Deligne’s construction for extending connections.

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 .

1. Preliminaries on connections.

Throughout let {X} be a complex manifold, and {F \rightarrow X} a holomorphic vector bundle.

Definition.

  • A connection on {F} is a {\mathbb{C}_{X}}-linear map \displaystyle \nabla : F \rightarrow F \otimes \Omega_{X}^{1},
    satisfying the Liebniz rule: for every function {\phi \in \mathcal{O}_{X}} and section {f \in F} \displaystyle \nabla(\phi f) = d \phi \otimes f + \phi \nabla f.
  • Given a vector field {V \in \mathcal{T}_{X}}, we can hook it into the 1-form part of the connection to get a {\mathbb{C}}-linear map {\nabla_{V} : F \rightarrow F}. The connection is flat if for all vector fields {V,U \in \mathcal{T}_{X}}\displaystyle [ \nabla_{V}, \nabla_{U}] = \nabla_{[V,U]}.

Remark. By hooking vector fields into the connection, we see that a connection defines an action of the tangent sheaf of {X} on the vector bundle :  \nabla : \mathcal{T}_{X} \to End_{\mathbb{C}}(F). The condition that {\nabla} be flat says simply that this is a morphism of sheaves of Lie algebras. In other words, flat connections are representations of the Lie algebroid {\mathcal{T}_{X}}.

Locally, we can pick a frame for {F}, allowing us to write a connection as

\displaystyle \nabla = d + \omega,

for a matrix valued 1-form {\omega}. More precisely let {f_{1}, ..., f_{m}} be a local frame. Then {\nabla f_{j} = \sum_{i} \omega_{ij} f_{i}}, for 1-forms {\omega_{ij}}. Then writing a general section in terms of the frame as a column vector {\phi} we have

\displaystyle \nabla \phi = d\phi + \omega \phi,

where {\omega = (\omega_{ij})_{ij}}.

It turns out that, despite being defined in terms of a complex structure on {X} and a holomorphic structure on {F}, flat connections are purely topological objects. This is the content of the following theorem:

Theorem. (Riemann-Hilbert Correspondence) There is an equivalence of categories between the category of holomorphic flat connections and the category of finite dimensional representations of the fundamental group of {X}:

\displaystyle \text{Rep}(\mathcal{T}_{X}) \longleftrightarrow \text{Rep}(\pi_{1}(X)).

The equivalence is given by sending a flat connection to its monodromy representation.

Sketch of Proof. Given a bundle with flat connection {(F, \nabla)}, we use the flat sections (i.e. {f \in F} such that {\nabla f = 0}) to identify nearby fibers of {F}. This allows us to define a parallel transport operator {P}, which associates a linear isomorphism to each homotopy class of paths:

\displaystyle [\gamma] \mapsto P_{\gamma} : F_{\gamma(0)} \xrightarrow{\sim} F_{\gamma(1)}.

Picking a base-point {x \in X} and applying the parallel transport operator to the loops based at {x} then gives the monodromy representation:

\displaystyle P|_{\pi_{1}(X,x)} : \pi_{1}(X,x) \rightarrow Gl(F_{x}).

Remark. Using the technology of Lie groupoids, this sketch can be turned into a proof: as argued above, a flat connection is a representation of the tangent algebroid. So, using an analogue of Lie’s second theorem for Lie groupoids/algebroids, it can be integrated to a representation of its source simply connected Lie groupoid, which in this case is the fundamental groupoid of {X}

\displaystyle \Pi_{1}(X) := \{ \text{homotopy classes of paths in } X\}

As it turns out, this representation is nothing but the parallel transport operator {P} defined above. Finally, since {\Pi_{1}(X)} is transitive, it is Morita equivalent to the fundamental group of {X} (its isotropy group), and therefore we can restrict the parallel transport operator to this group without losing any information. This is summarized in the following two equivalences

\displaystyle \text{Rep}(\mathcal{T}_{X}) \underset{\text{Lie}}{\longleftrightarrow} \text{Rep}(\Pi_{1}(X)) \underset{\text{Morita}}{\longleftrightarrow} \text{Rep}(\pi_{1}(X)).

The technology of Lie groupoids guarantees that both arrows are equivalences. This perspective is taken up in the following paper.

2. Connections with logarithmic poles.

Let {Y \subset X} be a divisor which is either smooth or has normal crossing singularities (i.e. it is locally a union of coordinate hyperplanes: there exist coordinates {x_{1}, ..., x_{n}} such that {Y = \{x_{1}...x_{p} = 0 \}}). Let {Y_{i}} denote the irreducible components of {Y}.

Definition.

  • The sheaf of logarithmic tangent vectors {\mathcal{T}_{X}(-\log Y)} is the subsheaf of {\mathcal{T}_{X}} consisting of vector fields on {X} which are tangent to {Y} along {Y}. This sheaf is locally free and hence defines a vector bundle {T_{X}(-\log Y)}. Note also that it inherits a bracket on its sections from the Lie bracket of vector fields. Hence it defines a Lie algebroid, with anchor map given by the inclusion of sheaves.
  • Dual to this is the sheaf of 1-forms with logarithmic poles along Y \displaystyle \Omega_{X}^{1}(\log Y) := (\mathcal{T}_{X}(- \log Y))^{\vee}.

Given local coordinates {x_{1}, ..., x_{n}} such that {Y = \{ x_{1} = 0 \}} we have

\displaystyle \mathcal{T}_{X}(-\log Y) = \langle x_{1} \partial_{x_{1}}, \partial_{x_{2}}, ... \rangle, \ \Omega_{X}^{1}(\log Y) = \langle \frac{dx_{1}}{x_{1}}, dx_{2}, ... \rangle.

Proposition. Suppose {Y} is smooth. The sheaf of 1-forms with logarithmic poles lies in the following short exact sequence

\displaystyle 0 \rightarrow \Omega_{X}^{1} \rightarrow \Omega_{X}^{1}(\log Y) \overset{\text{Res}}{\rightarrow} \mathcal{O}_{Y} \rightarrow 0,

where {\text{Res} : \Omega_{X}^{1}(\log Y) \rightarrow \mathcal{O}_{Y}} is the residue map, given locally by

\displaystyle \text{Res}(a_{1} \frac{dx_{1}}{x_{1}} + a_{2} dx_{2} + ... ) = a_{1}|_{Y}.

Remark. In the case that {Y} is a divisor with normal crossings we can define a residue map {\text{Res}_{i}} for each component {Y_{i}}.

Definition.

  • A connection on {F \rightarrow X} with logarithmic poles along {Y} is a {\mathbb{C}_{X}}-linear map \displaystyle \nabla : F \rightarrow F\otimes \Omega_{X}^{1}(\log Y)  satisfying the Liebniz rule. The notion of flatness still makes sense in this context. Note that a flat connection with logarithmic poles along {Y} defines a representation of the logarithmic tangent sheaf.
  • The residue of a logarithmic connection {\nabla} along {Y} is the endomorphism {\text{Res}{\nabla} \in End_{\mathcal{O}_{Y}}(F|_{Y})} defined by composing the connection with the residue on logarithmic forms: \displaystyle F \overset{\nabla}{\rightarrow} F \otimes \Omega_{X}^{1}(\log Y) \overset{\text{Res}}{\rightarrow} F \otimes \mathcal{O}_{Y}.

Remark. When {Y} is a normal crossing divisor we define the residue for each component {Y_{i}}.

Given coordinates for {X} and a choice of frame for {F} as above we can write a logarithmic connection as

\displaystyle \nabla = d + \frac{M_{1}}{x_{1}}dx_{1} + \sum_{i = 2}^{n} M_{i}dx_{i},

for holomorphic matrix-valued functions {M_{i}}. The residue for this connection is given by {M_{1}|_{Y}}.

3. Deligne’s construction for extending connections

Consider the exponential sequence

\displaystyle 0 \rightarrow \mathbb{Z} \rightarrow \mathbb{C} \overset{e^{2 \pi i }}{\rightarrow} \mathbb{C}^{\times} \rightarrow O,

and choose a set-theoretical splitting {\tau : \mathbb{C}^{\times} \rightarrow \mathbb{C}}. Then we have the following theorem which is due to Deligne:

Theorem. Let {F \rightarrow (X \setminus Y)} be a holomorphic vector bundle, and {\nabla} a holomorphic flat connection on {F}. Then there exists a holomorphic vector bundle with flat logarithmic connection {(G,\tilde{\nabla})} extending {(F, \nabla)} to {X}, unique up to unique isomorphism, such that

  • {\tilde{\nabla}} has logarithmic poles along {Y},
  • the eigenvalues of the residues of {\tilde{\nabla}} lie in the image of the splitting {\tau}.

Proof. This proof essentially follows the chapter by Malgrange in the book Algebraic D-modules.

We begin with a simplification: by the uniqueness part of the theorem, it only needs to be proven locally: once extensions are constructed on local neighbourhoods there is a unique way of gluing them together to get a global extension. So assume that {X} is a polydisc:

\displaystyle X = \{ (x_{1},...,x_{n}) \ | \ |x_{i}| < r_{i} \} = D_{r_{1}} \times ... \times D_{r_{n}},

and that {Y = \{ x_{1}...x_{p} = 0 \}}, with ith component {Y_{i} = \{ x_{i} = 0 \}}. The proof now proceeds in two parts: existence and uniqueness.

3.1. Existence

{(F, \nabla )} is a holomorphic flat connection on {X \setminus Y} and so by the Riemann-Hilbert correspondence it is determined by it’s monodromy representation. First note that since

\displaystyle X \setminus Y = (D_{r_{1}} \setminus \{ 0 \}) \times ... \times (D_{r_{p}} \setminus \{ 0 \})\times D_{r_{p+1}} \times ... \times D_{r_{n}},

it follows that {\pi_{1}(X \setminus Y) \cong \mathbb{Z}^{p}}. Let us make some choices in order to exhibit this isomorphism: choose a point {a \in X \setminus Y}. Then the fundamental group is generated by {p} loops, and we take the ith loop to be the loop {\gamma_{i}} which starts at {a} and goes around {Y_{i}} counterclockwise in the plane {\{x_{1} = a_{1}, ..., x_{i-1} = a_{i-1}, x_{i+1} = a_{i+1}, ..., x_{n} = a_{n}\}}. The monodromy representation is given by taking the parallel transport around each of these loops. Choosing an identification {\mathbb{C}^{m} \overset{\sim}{\rightarrow} F_{a}}, this is given by {p} mutually commuting invertible matrices {C_{1}, ..., C_{p} \in Gl(\mathbb{C}^m)}. In order to extend {(F, \nabla)} we need to construct a logarithmic connection on the trivial bundle {G = \mathbb{C}^{m} \times X} which has the same monodromy when restricted to {X \setminus Y} (since this will allow the map {\mathbb{C}^{m} \overset{\sim}{\rightarrow} F_{a}} to be extended to a well-defined isomorphism of flat bundles {G|_{X \setminus Y} \overset{\sim}{\rightarrow} F}).

To this end we make use of the theorem on matrix logarithms which states that there are unique matrices {\Gamma_{1}, ..., \Gamma_{p} \in End(\mathbb{C}^{m})} such that

  1. {\exp(-2 \pi i \Gamma_{i}) = C_{i}},
  2. the eigenvalues of {\Gamma_{i}} lie in the image of {\tau}.

Note that the matrices {\Gamma_{i}} mutually commute since the {C_{i}} do.

Now define the connection on {G} as follows

\displaystyle \tilde{\nabla} = d + \sum_{i = 1}^{p} \Gamma_{i} \frac{dx_{i}}{x_{i}},

which has logarithmic poles along {Y}. The residue along the component {Y_{i}} is given by

\displaystyle \text{Res}_{i}\tilde{\nabla} = \Gamma_{i},

which by hypothesis has its eigenvalues in the image of {\tau}. A fundamental solution to {\tilde{\nabla}} (i.e. a matrix-valued function whose columns are flat sections; this can be used to define the parallel transport operator) is given by

\displaystyle \Phi(x) = \exp(-\sum_{i = 1}^{p} \Gamma_{i} \log(x_{i})) = \Pi_{i = 1}^{p} x_{i}^{-\Gamma_{i}},

and so the monodromy around loop {\gamma_{i}} is given by

\displaystyle \Phi((a_{1},..., a_{i}e^{2 \pi i}, ...)) \Phi(a)^{-1} = \exp(-2 \pi i \Gamma_{i}) = C_{i}.

Hence {(G, \tilde{\nabla})} is the desired extension.

3.2. Uniqueness

Let {(G', \tilde{\nabla}')} be another connection satisfying the properties of the theorem. We want to show that there is a unique isomorphism between this flat vector bundle and the extension {(G, \tilde{\nabla})} constructed above. Since both {(G, \tilde{\nabla})} and {(G', \tilde{\nabla}')} extend {(F,\nabla)} from {X \setminus Y} to {X} they are already identified over {X \setminus Y}:

\displaystyle S : (G, \tilde{\nabla})|_{X \setminus Y} \overset{\sim}{\rightarrow} (G', \tilde{\nabla}')|_{X \setminus Y}.

Hence an isomorphism between these flat bundles must be an isomorphism extending {S} to {X}, and this is clearly unique if it exists. Therefore we only need to show that we can holomorphically extend {S} (and {S^{-1}}) to {X}. We will only show that {S} extends since the situation with {S^{-1}} is identical.

We note a further simplification: if we can show that {S} extends holomorphically to {Y_{i} \setminus \cup_{j \neq i} Y_{j}} then by Hartogs’ theorem it automatically extends to {Y_{i}}. Hence it suffices to assume that {Y} has a single component: {Y = \{ x_{1} = 0 \}}.

Now in order to show that {S} extends holomorphically we will derive a differential equation that it must satisfy. Recall what it means for {S} to be a morphism of flat bundles: {S \tilde{\nabla} = \tilde{\nabla}' S}. Writing {\tilde{\nabla} = d + \omega} and {\tilde{\nabla}' = d + \omega'} this condition can be expressed as follows:

\displaystyle dS = S \omega - \omega' S. \ \ \ \ \ (1)

In a neighbourhood of a point on the divisor we can write

\displaystyle \omega' = M_{1}' \frac{dx_{1}}{x_{1}} + \sum_{i = 2}^{n} M_{i}'dx_{i}

for holomorphic matrix valued functions {M_{i}'}. Plugging this, and the expression for {\omega}, into equation 1 and taking the coefficient of {dx_{1}} we get

\displaystyle x_{1} \frac{\partial S}{\partial x_{1}} = S \Gamma_{1} - M_{1}' S. \ \ \ \ \ (2)

Taking ({l^2}) norms of both sides of this equation, and noting that since {\Gamma_{1}} and {M_{1}'} are holomorphic they are bounded in some neighbourhood of every point on the divisor, we get

\displaystyle |x_{1}| ||\frac{\partial S}{\partial x_{1}}|| \leq ||S \Gamma_{1} || + ||M_{1}' S|| \leq C ||S||,

for some constant {C}. For points away from the divisor we can write {x_{1} = r e^{i \theta}} and we have

\displaystyle |\frac{\partial S_{ij}}{\partial r}| = |\frac{\partial S_{ij}}{\partial x_{1}}| \leq ||\frac{\partial S}{\partial x_{1}}|| \leq \frac{C}{r} ||S||,

where \displaystyle S_{ij} is the \displaystyle ij entry of the matrix \displaystyle S . By the fundamental theorem of calculus we have for {0 < r < r_{0}}

\displaystyle S_{ij}(r e^{i \theta}) = S_{ij}(r_{0} e^{i \theta}) + \int_{r_{0}}^{r} \frac{\partial S_{ij}}{\partial s} ds,

and taking absolute values and using the above inequality gives

\displaystyle |S_{ij}(r e^{i \theta})| \leq |S_{ij}(r_{0} e^{i \theta})| + \int_{r_{0}}^{r} \frac{C}{s} ||S(s e^{i \theta})|| ds.

Finally, if we add up the contributions from all components of {S} we end up with the following inequality for the norm of {S}:

\displaystyle ||S(r e^{i \theta}) || \leq K_{1} + \int_{r}^{r_{0}} \frac{K_{2}}{s} ||S(s e^{i \theta})|| ds,

for positive constants {K_{1}, K_{2}}. We can now apply the integral form of Grönwall’s inequality to deduce that there exists a positive constant {K} and a positive integer {N} such that for {|x_{1}|} small enough we have

\displaystyle ||S(x)|| \leq K |x_{1}|^{-N}.

By a version of the Riemann removable singularities theorem for many variables this implies that {S} is meromorphic at {Y}. Hence we can write a series expansion for {S} as follows:

\displaystyle S(x) = \sum_{k = q}^{\infty} x_{1}^{k} S_{k}(x_{2},...,x_{n}),

where {S_{k}} are holomorphic matrix-valued functions on {Y} and {S_{q}} is non-zero (i.e. we take {q} to be the smallest integer such that {S_{q} \neq 0}). Plugging this expression for {S} back into equation 2 and taking the coefficient of {x_{1}^{q}} yields the following equation

\displaystyle q S_{q} = S_{q} \Gamma_{1} - M_{1}'|_{Y} S_{q},

which, upon rearranging and observing that {\Gamma_{1} = \text{Res} \tilde{\nabla}} and {M_{1}'|_{Y} = \text{Res} \tilde{\nabla}'}, becomes

\displaystyle (\text{Res} \tilde{\nabla}' + q I) S_{q} = S_{q} \text{Res} \tilde{\nabla}.

If {S_{q}} was invertible then {\text{Res} \tilde{\nabla}} and {(\text{Res} \tilde{\nabla}' + q I)} would be conjugate and hence have the same spectrum. At present we only know that {S_{q} \neq 0} and so we are only guaranteed that they have an eigenvalue in common. But both residues have their eigenvalues in the image of {\tau}, and so this implies that {q = 0}. This shows that {S} extends holomorphically. {\Box}

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