Deformation of a vector bundle and cocycles

Question

This question is motivated by the question Tangent Space to Moduli Space of Vector Bundles on Curve which is about the computation of first-order deformations of a vector bundle, A.K.A, the tangent space at a rational point of the moduli space of vector bundles over a curve. It is well known that the first order deformations of a vector bundle $E$ are given by the elements of $H^1(C,End(E))$, but I don't quite understand the proof. By definition, we have a covering $\{U_{i}\}$ of $C$ where $E$ trivializes, and in addition, we have transition maps $$g_{ij}:U_{i}\cap U_{j}\rightarrow Gl(r)$$ satisfying the cocycle condition. And the big trick comes next where they said that the deformations of $E$ (vector bundles over $C_{\epsilon}$ being $\epsilon^2=0$ ) are given by transition functions of the form $g_{ij}(1+\epsilon a_{ij})$ with $a_{ij}\in \Gamma(U_{i}\cap U_{j}, \mathfrak{Gl}(r))$ being $\mathfrak{Gl}(r))$ the Lie algebra of $Gl(r)$. Can anybody explain it to me with patience this last part. I really want to understand it.

Answer 1

Question: "Can anybody explain it to me with patience this last part. I really want to understand it."

Answer: If $C$ is a projective curve over a field $k$ and $E$ is a coherent $\mathcal{O}_C$-module, you may (following Serenesi "Deformations of algebraic schemes") define the Quot scheme $Q:=Quot^C_E$, parametrizing coherent quotients $f:E \rightarrow F$ of $E$. Let $K:=ker(f)$. The tangent space of $Q$ at $(F,f)$ is given by the following formula:

$$T_{[f]}(Quot^C_E):=H^0(C, Hom_{\mathcal{O}_C}(K,F)).$$

The obstruction space $Ob_{[f]}(Quot^C_E)$ is given by the following formula:

$$Ob_{[f]}(Quot^C_E) \cong Ext^1_{\mathcal{O}_C}(K,F).$$

If $E$ and $F$ are locally trivial and of finite rank it follows $K$ is locally trivial of finite rank and

$$Ob_{[f]}(Quot^C_E) \cong Ext^1_{\mathcal{O}_C}(K,F) \cong Ext^1_{\mathcal{O}_C}(\mathcal{O}_C, K^*\otimes F) \cong $$

$$H^1(C, K^* \otimes F).$$

Hence if $E,F$ are of finite rank and locally trivial you get the tangent space

$$T_{[f]}(Quot^C_E):=H^0(C, K^* \otimes F)$$

and the obstruction space

$$Ob_{[f]}(Quot^C_E) \cong H^1(C, K^*\otimes F).$$

As I recall it: There is a detailed proof in the above mentioned book but it takes several pages. The book gives an introdution to the Hilbert and Quot schemes in the "language of schemes". Maybe this helps - a book is more reliable than online notes.

Note: In your case you should try to construct your moduli space $M_C$ using the quot-scheme. You will need a locally trivial (or coherent) sheaf $F$ on $C$ such that your vector bundles $[E]\in M_C$ are quotients of $F$. Then the above formula(s) give the tangent and obstruction space. If $E=K$ is finite rank locally trivial, it follows $E^*\otimes E \cong End_{\mathcal{O}_C}(E)$ hence

$$Ob_{[f]}(Quot^C_F)\cong H^1(C, E^*\otimes E) \cong H^1(C, End_{\mathcal{O}_C}(E)).$$

Hence you must construct a bundle $F$ and a quotient $f: F \rightarrow E$ with $ker(f)=E$.

Example: One choice is $F:=E \oplus E$. There is a canonical surjective map onto $E$ with kernel $E$. Hence

$$M_C := Quot^C_{E\oplus E}.$$

Any coherent sheaf $F\in Ext^1_{\mathcal{O}_C}(E,E)$ can be used.