I. The map \widetilde{\rho}
All throughout this talk, B is a \mathscr{C}^\infty manifold (resp. \mathbb{R}-analytic, resp. \mathbb{C}-analytic); \pi\colon V\to B denotes a proper mixed manifold; b_0 is a point of B; and V_0=\pi^{-1}(b_0) is thus a compact \mathbb{C}-analytic manifold.
Let \widetilde{\Theta} (resp. \widetilde{\Pi}) be the sheaf of germs of vertical holomorphic (resp. locally projectable holomorphic) vector fields on V. The quotient sheaf \widetilde{\Lambda}=\widetilde{\Pi}/\widetilde{\Theta} is exactly the inverse image under \pi of the sheaf \widetilde{T} of germs of \mathscr{C}^\infty fields (resp. …) of tangent vectors on B.
For every open subset U of B, set V_U=\pi^{-1}(U). The exact sequence 0 \to \widetilde{\Theta} \to \widetilde{\Pi} \to \widetilde{\Lambda} \to 0 of sheaves on V_U gives rise to a homomorphism \widetilde{\rho}_U\colon \mathrm{H}^0(U;\widetilde{T}) \xrightarrow{\pi_*} \mathrm{H}^0(V_U;\widetilde{\Lambda}) \xrightarrow{\delta} \mathrm{H}^1(V_U;\widetilde{\Theta}). Let \mathrm{R}^1\pi_*\widetilde{\Theta} be the sheaf on B defined by the presheaf U\mapsto\mathrm{H}^1(V_U;\widetilde{\Theta}). Then \widetilde{\rho} becomes a homomorphism of sheaves on B: \widetilde{\rho}\colon \widetilde{T} \to \mathrm{R}^1\pi_*\widetilde{\Theta}. In particular, we have a homomorphism \widetilde{\rho}_0\colon \widetilde{T}_0 \to \mathrm{R}^1\pi_*\widetilde{\Theta} = \mathrm{H}^1(V_0;\widetilde{\Theta}) where \widetilde{T}_0 is the vector space of germs at b_0 of fields of tangent vectors to B. Finally, we have a commutative diagram \begin{CD} \widetilde{T}_0 @>{\widetilde{\rho}_0}>> \mathrm{H}^1(V_0;\widetilde{\Theta}) \\@V{\varepsilon}VV @VV{\varepsilon}V \\T_0 @>>{\rho_0}> \mathrm{H}^1(V_0;\Theta_0) \end{CD} where \rho_0 is the Spencer–Kodaira map [2].
For the proper mixed manifold \pi\colon V\to B to be locally trivial in a neighbourhood of the point b_0\in B, it is necessary and sufficient for the map \widetilde{\rho}_0\colon\widetilde{T}_0\to\mathrm{H}^1(V_0;\widetilde{\Theta}) to be zero.
Proof. —
(Necessity). If \pi\colon V\to B is locally trivial at b_0, then, for every open subset U of B over which V is trivial, we have \widetilde{\Pi}=\widetilde{\Lambda}\oplus\widetilde{\Theta} on V_U, and so \delta\colon\mathrm{H}^0(V_U;\widetilde{\Lambda})\to\mathrm{H}^0(V_U;\widetilde{\Theta}) is zero.
(Sufficiency). Let (\eta_1,\ldots,\eta_p) be \mathscr{C}^\infty vector fields (resp. …) on a neighbourhood of b_0 in B, such that (\eta_1(b_0),\ldots,\eta_p(b_0)) forms a basis of the tangent space T_0 to B at b_0. It then follows from the hypothesis that the map \mathrm{H}^0(V_0;\widetilde{\Pi}) \to \mathrm{H}^0(V_0;\widetilde{\Lambda}) is surjective. So let (\xi_1,\ldots,\xi_p) be projectable holomorphic vector fields on a neighbourhood of V_0 in V, that project to (\eta_1,\ldots,\eta_p). Let f be the map defined on a neighbourhood of \{0\}\times V_0 in \mathbb{R}^p\times V_0 (resp. \mathbb{C}^p\times V_0) by f(t_1,\ldots,t_p,y) = e^{\xi_1}(t_1,e^{\xi_2}(\ldots,e^{\xi_p}(t_p,y)\ldots)). It follows from the proposition stated in [1] that f induces an isomorphism of mixed manifolds from U\times V_0 to \pi^{-1}(f_1(U)) over f_1, where U is a sufficiently small cubical neighbourhood of 0 in \mathbb{R}^p, and f_1 is the map from U to B defined by f_1(t_1,\ldots,t_p) = e^{\eta_1}(t_1,e^{\eta_2}(\ldots,e^{\eta_p}(t_p,b_0)\ldots)), which proves the theorem.
II. The regular case
For all b\in B, set V_b=\pi^{-1}(b). Consider the family \{\mathrm{H}^1(V_b;\Theta_b)\}_{b\in B} of finite-dimensional \mathbb{C}-vector spaces, and, for all b\in B, the map \varepsilon_b\colon \mathrm{H}^1(V_b;\widetilde{\Theta}) \to \mathrm{H}^1(V_b;\Theta_b).
For every open subset U\subset B, we have a map \widetilde{\varepsilon}_U\colon \mathrm{H}^1(V_U;\widetilde{\Theta}) \to \prod_{b\in U}\mathrm{H}^1(V_b;\Theta_B) that defines, by varying U, a homomorphism from the sheaf \mathrm{R}^1\pi_*\widetilde{\Theta} to the sheaf \Phi on B defined by \Phi(U)=\prod_{b\in U}\mathrm{H}^1(V_b;\Theta_b).
We say that the proper mixed manifold \pi\colon V\to B is regular if
- the dimension of \mathrm{H}^1(V_b;\Theta_b) does not depend on the point b\in B; and
- we can endow E=\bigcup_{b\in B}\mathrm{H}^1(V_b;\Theta_b) with the structure of a \mathscr{C}^\infty vector bundle (resp. …) such that \widetilde{\varepsilon} is an isomorphism from the sheaf \mathrm{R}^1\pi_*\widetilde{\Theta} to the sheaf of germs of \mathscr{C}^\infty sections (resp. …) of the bundle E.
In fact, Kodaira and Spencer have shown [2] that, by identifying the \mathrm{H}^1 spaces with spaces of harmonic forms, condition (2) is a consequence of condition (1).
Then Theorem 1 has the following corollary:
For the proper mixed manifold \pi\colon V\to B to be locally trivial, it is necessary and sufficient for it to be regular and, for all b\in B, for the Spencer–Kodaira map \rho_b\colon T_b \to \mathrm{H}^1(V_b;\Theta_b) to be zero.
Indeed, since \widetilde{\varepsilon} is injective, this condition implies that the map \widetilde{\rho}_b\colon \widetilde{T}_b \to \mathrm{H}^1(V_b;\widetilde{\Theta}) is zero for all b.
At the end of this talk, we will construct a counter-example which shows that it is necessary to assume that the mixed manifold is regular.
III. An example of non-regular deformation: Hopf manifolds
1. Hopf manifolds
Let n\geqslant 2 be an integer, and let b be an (n\times n) matrix with coefficients in \mathbb{C}, whose eigenvalues are all of modulus >1. The free group L(b) generated by b acts freely on \widetilde{V}=\mathbb{C}^n\setminus\{0\}, and the quotient space \widetilde{V}/L(b), which we call the Hopf manifold defined by b, is a compact \mathbb{C}-analytic manifold that is homeomorphic to S^{2n-1}\times S^1.
Note that V_b and V_{b'} are isomorphic if and only if there exists some a such that b'=aba^{-1} or b'=ab^{-1}a^{-1} (cf. Appendix).
Let \Theta be the sheaf of germs of holomorphic fields of tangent vectors on V_b.
We can identify \mathrm{H}^0(V_b;\Theta) with the vector space of matrices that commute with b, and \mathrm{H}^1(V_b;\Theta) has the same dimension as this vector space.
Proof. If X is a vector field on an open subset U\subset\widetilde{V}, then b_*(X) is the vector field on the open subset b(U) given by transporting via b, i.e. b_*X(u)=bX(b^{-1}u). Let \mathscr{U}=\{U_i\} be a cover of V by simply connected Stein open subsets; for all i, set \widetilde{U}_i=\chi^{-1}\{U_i\}, where \chi is the canonical map from \widetilde{V} to V_b. The cover \widetilde{\mathscr{U}}=\{\widetilde{U}_i\} of \widetilde{V} consists of Stein open subsets that are invariant under b (not necessarily connected, but this doesn’t matter). Then b_* defines a map, again denoted by b_*, from the group of cochains C^\bullet(\widetilde{V},\widetilde{U};\Theta) to itself.
We have the exact sequence 0 \to C^\bullet(V_b,\mathscr{U};\Theta) \xrightarrow{\chi^*} C^\bullet(\widetilde{V},\widetilde{U};\Theta) \xrightarrow{1-b_*} C^\bullet(\widetilde{V},\widetilde{U};\Theta) \to 0.
Proof. The only thing that we need to verify is that the map 1-b_* is surjective. For all (i_0,\ldots,i_q), let U'_{i_0,\ldots,i_q} be an open subset of \widetilde{V} such that \chi\colon U'_{i_0,\ldots,i_q} \to U_{i_0,\ldots,i_q} is a homeomorphism. The \widetilde{U}_{i_0,\ldots,i_q} is a disjoint union of the b_*^p U'_{i_0,\ldots,i_q}, where p\in\mathbb{Z}, and every \gamma\in C^q(\widetilde{V},\widetilde{U};\Theta) can be written in the form \gamma=\gamma_1-\gamma_2, with \gamma_1=0 on b^p(U'_{i_0,\ldots,i_q}) for p<0, and \gamma_2=0 for p\geqslant 0. Set \beta = \sum_{p\geqslant 0} b_*^p\gamma_1 + \sum_{p<0} b_*^p\gamma_2 (which is a locally finite sum). Then \beta-b_*\beta=\gamma, whence Lemma 1.
Now, to finish the proof of Proposition 2. From Lemma 1, we have the following exact sequence: 0 \to \mathrm{H}^0(V_b;\Theta) \xrightarrow{\chi^*} \mathrm{H}^0(\widetilde{V};\Theta) \xrightarrow{1-b_*} \mathrm{H}^0(\widetilde{V};\Theta) \xrightarrow{\delta_*} \mathrm{H}^1(V_b;\Theta) \xrightarrow{\chi^*} \mathrm{H}^1(\widetilde{V}y\Theta) \xrightarrow{1-b_*} \mathrm{H}^1(\widetilde{V};\Theta). We can show that \chi^*\colon \mathrm{H}^1(V_b;\Theta) \to \mathrm{H}^1(\widetilde{V};\Theta) is zero: if n>2, it is evident, since \mathrm{H}^1(\widetilde{V};\Theta)=0; if n=2, then a direct calculation on the cochains of a cover of \widetilde{V} by two Stein open subsets shows that 1-b_*\colon \mathrm{H}^1(\widetilde{V};\Theta) \to \mathrm{H}^1(\widetilde{V};\Theta) is bijective.
Now \mathrm{H}^0(\widetilde{V};\Theta) is the space of holomorphic vector fields on \widetilde{V}, but such a field extends to a holomorphic vector field on \mathbb{C}^n, and \mathrm{H}^0(\widetilde{V},\Theta)=L\oplus M, where L is the space of fields of linear vectors, and M is the space of fields of second-order vectors at 0. The subspaces L and M are invariant under b_*, and 1-b_*\colon M\to M is an isomorphism. Then Proposition 2 follows from remarking that, if an element of L is represented by a matrix a, then b_*a=bab^{-1}.
2. Mixed manifolds whose fibres are Hopf manifolds
Let B be the set of all (n\times n) matrices with coefficients in \mathbb{C} with eigenvalues all of modulus >1. This is an open subset of \mathbb{C}^{n^2}. Let \alpha be the transformation from B\times\widetilde{V} to itself defined by \alpha(b,x)=(b,b(x)). The free group L(\alpha) generated by \alpha acts linearly on B\times\widetilde{V}, and the quotient V=B\times\widetilde{V}/L(\alpha) is a \mathbb{C}-analytic manifold. By endowing it with the projection \pi\colon V\to B induced by the projection \pi_1\colon B\times\widetilde{V}\to B after passing to the quotient, we obtain a \mathbb{C}-analytic mixed manifold that is proper, but not regular. Indeed, condition 1 of the definition of regular mixed manifolds is not satisfied: for example, for n=2, the dimension of \mathrm{H}^1(V_b;\Theta) is 4 if b is a scalar matrix, but 2 in all other cases.
Note that the dimension of \mathrm{H}^1(V_b;\Theta_b) is an upper semi-continuous function of b, and that the set of b such that \dim\mathrm{H}^1(V_b;\Theta_b)\geqslant k is a closed analytic subspace of B. This is a general result, that we hope to be able to prove in a later talk of this seminar.
3. Calculation of \rho
We have T_b=\operatorname{Hom}(\mathbb{C}^n,\mathbb{C}^n)=L\subset\mathrm{H}^0(\widetilde{V};\Theta), and we defined, to prove Proposition 2, a surjective map \delta_*\colon L\to\mathrm{H}^1(V_b;\Theta).
The Spencer–Kodaira map \rho is given, for the mixed manifold studied in this section, by \rho(a) = \delta_*(ab^{-1}). In particular, it is surjective, and its kernel is the space of matrices of the form [\ell,b] for \ell\in L.
Proof. Let a\in T_b=L. Let \{U_i\} be a cover of V_b by simply connected Stein open subsets, and, for each i, let U'_i be a connected component of \widetilde{U}_i.
Let \eta'_i be the projectable holomorphic field on U'_i defined by \eta'_i(x)=(a,0); let \widetilde{\eta}_i be the projectable holomorphic field on \widetilde{U}_i defined by \widetilde{\eta}_i=\alpha_*^k\eta'_i on b^k(U'_i); and let \eta_i be the projectable holomorphic field on U_i corresponding to \widetilde{\eta}_i. By definition, \rho(a) is the cohomology class of the cochain \{\theta_{ij}\}, where \theta_{ij}=\eta_j-\eta_i is a vertical holomorphic field on U_{ij}.
Set \widetilde{\eta}_i(x)=(a,\beta_i(x)). Then \beta\in C^0(\widetilde{V};\Theta), and we have (1-b_*)\beta=ab^{-1}\in L\subset\mathrm{H}^0(\widetilde{V};\Theta). Indeed, \alpha_*\eta=\eta, \alpha_*\eta_i(b_{-1}x)=\eta_i(x), and \alpha_*(a,\beta(b^{-1}x)) = (a,\beta(x)), whence ab^{-1}x + b\cdot\beta(b^{-1}x) = \beta(x). We thus deduce that \theta=\delta_*(ab^{-1}), which proves Proposition 3.
4. A counter-example
Take n=2, and \sigma\in\mathbb{C} such that |\sigma|>1. Let B'\subset B be the set of matrices of the form \begin{pmatrix} \sigma & t \\0 & \sigma \end{pmatrix} where t\in\mathbb{C}, and let V'=\pi^{-1}(B') be the mixed manifold induced by V over V'; now B' is a line, and its tangent space T'_b at b is generated, for all b, by a=\begin{pmatrix}0&1\\0&0\end{pmatrix}. It follows from Proposition 3 that the Spencer–Kodaira map \rho'\colon T_b(B') \to \mathrm{H}^1(V_b;\Theta) is zero if and only if b \neq b_0 = \begin{pmatrix} \sigma & 0 \\0 & \sigma \end{pmatrix} since, if b\neq b_0, then a=[\ell,b], where \ell=\begin{pmatrix}t^{-1}&0\\0&0\end{pmatrix}; and if b=b_0, then \rho' is injective.
We can also see that V' is trivial on B'\setminus\{b_0\}.
Let \varphi\colon\mathbb{C}\to B'\subset B be the map defined by \varphi(t) = \begin{pmatrix} \sigma & t^2 \\0 & \sigma \end{pmatrix} and let V^\varphi be the mixed manifold given by the inverse image of V under \varphi. The Spencer–Kodaira map \rho_t^\varphi from \mathbb{C} to \mathrm{H}^1(V_{\varphi(t)};\Theta) is the composition \rho'_{\varphi(t)}\circ\mathrm{D}\varphi\colon \mathbb{C} \to T'_{\varphi(t)} \to \mathrm{H}^1(V_{\varphi(t)};\Theta), and this is zero for all t, since, if t\neq0, then \rho'_{\varphi(t)} is zero; and, if t=0, then \mathrm{D}\varphi is zero.
However, the mixed manifold V^\varphi is not locally trivial, since V_0^\varphi is not isomorphic to V_t^\varphi for t\neq0.
5. Question (K. Srinivasacharyulu)
We know that the Hopf manifolds are non-K"{a}hler, and thus non-algebraic. For n=2, the manifold V_b admits non-constant meromorphic functions if and only if b can be diagonalised with eigenvalues \sigma_1 and \sigma_2 satisfying \sigma_1^p=\sigma_2^q for some integers p and q (and there is then the function x_1^px_2^{-q}). The set of b satisfying this property is neither open nor closed, but it is a countable union of closed analytic subspaces. An analogous phenomenon arises for deformations of complex tori. Is this result general?
Appendix
With the notation of §III.1, let f\colon V_b\to V_{b'} be an isomorphism of \mathbb{C}-analytic manifolds. This lifts to an isomorphism of universal coverings \widetilde{f}\colon \mathbb{C}^n\setminus\{0\} \to \mathbb{C}^n\setminus\{0\}. By Hartog, \widetilde{f} extends to an isomorphism g\colon\mathbb{C}^n\to\mathbb{C}^n. We necessarily have g(bz) = (b')^kg(z) \tag{$*$} where z\in\mathbb{C}^n, and k is an integer; the same property, applied to the inverse map of g, shows that k=\pm1. Let a be the linear map that is tangent to g at the origin; the identity (*) then gives \begin{aligned} ab &= (b')^ka \\k &= \pm1 \end{aligned} whence b' = aba^{-1} \quad\text{or}\quad b'= ab^{-1}a^{-1}.