Flat parabolic vector bundles on elliptic curves

A.1 Parabolic vector bundles

Let C be a smooth irreducible projective complex curve and D=t1+⋯+tn a reduced divisor supported on n distinct points {t1,…,tn}⊂C. A rank 2 quasi-parabolic vector bundle on (C,D) is the data (E,𝐩), where E is a rank 2 holomorphic vector bundle over C and 𝐩={p1,…,pn} are given one-dimensional subspaces pk⊂Etk for k∈{1,…,n}. An isomorphism between quasi-parabolic vector bundles is, by definition, an isomorphism between underlying vector bundles preserving parabolic directions. A parabolic vector bundle is a quasi-parabolic vector bundle together with a collection of weights μ=(μ1,…,μn)∈[0,1]n. It allows us to introduce a notion of stability in order to introduce a good moduli space. Given a line bundle L⊂E, the μ-stability index of L is the real number

It is well known, see [16], that the moduli space of μ-semistable parabolic vector bundles with fixed determinant line bundle is a normal irreducible projective variety. The open subset of μ-stable parabolic bundles is smooth. We note that the stability index of L⊂E is zero if, and only if, the weights lie along the following hyperplane in [0,1]n:

Each one of these hyperplanes is called a wall. If we cut out [0,1]n by all possible walls, one gets in the complement of finitely many irreducible connected components, which are called chambers. In each chamber, any μ-semistable parabolic vector bundle is μ-stable, and the moduli space is constant, that is, it is independent of μ. Nevertheless, it can be empty. When we have two adjacent chambers separated by a wall, then there is a locus of μ-stable parabolic bundles in each chamber that became unstable when we cross the wall. Along the wall, we identify strictly semistable parabolic bundles (E,𝐩) and (E′,𝐩′) with gr⁢(E,𝐩)=gr⁢(E′,𝐩′), see [16, Section 4]. Over the projective line, the description of the moduli space of quasi-parabolic bundles has been done in [3].

A.2 Elementary transformations

In the construction of the moduli space of quasi-parabolic bundles, the determinant line bundle is fixed. Actually, up to twists and elementary transformations, we can choose the determinant bundle arbitrarily, for instance the trivial line bundle 𝒪C. Twists preserve the parity of the determinant and elementary transformations change it. We start by recalling what is an elementary transformation as well as its main properties. Given t∈C and a direction p⊂Et, the vector bundle E- is defined by the following exact sequence of sheaves:

where p appearing above is considered as a sky-scrapper sheaf. The new parabolic direction p-⊂E- is the kernel of the morphism E-→E. By identifying sections of E and E- outside of t, one obtains a birational bundle transformation

with center at p, which is an isomorphism outside t. We shall say that it is a negative elementary transformation. At a neighborhood of t, it can be described as follows. We can choose a local trivialization E|U≃U×ℂ2 such that

and elemt-:E|U⇢E-|U is given by

From the point of view of ruled surfaces, it corresponds to a flip with center at [p]∈ℙ⁢E, that is, a blow up at [p] followed by a contraction of the old fiber. If the direction p is contained in a line bundle L⊂E, then it is left unchanged and one obtains a line bundle L⊂E-. If L⊂E does not contain p, then we get L-⊂E, where

In addition, we have the following property:

When we perform an elementary transformation, the stability condition is preserved after an appropriate modification of weights. If (E,𝐩) is μ-stable and we perform an elementary transformation elemtk-, then (E-,𝐩-) is μ′-stable, where

If BunLμ⁢(C,D) denotes the moduli space of μ-semistable parabolic vector bundles with fixed determinant line bundle L, then elemtk- defines an isomorphism between moduli spaces

We can define a positive elementary transformationelemt+ as

where E+=E-⊗𝒪C⁢(t). It is the inverse of elemt-. As before, the stability condition is preserved by elementary positive transformations, with the same modification of weights.

A.3 Endomorphisms of quasi-parabolic vector bundles

The space of global endomorphisms of a rank 2 vector bundle as well as the automorphism group are well known, see for example [15, Theorem 1]. In this subsection, we study the space of traceless endomorphisms of a quasi-parabolic bundles over an elliptic curve.

Let (E,𝐩) be a quasi-parabolic bundle. We say that an endomorphism σ∈End⁢(E) fixes the parabolic structure 𝐩 if σ⁢(pk)⊂pk for all k=1,…,n. Let End⁢(E,𝐩) be the vector space of endomorphisms fixing the parabolic structure 𝐩 and End0⁢(E,𝐩) its subspace of traceless endomorphisms. We notice that we have a canonical decomposition

where Id∈End⁢(E,𝐩) is the identity. This follows from the following simple remark: if A is a 2×2 matrix, then

We recall that a quasi-parabolic bundle (E,𝐩) is decomposable if there exists a decomposition E=L1⊕L2 such that each parabolic direction is contained in L1 or L2. In this case, we write

A.4 The case of elliptic curves

In what follows we will determine the traceless endomorphisms of an indecomposable quasi-parabolic bundle over an elliptic curve. This will be useful to assure existence of logarithmic connections. Before that, we shall give one example.

Observe that, by Lemma A.2, one can find an embedding 𝒪C↪E passing through all but one parabolics over D1. In particular, given the decomposition D=D0+D1, there is a unique such parabolic bundle (E,𝐩) up to isomorphism.

Let E0 be the unique indecomposable rank 2 bundle, over an elliptic curve, with trivial determinant and having 𝒪C as maximal subbundle. It corresponds to the non-trivial extension defined by the exact sequence 0→𝒪C→E0→𝒪C→0.

A.5 Moduli space of connections

Let C be a smooth projective curve and let

be a reduced divisor on C, n≥1. We will fix the data in order to introduce the moduli space of connections. Firstly, let us fix a degree d line bundle L0 over C. We also set a local exponent ν∈ℂ2⁢n satisfying the Fuchs relation

and the generic condition ν1ϵ1+⋯+νnϵn∉ℤ for any ϵk∈{+,-}, to avoid reducible connections. Let ζ:L0→L0⊗ΩC1⁢(D) be any fixed rank 1 logarithmic connection on L0 satisfying

for all k=1,…,n. We denote by Conν⁢(C,D) the moduli space of triples (E,∇,𝐩), where

1. (E,𝐩) is a rank 2 quasi-parabolic vector bundle over (C,D) having L0 as determinant bundle,

2. ∇:E→E⊗ΩC1⁢(D) is a logarithmic connection on E with polar divisor D, having ν as local exponents and tr⁢(∇)=ζ,

3. two triples (E,∇,𝐩) and (E′,∇′,𝐩) are equivalent when there is an isomorphism between quasi-parabolic bundles (E,𝐩) and (E′,𝐩′) conjugating ∇ and ∇′.

Actually, in order to obtain a good moduli space, we need a stability condition. A triple (E,∇,𝐩) is called μ-stable (resp. μ-semistable) if for any ∇-invariant line bundle L⊂E we have

(see Definition A.1). But an invariant line bundle L would force a relation

which is obtained by applying the Fuchs relation to the restriction ∇|L. This contradicts our hypothesis on ν. Therefore under the generic condition on the local exponent all the connections arising in our moduli space are stable. It follows from [17, Theorem 3.5] that Conν⁢(C,D) is a quasi-projective variety.

A priori, Conν⁢(C,D) depends on the choice of L0. But up to twists, we can go into either the even case L0=𝒪C or the odd case L0=𝒪C⁢(t). In fact, given a rank 1 logarithmic connection η:L→L⊗ΩC1⁢(D) with local exponents (k1,…,kn), we can define a twisting map

where ν′=(ν1±+k1,…,νn±+kn). Such a map is an isomorphism between moduli spaces, in particular our moduli space depends only on the differences νk+-νk-. In the even case, we can assume that (L0,ζ)=(𝒪C,d), where d means the trivial rank 1 connection. In the odd case, one may suppose that (L0,ζ)=(𝒪C⁢(t),d-d⁢xx-t).

Now let us deal with elementary transformations. When we perform a transformation elemtk-:(E,𝐩)⇢(E-,𝐩-), the new connection ∇- on E- has local exponents

and the other νj, j≠k, are left unchanged. Finally, we can go from the odd to the even case by performing one negative elementary transformation elemtn-.

In the case we are interested in, C is supposed to be an elliptic curve and D=t1+t2. For computation, we can assume that C⊂ℙ2 is the smooth projective cubic curve

with λ∈ℂ, λ≠0,1. By the above digression one can set L=𝒪C⁢(w∞), w∞=(0:1:0). As local exponents, we can take

We note that for each k∈{1,2}, the condition νk+=νk- is equivalent to νk=0.