Analytic semi-universal deformations in logarithmic complex geometry

We show that every compact complex analytic space endowed with a fine logarithmic structure and every morphism between such spaces admit a semi-universal deformation. These results generalize the analogous results in complex analytic geometry first independently proved by A. Douady and H. Grauert in the ’70. We follow Douady’s two steps process approach consisting of an infinite-dimensional construction of the deformation space followed by a finite-dimensional reduction.

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 0.3.Let X 0 be a compact complex analytic space.A deformation of X 0 is a triple ((S, 0), X, i) consisting of a flat and proper morphism of complex spaces π : X → S and an isomorphism i : X 0 → X(0), where X(0) := π −1 (0).
A deformation ((S, 0), X, i) of a compact complex space X 0 is called complete, if it contains, in a small neighborhood of the base point 0 ∈ S, all possible deformations of X 0 .Technically, this means that if ((T, 0), Y, j) is another deformation of X 0 , then there exists a morphism of germs ϕ : (T, 0) → (S, 0) and an isomorphism α : Y → ϕ * X, such that α • j = ϕ * i.
Let D := ({•}, C[ǫ]/ǫ 2 ) be the double point and (S, 0) a germ of complex spaces.Denote with Hom(D, (S, 0)) the set of morphisms of germs D → (S, 0).We have a bijection Hom(D, (S, 0)) → T 0 S sending u : D → (S, 0) to du(v) ∈ T 0 S, where v ∈ TD is a basis element.If we denote with Ex 1 (0) the set of isomorphism classes of deformations of X 0 over D, we get a natural morphism (1) ks : T 0 S → Ex 1 (0), via u → u * π.This morphism is called the Kodaira-Spencer map.If ((S, 0), X, i) is a complete deformation of X 0 , then ks is an epimorphism.If 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 ; 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 ; 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.
In literature, a versal and effective deformation is called semi-universal or miniversal.By a general result of H. 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∈I 0 , such that, for each i ∈ I 0 , there exists a closed subset Z i ⊂ W i , for some W i open in C n i , and an isomorphism (2) f i : Moreover, we can find an isomorphism of the form (2) for any double U ij := U i ∩ U j and triple U ijk := U i ∩ U j ∩ 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 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 ⊂ 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 ⊂ C n i , we can consider the Banach algebra B(K) := {h : K i → C|h is continuous on K i and analytic on its interior}.
An ideal I ⊂ B(K) is called direct if there exists a C-vector subspace J of B(K), such that B(K) = I ⊕ J as C-vector spaces.Douady showed in [3, p. 34], that the set G(B(K)) := {I ⊂ B(K)|I is direct} can be endowed with the structure of a Banach manifold (see [3, p. 16]; [1,p. 38,Example 3.15]).The space G(B(K)) is called the Grassmannian of B(K).Furthermore, if we consider B(K) as a module over itself, the set 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 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 → 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 (4) Ki ⊂ Ki for double intersections and three polycylinders (5) K ′ i ⊂ Ki , Ki ⊂ Ki for triple intersections.We rewrite (4) and (5) using the following notation (6) Ki ⋐ K i and ( 7) , where I • 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 I on a compact complex space X 0 is a disassembly of X 0 given by a collection of pairs 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 I on a compact complex space X 0 (9) Q(X 0 ) := {q is a cuirasse on X 0 } can be endowed with the structure of a Banach analytic space.Moreover, if X → 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 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 ( 11) 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∈I,j∈∂i , where Y i ⊂ K i is a privileged subspace and g j i : Y j → Y i is a morphism.This collection of data satisfies gluing axioms ([4, p. 589]).The collection of puzzles (12) Z := {(Y i , g j i ) i∈I,j∈∂i } form a Banach analytic space, each puzzle z glues to a compact complex space X z and the collection of compact complex spaces (X z ) z∈Z glues to a proper Banach analytic family X over Z (see [4, p. 591]), which is anaflat (see [3, p. 66, Definition and Proposition 1]).Now, let X → S be a deformation of X 0 .The aim is to produce a map ϕ : S → Z, such that, in a neighborhood of some base point z 0 ∈ Z, with X z 0 ≃ X 0 , we have ϕ * X ≃ 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]) (13) q 0 ∈ Q(X 0 ).This means that if X → S is a deformation of X 0 , with base point 0 ∈ 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∈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 → 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 ( 14) we get a morphism ([4, p. 591]) Because a cuirasse q s is a disassembly of a compact complex space X s and the associated puzzle z qs glues to a compact complex space X zq s , it is reasonable to expect that X zq s is isomorphic to X s .In fact, we have an S-isomorphism ([4, p. 592]) (16) α q : ϕ * q X → X.In other words, the Banach analytic family X → 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 X → Z (see [4, pp. 593-599], [34, pp. 20-46] and subsection 1.3).This ends our survey about Douady's construction of a semiuniversal deformation of a compact complex space.Now, we assume that X 0 comes endowed with a fine log structure M X 0 .We view X 0 as a log space over the point Spec C with trivial log structure.Definition 0.5.A deformation of a compact fine log complex space (X 0 , M X 0 ) is a triple ((S, s 0 ), (X, M X ), i), where S is a complex space endowed with trivial log structure, s 0 ∈ S, p : (X, M X ) → (S, O × S ) is a log morphism between fine log complex spaces with underlying map of complex spaces X → S proper and flat, and i : (X 0 , M X 0 ) → (X, M X )(s 0 ) := p −1 (s 0 ) is a log isomorphism.
A deformation is complete if for any other deformation ((T, t 0 ), (X, M X ), j) of (X 0 , M X 0 ), there exists a morphism ψ : (T, O × T ) → (S, O × S ), sending t 0 to s 0 , and a log T -isomorphism For the sake of readability, in what follows, we shall mostly denote a complex space endowed with trivial log structure (S, O × S ) just by S. One of the key points, in the construction of deformations of log spaces, is to find a proper way to deform the log structure M X 0 coherently with the deformation of the underlying analytic space X 0 .We show, in subsection 1.1, that we can disassemble 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 M a X 0 on X 0 isomorphic to M X 0 .We call this collection of log charts a set of directed log charts (Definition 1.1).This insight leads to the notion of log cuirasse (Definition 1.10) and log puzzle (Definition 1.5).
In subsection 1.2, we construct an infinite-dimensional log family (X, M X ) → Z log (Proposition 1.8).Given a log deformation (Y, M Y ) → T of (X 0 , M X 0 ), with base point t 0 , an essential point is to show that a triangularly privileged log cuirasse q † 0 exists on (X 0 , M X 0 ) ≃ (Y, M Y )(t 0 ) and it extends to a log cuirasse q † t on the fibre (Y t , M Yt ), for t in a neighborhood of t 0 (Propositions 1. 18 and 1.19).This allows us to show the completeness of the log family (X, M X ) → Z log .
In subsection 1.3, we proceed with a finite-dimensional reduction procedure, which produces a semi-universal deformation of (X 0 , M X 0 ) out of the complete log family (X, M X ) → Z log .The finite-dimensionality is achieved with the exact same procedure used by Douady in the classical case.This is because the space Z log of log puzzles does not come endowed with a non-trivial log structure.We prove Theorem 0.6.(Theorem 1.32) Every compact fine log complex space (X 0 , M X 0 ) admits a semi-universal deformation ((S, s 0 ), (X, M 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 Spec an C[P ], with its canonical divisorial log structure, is log smooth over Spec 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 Spec an C[P ] of a monoid ring simply by Spec C[P ].
In section 2, we prove the following Theorem 0.7.(Theorem 2.4 and Proposition 2.12) Every morphism of compact fine log complex spaces f 0 : (X 0 , M X 0 ) → (Y 0 , 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 2.6), we obtain a relative semi-universal deformation of a compact fine log complex space (X 0 , M X 0 ) over a fine log complex space (Y 0 , M Y 0 ) (Definition 2.5).Notice that, in this case, Y 0 needs not to be compact.If (X 0 , M X 0 ) is a log subspace of (Y 0 , M Y 0 ), we get a semi-universal deformation of a log subspace in a fixed ambient log space (Remark 2.7).
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.
thank Siegmund Kosarew for very helpful comments, Helge Ruddat and Simon Felten for the hospitality at the Johannes Gutenberg University of Mainz during spring 2019 and Mark Gross for the hospitality at the University of Cambridge during spring 2020.I thank Bernd Siebert and The University of Texas at Austin for financial support.At the University of Hamburg, I was supported by the Research Training Group 1670 "Mathematics inspired by String Theory and Quantum Field Theory", funded by the German Research Foundation -Deutsche Forschungsgemeinschaft (DFG).

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 M X 0 .
1.1.Gluing log charts.Let (X 0 , M X 0 ) be a compact fine log complex space.Denote by α : M X 0 → O X 0 the structure map.The sheaf of monoids 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 We identify θ i with the corresponding map of monoid sheaves We get maps d m : J l → J l−1 , for 0 ≤ m ≤ l and 1 ≤ l ≤ 2, sending (i 0 , .., i m , .., i l ) to (i 0 , .., i m−1 , i m+1 , .., i l ).We set The set J, together with the maps (d m ), is called a simplicial set of order 2.
For each j := (i 0 , i 1 ) ∈ J 1 , assume that there is a log chart and comparison maps Each θ i defines an isomorphism of M U i with the log structure M i associated to the pre-log structure β i := α • θ i .Similarly, the pre-log structure β j := α • θ j defines a log structure M j and θ j defines an isomorphism of log structures M U j ≃ M j .
From this point of view, equation (17) means that ϕ i j provides an isomorphism between M i | U j and M j , and this isomorphism is compatible with the isomorphisms Now, if we have θ i , θ j , ϕ i j , fulfilling (17), we need compatibility on triple intersections for the patching of the 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 ) ∈ J 2 , a third system of charts and comparison maps for j ∈ ∂k.The analogue of the compatibility condition ( 17) is ( 18) Again, the ϕ j k define an isomorphism between the log structure M j | U k on U k and the log structure M k associated to the pre-log structure β k := α • θ k .In particular, all the isomorphisms of log structures are compatible and the (M i ) i∈J 0 glue in a well-defined fashion, as do their structure maps, to a log structure on X 0 isomorphic to M X 0 .This is just standard sheaf theory, for sheaves of monoids.
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 (θ i ) i∈J 0 , (θ j ) j∈J 1 and (θ k ) k∈J 2 are charts for the given log structure.Let (U i ) i∈J 0 be an open cover of X 0 and J as above.Assume we have prelog structures (β i ) i∈J and comparison maps (ϕ i j ) j∈J 1 ∪J 2 ,i∈∂j satisfying equations ( 17) and (18).Then the log structures (M i ) i∈J 0 glue to a log structure M X 0 on X 0 in such a way that the gluing data (β j ) j∈J 1 and compatibility (β k ) k∈J 2 arise from identifying M j and M k with restrictions of M X 0 to U j and U k respectively.Definition 1.2.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 1.2.Infinite dimensional construction.The notion of log structure can be naturally extended to the category of Banach analytic spaces.Indeed, let (X, Φ) be a Banach analytic space (see [3, pp. 22-25]; [1, p. 38, Definition 3.16]).Setting O X := Φ(C), we get a ringed space (X, O X ).

Definition 1.3.
A pre-log structure on a Banach analytic space (X, Φ) is a sheaf of monoids M X on X together with a homomorphism of sheaves of monoids: where the monoid structure on O X is given by multiplication.A pre-log structure is a called a log structure if 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, O × S ) just by S.Moreover, for the sake of readability, we shall often write Banach analytic morphisms just set-theoretically.
Let (X 0 , M X 0 ) be a compact fine log complex space.

Let
I := (I • , (K i ) i∈I , ( Ki ) i∈I , (K ′ i ) i∈I 0 ∪I 1 ) be as in (8) and Z the space of puzzles (12).Without loss of generality, we assume that the index sets I and J (Definition 1.4) 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 1.5.A log puzzle is a pair (z, l), where z := (Y i , g i j ) ∈ Z is a puzzle and l is a collection of data ((β i : , where ϕ i j := (φ i j,0 , η i j ), with φ i j,0 : P i → P j given by Definition 1.4.Definition 1.6.We denote the set of log puzzles by Z log .
The set of log puzzles Z log can be endowed with a Banach analytic structure.Indeed, for each polycylinder K i , let us consider the Grassmannian G(K i ) (3) and let Id : G(K i ) → G(K i ) be the identity map.Identifying Id with its graph, we get a universal [4, p. 579], [25, pp. 258-259] and [30,p. 183,Theorem 4.13]).Let us consider the Banach analytic space Each morphism η i j (z) gp : induces a morphism η i j (z) : . Hence, a point in M can be written as (z, (β i (z) : ).Thus, we naturally get an injective map (21) Proposition 1.7.The universal space of log puzzles Z log is Banach analytic.
Thus, we can define a double arrow Then Z log is given by the kernel of the double arrow defined by ρ 1 and ρ 2 : Let p : Z log → Z be the canonical projection and consider the Banach analytic space X log := p * X over Z log .Proposition 1.8.The Banach analytic space X log comes naturally endowed with a fine log structure M X log .
On the other hand, we have that the space X is canonically isomorphic to Hence, the collection of universal morphisms ((β i ), (ϕ i j )) defines a pre-log atlas (see Definition 1.2) on X log , which glues to a fine log structure M X log on X log (see Subsection 1.1).
We show that the universal family of log puzzles (X log , M X log ) → Z log gives a complete deformation of (X 0 , 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 ϕ q : S → Z (15), a Banach analytic space X ϕq over S obtained by gluing the pieces of the puzzle z q associated to q (14), and an S-isomporphism α q : X ϕq → X (16).Now, let (X 0 , M X 0 ) be a compact fine log complex space admitting a collection of directed log charts ((θ i : P i → M U i ) i∈I , (ϕ i j := (φ i j , η i j ) : P i → P j ⊕ O × U j ) j∈I 1 ∪I 2 ,i∈∂j ) (see Definition 1.1).We assume that φ i j coincide with the φ i j,0 given by Definition 1.4.Let q 0 ∈ Q(X 0 ) be a cuirasse on X 0 .We have an isomorphism (16) α q 0 : X ϕq 0 → X 0 .Definition 1.9.We naturally get a fine log structure on X ϕq 0 via Definition 1.10.A log cuirasse q † 0 on (X 0 , M X 0 ) is a pair given by a cuirasse q 0 = (Y i , f i ) i∈I on X 0 and a collection of directed log charts ((θ i : ) on (X ϕq 0 , M Xϕ q 0 ) (Definition 1.1).We denote the set of log cuirasses on (X 0 , M X 0 ) by Q(X 0 , M X 0 ).Remark 1.11.In Definition 1.10 we need to give the set of comparison morphisms (η i j ) in order to define, in Definition 1.20, 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, 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, M X ) can be covered by finitely many log charts (θ i : Definition 1.12.Let S be a Banach analytic space and (X, M X ) a fine log Banach analytic space proper and anaflat over S. We define the set of relative log cuirasses on (X, M X ) over S by Definition 1.13.We call a section q † : S → Q S (X, M X ), of the canonical projection π : Q S (X, M X ) → S, a relative log cuirasse on (X, M X ) over S. Definition 1.14.Let (X 0 , M X 0 ) be a compact fine log complex space.A log cuirasse q † 0 on (X 0 , M X 0 ) is called triangularly privileged if the underlying cuirasse q 0 ∈ Q(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 , 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 1.15.Let (X, 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 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 ∈ I and for each y ∈ Y i,0 there exists an open set Since each Y i is compact, we can find a finite set J and finitely many subsets . Hence, possibly after shrinking S ′ , we can assume that for each j ∈ J, V j = S ′ and f −1 i (X) ⊂ j∈J U j .Thus, we get that for each i ∈ I, A holomorphic line bundle with c 1 = 0 is topologically trivial, hence analytically isomorphic to the trivial line bundle by the following Lemma 1.16.Lemma 1.17.Let (X, M X ) be a fine log Banach analytic space.Assume that P := Γ(X, M X ) is globally generated and torsion free.Assume that for each m ∈ Γ(X, M X ), the torsor L m = κ −1 (m), with κ : M X → M X the canonical map, is trivial.Then there exists a chart P → Γ(X, M X ).
Proof.Let p 1 , ..., p r ∈ P be generators, that is we have a surjective map N r → Γ(X, M X ) sending e i to p i .For each i ∈ {1, ..., r}, choose a section m i ∈ L m .We obtain a chart φ : N r → Γ(X, M X ).Now, we want to modify φ so that it factors through P .Let K := ker(Z r → P gp ), we have P = N r /K.We get the following exact sequence 0 → K → Z r → P gp → 0.
Since, by assumption, P is torsion free, we can find a section π : Z r → K. Set in Γ(X, M gp X ).We get a chart by Now, let ψgp : Z r → Γ(X, M gp X ) and a i e i ∈ K.If ψgp ( a i e i ) = 1, we get that ψgp induces a chart ψ : P → Γ(X, M X ).Hence, assume Proposition 1.18.Let (X, M X ) be a fine log Banach analytic space over a Banach analytic space S. Let s 0 ∈ S and q † 0 a triangularly privileged log cuirasse on (X(s 0 ), M X(s 0 ) ).Then the set of log cuirasses Q S (X, M X ) on (X, M X ) over S can be endowed with the structure of a Banach analytic space in a neighborhood of (s 0 , q † 0 ).Proof.Let us consider the projection π : Q S (X, M X ) → Q S (X).By Lemma 1.15 and Lemma 1.16, we can use Lemma 1.17 and get the existence around (s 0 , q 0 ) of a local section ρ : Q S (X) → Q S (X, M X ), such that ρ(s 0 , q 0 ) = q † 0 .Now, let (s, q) ∈ Q S (X), in a small neighborhood of (s 0 , q 0 ), and consider ρ(s, q) ∈ Q S (X, M X ).We have that ρ(s, q) = (s, q = (Y i , f i ), (θ i ), (η i j )), where (θ i ), (η i j ) is a directed collection of log charts on (X ϕq , M Xϕ q ) (see Definition 1.10).Any other directed set of log charts ((θ ′ i ), (η [25, pp. 258-259] and [30, p. 183, Theorem 4.13]).We can define a map ), which defines a structure of Banach analytic space on Q S (X, M X ) in a neighborhood of (s 0 , q † 0 ).
Proposition 1.19.Let (X, M X ) be a fine log Banach analytic space proper and anaflat over a Banach analytic space S. Let s ∈ S and q † (s) a triangularly privileged log cuirasse on (X(s), M X(s) ).Then is smooth in a neighborhood of q † (s).
Proof.Let q † (s) be a triangularly privileged log cuirasse on (X(s), M X(s) ).By Proposition 1.18, we have that in a neighborhood of (s, q † (s)), the space Q S (X, M X ) is isomorphic to Let q(s) = (Y i , f i ) be the triangularly privileged cuirasse on X(s) underlying q † (s).By [4, p. 589, Corollary 2], π : Q S (X) → S is smooth in a neighborhood of (s, q(s)).Furthermore, by [4, p. 585, Proposition 2], we have that Hence, the statement follows.
Analogously to the classical case ( 14), we define the notion of log puzzle associated to a log cuirasse.Let (X 0 , M X 0 ) be a compact fine log complex space.Let q 0 be a cuirasse on X 0 .By Definition 1.9, we get a compact fine log complex space (X ϕq 0 , M Xϕ q 0 ), which is isomorphic to (X 0 , M X 0 ).Let q † 0 = (q 0 , (θ i ), (η i j )) be a log cuirasse on (X 0 , M X 0 ) (see Definition 1.10).Let α Xϕ q 0 : M Xϕ q 0 → O Xϕ q 0 be the structure log morphism and z q 0 ∈ Z the puzzle associated to q 0 .Definition 1.20.We call the log puzzle associated to q † 0 .Clearly, z q † 0 ∈ Z log (Definition 1.5).Let (X, M X ) be a fine log Banach analytic space proper and anaflat over a Banach analytic space S. Let q † be a relative log cuirasse on (X, M X ) over S. Definition 1.21.We can define a morphism Let q † = (q, (θ i ), (η i j )) be a log cuirasse on (X, M X ) over S. Let (X ϕq , M Xϕ q ) given by Definition 1.9 and α Xϕ q : M Xϕ q → O Xϕ q the structure log morphism.For each i ∈ I, let M a Xϕ q ,i be the log structure associated to the pre-log structure α Xϕ q • θ i .The collection of log structures (M a Xϕ q ,i ) glues to a log structure M a Xϕ q on X ϕq (Subsection 1.1).
Definition 1.22.We set The fine log Banach analytic space (X ϕ q † , M Xϕ q † ) is obtained by gluing the pieces of the log puzzle z q † associated to the cuirasse q † .Proposition 1.23.Let (X, M X ) be a fine log Banach analytic space proper and anaflat over a Banach analytic space S. Let q † be a relative log cuirasse on (X, M X ) over S.Then, there exists a log S-isomorphism Proof.By Proposition 16, we have an S-isomorphism α q : X ϕq → X.Moreover, we have M Xϕ q := α * q M X (see Definition 1.10).Hence, α q induces a S-log isomorphism α q : (X ϕq , M Xϕ q ) → (X, M X ).Now, the log cuirasse q † gives us a collection of directed log charts ((θ i ), (η i j )) for M Xϕ q .Let α Xϕ q : M Xϕ q → O Xϕ q be the structure log morphism and M a Xϕ q ,i the log structure associated to the pre-log structure α Xϕ q • θ i , for each i ∈ I.By the definition of log chart ([26, p. 249]), we have an isomorphism α ♭ i : M a Xϕ q ,i → M Xϕ q .Then the collection of log structures (M a Xϕ q ,i ), together with the isomorphisms (α ♭ i ), glues to a log structure M a Xϕ q on X ϕq , together with an isomorphism α ♭ : M a Xϕ q → M Xϕ q (see Subsection 1.1).Hence, set (X ϕ q † , M Xϕ q † ) := (X ϕq , M a Xϕ q ) and α := (Id, α ♭ ), we get an isomorphism α : We are ready to prove the existence of an infinite-dimensional complete deformation of a fine compact log complex space (X 0 , M X 0 ).With the due modifications, the proof of Theorem 1.25 is identical to the proof of Theorem 0.1 ([4, p. 592]).Let q 0 = (Y i,0 , f i,0 ) be a triangularly privileged cuirasse on X 0 and ((θ i,0 ), (η i j,0 )) the collection of directed log charts on (X 0 , M X 0 ) as in Definition 1.4.Then, ) is a triangularly privileged log cuirasse on (X 0 , M X 0 ) (see Definition 1.14).Let z q † 0 be the log puzzle associated to q † 0 (see Definition 1.20).Let (X log , M X log ) → Z log be the universal space of log puzzles (see Proposition 1.8) and the log isomorphism given by Proposition 1.23.
Proof.Let (X, M X ) be a fine log Banach analytic space proper and anaflat over a Banach analytic space S. Let s 0 ∈ S and i : (X(s 0 ), M X(s 0 ) ) → (X 0 , M X 0 ) a log isomorphism.Since i * q † 0 is a triangularly privileged log cuirasse (see Definition 1.14) on (X(s 0 ), M X(s 0 ) ), we have that Q S (X, M X ) is smooth over S in a neighborhood of i * q † 0 (Proposition 1.19).Therefore there exists a local relative log cuirasse q † on (X, M X ) defined in a neighborhood S ′ of s 0 in S. Hence, taking ϕ q † : S → Z log (Definition 1.21) and the S ′ -isomorphism 23), the statement follows.
1.3.Finite dimensional reduction.Let (X log , M X log ) → Z log be the complete deformation of (X 0 , M X 0 ) given by Theorem 1.25 and Q Z log (X log , M X log ) the space of relative log cuirasses on (X log , M X log ) over Z log (Definition 1.10).Since the finitedimensional reduction is performed on the Banach analytic space Q Z log (X log , M 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 1.2 that the space Z log (Definition 1.6) parametrizes all log puzzles z † of type I (Definition 1.5).Each fibre (X log , M X log )(z † ) of the map (X log , M X log ) → Z log is obtained by gluing the "pieces", (Y i ) i∈I and (β i : , where z † ∈ Z log is a log puzzle and q † is a log cuirasse on the fibre (X log , M X log )(z † ).To the log cuirasse q † we can naturally associate another log puzzle z q † ∈ Z † (Definition 1.20).In principle, (Proposition 1.23).However, we can consider the subspace Z log ⊂ Q Z log (X log , M X log ) defined by selecting, in each fibre Q((X log , M X log )(z † )) of the canonical projection π : Q Z log (X log , M X log ) → Z log , all log cuirasses q † on (X log , M X log )(z † ) whose associated log puzzle z q † coincides exactly with z † .More precisely, there exists a canonical relative log cuirasse q † on (26) [25, p.267] and [1, p. 90].By Definition 1.21, we get an associated morphism Then, the subspace Z log is obtained as the kernel of the double arrow (π, ϕ q † ): The space Z log parametrizes all log cuirasses on compact fine log complex spaces "close" to (X 0 , M X 0 ).This space is not as pathological as Z log (see [4, p. 590, Remark]) and it still gives a complete deformation of (X 0 , M X 0 ).Indeed, given any log Banach analytic space (X, M X ) proper and anaflat over a Banach analytic space S, we get a map from the space of relative log cuirasses Q S (X, M X ) into the space of log puzzles Z log by ϕ is a relative log cuirasse on (X, M X ) over S, that is a section of the projection π : Q S (X, M X ) → S, the composition is a morphism satisfying the completeness property (see Definition 1.21 and Theorem 1.25).Indeed, for each s ∈ S, the fibre (X log , M X log )(ϕ σ † (s)) is isomorphic to the fibre (X, M X )(s) via an isomorphism α σ † (Proposition 1.23).Identifying these two isomorphic fibres, we get a map For more details, see [1, pp.70-71 and pp.90-91].We can draw the following commutative diagram: In fact, ψ σ † is the unique morphism from S to Q Z log (X log , M X log ) making the above diagram commutative (see [4, p. 593]).By construction, ψ σ † factors through Z log ⊂ Q Z log (X log , M X log ) and it is used to prove Proposition 1.26.Let be the canonical injection, we set Let q † 0 be a triangularly privileged log cuirasse on (X 0 , M X 0 ) (see Definition 1.14) and z q † 0 ∈ Z log the associated log puzzle (Definition 1.20).We get a point (z In what follows, we are going to decompose Z log into a product Σ log × R log , where Σ 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 1.27.([4, p. 594]) Let us consider two types of cuirasses, namely I = (I • , (K i ), ( Ki ), (K ′ i )) and Î = (I • , ( Ki ), ( Ki ), ( K′ i )) (8), which have the same underlying simplicial set.We write I ⋐ Î, if K i ⋐ Ki , Ki ⊂ Ki and K ′ i ⊂ K′ i .Let Î be a type of cuirasse and q † a relative log cuirasse of type Î on a log Banach analytic space (X, M X ) proper and anaflat over S.Then, by slightly shrinking each polycylinder Ki , Ki and K′ i in Î, we can get polycylinders K i , Ki and K ′ i respectively and hence a type of cuirasse I, such that I ⋐ Î.Then, If I ⋐ Î are two types of cuirasses, then we can construct the spaces of log puzzles Z log and Ẑlog of type I and Î respectively (see Definition 1.6).It can be shown (see [4, p. 595] and [34, p.44]), that the restriction morphism (31) j † : Ẑlog → Z log 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 1.28.Set Q log 0 := Q(X 0 , M X 0 ), the space of log cuirasses on (X 0 , M X 0 ) (see Definition 1.10).By Proposition 1.19 the projection π : Q Z log (X log , M X log ) → Z log is smooth in a neighborhood of (z q † 0 , q † 0 ), hence we can opportunely choose (see [4, p. 595], [25, p. 269] and [1, p. 96]) a local trivialization (32) (π, ρ † ) : Proposition Thus, we get the existence of an embedding ι † : Z log ֒→ Q log 0 × C m making the following diagram commutative: (34) .
Proposition 1.30.Let S be a Banach analytic space and f, g : S → Z log morphisms.Then f * (X Z log , M X Z log ) ≃ g * (X Z log , M X Z log ), if and only if there exists h † : S → Q log 0 such that the following diagram commutes (39) In other words, if and only if, for each s ∈ S, g(s) is obtained "changing" f (s) by a log cuirasse q † on the central fibre (X 0 , M X 0 ).Notice that, by Proposition 1.30 Let us denote with Ex 1 (X 0 , M X 0 ) the set of equivalence classes of infinitesimal deformations of (X 0 , M X 0 ), that is deformations over the double point D = ({•}, C[ǫ]/ǫ 2 ).For the sake of clarity, set r † The kernel ker ks corresponds to the trivial deformations of (X 0 , M X 0 ) over D. By Proposition 1.30, with S = D, we see that the trivial deformations of (X 0 , M X 0 ) over D are given by Im T q † 0 δ † .Hence, Let us identify Z log with its image in Q log 0 × C m under ι † .By Proposition 1.29, let Σ log be the Banach submanifold of Q log 0 such that (42) Let Y be another subspace of Σ 2 , containing Σ 1 , and set Let φ : Σ 1 × R → Y be a morphism inducing the identity on Σ 1 ×0 and 0×R.Then, φ is an isomorphism.
From Lemma 1.31, we obtain that the restriction of the morphism (37) is an isomorphism.This fact, together with Proposition 1.30 and (44), is used to prove Theorem 1.32.
Let i : R log ֒→ Z log be the canonical injection.Set ) is a complete deformation of of (X 0 , M X 0 ).Hence, there exists a morphism ψ † : S → Z log such that (X, M X ) ≃ ψ * † (X Z log , M X Z log ).Let Σ log and R log given by ( 42) and (43) respectively.Let π R log : R log × Σ log → R log and π Σ log : R log × Σ log → Σ log be the projections.By Lemma 1.31, the morphism ω † | R log ×Σ log : R log × Σ log → Z log (46) is an isomorphism.Thus, setting g : Hence, by Proposition 1.30 Thus, the deformation ((X R log , M X R log ) → R log , r † 0 ) is complete and effective.Now, let ((S, s 0 ), (X, M X ), i) be a deformation of (X 0 , 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 Let q † be the canonical relative log cuirasse on (X Z log , M X Z log ) over Z log (26).Then, h ′ * q † is a relative log cuirasse on (X, M X )| S ′ over S ′ , whose associated morphism (27) coincides with h ′ .Since, by Proposition 1.19, Q S (X, M X ) is smooth over S in a neighborhood of q † 0 ∈ Q((X(s 0 ), M X(s 0 ) ), there exists a relative cuirasse q † on (X, M X ) over S, such that q † | S ′ = h ′ * q † .Let h : S → Z log be the morphism associated to q † (30) and π R log : Z log → R log the projection.Then, h : Thus, the deformation (X R log , M X R log ) → R log is also versal and, therefore, semiuniversal.

Semi-universal deformations of log morphisms
In what follows, we construct a semi-universal deformation of a morphism f 0 : (X 0 , M X 0 ) → (Y 0 , M Y 0 ) of compact fine log complex spaces.Let X be a complex space and α i : M i → O X , i = 1, 2, two fine log structures on X.Let γ : M 1 → M 2 be a morphism of the ghost sheaves.Let f : T → X be a morphism of complex spaces and set γ T : (M 1 ) T → (M 2 ) T , the pull-back of γ via f .
By the universal property, the statement is local in X.Hence, let β i : P i → Γ(X, M i ), i = 1, 2, be two log charts for M 1 and M 2 respectively.Let p 1 , ..., p n ∈ P 1 be a generating set for P 1 as monoid.Consider the sheaf of finitely generated O Xalgebras F ) := Spec an F X , the relative analytic spectrum of F X over X.Now, we check the universal property.Let f : T → X be given.We want to show that giving a commutative diagram of complex spaces , which is the identity on X and such that ϕ ♭ = γ T .Giving a morphism g is equivalent to giving a section of (Spec an F X ) × X T over T .But ))|1 ≤ i ≤ n , and the latter complex space is Spec an F T associated to the data (T, Thus, without loss of generality, we can assume T = X and f is the identity.Now, giving ϕ : (X, M 1 ) → (X, M 2 ), with ϕ ♭ = γ, is equivalent to specifying ϕ ♭ .From ϕ ♭ we obtain a map η : P 1 → Γ(X, O × X ) with the property that for all p ∈ P 1 , Conversely, η completely determines ϕ ♭ .In addition, ϕ ♭ is a homomorphism of monoids if and only if η is a homomorphism, and since η takes values in the group O × X , specifying ϕ ♭ is equivalent to specifying a section of Spec an O X [P gp 1 ].Indeed, a section of Spec an O X [P gp 1 ] over X is the same as a morphism X → Spec C[P gp 1 ], which in turn is the same as an element of Hom(P 1 , Γ(X, O × X )).Second, since ϕ * = id, we must have α 1 = α 2 • ϕ ♭ , so for each p ∈ P 1 , we must have If this holds for each p i , it holds for all p.Thus a section of Spec an 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 α 1 (β 1 (p i )) − z p i α 2 (β 2 (γ(p i ))), demonstrating the result.Now, assume the complex space X is proper over a germ of complex spaces (S, s 0 ).This is exactly the functor of sections X/S (Z/X) discussed, in the algebraicgeometric 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]).
Proof.Let ((X, M X ) → R, r 0 ) and ((Y, M Y ) → R, r 0 ) be the semi-universal deformations of (X 0 , M X 0 ) and (Y 0 , M Y 0 ) respectively given by Theorem 1.32.By pullingback 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 Mor R (X, Y) given by Proposition 2.3.Let p : Mor R (X, Y) → R be the projection and set m 0 := (r 0 , f 0 ).By Proposition 2.3, we get a universal morphism f : p * X → p * Y, such that the restriction of f to the central fibre p * X(m 0 ) equals f 0 .We can consider two fine log structures on p * X, namely Assume F x = 0. Then the following are equivalent: (1) F is flat over S at x and F s is flat over Y s at x; (2) Y is flat over S at y and F is flat over Y at x. Proposition 2.9.([6, p. 159]) Let f : X → Y be a morphism of complex spaces.Let p ∈ X.Then the following are equivalent (1) f is smooth (submersion) at p ∈ X; (2) f is flat at p and the fibre X f (p) is a manifold.
Proposition 2.8 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, O X ) and p ∈ X, the stalk O X,p is a Noetherian local ring (see [22, p. 80]).The following Lemma 2.11 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 2.11.Any log smooth morphism of fine log complex spaces is log flat.
Let f : (X, M X ) → (Y, M Y ) be the semi-universal deformation of f 0 : (X 0 , M X 0 ) → (Y 0 , M Y 0 ), over a germ of complex spaces (S, s 0 ), given by Theorem 2.4 or Corollary 2.6.Denote with π 1 and π 2 the morphisms of (X, M X ) and (Y, M Y ) into (S, s 0 ) respectively.Proposition 2.12.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 X such that f | (U ′ ,M U ′ ) is log flat (log smooth).Then, since π 1 : X → S is a proper map between locally compact Hausdorff spaces, it is closed.Hence, by Lemma 2.10, we can find an open neighborhood W of s 0 such that π −1 1 (W ) is contained in U ′ .This ensures us that f is log flat (log smooth) as relative morphism over (W, s 0 ) ⊂ (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

Lemma 2 . 10 .
Let f : X → Y be a continuous map between topological spaces.If f is closed, then for all y ∈ Y and open subset U ⊂ X satisfying f −1 (y) ⊂ U, there exists an open neighborhood V of y satisfying f −1 (V ) ⊂ U. Proof.Let us consider the closed subset X\U.Since f is closed, f (X\U) is closed in Y .Therefore, Y \f (X\U) is open in Y and it contains y as f −1 (y) ⊂ U. Take V := f −1 (Y \f (X\U)).