Complex Geometry – Francis Bischoff

Let $L \to X$ be a holomorphic line bundle over a complex manifold $X$ . The total space of this bundle is a complex manifold in its own right, and it contains the smooth hypersurface $X$ , which is embedded as the set of zero vectors. Let $U = L \setminus X$ denote the complement.

Let $G$ be a complex Lie group. In this post we will study principal $G$ -bundles $P \to L$ that are equipped with connections $\nabla$ which are flat and have logarithmic singularities along $X$ . Denote the category of these connections $FC(L,X,G)$ . Given any connection $(P, \nabla) \in FC(L,X,G)$ , we can restrict it to $U$ , where it defines a non-singular flat connection. By the Riemann-Hilbert correspondence, $\nabla|_{U}$ is equivalent to a homomorphism

$F : \pi_{1}(U) \to G.$

In fact, we get a ‘restriction functor’

$R: FC(L,X,G) \to Rep(\pi_{1}(U), G).$

Question : Does every $G$ -representation of $\pi_{1}(U)$ come from an object of $FC(L,X,G)$ ?

When the group $G = GL(m, \mathbb{C})$ , Deligne’s construction answers this question in the affirmative. In a previous blog post, I explained this construction, as well as some of the required background on flat logarithmic connections. A key tool in this construction is the fact, explained in another blog post, that a set-theoretic logarithm determines canonical matrix logarithms. This fact about $GL(m, \mathbb{C})$ is not shared by other groups, and hence Deligne’s construction does not extend. The purpose of this post is to give a counterexample to the Question when $G = SL(2, \mathbb{C})$ .

In order to construct the counterexample, I will use a description of the category $FC(L,X,G)$ that I provided in my paper. This applies in the case that $G$ is complex and reductive, but for this post, I will stick to the case of $SL(2, \mathbb{C})$ .

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In this post I will discuss a construction, due to Deligne, for extending a flat vector bundle to a flat logarithmic vector bundle. The setting of the construction is as follows: let $X$ be a smooth complex manifold, and let $Y \subseteq X$ be a divisor (i.e. a codimension 1 subvariety). Then given a branch of the logarithm $\tau$ (it is in fact sufficient to choose a ‘set theoretical’ logarithm) Deligne’s construction gives a canonical way of extending a flat holomorphic vector bundle on the complement of $Y$ to a holomorphic vector bundle on $X$ with a flat logarithmic connection having poles along $Y$ :

Theorem. Let $(F,\nabla)$ be a holomorphic vector bundle with flat connection on $X \setminus Y$ . There exists a unique extension $(G,\tilde{\nabla})$ to all of $X$ such that $\tilde{\nabla}$ has logarithmic singularities along $Y \subseteq X$ and the eigenvalues of the residue of $\tilde{\nabla}$ lie in the image of $\tau$ .