A ‘counterexample’ to Deligne’s construction

Posted on Categories Algebraic topology, Complex Geometry, Mathematics

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}) .

A few preliminary concepts to start. First, the fundamental group of the complement U = L \setminus X . It is a \mathbb{C}^{\times} -bundle over X , and hence the fundamental groups of U , X , and \mathbb{C}^{\times} are related to each other through an exact sequence:

\displaystyle \pi_{1}(\mathbb{C}^{\times}) \to \pi_{1}(U) \to \pi_{1}(X) \to 1.

This sequence is a portion of a long exact sequence relating all the homotopy groups. Such a sequence exists for any fibration. In this sequence, \pi_{1}(\mathbb{C}^{\times}) \cong \mathbb{Z} , and the generator 1 corresponds to a loop that goes once around the fiber. It maps to a loop l \in \pi_{1}(U) which is a central element of the group. In the picture above, I drew it as a purple loop.

Next, the residue of a connection. Let (P, \nabla) \in FC(L,X,SL(2, \mathbb{C})) be a connection. In a neighbourhood of a point of X , we can trivialize the bundle P and express the connection as follows:

\nabla = d + B(z,x) \frac{dz}{z} + \sum_{i}C_{i}(z,x)dx_{i}.

Here (z,x) are local coordinates on L , with z the coordinate on the fibre which vanishes along X , and B(z,x) and C_{i}(z,x) are holomorphic functions valued in \mathfrak{sl}(2, \mathbb{C}) , the matrices with zero trace. The residue of \nabla is the element

A = B(0,x) \in \mathfrak{sl}(2, \mathbb{C}).

The way I’ve stated it doesn’t quite make sense, since B(0,x) apparently depends on x , and on the choice of a trivialization of P . There is a more invariant definition of the residue which deals with all these issues, but all you need to know for now is that A is well-defined up to conjugation. The residue is an invariant of the connection, and it makes sense to fix it (i.e. its conjugacy class). So let FC_{A}(L,X,SL(2, \mathbb{C})) denote the full subcategory of connections whose residue is conjugate to A .

Now consider an element A \in \mathfrak{sl}(2, \mathbb{C}) . By conjugating it, we can put it into Jordan normal form:

gAg^{-1} = \begin{pmatrix} r+ is & t \\ 0 & -r-is \end{pmatrix},

where r and s are real. This leads to a well-defined decomposition A = a + ib + N , where

\displaystyle gag^{-1} = \begin{pmatrix} r & 0 \\ 0 & -r \end{pmatrix}, \ \ gbg^{-1} = \begin{pmatrix} is & 0 \\ 0 & -is \end{pmatrix}, \ \ gNg^{-1} = \begin{pmatrix} 0 & t \\ 0 & 0 \end{pmatrix}.

We will associate a bunch of subgroups of SL(2, \mathbb{C}) to the element A . First, we define the parabolic subgroup P(a) associated to a . If a = 0 , then P(a) = SL(2, \mathbb{C}) . If a \neq 0 , then it has a unique positive eigenvalue r > 0 , and a negative eigenvalue -r . The corresponding eigenspaces give a decomposition of \mathbb{C}^2 :

\mathbb{C}^2 = L_{r} \oplus L_{-r}.

The parabolic P(a) is defined to be the subgroup of elements g \in SL(2, \mathbb{C}) which preserve the positive eigenline: gL_{r} \subseteq L_{r} . Using the eigenspace decomposition as a basis of \mathbb{C}^2 , the parabolic consists of elements of the following form

\displaystyle \begin{pmatrix} \lambda & t \\ 0 & \lambda^{-1} \end{pmatrix}.

In this basis, the diagonal matrices define the centralizer of a : C(a) . The elements g \in P(a) which restrict to the identity on the positive line define the unipotent radical of P(a) :

U(a) = \{ g \in P(a) \ | \ g|_{L} = id \} = \{ \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \}.

You should check that U(a) is a normal subgroup of P(a) , that it has trivial intersection with C(a) , and that any element of P(a) can be written as a product of elements from these two subgroups. Therefore, the parabolic is a semidirect product of the two groups:

P(a) = U(a) \rtimes C(a),

and there is a canonical projection map \pi: P(a) \to C(a) . By the way, if a = 0, then C(a) = SL(2, \mathbb{C}) , and U(a) is trivial.

Okay, with all of this set-up out of the way, I can define a fairly simple category. First, given the element A \in \mathfrak{sl}(2, \mathbb{C}) , the exponential \exp(2 \pi i A) \in C(a) . Let \mathcal{O}_{A} \subseteq C(a) denote its conjugacy class. Now define the set

K_{A} = \{ F: \pi_{1}(U) \to P(a) \ | \ \pi(F(l)) \in \mathcal{O}_{A} \}.

Recall that l \in \pi_{1}(U) is the central element corresponding to the loop in the fibre of L . The parabolic group P(a) acts on K_{A} by conjugation. Hence we can define the following action groupoid

P(a) \ltimes K_{A}.

The objects of this category are the elements F \in K_{A} , and the morphisms are the pairs (g, F) \in P(a) \times K_{A} . More precisely, (g,F) is a morphism going between F and gFg^{-1} .

Now I can finally state the classification theorem, which is adapted from Theorem 5.2 of my paper.

Theorem. Let A \in \mathfrak{sl}(2, \mathbb{C}) . There is an equivalence of categories

FC_{A}(L,X,SL(2, \mathbb{C})) \cong P(a) \ltimes K_{A}.

Let’s look at a few simple examples. First, if A = 0 , then P(a) = C(a) = SL(2, \mathbb{C}) , \pi = id , and \mathcal{O}_{A} = \{id \} . In this case, K_{A} consists of homomorphisms F: \pi_{1}(U) \to SL(2, \mathbb{C}) satisfying F(l) = id . These homomorphisms factor through \pi_{1}(U)/\mathbb{Z} \cong \pi_{1}(X) . In other words, these are precisely the representations that are pulled back from homomorphisms F : \pi_{1}(X) \to SL(2, \mathbb{C}) . Hence, in this case we get

FC_{0}(L, X, SL(2, \mathbb{C})) \cong Rep(\pi_{1}(X), SL(2, \mathbb{C})) \cong Rep(\pi_{1}(L), SL(2, \mathbb{C})).

For the second isomorphism, we’ve used the fact that L retracts onto X and so has isomorphic fundamental group. This makes sense: looking back at the local expression for a connection, we see that if the residue vanishes, then the connection is actually smooth on all of L .

Next, let’s take A = \begin{pmatrix} r & 0 \\ 0 & -r \end{pmatrix}, with r > 0 . Then

P(a) = \{ \begin{pmatrix} \lambda & t \\ 0 & \lambda^{-1} \end{pmatrix} \} , \ \ C(a) = \{ \begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix} \} \cong \mathbb{C}^{*}.

And since C(a) is commutative, the conjugacy class \mathcal{O}_{A} consists of a single element

\mathcal{O}_{A} = \{ \begin{pmatrix} e^{2 \pi i r} & 0 \\ 0 & e^{-2 \pi i r} \end{pmatrix} \}.

The set K_{A} therefore consists of homomorphisms F: \pi_{1}(U) \to P(a) such that

F(l) = \begin{pmatrix} e^{2 \pi i r} & t \\ 0 & e^{-2 \pi i r} \end{pmatrix}.

Okay, back to the main question. Given our new description of the category of representations, our restriction functor R is simply given by forgetting the parabolic P(a) . More precisely, an object F \in K_{A} is sent to F , viewed now as a homormophism F: \pi_{1}(U) \to SL(2, \mathbb{C}) . So to provide a counterexample to the question, we need to give a representation F of \pi_{1}(U) , and then somehow argue that it could not have come from any of the categories P(a) \ltimes K_{A} . To simplify things, let’s assume that the line bundle is trivial L = X \times \mathbb{C} , so that U = X \times \mathbb{C}^{\times} , and \pi_{1}(U) = \pi_{1}(X) \times \mathbb{Z} . And to make our life even more simple, let’s postpone choosing the representation of \pi_{1}(X) , and focus first on the monodromy M = F(l). If this is going to work, M must centralize the representation of \pi_{1}(X) , and this is guaranteed if M lies in the centre of SL(2, \mathbb{C}) . This centre is given by

Z(SL(2,\mathbb{C})) = \{ id, -id \}.

Choosing M=id won’t work, because then we can always solve the problem with A = 0 . So lets take M = -id . No matter what A turns out to be, we must have \pi(M) = -id and \exp(2\pi i A) = -id . This implies that A is diagonalizable, and can be taken to be of the form

A = \begin{pmatrix} t & 0 \\ 0 & -t \end{pmatrix},

with t \in \mathbb{C} satisfying \exp(2 \pi i t) = -1 . Hence t \in \frac{1}{2} + \mathbb{Z} . In particular, t must be non-vanishing and real, and we may assume that it is positive. The group P(a) is therefore the group of elements preserving the line L_{t} . Hence we obtain the following result.

Theorem. Let F: \pi_{1}(X) \times \mathbb{Z} \to SL(2, \mathbb{C}) be a homomorphism such that F(l) = -id . If F is the monodromy of a flat SL(2, \mathbb{C}) connection on X \times \mathbb{C} with logarithmic singularities along X \times 0 , then there exists a line L \subset \mathbb{C}^2 which is preserved by F(p) , for all p \in \pi_{1}(X) . In particular, \mathbb{C}^2 is not an irreducible representation of \pi_{1}(X) . 

This theorem provides a way of constructing a counterexample. Namely, let \rho: \pi_{1}(X) \to SL(2, \mathbb{C}) be an irreducible representation, and define

F: \pi_{1}(X) \times \mathbb{Z} \to SL(2, \mathbb{C}), \qquad (p, n) \mapsto (-1)^n\rho(p).

Then the theorem says that F does not extend to a logarithmic connection on X \times \mathbb{C} .

It is easy to produce such irreducible representations. Here’s a fun example. Let B_{3} be the braid group on three strands. It has the following presentation:

B_{3} = \{ u, v \ | \ uvu=vuv \}.

This group has a representation into SL(2, \mathbb{C}) given as follows:

u \mapsto \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \qquad v \mapsto \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}.

To see this, just check that the relation uvu=vuv is satisfied. It’s also easy to see that this representation does not fix any line L \subset \mathbb{C}^2 . Indeed, the only line fixed by u is the span of (1,0) , and the only line fixed by v is the span of (0,1) .

I know of at least two ways of realizing B_{3} as the fundamental group of a complex manifold. First, let’s think of elements of B_{3} as braids on three strands. Namely, we have three bits of string, and we twist them up as follows.

A picture of an element of the braid group: it is a braid on three strands. Viewing this as a movie with time going upwards, it also describes the motion of 3 distinct points on the plane that are being permuted.

Viewing this as a movie where time goes upwards, this is the motion of three distinct points in the plane. In other words, this is a path in the space of triples of distinct points in the complex plane \mathbb{C} . Thinking of these points as the components of a vector in \mathbb{C}^3 , we can think of this as a path in the following space

Y = \{ (x, y, z) \in \mathbb{C}^3 \ | \ x \neq y, x\neq z, y \neq z \}.

A loop in this space corresponds to those braids where the points end up where they started. These are the pure braids, and hence \pi_{1}(Y) is the pure braid group. In order to get the full braid group, we need all permutations (x,y,z), (x,z,y), etc. to count as the same, so that any braid gives a loop. So we simply take the quotient by the permutation action

X = Y/S_{3}.

Then \pi_{1}(X) \cong B_{3} .

The second way of realizing B_{3} as a fundamental group comes from an alternate presentation:

B_{3} = \{ (x, y) \ | \ x^2 = y^3 \}.

This is the presentation of B_{3} as a knot group: it is the fundamental group of the complement of the trefoil knot in the three-dimensional sphere. It takes a bit of thinking to see this, but this knot complement is homotopic to the complement of the complex curve x^2 = y^3 in \mathbb{C}^2 . Hence,

B_{3} \cong \pi_{1}( \mathbb{C}^2 \setminus \{ x^2 = y^3 \} ).

In fact, there is yet another way of realizing B_{3} as a fundamental group. This paper proves that every finitely presented group can be realized as the fundamental group of a compact complex 3-manifold.

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