Introduction
Let V_0 be a compact complex-analytic manifold, and let \Theta be the sheaf of germs of holomorphic fields of tangent vectors. We ask the following question: given an element a\in\mathrm{H}^1(V_0,\Theta), does there exists a deformation of V_0, with a non-singular base (i.e. a fibred mixed manifold \pi\colon V\to B, with b_0\in B, along with an isomorphism V_0\xrightarrow{\cong}\pi^{-1}(b_0)), such that a is the image, under the map \rho defined in [Talk no. 2], of a vector v that is tangent to B at b_0? An element a\in\mathrm{H}^1(V_0,\Theta) for which the answer is positive is called a deformation vector. We will give a necessary condition for a to be a deformation vector; this condition is written [a\smile a]=0. We will then give an example where this condition is not satisfied.
I. Exact sequences of sheaves of algebras
Let K be a commutative ring, and let \Phi, \Phi_1, and \Phi_2 be sheaves of K-modules on some space X, and suppose that we have some given homomorphism \Phi_1\otimes\Phi_2\to\Phi, written as a product. We define, for any cover {\mathscr{U}} of X, the cup product \smile\colon C^p(X,{\mathscr{U}};\Phi_1) \otimes C^q(X,{\mathscr{U}};\Phi_2) \to C^{p+q}(X,{\mathscr{U}};\Phi) by the formula (\alpha\smile\beta)_{i_0,\ldots,i_{p+q}} = \alpha_{i_0,\ldots,i_p}\cdot\beta_{i_p,\ldots,i_{p+q}}. We have the relation \mathrm{d}(\alpha\smile\beta) = \mathrm{d}\alpha\smile\beta + (-1)^p\alpha\smile\mathrm{d}\beta. This induces a cup product on the cohomology of the cover {\mathscr{U}}, and, by passing to the inductive limit over open covers, a cup product \smile\colon \mathrm{H}^p(X;\Phi_1) \otimes \mathrm{H}^q(X;\Phi_2) \to \mathrm{H}^{p+q}(X;\Phi).
A sheaf of algebras on X is a sheaf of modules \Phi on X endowed with a product \Phi\otimes\Phi\to\Phi (which we do not assume to be either commutative nor associative).
If f\colon\Phi\to\Psi is a homomorphism of sheaves of algebras, then the kernel \Phi' of f is a sheaf of two-sided ideals of \Phi, i.e. we have products \Phi'\otimes\Phi\to\Phi' and \Phi\otimes\Phi'\to\Phi' such that the two diagrams \begin{CD} \Phi'\otimes\Phi @>>> \Phi' \\@VVV @VVV \\\Phi\otimes\Phi @>>> \Phi \end{CD} \qquad \begin{CD} \Phi\otimes\Phi' @>>> \Phi' \\@VVV @VVV \\\Phi\otimes\Phi @>>> \Phi \end{CD} both commute.
Let 0\to\Phi'\to\Phi\to\Phi''\to0 be an exact sequence of sheaves of algebras on X; let a\in\mathrm{H}^p(X;\Phi''). Then \delta a\in\mathrm{H}^{p+1}(X;\Phi'), and, for any class b\in\mathrm{H}^q(X;\Phi'), we have \delta a\smile b=0.
Proof. Let {\mathscr{U}} be a cover of X such that a and b are represented by cocycles \alpha and \beta (respectively), and such that \alpha lifts to a cochain \eta\in C^p(X,{\mathscr{U}};\Phi). Then \delta\eta is a cocycle in C^{p+1}(X,{\mathscr{U}};\Phi') whose class in \mathrm{H}^{p+1}(X;\Phi') is, by definition, \delta a, and \delta a\smile b is the class of \delta\eta\smile\beta. But \delta(\eta\smile\beta)=\delta\eta\smile\beta, and \eta\smile\beta is a cochain in C^{p+q}(X,{\mathscr{U}};\Phi'), since \Phi' is a sheaf of ideals. So the cocycle \delta\eta\smile\beta is cohomologous to 0 in \mathrm{H}^{p+q+1}(X;\Phi'), which proves the proposition.
II. The primary obstruction
Let V_0 be a complex-analytic manifold, and \Theta_0 the sheaf of germs of holomorphic fields of tangent vectors. Then \Theta_0 is a sheaf of Lie algebras, and, if a,b\in\mathrm{H}^\bullet(V_0,\Theta_0), then we denote by [a\smile b] the cup product defined by the bracket [-,-]\colon\Theta_0\otimes\Theta_0\to\Theta_0. It satisfies [b\smile a] = (-1)^{pq+1}[a\smile b] for a\in\mathrm{H}^p(V_0,\Theta_0) and b\in\mathrm{H}^q(V_0,\Theta_0).
Let \pi\colon V\to B be a mixed manifold, b_0 a point of B, V_0=\pi^{-1}(b_0), and let \rho_0\colon T_0\to\mathrm{H}^1(V_0,\Theta_0) be Spencer–Kodaira map. Then, if u and v are tangent vectors of B at b_0, we have [\rho_0(u)\smile\rho_0(v)] = 0.
Let V_0 be a complex-analytic manifold, and \Theta the sheaf of germs of holomorphic fields of tangent vectors of V_0. If a\in\mathrm{H}^1(V_0,\Theta) is a deformation vector, then [a\smile a]=0.
Proof. (Proof of the Corollary). This is simply a particular case of Theorem 1; note that [a\smile b] is a symmetric bilinear map from \mathrm{H}^1\otimes\mathrm{H}^1 to \mathrm{H}^2, and that we are in characteristic 0\neq2.
Proof. (Proof of Theorem 1). Consider the following sheaves on V_0:
- \Theta_0: the sheaf of germs of vertical holomorphic fields on V_0;
- \widetilde{\Theta}_0: the sheaf of germs of vertical holomorphic fields on V;
- \Pi_0: the sheaf of germs of locally projectable holomorphic fields on V_0;
- \widetilde{\Pi}_0: the sheaf of germs of locally projectable holomorphic fields on V;
- \Lambda_0: the sheaf \pi^*T_0, where T_0 is the tangent space of B at b_0; and
- \widetilde{\Lambda}_0: the sheaf \pi^*\widetilde{T}_0, where \widetilde{T}_0 is the space of germs at b_0 of fields on B of tangent vectors of B.
We have the following diagram: \begin{CD} 0 @>>> \widetilde{\Theta}_0 @>>> \widetilde{\Pi}_0 @>>> \widetilde{\Lambda}_0 @>>> 0 \\@. @V{\varepsilon}VV @V{\varepsilon}VV @V{\varepsilon}VV @. \\0 @>>> \Theta_0 @>>> \Pi_0 @>>> \Lambda_0 @>>> 0 \end{CD} whence we obtain the following commutative diagram: \begin{CD} \widetilde{T}_0 @>{\widetilde{\rho}}>> \mathrm{H}^1(V_0;\widetilde{\Theta}) \\@V{\varepsilon}VV @VV{\varepsilon}V \\T_0 @>>{\rho}> \mathrm{H}^1(V_0;\Theta_0) \end{CD}
Let u,v\in T_0 be fixed tangent vectors of B at b_0. We can always find vector fields \widetilde{u} and \widetilde{v} on B that take the values u and v (respectively) at b_0; \epsilon(\widetilde{u})=u and \epsilon(\widetilde{v})=v. The exact sequence 0 \to \widetilde{\Theta}_0 \to \widetilde{\Pi}_0 \to \widetilde{\Lambda}_0 \to 0 is a sequence of homomorphisms of sheaves of Lie algebras, and so [\widetilde{\rho}(\widetilde{u})\smile\widetilde{\rho}(\widetilde{v})] = 0 by Proposition 1. But \epsilon\colon\widetilde{\Theta}_0\to\Theta_0 is also a homomorphism of sheaves of Lie algebras, and the diagram \begin{CD} \mathrm{H}^1(V_0,\widetilde{\Theta}_0)\otimes\mathrm{H}^1(V_0,\widetilde{\Theta}_0) @>{[-\smile-]}>> \mathrm{H}^2(V_0,\widetilde{\Theta}_0) \\@V{\varepsilon\otimes\varepsilon}VV @VV{\varepsilon}V \\\mathrm{H}^1(V_0,\Theta_0)\otimes\mathrm{H}^1(V_0,\widetilde{\Theta}_0) @>>{[-\smile-]}> \mathrm{H}^2(V_0,\Theta_0) \end{CD} commutes. We thus deduce that [\rho(u)\smile\rho(v)]=0.
—
- We make essential use of the fact that \epsilon\colon\widetilde{T}_0\to T_0 is surjective, and thus of the fact that B has no singularities.
- We actually have [\rho(u)\smile b]=0 for all u\in T_0, for any class b\in\mathrm{H}^1(V_0,\Theta_0) that is in the image of \mathrm{H}^1(V_0,\widetilde{\Theta}_0) under \epsilon. In particular, for an element a\in\mathrm{H}^1(V_0,\Theta_0) to be a regular deformation vector (in the sense of [Talk no. 3]), it is necessary and sufficient for [a\smile b]=0 for all b\in\mathrm{H}^1(V_0,\Theta_0).
If V_0 is a compact complex-analytic manifold, and a\in\mathrm{H}^1(V_0,\Theta), then we call [a\smile a]\in\mathrm{H}^2(V_0,\Theta) the primary obstruction to the deformation of V_0 along a. For a to be a deformation vector, it is necessary that this primary obstruction be zero; but it is not sufficient: we can define a sequence of set-theoretic maps \omega_n, called obstructions, with \omega_1\colon\mathrm{H}^1(V_0,\Theta)\to\mathrm{H}^2(V_0,\Theta) given by \omega_1(a)=[a\smile a], and with \omega_{k+1} defined on the subset of \mathrm{H}^1(V_0,\Theta) where \omega_k vanishes, with values in varying quotients1 of \mathrm{H}^2(V_0,\Theta), and a necessary condition for a to be a deformation vector is that all the \omega_k(a) be defined and real. I do not know if this condition is sufficient. Kodaira, Spencer, and Nijenhuis [3] have shown that, if \mathrm{H}^2(V_0,\Theta)=0, then every element of \mathrm{H}^1(V_0,\Theta) is a deformation vector. In this case, we even have a locally universal deformation whose base is a manifold, and \rho is an isomorphism from the tangent space of this manifold to \mathrm{H}^1(V_0,\Theta)
III. An example of obstruction
1. The manifold V_0
Let X=E/\Gamma be a 2-dimensional complex torus, i.e. E\cong\mathbb{C}^2 and \Gamma\cong\mathbb{Z}^4, and let D the be projective line \mathbb{P}^1\mathbb{C}. Set V_0=X\times D. The sheaf \Theta of holomorphic fields of tangent vectors of V_0 is the direct sum of the sheaves of Lie algebras \Theta_1 and \Theta_2, where \begin{aligned} \Theta_1 &= {\mathcal{O}}\otimes_{{\mathcal{O}}_X}\pi_1^*\Theta_X \\\Theta_2 &= {\mathcal{O}}\otimes_{{\mathcal{O}}_D}\pi_2^*\Theta_D \end{aligned} where \pi_1\colon V_0\to X and \pi_2\colon V_0\to D are the projections, {\mathcal{O}}, {\mathcal{O}}_X, and {\mathscr{D}} are the structure sheaves (sheaves of local rings), and \Theta_X and \Theta_D are the sheaves of germs of holomorphic fields of tangent vectors of X and D (respectively). We are mostly interested in \Theta_2. Also, \mathrm{H}^1(V_0,\Theta_2) is given by the Künneth exact sequence: 0 \to \mathrm{H}^0(X,{\mathcal{O}}_X)\otimes\mathrm{H}^1(D,\Theta_D) \to \mathrm{H}^1(V_0,\Theta_2) \to \mathrm{H}^1(X,{\mathcal{O}}_X)\otimes\mathrm{H}^0(D,\Theta_D) \to 0. But we know that \mathrm{H}^0(D,\Theta_D) is the Lie algebra {\mathfrak{a}} of the group A = \operatorname{GL}(2,\mathbb{C})/\mathbb{C}^* = \operatorname{SL}(2,\mathbb{C})/\{\pm1\} of automorphisms of D, and that \mathrm{H}^1(D,\Theta_D)=0, as we can easily see by taking a cover of D by two open subsets. We have already seen (in [Talk no. 1]) that, if X=E/\Gamma, then \mathrm{H}^1(X,{\mathcal{O}})=\operatorname{Hom}(\Gamma,\mathbb{C})/\operatorname{Hom}_{\mathbb{C}}(E,\mathbb{C}) is of dimension 2. So \mathrm{H}^1(V_0,\Theta_2)=\mathrm{H}^1(X,{\mathcal{O}})\otimes{\mathfrak{a}} is of dimension 6. The cup product \mathrm{H}^1(V_0,\Theta_2)\otimes\mathrm{H}^1(V_0,\Theta_2) \to \mathrm{H}^2(V_0,\Theta_2) is given by the formula [(\gamma\otimes\alpha)\smile(\gamma'\otimes\alpha')] = (\gamma\smile\gamma')\otimes[\alpha,\alpha']. The cone of elements \varphi\in\mathrm{H}^1(V_0,\Theta_2) such that [\varphi\smile\varphi]=0 can be identified with the cone of rank 1 tensors in \mathrm{H}^1(X,{\mathcal{O}})\otimes{\mathfrak{a}}. Indeed, if \varphi=\gamma\otimes\alpha, then [\varphi\smile\varphi] = (\gamma\smile\gamma)\otimes[\alpha,\alpha] = 0\otimes0 = 0 and, if \varphi is not a simple tensor, then we have \varphi = \gamma\otimes\alpha + \gamma'\otimes\alpha' with \gamma and \gamma' independent, and \alpha and \alpha' independent, so [\varphi\smile] = 2(\gamma\smile\gamma')\otimes[\alpha,\alpha'] \neq 0.
2. The mixed space V
In this example, every element of \mathrm{H}^1(V_0,\Theta_2) whose primary obstruction is zero is a deformation vector. More precisely:
There exists a mixed space \pi\colon V\to B and a point b_0\in B such that
- \pi^{-1}(b_0)=V_0 (the manifold defined in §III.1);
- there exists an isomorphism \sigma from a \mathbb{C}-analytic space B to the cone of elements \varphi\in\mathrm{H}^1(V_0,\Theta_2) such that [\varphi\smile\varphi]=0; and
- for every subspace B' of B that has no singularities at b_0, the Spencer–Kodaira map \rho from the tangent space of B' at b_0 to \mathrm{H}^1(V_0,\Theta) agrees with \sigma\colon B'\to\mathrm{H}^1(V_0,\Theta_2).
Let H be the analytic space of homomorphisms from \Gamma to {\mathfrak{a}} whose images are contained in a vector subspace of {\mathfrak{a}} that is 1-dimensional over \mathbb{C} (i.e. (4\times2) matrices of rank 1 with coefficients in \mathbb{C}). For every h\in H, e\circ h is a homomorphism from \Gamma to A, where e\colon{\mathfrak{a}}\to A denotes the exponential map, and we construct a manifold V_h that is fibred over X with fibre D as follows: V_h is the quotient of E\times D by the equivalence relation defined by \Gamma acting via \gamma\star(x,y) = (x+\gamma,((e\circ h)(\gamma))\cdot y). These manifolds are the fibres of a mixed space W\to H, where W is the quotient of H\times E\times D by the equivalence relation defined by \Gamma acting via \gamma\star(h,x,y) = (h,x+y,(e\circ h(y))\cdot y). We now place the following equivalence relation on H: we have h'\sim h if and only if (h'-h) extends to an \mathbb{C}-linear map f\colon E\to{\mathfrak{a}}. Note that, if h'(\Gamma) and h(\Gamma) are contained in the same subspace L of {\mathfrak{a}} of dimension 1 over \mathbb{C} (or if h'\sim h), then we also have f(E)\subset L (or h\sim0 and h'\sim0). In both cases, V_h and V_{h'} are isomorphic, and we have an isomorphism i_{h',h}\colon V_h\to V_{h'} defined by i_{h',h}(x,y) = (x,e\circ f(x)\cdot y) (in the first case), or i_{h',h} = i_{h',0}\circ i_{0,h} (in the second case). If h, h', and h'' are in the same class, then we have i_{h''h}=i_{h''h'}\circ i_{h'h}, and we can place on W the equivalence relation (h',z') \sim (h,z) \iff h'\sim h\text{ or }z'=i_{h'h}z for h,h'\in H, z\in V_h, and z'\in V_{h'}.
Let B and V be the quotients of H and W (respectively) by these equivalence relations. We have a projection V\to B. To show that the structures of a \mathbb{C}-analytic space on H and W induce structures of a \mathbb{C}-analytic space on their quotients B and V, it suffices to remark that we can lift B to a analytic subspace of H: let, for example, (\gamma_1,\gamma_2,\gamma_3,\gamma_4) be a basis of \Gamma such that (\gamma_1,\gamma_2) is a basis of E over \mathbb{C}; then each class b\in B contains exactly one element h\in H such that h(\gamma_1) = h(\gamma_2) = 0.
3. Calculating \rho_0
Let T be the Zariski tangent space of B at b_0, i.e. the dual of {\mathfrak{I}}/{\mathfrak{I}}^2, where {\mathfrak{I}} is the ideal of germs at b_0 of analytic functions on B that are zero at b_0. Then T_0 can be identified with \operatorname{Hom}(\Gamma,a)/\operatorname{Hom}_{\mathbb{C}}(E,a). Also, \begin{aligned} \mathrm{H}^1(V_0,\Theta) &= \mathrm{H}^1(V_0;\Theta_1) \oplus \mathrm{H}^1(V_0;\Theta_2) \\&= \big(\mathrm{H}^1(X;{\mathcal{O}}) \otimes E\big) \oplus \big(\mathrm{H}^1(X;{\mathcal{O}})\otimes a\big), \end{aligned} and the second term of this term can be identified with the quotient \operatorname{Hom}(\Gamma,a)/\operatorname{Hom}_{\mathbb{C}}(E,a). We are going to show that the map \rho_0\colon T_0\to\mathrm{H}^1(V_0;\Theta) is exactly the canonical injection defined by these identifications.
Let u\in T_0=\operatorname{Hom}(\Gamma,\alpha)/\operatorname{Hom}(E,\alpha) be the class of an element h\in\operatorname{Hom}(\Gamma,\alpha), which we suppose to be of rank 1. Then we can write h in the form \eta\otimes\sigma, where \eta\in\operatorname{Hom}(\Gamma,\mathbb{C}), \sigma\in\alpha, and we can consider h as a tangent vector to H at 0. Let \overline{h} be the field of tangent vectors to H\times E\times D at 0\times E\times D that projects onto h, and thus whose components over E\times D are zero. Let (U_i) be a cover of X=E/\Gamma by simply connected open subsets, and choose, for each i, a component \widetilde{U}_i of the inverse image of U_i in E. We will denote by v_i the image over U_i\times D of the field \overline{h}|\widetilde{U}_i\times D. This is a projectable holomorphic field on 0\times U_i\times D of tangent vectors of H\times U_i\times D, and we set w_{ij}=v_j-v_i, so that w_{ij} is a vertical holomorphic field on U_{ij}\times D, and these fields form a cocycle whose cohomology class will be, by definition, \rho_0(u).
Let x\in U_{ij}, and let \widetilde{x}_i and \widetilde{x}_j be its inverse image in \widetilde{U}_i and \widetilde{U}_j (respectively). We have that \widetilde{x}_j=\widetilde{x}_i+\gamma_{ij}(x), where \gamma_{ij}(x)\in\Gamma, and w_{ij}(x) = \overline{h}(\widetilde{x}_j) - [\gamma_{ij}(x)]_*(\overline{h}(\widetilde{x}_i)) = -h(\gamma_{ij}(x)) \in\alpha. Now w_{ij} is a vector field on D, and so (w_{ij}) \in \mathrm{Z}^1(V_0,(U_i\times D);\Theta_2), and w_{ij} is of the form \zeta\otimes\alpha, where \zeta\in\mathrm{Z}^1(V_0,(U_i\times D);{\mathcal{O}}) is the cocycle defined by \zeta_{ij}(x)=-\eta(\gamma_{ij}(x)). This is a cocycle whose cohomology class is (up to a sign) the element of \mathrm{H}^1(V_0,{\mathcal{O}}) that is identified with the class \eta in \operatorname{Hom}(\Gamma,\mathbb{C})/\operatorname{Hom}_{\mathbb{C}}(E,\mathbb{C}). QED.
Appendix: Higher obstructions
I. Definition of obstructions
1. The sheaf of germs of vertical automorphisms
Let V_0 be a \mathbb{C}-analytic manifold, which we assume to be compact, and B a \mathbb{C}-analytic space, and let b_0\in B. We are going to define a sheaf \Gamma of non-abelian groups on V_0. For every open subset U of V_0, consider the isomorphisms of analytic varieties \gamma\colon W\to W', where W and W' are open subsets of B\times V_0 that contain \{b_0\}\times U, such that the following conditions are satisfied:
- \pi_1\gamma=\pi_1 is the projection B\times V_0 to B;
- \gamma is the identity on \{b_0\}\times U.
Then \Gamma(U) consists of equivalence classes of these isomorphisms, where we identify \gamma_1 with \gamma_2 if they agree on a neighbourhood of \{b_0\}\times U.
It is clear that \Gamma(U) is a group under composition of isomorphisms, and that the \Gamma(U) form a sheaf \Gamma of non-abelian groups.
We can identify \mathrm{H}^1(V_0,\Gamma) with the set of classes of deformation germs of V_0 over (B,b_0).
Recall that a deformation germ of V_0 over (B,b_0) is a deformation of V_0 over a neighbourhood of b_0 in B, and that two such deformations (B',b_0,V',\pi',\iota') and (B'',b_0,V'',\pi'',\iota'') are locally equivalent if there exists a neighbourhood W' of (\pi')^{-1}(b_0) in V', a neighbourhood W'' of (\pi'')^{-1}(b_0) in V'', and an isomorphism \varphi from W' to W'' such that the diagram \begin{CD} V_0 @= V_0 \\@VVV @VVV \\W' @>{\varphi}>> W'' \\@V{\pi'}VV @VV{\pi''}V \\B @= B \end{CD} commutes.
Proof. (Proof of Proposition 1). Let (B',b_0,V,\pi,\iota) be a deformation of V_0 over a neighbourhood V' of b_0 in B. Then we can find a cover \{U_i\} of V_0 and a cover \{W_i\} of a neighbourhood of \iota(V_0) in V, along with isomorphisms \{h_i\}, where h_i is an isomorphism from a neighbourhood of \{b_0\}\times U_i in B\times V_0 to W_i that agrees with \iota on \{b_0\}\times U_i, and such that \pi\circ h_i=\pi_1.
Set \gamma_{ij}=h^{-1}_i\circ h_j. We can show that the \gamma_{ij} define an element of \Gamma(U_i\cap U_j), and that \gamma_{ij}\circ\gamma_{jk}=\gamma_{ik}. The \gamma_{ij} thus form a cocycle \gamma\in\operatorname{Z}^1(V_0,\{U_i\};\Gamma). Such a cocycle is said to be associated to the deformation. It will still be associated to the deformation if pass to a finer cover. Let (B',b_0,V',\pi',\iota') be a deformation that is locally equivalent to the first, and let \gamma' be a cocycle associated to this deformation. We can suppose, by refining the covers if necessary, that the cocycles \gamma and \gamma' are defined with respect to the same cover \{U_i\} of V_0. Let f be an isomorphism from a neighbourhood of \iota(V_0) in V to a neighbourhood of \iota'(V_0) in V'. Set f_i=(h'_i)^{-1}\circ f\circ h_i. Then f_i\in\Gamma(U_i), and f_i\circ\gamma_{ij} = \gamma'_{ij}\circ f_j. We thus conclude that the cocycles associated to a deformation form a cohomology class that depends only on the local class of the deformation.
Conversely, suppose we have a locally finite cover \{U_i\} of V_0 and a cocycle \gamma\in\operatorname{Z}^1(V_0,\{U_i\};\Gamma). Then \gamma_{ij} can be represented by an isomorphism from an open W_{ij} of B\times V_0 to another open W_{ji}, with the two open subsets both containing \{b_0\}\times U_{ij}. Pick a refinement \{U'_i\} of the cover \{U_i\}, and take some neighbourhood B'' of b_0 in B small enough such that B''\times U'_{ij}\subset W_{ij} for all (i,j), and such that the equality \gamma_{ij}\circ\gamma_{jk}=\gamma_{ik} holds wherever it is defined in B''\times U'_{ijk}. We thus obtain a deformation V of V_0 on B'' by gluing the B''\times U'_i via the \gamma_{ij}.
Finally, we can show that all the above does indeed define a bijection between the set of local classes of deformations of V_0 over (B,b_0) and \mathrm{H}^1(V_0;\Gamma).
2. Higher obstructions
For every open subset U\subset V_0, the group \Gamma(U) is naturally filtered: denote by {\mathscr{F}}_k(U) the group of vertical automorphisms that are tangent to the identity up to order k-1. Then \Gamma becomes a filtered sheaf: \Gamma = {\mathscr{F}}_1 \supset {\mathscr{F}}_2 \supset \ldots \qquad\text{and }\bigcap {\mathscr{F}}_k=\{0\}. Set \begin{aligned} {\mathscr{Q}}_k &= \Gamma/{\mathscr{F}}_{k+1} \\{\mathscr{G}}_k &= {\mathscr{F}}_k/{\mathscr{F}})_{k+1} = \operatorname{Ker}({\mathscr{Q}}_k\to{\mathscr{Q}}_{k-1}). \end{aligned} For all k, {\mathscr{G}}_k is a sheaf of abelian groups, which we will write additively. If B=\mathbb{C} and b_0=0 (we then speak of the deformation in one parameter), for all k, {\mathscr{G}}_k can be identified with the sheaf \Theta of germs of vector fields tangent to V_0. In the general case, {\mathscr{G}}_k = {\mathfrak{m}}^k/{\mathfrak{m}}^{k+1}\otimes\Theta where {\mathfrak{m}} is the maximal ideal of the point b_0 in B.
Now, if a\in{\mathscr{F}}_p and b\in{\mathscr{F}}_q, then the commutator aba^{-1}b^{-1} is in {\mathscr{F}}_{p+q}, and this defines a map {\mathscr{G}}_p\otimes{\mathscr{G}}_q\to{\mathscr{G}}_{p+q} which endows {\mathscr{G}}_\bullet=\bigoplus{\mathscr{G}}_k with the structure of a sheaf of Lie algebras that is isomorphic to the tensor product of \Theta with the graded algebra associated to the maximal ideal {\mathfrak{m}} of b_0 in B filtered by powers.
The exact sequence of non-abelian groups 0 \to {\mathscr{G}}_{k+1} \to {\mathscr{Q}}_{k+1} \to {\mathscr{Q}}_k \to 0 in which {\mathscr{G}}_{k+1} is a subgroup of {\mathscr{Q}}_{k+1} contained in its centre gives rise [1] to an exact sequence of pointed sets \mathrm{H}^1(V_0;{\mathscr{Q}}_{k+1}) \to \mathrm{H}^1(V_0;{\mathscr{Q}}_k) \xrightarrow{\delta_k} \mathrm{H}^2(V_0;{\mathscr{G}}_{k+1}) i.e. for an element q\in\mathrm{H}^1(V_0,{\mathscr{Q}}_k) to be in the image of \mathrm{H}^1(V_0;{\mathscr{Q}}_{k+1}), it is necessary and sufficient for \delta_k q=0 in \mathrm{H}^2(V_0;{\mathscr{G}}_{k+1}). A necessary condition for q to be in the image of \mathrm{H}^1(V_0;\Gamma)\to\mathrm{H}^1(V_0;{\mathscr{Q}}_k) is thus \delta_k q=0 in \mathrm{H}^2(V_0;{\mathscr{G}}_{k+1}).
Let q\in\mathrm{H}^1(V_0;{\mathscr{Q}}_i), and let k\geqslant i. We define an obstruction of order k of the element q to be the direct image in \mathrm{H}^2(V_0;{\mathscr{G}}_{k+1}) under \delta_k of the inverse image of q in \mathrm{H}^1(V_0;{\mathscr{Q}}_k). It is thus a subset of \mathrm{H}^2(V_0;{\mathscr{G}}_{k+1}). The obstruction is said to be trivial if the identity element belongs to this subset. Being trivial is a necessary and sufficient condition for q to be in the image of \mathrm{H}^1(V_0;{\mathscr{Q}}_{k+1}), and a necessary condition for q to be in the image of \mathrm{H}^1(V_0;\Gamma).
If q is not in the image of \mathrm{H}^1(V_0,{\mathscr{Q}}_k), then its obstruction of order k is empty, and thus non-trivial.
This definition is used most of all in the case of deformations in one parameter (B=\mathbb{C} and b_0=0), where {\mathscr{G}}_{k+1}=\Theta for all k, and {\mathscr{Q}}_1={\mathscr{G}}_1=\Theta. The successive obstructions of an element a\in\mathrm{H}^1(V_0;\Theta) are thus subsets of \mathrm{H}^2(V_0;\Theta), and for a to be a deformation vector, it must be the case that all of its obstructions are trivial. Indeed, the element of \mathrm{H}^1(V_0;\Theta) that corresponds, under the identifications we have made (\Theta={\mathscr{Q}}_1=\Gamma/{\mathscr{F}}_2, and Proposition 1), to a deformation germ is exactly the image under the Spencer–Kodaira map \rho of the canonical basis vector of the tangent space to \mathbb{C} at 0.
II. Calculation of obstructions
1. Relation to the sheaf \Omega
From now on, we work in the case of deformations in one parameter, i.e. B=\mathbb{C} and b_0=0.
Let \Omega be the sheaf of universal enveloping algebras of the Lie algebras of the sheaf \Theta (i.e. \Omega(U) is the universal enveloping algebra of \Theta(U)).
Then \Omega contains \Theta as a subsheaf, and even as a direct factor (by the Poincaré–Birkhoff–Witt Theorem in characteristic 0). For all k, consider the sheaf of algebras \Omega_k=\Omega[t]/(t^{k+1}). For i\leqslant k, we have a map of sheaves of sets \exp_i\colon \Theta \to \Omega_k defined by \exp_i(\Theta) = \sum_p\frac{1}{M} \Theta^p t^p
(Campbell–Hausdorff). We can identify {\mathscr{Q}}_k with the sheaf of multiplicative subgroups of \Omega_k generated by the images of the \exp_i for i\leqslant k.
The proof of this proposition will not be given here. We denote by \Omega_k^\times the sheaf of multiplicative subgroups of \Omega_k consisting of the elements whose constant terms is 1. The commutative diagram of sheaves of (non-abelian) groups \begin{CD} 0 @>>> \Theta @>>> {\mathscr{Q}}_{k+1} @>>> {\mathscr{Q}}_k @>>> 0 \\@. @VVV @VVV @VVV @. \\0 @>>> \Omega @>>> \Omega_{k+1}^\times @>>> \Omega_k^\times @>>> 0 \end{CD} gives rise to a commutative diagram of sets \begin{CD} \mathrm{H}^1(V_0;{\mathscr{Q}}_k) @>{\delta_k}>> \mathrm{H}^2(V_0;\Theta) \\@VVV @VVV \\\mathrm{H}^1(V_0;\Omega_k^\times) @>>{\delta_k}> \mathrm{H}^2(V_0;\Omega) \end{CD} in which \mathrm{H}^2(V_0;\Theta) is a vector subspace of \mathrm{H}^2(V_0;\Omega).
2. Calculation of the primary obstruction
Now let a\in\mathrm{H}^1(V_0;\Theta), and let \alpha=(\alpha_{ij}) be a cocycle of the class a (the choice of the cocycle \alpha does not matter, since every cocycle that is cohomologous to a deformation cocycle is itself a deformation cocycle). The corresponding multiplicative cocycle in \Omega_1^\times is (1+\alpha_{ij}t). This cocycle can be lifted to \Omega_i^\times as the cochain (1+\alpha_{ij}t), and we have \begin{aligned} (1+\alpha_{ij}t) (1+\alpha_{jk}t) &= 1 + (\alpha_{ij}+\alpha_{jk})t + \alpha_{ij}\alpha_{jk}t^2 \\&= (1 + \alpha_{ik}t + \alpha_{ij}\alpha_{jk}t^2) \\&= (1+\alpha_{ik}t) \{1 + \alpha_{ij}\alpha_{jk}t^2). \end{aligned} Finally, let \delta_1 a=a\smile a where the cup product is taken in the sheaf of algebras \Omega.
Note that, if we denote by \bar{\smile} the cup product taken in the sheaf of algebras opposite to \Omega, i.e. defined on the level of cochains by (\alpha\bar{\smile}\beta)_{ijk}=\beta_{jk}\alpha_{ij}, we always have that a\bar{\smile}b=-b\smile a in cohomology.
Consequently, [a\smile a] = (a\smile a)-(a\bar{\smile}a) = 2a\smile a and \delta_1a=a\smile a=\frac12[a\smile a]. We thus recover, up to a factor of \frac12, the obstruction defined earlier in this talk.
3. Calculation of the secondary obstruction
Now suppose that a\smile a=0, so that we can find a cochain \beta=(\beta_{ij}) such that \delta\beta+\alpha\smile\alpha=0, i.e. \beta_{ik} = \beta_{ij}+\beta_{jk}+\alpha_{ij}\alpha_{jk}. Then (1+\alpha_{ij}t+\beta{ij}t^2) is a cocycle in \Omega_2^\times, and we can choose the cochain \beta to be a cocycle in {\mathscr{Q}}_2.
This cocycle can be lifted to \Omega_3^\times as the cochain (1+\alpha_{ij}t+\beta_{ij}t^2), and we have that \begin{aligned} &(1+\alpha_{ij}t+\beta_{ij}t^2) (1+\alpha_{jk}t+\beta_{jk}t^2) \\=\,\,& 1+ (\alpha_{ij}+\alpha_{jk})t + (\beta_{ij}+\beta_{jk}+\alpha_{ij}\alpha_{jk})t^2 + (\alpha_{ij}\beta_{jk}+\beta_{ij}\alpha_{jk})t^3 \\=\,\,& (1 + \alpha_{ik}t + \beta{ik}t^2) (1+ (\alpha_{ij}\beta_{jk}+\beta_{ij}\alpha_{jk})t^3). \end{aligned} The secondary obstruction of a is thus the cohomology class of the cocycle (\alpha_{ij}\beta_{jk}+\beta_{ij}\alpha_{jk})\in\operatorname{Z}^2(V_0;\Omega). This class depends on the choice of the cochain \beta: if we choose some other \beta'=\beta+\theta, where \Theta\in\operatorname{Z}^1(V_0;\Theta), then the cocycle is modified by \alpha\smile\theta+\theta\smile\alpha, and its class by an element of [a\smile\mathrm{H}^1(V_0;\Theta)]. We recover the Massey triple product (a,a,a) taken in the algebra \Omega, but with a slightly more restrictive indetermination.
We can try to calculate this secondary obstruction without leaving the sheaf \Theta, but the calculations are then much more complicated: we must take a cochain \beta=(\beta_{ij}) such that \delta\beta+\frac12[a\smile a]=0. Then the secondary obstruction of \alpha is the class of the cocycle [\alpha_{ij},\beta_{jk}] + \frac16[[\alpha_{ij},\alpha_{jk}],\alpha_ij+2\alpha_{jk}]. The calculation done in the sheaf of enveloping algebras \Omega can be generalised to obstructions of order r: we are led to determining, by induction, cochains \omega_r such that \begin{cases} \omega_1 = \alpha \\\delta\omega_r + \sum_{p+q=r}\omega_p\smile\omega_q = 0 \\1 + \sum_{1\leqslant p\leqslant r}\omega_p t^p \in \operatorname{C}^1(V_0;{\mathscr{Q}}_r) \end{cases}
4. Using spectral sequences
Let \varphi\colon V_0\to X be an arbitrary map, which gives rise to a spectral sequence of graded Lie algebras \mathrm{H}^\bullet(X;\mathbb{R}^\bullet\varphi\Theta) \Rightarrow \mathrm{H}^\bullet(V_0;\Theta). Let a \in \mathrm{H}^1(X;\varphi_*\Theta) \subset \mathrm{H}^1(V_0;\Theta). If the element -\frac12[a\smile a] \in \mathrm{H}^2(X;\varphi_*\Theta) = E_2^{2,0} is non-zero, but is the image under the differential \mathrm{d}_2 of the spectral sequence of an element b\in E_2^{0,1}, then the image of the secondary obstruction of a in E_\infty^{1,1} consists of the elements of the form [a,b]. In particular, if, for all b such that \mathrm{d}_2 b=-\frac12[a,a], we have that [a,b]\neq0, then the secondary obstruction is non-trivial.
However, if [a,b]=0 in E^{1,1}, then we can only say that the secondary obstruction comes from E_\infty^{2,0}, and if this group is non-zero, then we cannot conclude anything.
Proof. Let \alpha be a cocycle on V_0 representing the class a. The element b\in E_2^{0,1} can be represented by a cochain \beta = (\beta_{ij}) \in \operatorname{C}^1(V_0;\Theta) such that \delta\beta + \frac12[a\smile a] = 0. We thus obtain a cochain \beta' \in \operatorname{C}^1(V_0;\Omega) such that 1 + \alpha t + \beta' t^2 \in \operatorname{C}^1(V_0;{\mathscr{Q}}_2) by setting \beta'_{ij}=\beta_{ij}+\frac12\alpha_{ij}^2; this cochain satisfies \delta\beta'+\alpha\smile\alpha=0. But this new cochain represents, in the E_2^{0,1} term of the spectral sequence of the sheaf \Omega, the same element b as the cochain \beta, since it differs from it by a cochain that comes from X. The secondary obstruction is thus the class of the cocycle \alpha\smile\beta'+\beta'\smile\alpha, which represents in the E^{1,1} term of the spectral sequence the element [a,b].
This proposition allows us to construct non-trivial examples of secondary obstructions. Consider the group N of matrices of the form \begin{pmatrix} 1 & x & y \\0 & 1 & z \\0 & 0 & 1 \end{pmatrix} where x,y,z\in\mathbb{C}, and let Y=N/\Gamma, where \Gamma is the subgroup of N consisting of elements where x,y,z\in\mathbb{Z}+i\mathbb{Z}. Then Y is fibred over a complex torus of dimension two T^2\cong\mathbb{C}^2/\mathbb{Z}^4. We find non-trivial secondary obstruction elements in \mathrm{H}^1(V_0;\Theta), where V_0 is the product of Y with a projective line D. (We use the spectral sequence obtained by projecting onto T^2\times D). This variety has a “versal” deformation whose Zariski tangent space of the base B can be identified via the Spencer–Kodaira map \rho with \mathrm{H}^1(V_0;\Theta). Further, B has, at its base point b_0, a conic singularity of degree 3, whose equation is given by the secondary obstruction.
I do not know of any examples of non-trivial secondary obstructions on varieties V_0 that satisfy \mathrm{H}^0(V_0;\Theta)=0, but some very likely exist.