I. Category of models
Let B be a topological space. We define the category \mathscr{S}_B^n in the following manner: the objects of \mathscr{S}_B^n are the open subsets of B\times\mathbb{C}^n, and a morphism f\colon U\to U' from an open subset U\subset B\times\mathbb{C}^n to an open subset U'\subset B\times\mathbb{C}^n is a continuous map f\colon U\to U' satisfying the following two conditions:
- the diagram \begin{CD} U @>f>> U' \\@V{\pi_1}VV @VV{\pi_1}V \\B @= B \end{CD} commutes, where \pi_1 denotes the projection of B\times\mathbb{C}^n to B; and
- for all x\in B, the map f_x\colon U_x\to U'_x is holomorphic, where U_x = \{z\in\mathbb{C}^n \mid (x,z)\in U\} (and similarly for U').
If B is endowed with the structure of a \mathscr{C}^\infty manifold (resp. an \mathbb{R}-analytic manifold, resp. \mathbb{C}-analytic manifold), then we obtain a category \mathscr{C}^\infty\mathscr{S}_B (resp. \mathbb{R}\mathscr{S}_B, resp. \mathbb{C}\mathscr{S}_B) by requiring the morphisms to be \mathscr{C}^\infty (resp. \mathbb{R}-analytic, resp. \mathbb{C}-analytic).
More generally, if f_1\colon B\to B' is a continuous map from one topological space to another, then a morphism of \mathscr{S}_{f_1} is a continuous map f from an object U of \mathscr{S}_B to an object U' of \mathscr{S}_{B'} such that
- the diagram \begin{CD} U @>f>> U' \\@V{\pi_1}VV @VV{\pi_1}V \\B @>>{f_1}> B' \end{CD} commutes; and
- f_x\colon U_x\to U'_{f_1(x)} is holomorphic for all x\in B.
If f_1 is a \mathscr{C}^\infty map from one \mathscr{C}^\infty manifold to another, then f will be a morphism of \mathscr{C}^\infty\mathscr{S}_{f_1} if, further, it is a \mathscr{C}^\infty map (resp. …). We thus obtain, for every category of topological spaces, a fibred category \mathscr{S}^n (resp. \mathscr{C}^\infty\mathscr{S}^n, resp. …).
II. The definition of mixed spaces and mixed varieties
1. First definition
Let B and V be separated spaces, and let \pi\colon V\to B be a continuous map. The structure of a mixed space over B is defined on V by a system of charts \varphi_i\colon U_i\to V, where the (U_i) are objects of \mathscr{S}_B^n; for each i, \varphi_i is a homeomorphism from U_i to an open subset of V such that the diagram \begin{CD} U_i @>{\varphi_i}>> V \\@V{\pi_1}VV @VV{\pi}V \\B @= B \end{CD} commutes; finally, for all i and all j, the “change of chart” \varphi_j^{-1}\circ\varphi_i is an isomorphism of \mathscr{S}_B from an open subset of U_i to an open subset of U_j.
The structure thus defined is that of a (\mathscr{C}^0,\mathbb{C})-mixed space. If B is a \mathbb{C}-analytic space, and if the change of chart maps are all \mathbb{C}-analytic, then we have a \mathbb{C}-analytic mixed space. In this case, V itself is a \mathbb{C}-analytic space, and the fibres V_x=\pi^{-1}(x) are \mathbb{C}-analytic sub-manifolds.
If B is a \mathscr{C}^\infty manifold (resp. \mathbb{R}-analytic, resp. \mathbb{C}-analytic), and if the change of chart maps are all \mathscr{C}^\infty (resp. …), then we have a (\mathscr{C}^\infty,\mathbb{C})-mixed manifold (resp. (\mathbb{R},\mathbb{C}), resp. (\mathbb{C},\mathbb{C})). In this case, V itself is a manifold. Note that the notion of a (\mathbb{C},\mathbb{C})-mixed manifold, or a \mathbb{C}-analytic mixed manifold, reduces to simply having a \mathbb{C}-analytic manifold V endowed with a projection \pi\colon V\to B onto another \mathbb{C}-analytic manifold such that \pi is of maximal rank at every point.1
Let \pi\colon V\to B and \pi'\colon V'\to B' be mixed spaces, and let f_1\colon B\to B' be a continuous (resp. …) map. Then a morphism from V to V' over f_1 is a continuous map f\colon V\to V' such that the diagram \begin{CD} V @>f>> V' \\@V{\pi}VV @VV{\pi'}V \\B @>>{f_1}> B' \end{CD} commutes, and such that, for any charts \varphi_i\colon U_i\to V and \varphi'_j\colon U'_j\to V', the map {\varphi'_j}^{-1}\circ f\circ\varphi_i is a morphism of \mathscr{S}_{f_1} (resp. …) from an open subset of U_i to U_j.
2. An equivalent definition
We now give another way of defining mixed spaces, equivalent to the above.
Given separated spaces B and V, along with a continuous map \pi\colon V\to B, the structure of a pre-mixed space consists of the structure of a \mathbb{C}-analytic manifold on each fibre V_x=\pi^{-1}(x). Given pre-mixed spaces \pi\colon V\to B and \pi'\colon V'\to B', along with a continuous map f_1\colon B\to B', a morphism of pre-mixed spaces over f_1 is a continuous map f\colon V\to V' such that the diagram \begin{CD} V @>f>> V' \\@V{\pi}VV @VV{\pi'}V \\B @>>{f_1}> B' \end{CD} commutes and induces a \mathbb{C}-analytic map on each fibre.
A mixed space is a pre-mixed space \pi\colon V\to B such that every point y\in V admits a neighbourhood W in V that is isomorphic as a pre-mixed space to an open subset of B\times\mathbb{C}^n, via an isomorphism over the identity. The morphisms of mixed spaces are the same: mixed spaces form a full subcategory.
3. Deformations
A mixed space \pi\colon V\to B is said to be proper if B is locally compact and the map \pi is proper (i.e. the inverse image of any compact subset is compact). If it is a mixed manifold, then we can show that it is a fibred manifold that is locally trivial with respect to the underlying \mathscr{C}^\infty structure, but the previous talk shows that, in general, any two fibres are not isomorphic as \mathbb{C}-analytic manifolds.
Let V_0 be a compact \mathbb{C}-analytic manifold, B a locally compact space, and b_0\in B. Then a \mathbb{C}-analytic deformation of V_0 over (B,b_0) consists of a proper \mathbb{C}-analytic mixed space \pi\colon V\to B along with an isomorphism of \mathbb{C}-analytic manifolds i\colon V_0\to\pi^{-1}(b_0).
The goal of this seminar is the study, at least local, and an attempt at a classification of, \mathbb{C}-analytic deformations of a given compact \mathbb{C}-analytic manifold V_0.
Let V_0 be a compact \mathbb{C}-analytic manifold. A \mathbb{C}-analytic deformation (\pi\colon V\to B,i\colon V_0\to V) of V_0 is said to be locally complete if, for any other deformation (\pi'\colon V'\to B',i'\colon V_0\to V') of V_0, there exists a neighbourhood B'_1 of b'_0 in B', an analytic map f_1\colon B'_1\to B with f_1(b'_0)\to b_0, and a morphism of \mathbb{C}-analytic mixed spaces f\colon {\pi'}^{-1}(B'_1)\to V over f_1 such that f\circ i'=i. The deformation is said to be locally universal is furthermore the germ of f_1 at b'_0 is determined uniquely by this condition.
It seems that every compact \mathbb{C}-analytic manifold V_0 admits a locally complete \mathbb{C}-analytic deformation, and a locally universal one if the group of automorphisms of V_0 is discrete.
III. Vector fields
1. Study on models
Let B be a space, U an object of \mathscr{S}_B (i.e. an open subset of B\times\mathbb{C}^n), b_0 a point of B, and set U_0=\pi^{-1}(b_0).
A holomorphic field of tangent vectors on U_0 (i.e. a holomorphic map from U_0 to \mathbb{C}^n) is said to be a vertical holomorphic field on U_0. A vertical holomorphic field on U is a continuous (resp. …) map \theta\colon U\to\mathbb{C}^n that induces a vertical holomorphic field on each fibre U_x. If f\colon U\to U' is an isomorphism in \mathscr{S}_B, then the transport f_*\theta of \theta by f is defined by f_*\theta(f(x,z)) = \mathrm{D}_2 f_{x,z}\cdot\theta(x,z) where \mathrm{D}_2 f_{x,z} is the linear map from \mathbb{C}^n to itself that is tangent to f_x at the point z\in U_x. This is again a vertical holomorphic field, since it follows from a Cauchy integral that the matrix \mathrm{D}f_{x,z} depends continuously on the pair (x,z).
Now suppose that B is a \mathscr{C}^\infty manifold, just for simplicity, and let T_0 be the tangent space to B at b_0. A field of tangent vectors to U defined on U_0, i.e. a map \omega\colon U_0\to T_0\times\mathbb{C}^n, is said to be a projectable holomorphic field if \omega(b_0,z)=(t_0,\theta(z)) (where t_0\in T_0 is a vector that does not depend on z, called the projection of the field \omega) and \theta(z) is a holomorphic vector field. If B is a \mathbb{C}-analytic space, possibly with a singularity at b_0, then we give the same definition, but with T_0 then being the Zariski tangent space to B at b_0, i.e. the dual of \mathfrak{m}/\mathfrak{m}^2, where \mathfrak{m} is the ideal of germs at b_0 of holomorphic functions on B that vanish at b_0.
If f\colon U\to U' is an isomorphism of \mathscr{C}^\infty\mathscr{S}_B (resp. …), then then transport f_*\omega is defined by f_*\omega(f(b_0,z)) = \mathrm{D}f_{b_0,z}\omega(b_0,z) where \mathrm{D}f_{b_0,z}\colon T_0\times\mathbb{C}^n\to T_0\times\mathbb{C}^n is now the linear map that is tangent to f at the point (b_0,z). This is a projectable holomorphic field. Indeed, the matrix \mathrm{D}f_{b_0,z} can be written as \begin{pmatrix} I & 0 \\\mathrm{D}_1f & \mathrm{D}_2f \end{pmatrix} and \begin{aligned} \mathrm{D}_1f\colon T &\to \mathbb{C}^n \\\mathrm{D}_2f\colon \mathbb{C}^n &\to \mathbb{C}^n \end{aligned} both depend holomorphically on z (for \mathrm{D}_1f, this follows from the fact that f_x is holomorphic for every x). By setting f_*\omega(b_0,z')=(t_0,\theta'(z')), we have \begin{gathered} \theta'(z') = \mathrm{D}_1f_{b_0,z}(t_0) + \mathrm{D}_2f_{b_0,z}(\omega(z)) \\\text{if }z'=f_{b_0}(z) \end{gathered} which shows that f_*\omega is indeed a projectable holomorphic field.
A projectable holomorphic field on U is a \mathscr{C}^\infty field of vectors tangent to U that induces a projectable holomorphic field on each fibre.
2. Vector fields on a mixed manifold
Let \pi\colon V\to B be a (\mathscr{C}^\infty,\mathbb{C})-mixed manifold (resp. …, resp. a \mathbb{C}-analytic mixed space). By transporting along the charts, we define the notions of
- vertical holomorphic fields on an open subset of a fibre;
- vertical holomorphic fields on a open subset of V;
- projectable holomorphic fields on an open subset of a fibre; and
- projectable holomorphic fields on an open subset of V.
Let \xi be a \mathscr{C}^\infty vector field (resp. …) on V. By integrating \xi, we obtain a \mathscr{C}^\infty map, denoted by e^\xi, from an open subset W\subset\mathbb{R}\times V containing \{0\}\times V (resp. \mathbb{C}-analytic map from an open subset W\subset\mathbb{C}\times V) to V, characterised by
- e^\xi(t_1+t_2,y) = e^\xi(t_1,e^\xi(t_2,y)), with the left-hand side being defined whenever the right-hand side is; and
- \frac{\partial}{\partial t}e^\xi(t,y)|_{0,y} = \xi(y).
Note that W is a mixed manifold over \mathbb{R}\times B (resp. a mixed space over \mathbb{C}\times B).
For e^\xi\colon W\to V to be a morphism of mixed spaces over the projection \mathbb{R}\times B\to B, it is necessary and sufficient for \xi to be a vertical holomorphic field. For e^\xi\colon W\to V to be a morphism of mixed spaces over a map from an open subset of \mathbb{R}\times B containing \{0\}\times B to B, it is necessary and sufficient for \xi to be a projectable holomorphic field.
The proof is left to the reader.
IV. The Spencer–Kodaira map
Let \pi\colon V\to B be a mixed manifold (resp. a \mathbb{C}-analytic mixed space), b\in B, and V_0=\pi^{-1}(b_0). Let T_0 be the tangent space to B at b_0 (resp. the Zariski tangent space). We introduce the following sheaves on V_0:
- \Theta_0: the sheaf of germs of vertical holomorphic fields on V_0 ;
- \Pi_0: the sheaf of germs of locally projectable holomorphic fields on V_0 ; and
- \Lambda_0: the sheaf \pi^*T_0, i.e. the sheaf of germs of locally constant maps from V_0 to T_0.
We have an exact sequence of sheaves on V_0 0 \to \Theta_0 \to \Pi_0 \to \Lambda_0 \to 0 that gives rise to the long exact sequence in cohomology \ldots \to \mathrm{H}^0(V_0;\Pi_0) \to \mathrm{H}^0(V_0;\Lambda_0) \xrightarrow{\delta} \mathrm{H}^1(V_0;\Theta_0) \to \ldots. We also have a canonical map \iota\colon T_0 \to \mathrm{H}^0(V_0;\Lambda_0) that is injective if V_0 is non-empty, and surjective if V_0 is connected.
The Spencer–Kodaira map is the composition \rho_0 = \delta\circ\iota\colon T_0 \to \mathrm{H}^1(V_0;\Theta_0).
This map is an essential tool in the local study of deformations of \mathbb{C}-analytic varieties. Note that \Theta_0 is exactly the sheaf of germs of holomorphic fields of tangent vectors to V_0, and thus depends only on V_0, while T_0 depends only on the base. Also, \Theta_0 is a coherent analytic sheaf on V_0, and, if V_0 is compact, then \mathrm{H}^1(V_0;\Theta_0) is a finite-dimensional vector space over \mathbb{C} [1]. We thus see that, in this case (which is the only case where we can say anything non-trivial), \rho_0 might be possible to calculate.
It is clear that, if the given mixed manifold is trivial (i.e. if V=B\times V_0, with \pi being the projection to B), then the map \rho_0 is zero. The next talk aims to show that, in a certain sense, \rho indicates the non-triviality of V in a neighbourhood of V_0.
Bibliography
Footnotes
[Trans.] The more common modern nomenclature is to simply call such an object a family of complex manifolds.↩︎