X_0})$$ (see Definition 2.14) and $$z_{q^{\dagger }_0}\in {\mathfrak {Z}}^{\log }$$ the associated log puzzle (Definition 2.20). We get a point $$(z_{q^{\dagger }_0},q^{\dagger }_0)$$ in $$Z^{\log }$$.

Proposition 2.26

([4, p. 593, Theorem 1], [34, p. 31, Satz 1.15], [1, p. 91, Proposition 5.26]) The morphism $$({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}})\rightarrow Z^{log}$$ is a complete deformation of the compact fine log complex space $$(X_0,{\mathcal {M}}_{X_0})$$, in a neighborhood of $$(z_{q^{\dagger }_0},q^{\dagger }_0)$$.

In what follows, we are going to decompose $$Z^{\log }$$ into a product $$\Sigma ^{\log }\times R^{\log }$$, where $$\Sigma ^{\log }$$ is a Banach manifold and $$R^{\log }$$ is a finite dimensional complex analytic space, which will be our finite dimensional semi-universal deformation space.

To do that, let us start by introducing the notion of extendable log cuirasse ([1, pp. 95–96]) by adapting, to the log context, Douady’s notion of extendable cuirasse ([4, p. 594]). This is a fundamental tool to achieve finite dimensionality.

Definition 2.27

([4, p. 594]) Let us consider two types of cuirasses, namely $${\mathfrak {I}}=(I_{\bullet }, (K_{i}),({\tilde{K}}_{i}),(K'_{i}))$$ and $$\hat{{\mathfrak {I}}}=(I_{\bullet }, ({\hat{K}}_{i}),(\hat{{\tilde{K}}}_{i}),(\hat{K'}_{i}))$$ (8), which have the same underlying simplicial set. We write $${\mathfrak {I}}\Subset \hat{{\mathfrak {I}}}$$, if $$K_i\Subset {\hat{K}}_i$$, $${\tilde{K}}_{i}\subset \hat{{\tilde{K}}}_{i}$$ and $$K'_{i}\subset \hat{K'}_{i}$$.

Let $$\hat{{\mathfrak {I}}}$$ be a type of cuirasse and $${\hat{q}}^{\dagger }$$ a relative log cuirasse of type $$\hat{{\mathfrak {I}}}$$ on a log Banach analytic space $$(X,{\mathcal {M}}_X)$$ proper and anaflat over S. Then, by slightly shrinking each polycylinder $${\hat{K}}_i$$, $$\hat{{\tilde{K}}}_{i}$$ and $${\hat{K}}'_{i}$$ in $$\hat{{\mathfrak {I}}}$$, we can get polycylinders $$K_i$$, $${\tilde{K}}_{i}$$ and $$K'_{i}$$ respectively and hence a type of cuirasse $${\mathfrak {I}}$$, such that $${\mathfrak {I}}\Subset \mathfrak {{\hat{I}}}$$. Then,

\begin{aligned} q^{\dagger }:={\hat{q}}^{\dagger }|_{{\mathfrak {I}}} \end{aligned}

is an extendable relative log cuirasse on $$(X,{\mathcal {M}}_X)$$ over S.

If $${\mathfrak {I}}\Subset \hat{{\mathfrak {I}}}$$ are two types of cuirasses, then we can construct the spaces of log puzzles $${\mathfrak {Z}}^{\log }$$ and $$\hat{{\mathfrak {Z}}}^{\log }$$ of type $${\mathfrak {I}}$$ and $$\hat{{\mathfrak {I}}}$$ respectively (see Definition 2.6). It can be shown (see [4, p. 595] and [34, p.44]), that the restriction morphism

\begin{aligned} j^{\dagger }:\hat{{\mathfrak {Z}}}^{\log }\rightarrow {\mathfrak {Z}}^{\log } \end{aligned}
(31)

is compact (in the sense of [3, p. 28]). This fact, together with the finite dimensionality results [3, p. 29, Proposition 3] and [25, p. 271] (see, also, [1, pp. 43–44]), is used to prove Proposition 2.28.

Set

\begin{aligned} Q^{\log }_0:={\mathcal {Q}}(X_0,{\mathcal {M}}_{X_0}), \end{aligned}

the space of log cuirasses on $$(X_0,{\mathcal {M}}_{X_0})$$ (see Definition 2.10). By Proposition 2.19 the projection $$\pi :{\mathcal {Q}}_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log },{\mathcal {M}}_{{\mathfrak {X}}_{\log }})\rightarrow {\mathfrak {Z}}^{\log }$$ is smooth in a neighborhood of $$(z_{q^{\dagger }_0},q^{\dagger }_0)$$, hence we can opportunely choose (see [4, p. 595], [25, p. 269] and [1, p. 96]) a local trivialization

\begin{aligned} (\pi ,\rho ^{\dagger }):{\mathcal {Q}}_{{\mathfrak {Z}}^{\log }}({\mathfrak {X}}_{\log }, {\mathcal {M}}_{{\mathfrak {X}}_{\log }})\rightarrow {\mathfrak {Z}}^{\log }\times Q^{\log }_0. \end{aligned}
(32)

Set

\begin{aligned} p^{\dagger }:=\rho ^{\dagger }|_{Z^{\log }}. \end{aligned}
(33)

Proposition 2.28 is the log version of [4, p. 596, Proposition 4], [34, p.43, Satz 1.33] and [25, p. 269, Lemma 1]. The same proof applies here likewise. For further details, see [1, p. 96, Proposition 5.30].

Proposition 2.28

The morphism $$p^{\dagger }:Z^{\log }\rightarrow Q^{\log }_0$$ is of relative finite dimension in a neighborhood of $$(z_{q^{\dagger }_0},q^{\dagger }_0)$$.

Thus, we get the existence of an embedding $$\iota ^{\dagger }:Z^{\log }\hookrightarrow Q^{\log }_0\times {\mathbb {C}}^m$$ making the following diagram commutative:

(34)

By Proposition 2.19, the canonical projection

\begin{aligned} \pi _{Z^{\log }}:{\mathcal {Q}}_{Z^{\log }}({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}})\rightarrow Z^{\log } \end{aligned}

is smooth in a neighborhood of $$(z_{q_0^{\dagger }},q_0^{\dagger },q_0^{\dagger })$$, hence we can opportunely choose (see [34, p. 35], [25, p. 269] and [1, p. 94]) a local trivialization

\begin{aligned} \gamma ^{\dagger }:Z^{\log }\times Q^{\log }_0\rightarrow {\mathcal {Q}}_{Z^{\log }}({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}}). \end{aligned}
(35)

We notice that the restriction of the canonical relative log cuirasse $${\mathfrak {q}}^{\dagger }$$(26) to $$Z^{\log }$$ produces a canonical relative log cuirasse on $$({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}})$$ over $$Z^{\log }$$. Hence, by (30), we get a map

\begin{aligned} \psi _{{\mathfrak {q}}^{\dagger }}:{\mathcal {Q}}_{Z^{\log }}({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}})\rightarrow Z^{\log }. \end{aligned}
(36)

Thus, we can define morphisms:

\begin{aligned} \omega ^{\dagger }:=\psi _{{\mathfrak {q}}^{\dagger }}\circ \gamma ^{\dagger } \text { and } \delta ^{\dagger }:=\omega ^{\dagger }|_{\{(z_{q^{\dagger }_0},q^{\dagger }_0)\}\times Q^{\log }_0} \end{aligned}
(37)

By Proposition 2.28, we can draw the following commutative diagram:

(38)

Proposition 2.29 is the log version of [4, p. 595, Proposition 2] and [34, p. 40, Satz 1.31]. See also [1, p. 77, Proposition 4.40].

Proposition 2.29

The linear tangent map:

\begin{aligned} \text {T}_{q^{\dagger }_0}(p^{\dagger }\circ \delta ^{\dagger }):\text {T}_{q^{\dagger }_0}Q^{\log }_0\rightarrow \text {T}_{q^{\dagger }_0}Q^{\log }_0 \end{aligned}

is of the form $${{\,\textrm{Id}\,}}-v^{\dagger }$$, with $$v^{\dagger }$$ compact.

From Proposition 2.29, it follows that $${\text {ker}}\text {T}_{q^{\dagger }_0}(p^{\dagger }\circ \delta ^{\dagger })$$ is of finite dimension. Moreover, by (38), we have

\begin{aligned} {\text {ker}}\text {T}_{q^{\dagger }_0}(p^{\dagger }\circ \delta ^{\dagger })\supset {\text {ker}}\text {T}_{q^{\dagger }_0}\delta ^{\dagger }={\text {ker}}\text {T}_{q^{\dagger }_0}(\iota ^{\dagger }\circ \delta ^{\dagger }). \end{aligned}

Hence, $${\text {ker}}\text {T}_{q^{\dagger }_0}(\iota ^{\dagger }\circ \delta ^{\dagger })$$ is of finite dimension. Since $$\pi _1$$ (38) is a surjective map, we can conclude that $${{\,\textrm{Im}\,}}\text {T}_{q^{\dagger }_0}(\iota ^{\dagger }\circ \delta ^{\dagger })$$ has finite codimension in $$\text {T}_{q^{\dagger }_0}Q_0^{\log }$$ (see [34, p. 45]).

Let us consder $$\omega ^{\dagger }:Z^{\log }\times Q^{\log }_0\rightarrow Z^{\log }$$ given by (37). Proposition 2.30 is the log version of [34, p. 36, Satz 1.25]. The same proof applies here likewise. For further details see [1, p. 94, Proposition 5.28].

Proposition 2.30

Let S be a Banach analytic space and $$f,g:S\rightarrow Z^{\log }$$ morphisms. Then $$f^{*}({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}})\simeq g^{*}({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}})$$, if and only if there exists $$h^{\dagger }:S\rightarrow Q^{\log }_0$$ such that the following diagram commutes

(39)

In other words,

\begin{aligned}f^{*}({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}})\simeq g^{*}({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}}) \end{aligned}

if and only if, for each $$s\in S$$, g(s) is obtained “changing” f(s) by a log cuirasse $$q^{\dagger }$$ on the central fibre $$(X_0,{\mathcal {M}}_{X_0})$$. Notice that, by Proposition 2.30

\begin{aligned} \omega ^{\dagger }|_{Z^{\log }\times q_0^{\dagger }}={{\,\textrm{Id}\,}}_{Z^{\log }}. \end{aligned}
(40)

Let us denote with $${\text {Ex}}^1(X_0,{\mathcal {M}}_{X_0})$$ the set of equivalence classes of infinitesimal deformations of $$(X_0,{\mathcal {M}}_{X_0})$$, that is deformations over the double point $$D=(\{\cdot \},{\mathbb {C}}[\epsilon ]/\epsilon ^{2})$$. For the sake of clarity, set $$r^{\dagger }_0:=(z_{q^{\dagger }_{0}},q^{\dagger }_{0})$$.

Since $$({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}})\rightarrow (Z^{\log },r^{\dagger }_0)$$ is complete, the Kodaira–Spencer map (1) is surjective

\begin{aligned} \text {ks}:\text {T}_{r^{\dagger }_0}Z^{\log }\twoheadrightarrow {\text {Ex}}^{1}(X_0,{\mathcal {M}}_{X_0}). \end{aligned}

The kernel $${\text {ker}}\text {ks}$$ corresponds to the trivial deformations of $$(X_0,{\mathcal {M}}_{X_0})$$ over D. By Proposition 2.30, with $$S=D$$, we see that the trivial deformations of $$(X_0,{\mathcal {M}}_{X_0})$$ over D are given by $${{\,\textrm{Im}\,}}\text {T}_{q^{\dagger }_0}\delta ^{\dagger }$$. Hence,

\begin{aligned} {\text {Ex}}^1(X_0,{\mathcal {M}}_{X_0})=\text {T}_{r^{\dagger }_0}Z^{\log }/{{\,\textrm{Im}\,}}\text {T}_{q^{\dagger }_0}\delta ^{\dagger }. \end{aligned}
(41)

Let us identify $$Z^{\log }$$ with its image in $$Q^{\log }_0\times {\mathbb {C}}^{m}$$ under $$\iota ^{\dagger }$$. By Proposition 2.29, let $$\Sigma ^{\log }$$ be the Banach submanifold of $$Q_0^{\log }$$ such that

\begin{aligned} \text {T}_{q^{\dagger }_0}\Sigma ^{\log }\oplus {\text {ker}}\text {T}_{q^{\dagger }_0}\delta ^{\dagger }=\text {T}_{q^{\dagger }_0}Q^{\log }_0. \end{aligned}
(42)

Let $$r:Q^{\log }_0\times {\mathbb {C}}^m\rightarrow \delta ^{\dagger }(\Sigma ^{\log })$$ be a retraction and set

\begin{aligned} R^{\log }:=r^{-1}(q^{\dagger }_0)\cap Z^{\log }. \end{aligned}
(43)

By construction

\begin{aligned} \text {T}_{r^{\dagger }_0}R^{\log }={\text {Ex}}^1(X_0,{\mathcal {M}}_{X_0}) \end{aligned}
(44)

Lemma 2.31

([4, p. 598, Proposition 5] and [34, p. 37, Satz 1.28]) Let $$\Sigma _1, H$$ and $$\Sigma _2$$ be Banach manifolds, with $$\Sigma _1, H\subset \Sigma _2$$ and $$0\in \Sigma _1\cap H$$. Assume that $$\Sigma _1$$ is of finite codimension and

\begin{aligned} \text {T}_0\Sigma _1\oplus \text {T}_0 H =\text {T}_0\Sigma _2. \end{aligned}
(45)

Let Y be another subspace of $$\Sigma _2$$, containing $$\Sigma _1$$, and set

\begin{aligned} R:=H\cap Y. \end{aligned}

Let

\begin{aligned} \phi :\Sigma _1\times R\rightarrow Y \end{aligned}

be a morphism inducing the identity on $$\Sigma _1\times 0$$ and $$0\times R$$. Then, $$\phi$$ is an isomorphism.

From Lemma 2.31, we obtain that the restriction of the morphism (37)

\begin{aligned} \omega ^{\dagger }|_{R^{\log }\times \Sigma ^{\log }}:R^{\log }\times \Sigma ^{\log }\rightarrow Z^{\log } \end{aligned}
(46)

is an isomorphism. This fact, together with Proposition 2.30 and (44), is used to prove Theorem 2.32.

Let $$i:R^{\log }\hookrightarrow Z^{\log }$$ be the canonical injection. Set

\begin{aligned} ({\mathfrak {X}}_{R^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{R^{\log }}}):=i^{*}({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}}). \end{aligned}

Let $$\alpha _{q^{\dagger }_0}:({\mathfrak {X}}_{R^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{R^{\log }}})(r^{\dagger }_0)\rightarrow (X_0,{\mathcal {M}}_{X_0})$$ be the log isomorphism given by Proposition 2.23. The proof of Theorem 2.32 is identical to the proof of [4, p. 598, Théorème Principal and p. 601, Proposition 1] and to the proof of [34, p. 38, Satz 1.30].

Theorem 2.32

The triple $$((R^{\log },r^{\dagger }_0), ({\mathfrak {X}}_{R^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{R^{\log }}}), \alpha _{q^{\dagger }_0})$$, is a semi-universal deformation of $$(X_0,{\mathcal {M}}_{X_0})$$.

Proof

Let $$((S,s_0),(X,{\mathcal {M}}_X),i)$$ be a deformation of $$(X_0,{\mathcal {M}}_{X_0})$$. By Proposition 2.26, $$(({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}})\rightarrow Z^{\log },(z_{q^{\dagger }_0},q^{\dagger }_0))$$ is a complete deformation of of $$(X_0,{\mathcal {M}}_{X_0})$$. Hence, there exists a morphism $$\psi _{\dagger }:S\rightarrow Z^{\log }$$ such that

\begin{aligned} (X,{\mathcal {M}}_X)\simeq \psi _{\dagger }^{*}({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}}). \end{aligned}

Let $$\Sigma ^{log}$$ and $$R^{\log }$$ given by (42) and (43) respectively. Let $$\pi _{R^{\log }}:R^{\log }\times \Sigma ^{\log }\rightarrow R^{\log }$$ and $$\pi _{\Sigma ^{\log }}:R^{\log }\times \Sigma ^{\log }\rightarrow \Sigma ^{\log }$$ be the projections. By Lemma 2.31, the morphism $$\omega ^{\dagger }|_{R^{\log }\times \Sigma ^{\log }}:R^{\log }\times \Sigma ^{\log }\rightarrow Z^{\log }$$ (46) is an isomorphism. Thus, setting $$g:=\pi _{R^{\log }}\circ (\omega ^{\dagger }|_{R^{\log }\times \Sigma ^{\log }})^{-1}\circ \psi _{\dagger }$$ and $$h^{\dagger }:=\pi _{\Sigma ^{\log }}\circ (\omega ^{\dagger }|_{R^{\log }\times \Sigma ^{\log }})^{-1}\circ \psi _{\dagger }$$, we have

\begin{aligned} \omega ^{\dagger }\circ (g,h^{\dagger })=\psi _{\dagger }. \end{aligned}

Hence, by Proposition 2.30

\begin{aligned} g^{*}({\mathfrak {X}}_{R^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{R^{\log }}}) \simeq \psi _{\dagger }^{*}({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_ {Z^{\log }}})\simeq (X,{\mathcal {M}}_X). \end{aligned}

Moreover, by construction, $$T_{q_0^{\dagger }}R^{\log }={\text {Ex}}^1(X_0,{\mathcal {M}}_{X_0})$$ (44).

Thus, the deformation $$(({\mathfrak {X}}_{R^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{R^{\log }}})\rightarrow R^{\log },r^{\dagger }_0)$$ is complete and effective.

Now, let $$((S,s_0),(X,{\mathcal {M}}_X),i)$$ be a deformation of $$(X_0,{\mathcal {M}}_{X_0})$$ and $$(S',s_0)$$ a subgerm of $$(S,s_0)$$. Because of the just proved completeness, we can find a morphism $$h':(S',s_0)\rightarrow (R^{\log },r_0)$$ such that

\begin{aligned} (X,{\mathcal {M}}_X)|_{S'}\simeq h^{'*}({\mathfrak {X}}_{R^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{R^{\log }}}). \end{aligned}

Let $${\mathfrak {q}}^{\dagger }$$ be the canonical relative log cuirasse on $$({\mathfrak {X}}_{Z^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{Z^{\log }}})$$ over $$Z^{\log }$$ (26). Then, $$h^{'*}{\mathfrak {q}}^{\dagger }$$ is a relative log cuirasse on $$(X,{\mathcal {M}}_{X})|_{S'}$$ over $$S'$$, whose associated morphism (27) coincides with $$h'$$. Since, by Proposition 2.19, $${\mathcal {Q}}_S(X,{\mathcal {M}}_X)$$ is smooth over S in a neighborhood of $$q^{\dagger }_0\in {\mathcal {Q}}((X(s_0),{\mathcal {M}}_{X(s_0)})$$, there exists a relative cuirasse $$q^{\dagger }$$ on $$(X,{\mathcal {M}}_X)$$ over S, such that $$q^{\dagger }|_{S'}=h^{'*}{\mathfrak {q}}^{\dagger }$$. Let $${\tilde{h}}:S\rightarrow Z^{\log }$$ be the morphism associated to $$q^{\dagger }$$ (30) and $$\pi _{R^{\log }}:Z^{\log }\rightarrow R^{\log }$$ the projection. Then, $$h:=\pi _{R^{\log }}\circ {\tilde{h}}$$ satisfies $$(X,{\mathcal {M}}_X)\simeq h^{*}({\mathfrak {X}}_{R^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{R^{\log }}})$$ and $$h|_{S'}=h'$$.

Thus, the deformation $$({\mathfrak {X}}_{R^{\log }},{\mathcal {M}}_{{\mathfrak {X}}_{R^{\log }}})\rightarrow R^{\log }$$ is also versal and, therefore, semi-universal.$$\square$$

3 Semi-universal deformations of log morphisms

In what follows, we construct a semi-universal deformation of a morphism $$f_0:(X_0,{\mathcal {M}}_{X_0})\rightarrow (Y_0,{\mathcal {M}}_{Y_0})$$ of compact fine log complex spaces. Let X be a complex space and $$\alpha _i:{\mathcal {M}}_i\rightarrow {\mathcal {O}}_X$$, $$i=1,2$$, two fine log structures on X. Let $$\gamma :{\overline{{\mathcal {M}}}}_1\rightarrow {\overline{{\mathcal {M}}}}_2$$ be a morphism of the ghost sheaves. Let $$f:T\rightarrow X$$ be a morphism of complex spaces and set $$\gamma _T:({\overline{{\mathcal {M}}}}_1)_T\rightarrow ({\overline{{\mathcal {M}}}}_2)_T$$, the pull-back of $$\gamma$$ via f.

Lemma 3.1

([13, p. 474]) The functor

\begin{aligned} {\text {Mor}}^{\log }_{X}: \textit{An}_{X}\rightarrow \textit{Sets} \end{aligned}

defined on the objects by

\begin{aligned} (f:T\rightarrow X)\mapsto \{\varphi :(T,f^{*}{\mathcal {M}}_1)\rightarrow (T,f^{*}{\mathcal {M}}_2)|{\overline{\varphi }}^{\flat }=\gamma _T\} \end{aligned}

is represented by a complex space $$\mathfrak {Mor}^{\log }_X({\mathcal {M}}_1,{\mathcal {M}}_2)$$ over X.

Proof

By the universal property, the statement is local in X. Hence, let $$\beta _i:P_i\rightarrow \Gamma (X,{\mathcal {M}}_i)$$, $$i=1,2$$, be two log charts for $${\mathcal {M}}_1$$ and $${\mathcal {M}}_2$$ respectively. Let $$p_1,\ldots ,p_n\in P_1$$ be a generating set for $$P_1$$ as monoid. Consider the sheaf of finitely generated $${\mathcal {O}}_X$$-algebras

\begin{aligned} {\mathcal {F}}_X:={\mathcal {O}}_X[P^{gp}_1]/\langle \alpha _1(\beta _1(p_i))-z^{p_i}\alpha _2(\beta _2(\gamma (p_i)))|1\le i\le n\rangle . \end{aligned}

Set $$\mathfrak {Mor}^{\log }_X({\mathcal {M}}_1,{\mathcal {M}}_2):=\varvec{{{\,\mathrm{Spec_{an}}\,}}} {\mathcal {F}}_X$$, the relative analytic spectrum of $${\mathcal {F}}_X$$ over X. Now, we check the universal property. Let $$f:T\rightarrow X$$ be given. We want to show that giving a commutative diagram of complex spaces

is equivalent to giving a log morphism $$\varphi :(T,f^{*}{\mathcal {M}}_1)\rightarrow (T,f^{*}{\mathcal {M}}_2)$$, which is the identity on X and such that $${\overline{\varphi }}^{\flat }=\gamma _T$$. Giving a morphism g is equivalent to giving a section of $$(\varvec{{{\,\mathrm{Spec_{an}}\,}}}{\mathcal {F}}_X)\times _X T$$ over T. But

\begin{aligned} (\varvec{{{\,\mathrm{Spec_{an}}\,}}}{\mathcal {F}}_X)\times _X T=\varvec{{{\,\mathrm{Spec_{an}}\,}}}{\mathcal {O}}_T[P^{gp}_1]/ \langle f^{*}(\alpha _1(\beta _1(p_i)))-z^{p_i}f^{*}(\alpha _2(\beta _2(\gamma (p_i)))) |1\le i\le n\rangle , \end{aligned}

and the latter complex space is $$\varvec{{{\,\mathrm{Spec_{an}}\,}}}{\mathcal {F}}_T$$ associated to the data $$(T,f^{*}{\mathcal {M}}_1)$$,$$(T,f^{*}{\mathcal {M}}_2)$$ with charts $$f^{*}(\beta _i)=f^{\flat }\circ \beta _i:P_i\rightarrow \Gamma (Y,f^{*}{\mathcal {M}}_i)$$. Thus, without loss of generality, we can assume $$T=X$$ and f is the identity. Now, giving $$\varphi :(X,{\mathcal {M}}_1)\rightarrow (X,{\mathcal {M}}_2)$$, with $${\overline{\varphi }}^{\flat }=\gamma$$, is equivalent to specifying $$\varphi ^{\flat }$$. From $$\varphi ^{\flat }$$ we obtain a map $$\eta :P_1\rightarrow \Gamma (X,{\mathcal {O}}^{\times }_X)$$ with the property that for all $$p\in P_1$$,

\begin{aligned} \varphi ^{\flat }(\beta _1(p))=\eta (p)\cdot \beta _2(\gamma (p)). \end{aligned}

Conversely, $$\eta$$ completely determines $$\varphi ^{\flat }$$. In addition, $$\varphi ^{\flat }$$ is a homomorphism of monoids if and only if $$\eta$$ is a homomorphism, and since $$\eta$$ takes values in the group $${\mathcal {O}}^{\times }_X$$, specifying $$\varphi ^{\flat }$$ is equivalent to specifying a section of $$\varvec{{{\,\mathrm{Spec_{an}}\,}}}{\mathcal {O}}_X[P^{gp}_1]$$. Indeed, a section of $$\varvec{{{\,\mathrm{Spec_{an}}\,}}}{\mathcal {O}}_X[P^{gp}_1]$$ over X is the same as a morphism $$X\rightarrow {\text {Spec}}{\mathbb {C}}[P^{gp}_1]$$, which in turn is the same as an element of $${\text {Hom}}(P_1,\Gamma (X,{\mathcal {O}}^{\times }_X))$$. Second, since $$\varphi ^{*}={\text {id}}$$, we must have $$\alpha _1=\alpha _2\circ \varphi ^{\flat }$$, so for each $$p\in P_1$$, we must have

\begin{aligned} \alpha _1(\beta _1(p))=\alpha _2(\varphi ^{\flat }(\beta _1(p)))=\eta (p)\cdot \alpha _2(\beta _2(\gamma (p))). \end{aligned}

If this holds for each $$p_i$$, it holds for all p. Thus a section of $$\varvec{{{\,\mathrm{Spec_{an}}\,}}}{\mathcal {O}}_X[P^{gp}_1]$$ over X determines a morphism of log structures if and only if it lies in the subspace determined by the equations

\begin{aligned} \alpha _1(\beta _1(p_i))-z^{p_i}\alpha _2(\beta _2(\gamma (p_i))), \end{aligned}

demonstrating the result.$$\square$$

Now, assume the complex space X is proper over a germ of complex spaces $$(S,s_0)$$.

Proposition 3.2

([13, p. 475]) The functor

\begin{aligned}{\text {Mor}}^{\log }_{X/S}:(f:(T,t_0)\rightarrow (S,s_0))\mapsto \{\varphi :(X_T,({\mathcal {M}}_1)_T)\rightarrow (X_T,({\mathcal {M}}_2)_T)|{\overline{\varphi }}^{\flat }=\gamma _T\} \end{aligned}

is represented by a germ $$\mathfrak {Mor}^{\log }_{X/S}({\mathcal {M}}_1,{\mathcal {M}}_2)$$ of complex spaces over $$(S,s_0)$$.

Proof

Let $$Z=\mathfrak {Mor}^{\log }_X({\mathcal {M}}_1,{\mathcal {M}}_2)$$. By Lemma 3.1, $$\mathfrak {Mor}^{\log }_{X/S}({\mathcal {M}}_1,{\mathcal {M}}_2)$$ is isomorphic to the functor

\begin{aligned} (\psi :T\rightarrow S)\mapsto \{\text {sections of }\psi ^{*}Z\rightarrow \psi ^{*}X\}. \end{aligned}

This is exactly the functor of sections $$\prod _{X/S}(Z/X)$$ discussed, in the algebraic-geometric setting, in [15, p. 267] and here it is represented by an open subspace of the relative Douady space of Z over S (see [29]).$$\square$$

Proposition 3.3

([7, p. 130]) Let $$f_0:X_0\rightarrow Y_0$$ be a holomorphic map between compact complex spaces. Then $$f_0$$ admits a semi-universal deformation.

Theorem 3.4

Every morphism $$f_{0}:(X_0,{\mathcal {M}}_{X_0})\rightarrow (Y_0,{\mathcal {M}}_{Y_0})$$ of compact fine log complex spaces admits a semi-universal deformation parametrized by a germ of complex spaces $$(S,s_0)$$.

Proof

Let $$(({\mathfrak {X}},{\mathcal {M}}_{{\mathfrak {X}}})\rightarrow R,r_0)$$ and $$(({\mathcal {Y}},{\mathcal {M}}_{{\mathcal {Y}}})\rightarrow R,r_0)$$ be the semi-universal deformations of $$(X_0,{\mathcal {M}}_{X_0})$$ and $$(Y_0,{\mathcal {M}}_{Y_0})$$ respectively given by Theorem 2.32. By pulling-back to the product of the base spaces, we can assume that the two deformations are defined over the same base space. Let us consider the finite dimensional complex analytic space $$\mathfrak {Mor}_{R}({\mathfrak {X}},{\mathcal {Y}})$$ given by Proposition 3.3. Let $$p:\mathfrak {Mor}_{R}({\mathfrak {X}},{\mathcal {Y}})\rightarrow R$$ be the projection and set $$\underline{m}_0:=(r_0,\underline{f}_0)$$. By Proposition 3.3, we get a universal morphism $$\underline{f}:p^{*}{\mathfrak {X}}\rightarrow p^{*}{\mathcal {Y}}$$, such that the restriction of $$\underline{f}$$ to the central fibre $$p^{*}{\mathfrak {X}}(\underline{m}_0)$$ equals $$\underline{f}_0$$. We can consider two fine log structures on $$p^{*}{\mathfrak {X}}$$, namely

\begin{aligned}{\mathcal {M}}_1:=p^{*}{\mathcal {M}}_{{\mathfrak {X}}} \text { and } {\mathcal {M}}_2:=\underline{f}^{*}p^{*}{\mathcal {M}}_{{\mathcal {Y}}}. \end{aligned}

Set $$\gamma :={\overline{f}}^{\flat }_0$$ and $$m_0:=(\underline{m}_0,f^{\flat }_0)$$. For the sake of clarity, denote $${\mathfrak {M}}:= \mathfrak {Mor}_{R}({\mathfrak {X}},{\mathcal {Y}})$$. Now, consider the germ of complex spaces $$(\mathfrak {Mor}^{\log }_{p^{*}{\mathfrak {X}}/{\mathfrak {M}}}({\mathcal {M}}_1,{\mathcal {M}}_2),m_0)$$, together with the projection

\begin{aligned}\pi :(\mathfrak {Mor}^{\log }_{p^{*}{\mathfrak {X}}/{\mathfrak {M}}}({\mathcal {M}}_1,{\mathcal {M}}_2),m_0)\rightarrow (\mathfrak {Mor}_{R}({\mathfrak {X}},{\mathcal {Y}}),\underline{m}_0), \end{aligned}

provided by Proposition 3.2. Moreover, by Proposition 3.2, we get a morphism $$f^{\flat }:\pi ^{*}(p^{*}{\mathfrak {X}},{\mathcal {M}}_1)\rightarrow \pi ^{*}(p^{*}{\mathfrak {X}},{\mathcal {M}}_2)$$.

Hence, we get a log morphism

\begin{aligned}f:=(\pi ^{*}\underline{f},f^{\flat }):\pi ^{*}p^{*}({\mathfrak {X}},{\mathcal {M}}_{{\mathfrak {X}}})\rightarrow \pi ^{*}p^{*}({\mathcal {Y}},{\mathcal {M}}_{{\mathcal {Y}}})\end{aligned}

over $$(\mathfrak {Mor}^{\log }_{p^{*}{\mathfrak {X}}/{\mathfrak {M}}}({\mathcal {M}}_1,{\mathcal {M}}_2),m_0)$$. Set $$(S,s_0):=(\mathfrak {Mor}^{\log }_{p^{*}{\mathfrak {X}}/{\mathfrak {M}}}({\mathcal {M}}_1,{\mathcal {M}}_2),m_0)$$. Using the universal property of $$(S,s_0)$$ (see Proposition 3.2) and Theorem 2.32, the statement follows.$$\square$$

Moreover, we can deform $$(X_0,{\mathcal {M}}_{X_0})$$ as relative log space over $$(Y_0,{\mathcal {M}}_{Y_0})$$. That is, we can deform $$(X_0,{\mathcal {M}}_{X_0})$$ together with the morphism $$f_0$$ into $$(Y_0,{\mathcal {M}}_{Y_0})$$. In this case, $$Y_0$$ needs not to be compact. More precisely,

Definition 3.5

Let $$f_0:(X_0,{\mathcal {M}}_{X_0})\rightarrow (Y_0,{\mathcal {M}}_{Y_0})$$ be a log morphism of fine log complex spaces, with $$X_0$$ compact. A semi-universal deformation of $$(X_0,{\mathcal {M}}_{X_0})$$ over $$(Y_0,{\mathcal {M}}_{Y_0})$$, with base a germ of complex spaces $$(S,s_0)$$, is a commutative diagram

where p is a semi-universal deformation of $$(X_0,{\mathcal {M}}_{X_0})$$, together with an isomorphism $$i:(X_0,{\mathcal {M}}_{X_0})\rightarrow ({\mathfrak {X}},{\mathcal {M}}_{{\mathfrak {X}}})(s_0)$$, such that $$f\circ i=f_0$$.

The same proof of Theorem 3.4, with $$({\mathcal {Y}},{\mathcal {M}}_{{\mathcal {Y}}}):=(Y_0,{\mathcal {M}}_{Y_0})\times R$$, gives us the following

Corollary 3.6

Let $$f_0:(X_0,{\mathcal {M}}_{X_0})\rightarrow (Y_0,{\mathcal {M}}_{Y_0})$$ be a log morphism of fine log complex spaces, with $$X_0$$ compact. Then $$(X_0,{\mathcal {M}}_{X_0})$$ admits a semi-universal deformation over $$(Y_0,{\mathcal {M}}_{Y_0})$$.

Remark 3.7

If $$f_0$$ is a log embedding, then Corollary 3.6 gives us a semi-universal deformation of a log subspace $$(X_0,{\mathcal {M}}_{X_0})$$ in a fixed ambient log space $$(Y_0,{\mathcal {M}}_{Y_0})$$.

Now, we assume that $$f_0$$ is a log flat (log smooth) morphism. We show that, in this case, we get a log flat (log smooth) semi-universal deformation of $$f_0$$. We need the following two results in analytic geometry.

Proposition 3.8

(Critère de platitude par fibres)([14, p. 138] and [33, Tag 00MP]) Let S be a complex space. Let $$f: X \rightarrow Y$$ be a morphism of complex spaces over S. Let $${\mathcal {F}}$$ be a coherent $${\mathcal {O}}_X$$-module. Let $$x \in X$$. Set $$y = f(x)$$ and $$s \in S$$ the image of x in S. Let $$X_s$$ and $$Y_s$$ be the fibres of X and Y over s. Set:

\begin{aligned}{\mathcal {F}}_s = (X_s \hookrightarrow X)^*{\mathcal {F}}.\end{aligned}

Assume $${\mathcal {F}}_x \not = 0$$. Then the following are equivalent:

1. (1)

$${\mathcal {F}}$$ is flat over S at x and $${\mathcal {F}}_s$$ is flat over $$Y_s$$ at x;

2. (2)

Y is flat over S at y and $${\mathcal {F}}$$ is flat over Y at x.

Proposition 3.9

([6, p. 159]) Let $$f:X\rightarrow Y$$ be a morphism of complex spaces. Let $$p\in X$$. Then the following are equivalent

1. (1)

f is smooth (submersion) at $$p\in X$$;

2. (2)

f is flat at p and the fibre $$X_{f(p)}$$ is a manifold.

Proposition 3.8 is due to A. Grothendieck in the algebraic geometry setting. The result can be naturally extended to the analytic setting as for any complex space $$(X,{\mathcal {O}}_X)$$ and $$p\in X$$, the stalk $${\mathcal {O}}_{X,p}$$ is a Noetherian local ring (see [22, p. 80]).

Lemma 3.10

Let $$f: X\rightarrow Y$$ be a continuous map between topological spaces. If f is closed, then for all $$y\in Y$$ and open subset $$U\subset X$$ satisfying $$f^{-1}(y)\subset U$$, there exists an open neighborhood V of y satisfying $$f^{-1}(V)\subset U$$.

Proof

Let us consider the closed subset $$X\backslash U$$. Since f is closed, $$f(X\backslash U)$$ is closed in Y. Therefore, $$Y\backslash f(X\backslash U)$$ is open in Y and it contains y as $$f^{-1}(y)\subset U$$. Take $$V:=f^{-1}(Y\backslash f(X\backslash U)).$$ $$\square$$

The following Lemma 3.11 can be found, in the algebraic geometry setting, in [26, p. 424]. This is a local statement, which extends naturally to the analytic setting.

Lemma 3.11

Any log smooth morphism of fine log complex spaces is log flat.

Let $$f:({\mathfrak {X}},{\mathcal {M}}_{{\mathfrak {X}}}) \rightarrow ({\mathcal {Y}},{\mathcal {M}}_{{\mathcal {Y}}})$$ be the semi-universal deformation of $$f_{0}:(X_0,{\mathcal {M}}_{X_0})\rightarrow (Y_0,{\mathcal {M}}_{Y_0})$$, over a germ of complex spaces $$(S,s_0)$$, given by Theorem 3.4 or Corollary 3.6. Denote with $$\pi _1$$ and $$\pi _2$$ the morphisms of $$({\mathfrak {X}},{\mathcal {M}}_{{\mathfrak {X}}})$$ and $$({\mathcal {Y}},{\mathcal {M}}_{{\mathcal {Y}}})$$ into $$(S,s_0)$$ respectively.

Proposition 3.12

If $$f_0$$ is log flat (log smooth), then f is log flat (log smooth) in an open neighborhood of $$s_0$$.

Proof

Let us assume that there exists an open neighborhood $$U'$$ of $$X_0$$ in $${\mathfrak {X}}$$ such that $$f|_{(U',{\mathcal {M}}_{U'})}$$ is log flat (log smooth). Then, since $$\pi _1:{\mathfrak {X}}\rightarrow S$$ is a proper map between locally compact Hausdorff spaces, it is closed. Hence, by Lemma 3.10, we can find an open neighborhood W of $$s_0$$ such that $$\pi _1^{-1}(W)$$ is contained in $$U'$$. This ensures us that f is log flat (log smooth) as relative morphism over $$(W,s_0)\subset (S,s_0)$$. Since log flatness (log smoothness) is a local property, we choose a log chart for f. We have the following commutative diagram

Let us consider the universal morphism $$\underline{u}:U\rightarrow V\times _{{\text {Spec}}{\mathbb {C}}[Q]} {\text {Spec}}{\mathbb {C}}[P]$$. Let $$p:V\times _{{\text {Spec}}{\mathbb {C}}[Q]} {\text {Spec}}{\mathbb {C}}[P]\rightarrow V$$ be the projection. We get the following commutative diagram:

Assume $$f_0$$ log flat, then we have that $$\underline{u}$$ is flat at $$s_0$$. Moreover, by Theorem 2.32, $$\pi _1|_{U}$$ is flat too. For the sake of clarity, set $$A:=V\times _{{\text {Spec}}{\mathbb {C}}[Q]} {\text {Spec}}{\mathbb {C}}[P]$$.

We use Proposition 3.8 for $${\mathcal {F}}={\mathcal {O}}_{U}$$. Since condition 1 holds, by condition 2 we get that $${\mathcal {O}}_{U,x}$$ is a flat $${\mathcal {O}}_{A,\underline{u}(x)}$$-module, for each $$x\in \pi _1^{-1}|_{U}(s_0)$$. Since every flat holomorphic map is open, we get the existence of an open subset $$U'$$ of U, containing $$\pi _1^{-1}|_{U}(s_0)$$, such that $$\underline{u}_{\arrowvert U'}$$ is flat. This proves the first part of the statement. Now, assume $$f_0$$ log smooth. By Lemma 3.11, $$f_0$$ is log flat. Hence, by the first part of this proof, we get the existence of an open subset $$U'$$ in U such that $$\underline{u}_{\arrowvert U'}$$ is flat. Let $$x\in \pi _1^{-1}|_{U'}(s_0)$$ and set $$y:=\underline{u}(x)\in V\times _{{\text {Spec}}{\mathbb {C}}[Q]} {\text {Spec}}{\mathbb {C}}[P]$$. Since $$f_0$$ is log smooth, $$\underline{u}$$ is smooth at $$s_0$$. Hence, we get that the fibre $$U'_{y}$$ of $$\underline{u}_{\arrowvert U'}$$ over y is a manifold. Therefore, using Proposition 3.9, we get the second part of the statement.$$\square$$

Example 3.13

Let $$({\text {Spec}}{\mathbb {C}},Q)$$ be a log point. Let $$(X_0,{\mathcal {M}}_{X_0})$$ be a compact fine log complex space and $$f_0:(X_0,{\mathcal {M}}_{X_0})\rightarrow ({\text {Spec}}{\mathbb {C}},Q)$$ a log smooth morphism. Since $${\text {Hom}}((Q,+),({\mathbb {C}},\cdot ))={\text {Spec}}{\mathbb {C}}[Q]$$, a semi-universal deformation of the log point $$({\text {Spec}}{\mathbb {C}},Q)$$ is given by the affine toric variety $${\text {Spec}}{\mathbb {C}}[Q]$$ endowed with the canonical log structure. Let $$p_0\in {\text {Spec}}{\mathbb {C}}[Q]$$ be the base point. Let $$(({\mathfrak {X}},{\mathcal {M}}_{{\mathfrak {X}}})\rightarrow R,r_0)$$ be the semi-universal deformation of $$(X_0,{\mathcal {M}}_{X_0})$$ given by Theorem 2.32. Let $$R\times {\text {Spec}}{\mathbb {C}}[Q]$$ and consider the projections $$\pi _1$$, $$\pi _2$$ onto the first and second factor respectively. Then, $$\pi _1^{*}({\mathfrak {X}},{\mathcal {M}}_{{\mathfrak {X}}})$$ and $$\pi _2^{*}{\text {Spec}}(Q\rightarrow {\mathbb {C}}[Q])$$ are semi-universal deformations of $$(X_0,{\mathcal {M}}_{X_0})$$ and $$({\text {Spec}}{\mathbb {C}},Q)$$ over $$R\times {\text {Spec}}{\mathbb {C}}[Q]$$ respectively. Let $$(r_0,p_0)\in R\times {\text {Spec}}{\mathbb {C}}[Q]$$ be the base point. By Theorem 3.4, we get a germ of complex spaces $$(S,s_0)$$, together with a morphism of germs $$p:(S,s_0)\rightarrow (R\times {\text {Spec}}{\mathbb {C}}[Q], (r_0,p_0))$$, and a log S-morphism $$f:p^{*}\pi _1^{*}({\mathfrak {X}},{\mathcal {M}}_{{\mathfrak {X}}})\rightarrow p^{*}\pi _2^{*}{\text {Spec}}(Q\rightarrow {\mathbb {C}}[Q])$$, which is a semi-universal deformation of $$f_0$$. By Proposition 3.12, f is log smooth in a neighborhood of $$(X_0,{\mathcal {M}}_{X_0})$$.