Tangent Vectors are 1st Order Deformations

This blog tries to explain the following philosophy:

1. Parametrized curves at some object are just first order deformations.
2. Tangent vectors are just infinitesimal deformation of the 1st order deformations.
3. The obstructions of the 1st order deformations are just singularities of the moduli space of that object (hence they are contained in some cohomology groups).

It seems trivial that tangent vectors can be viewed by 1st order deformations, but this opinion will be very useful in studying the tangent space of some moduli spaces.

1. Tangent Space of Some Moduli Spaces

For example, let $\mathcal{M}$ be the moduli space of some special vector bundles over some manifold $X$ , for $E\in\mathcal{M }$ , what is $T_E\mathcal{M}$ ?

First, notice that the bundle $E$ can be determined by clutching functions $\{g_{ij}\}\in H^1(X,\mathcal{GL})$ for some Leray cover $\{U_i\}$ of $X$ , the 1st deformation of $E$ can be write by

$\tilde{g}_{ij}=g_{ij}(1+\epsilon a_{ij})$

where $(1+\epsilon a_{ij})\in k[\epsilon]/\langle\epsilon^2\rangle$ , here $k=\mathcal{E}nd(E)(U_i\cap U_j)$ , since $\tilde{g}_{ij}$ ‘s are also clutching functions, hence they satisfied cocycle condition:

$\tilde{g}_{ij}\tilde{g}_{jk}=\tilde{g}_{ik},\quad \tilde{g}_{ij}=\tilde{g}_{ji}^{-1}$

Also observe that $g_{ij}(1+\epsilon a_{ij})$ and $(1+\epsilon a_{ij})g_{ij}$ must define the same deformation, hence we can deduce that

$a_{ij}+a_{jk}-a_{ik}=0$

That just implies $\{a_{ij}\}$ is the cocycle in sheaf cohomology $H^1(X;\mathcal{E}nd(E))$ , and the tangent vectors are just infinitesimal deformation $d\tilde{g}_{ij}/d\epsilon$ . $\blacksquare$

Similarly, if $\mathcal{M}$ is the moduli space of some certain connections, what is the tangent space $T_{(E,\nabla)}\mathcal{M}$ at some connection $(E,\nabla)$ ? In fact, it is

$T_{(E,\nabla)}\mathcal{M}=H^1(X;\mathcal{E}nd_{fl}(E))$

where $\mathcal{E}nd_{fl}(E)$ means the sheaf of $\tilde{\nabla}-$ flat sections of $\mathcal{E}nd(E)$ , and $\tilde{\nabla}$ is the induced connection on the adjoint bundle $\mathrm{End}(E)$ .

To see this, a connection $(E,\nabla)$ can be expressed by $(g_{ij}, A_i)$ , where $g_{ij}$ ‘s are the transition functions of the bundle $E$ subordinate to some Leray cover, $A_i$ ‘s are connection matrices of $\nabla$ subordinate to the same cover. The the 1st order deformation of $(E,\nabla)$ can be expressed by

$(g_{ij}(1+\epsilon a_{ij}), A_i(1+t\nabla_i))$

However, the deformation of the connection can be viewed by two ways:

$g_{ij}(1+\epsilon a_{ij})A_i(1-\epsilon a_{ij})g_{ij}^{-1}+(d(g_{ij}(1+\epsilon a_{ij})))(1-\epsilon a_{ij})g_{ij}^{-1}=g_{ij}A_i(1+t\nabla_i)g_{ij}^{-1}+(dg_{ij})g_{ij}^{-1}$

After a brief calculation, we obtained

$\nabla_i=0, [A_i,a_{ij}]=0$

which is exactly what we claimed. $\blacksquare$

2. Singularities of Some Moduli Spaces.

Let’s see for example the moduli space of some bundles $\mathcal{M}$ , what are the singularities of $\mathcal{M}$ ?

Geometrically, the singularities are the points which will forbid doing the 1st order deformation along some directions, that is to say, something will obstruct you to do the deformation. How to characterize those obstructions?

First of all, let’s try to write down all possible 1st order deformations at $E=\{g_{ij}\}\in\mathcal{M}$ , let $\dim H^1(X;\mathcal{E}nd(E))=N$ , we will need $N$ deformation parameters: $\epsilon_1,...,\epsilon_N$ , hence all 1st order deformations can be write by:

$\begin{aligned}G_{ij}:=g_{ij}\left(1+\sum_{k=1}^N\epsilon_k a_{ij}^{(k)}\right)\end{aligned}$

where the cocycles $\{a_{ij}^{(k)}\}_{k=1}^N$ forms the basis of $H^1(X;\mathcal{E}nd(E))$ , and the tangent vectors are just infinitesimal deformations $\partial G_{ij}/\partial \epsilon_k$ .

However, these $G_{ij}$ may not always a cocycle, by a brief calculation, we can find a cocycle $o_{ijk}\in H^2(X;\mathcal{E}nd(E))$ such that

$\begin{aligned}G_{ij}G_{jk}G_{ki}=1+\sum_{\nu,\mu} o_{ijk}\epsilon_{\nu}\epsilon_{\mu}\end{aligned}$

The cocycle $\{o_{ijk}\}$ is exactly the obstruction of the deformation. So, $E$ is a smooth point of $\mathcal{M}$ , if $H^2(X;\mathcal{E}nd(E))=0$ .

Can we give a more detailed description?

Notice that from the asymptotic expansion of determinant in linear algebra, we have

$\begin{aligned}\mathrm{det} (G_{ij}G_{jk}G_{ki})=1+\sum_{\nu,\mu}\mathrm{Tr}(o_{ijk})\epsilon_{\nu}\epsilon_{\mu}\end{aligned}$

Therefore, the trace of obstruction $\{\mathrm{Tr}(o_{ijk})\}$ obstructs the 1st order deformation of the line bundle $\det E$ . Note that, the obstruction of deformation of $\det E$ lies in $H^2(X,\mathcal{O}_X)$ (the endomorphism bundle of any line bundle must be trivial).

However, there are no obstructions in the deformation of a line bundle $L\in H^1(X,\mathcal{O}_X^*)$ , since any deformation $\alpha\in H^1(X;\mathcal{E}nd(L))=H^1(X,\mathcal{O}_X)$ can be integrated by $\exp\alpha\in H^1(X,\mathcal{O}_X^*)$ (although this is Mukai’s explanation, but I think, the main reason is $\mathcal{O}_X^*$ is a sheaf of Abelian groups, hence obstruction can be cancellated), so we have: