PROJECTIVE STRUCTURES ON A RIEMANN SURFACE, II

INDRANIL BISWAS AND A. K. RAINA

1. Introduction

For a compact Riemann surface X, let Δ be the diagonal in X × X, and p_{i}, i = 1, 2, be the projection of X × X onto the i-th factor. We denote by Δ_{n} the n-th order infinitesimal neighborhood of Δ in X ×X, defined by the non-reduced divisor (n + 1)Δ, and by L a square-root of the holomorphic tangent bundle T_{X} of X.

We recall that a projective structure on X is an equivalence class of coverings by holomorphic coordinate charts such that all the transition functions are Möbius transformations. Deligne has shown in [De] that a projective structure can also be defined as the extension of a natural embedding of Δ_{2} in a certain projective bundle P_{tg} to an embedding of Δ_{3}. The projective bundle P_{tg} coincides with the projectivized jet bundle ℙ(J^{1}(L)). The line bundle M := p_{1}^{∗}L ⨂ p_{2}^{∗}L ⨂ O_{X×X}(−Δ) admits a natural trivialization over Δ_{1}. In part I [BR] it was shown that a projective structure on X can be viewed as a choice of an extension of the trivialization of the line bundle M on Δ_{1} to a trivialization on Δ_{2}.

The question that we address here is the interpretation of an embedding of higher order infinitesimal neighborhoods of Δ into P_{tg} and of the trivialization of M on higher order infinitesimal neighborhoods of Δ.

In Theorem 3.6 (cf. Remark 3.12) we prove that for n ≥ 3 the space of all trivializations of M on Δ_{n}, which restrict to the canonical trivialization over Δ_{1}, is canonically identified with the space of all projective structures on X together with a differential of order i for each i ∈ [3, n]. In Lemma 4.8 we prove that the latter space is also identified, in a canonical fashion, with the space of all embeddings of Δ_{n+1} in ℙ(J^{1}(L)). In other words, the two spaces, of trivializations and embeddings respectively, are canonically identified.

2 INDRANIL BISWAS AND A. K. RAINA

In Theorems 3.11 and 4.9 we establish that for n ≥ 2 these spaces, of trivializations or embeddings, can be identified with a certain natural class of differential operators on X of order n. The differential operators in question map sections of L^{⊗(n−1)} to sections of L^{⊗(−n−1)} and have the following form in local coordinates

\frac{d^{n}}{dx^{n}} + ∑_{i=2}^{n} f_{i} \frac{d^{n−i}}{dx^{n−i}}

The results of Appendices A and B are used in Sections 2 and 3.

One of the motivations of the present work has been from Mathematical Physics, viz. a continuing study of a certain model quantum field theory on a curve, known as the b − c system, which appears in conformal field theory and string theory studies (see [R1] for a review of earlier work). An important aspect of such a quantum field theory model is its so-called Operator Product Expansion (OP E). The OP E is basic to the theory of vertex operator algebras [K], but is usually restricted to the genus zero case. We can regard [R2] and [BR] as identifying the “one point functions” of the first and second terms of the holomorphic part of the OP E of the quantum fields b and c as the “one point function” of, respectively, the current and the energy-momentum tensor of the system. The present paper can be regarded as identifying the “one point function” of the first n terms of the holomorphic part of the OP E as an n-th order differential operator of the above type, for all n ≥ 2. Indeed, the interpretation can be extended to n=1 as well, since it was shown in [R2] that the current < j > is a one form and arises from the connection d + < j >, where d is the de Rham connection on X.

The first named author thanks Tim McCune for his remarks. The other (AKR) gratefully acknowledges the support and hospitality of the Max-Planck-Institut für Mathematik, Bonn, during this work.

2. Projective structures on a Riemann surface

A projective atlas on a Riemann surface X is a covering {U_{α}, ϕ_{α}}_{α∈I} of X by holomorphic coordinate charts, where ϕ_{α} is a biholomorphism from U_{α} to an open set in ℂ, such that any ϕ_{β} ∘ ϕ_{α}^{−1}, α, β ∈ I, is the restriction of a Möbius transformation to ϕ_{α}(U_{α}). Such an atlas on X gives a one cocycle on X with values in P SL(2, ℂ), the Möbius

PROJECTIVE STRUCTURES ON A RIEMANN SURFACE, II 3

group of automorphisms of ℂℙ^{1}. A projective structure on X is an equivalence class of compatible projective atlases and determines an element in H^{1}(X, P SL(2, ℂ)) [Gu].

Let X be a compact Riemann surface equipped with a projective structure 𝔓. Choosing a line bundle L with L^{⊗2} = T_{X}, is equivalent to choosing a lift of the element corresponding to 𝔓 in H^{1}(X, P SL(2, ℂ)), to an element in H^{1}(X, SL(2, ℂ)) [Ty]. The map from H^{1}(X, SL(2, ℂ)) to H^{1}(X, P SL(2, ℂ)) in question is the one induced by the natural projection of SL(2, ℂ) onto P SL(2, ℂ).

2a. A line bundle on the self-product. We shall first recall an alternative description of a projective structure from [BR].

Let X be a compact connected Riemann surface. Fix a line bundle L on X, which is a square-root of T_{X}, along with an isomorphism

Ψ : L^{⊗2} ⟶ T_{X}  (2.1)

The space of all projective structures on X has the following description in terms of the line bundle M := p_{1}^{∗}L ⨂ p_{2}^{∗}L ⨂ O_{X×X}(−Δ) on X × X [BR]:

Theorem 2.2. Take any nonzero integer n. The restriction of the line bundle M^{⊗n} on X×X to the (nonreduced) divisor 2Δ has a canonical trivialization, which is invariant under the involution of X × X, defined by switching coordinates if n is even, and it is anti-invariant if n is odd. To each projective structure on X there is a naturally associated trivialization of the formal completion of M^{⊗n} along Δ, invariant (resp. anti-invariant) under the involution of X ×X if n is even (resp. odd), with the property that the trivialization restricts to the natural trivialization of M^{⊗n} over 2Δ. Moreover, this association gives a bijective map between the space of all projective structures on X and the space of all trivializations of M^{⊗n} over 3Δ which restrict to the natural trivialization over 2Δ.

Remark 2.3.

1. A trivialization of the formal completion of M^{⊗n} along Δ means compatible trivializations of the line bundle M^{⊗n} over infinitesimal neighborhoods of Δ in X × X of every order([Ha], page 194). This is equivalent to trivializing M^{⊗n} over some analytic neighborhood of Δ.

4 INDRANIL BISWAS AND A. K. RAINA

2. The restriction of the line bundle O_{X×X}(−nΔ) to Δ is K_{Δ}^{⊗n}, the n-th tensor power of the cotangent bundle, and hence the restriction of M^{⊗n} to Δ is the trivial line bundle. If n is even then the trivialization is canonical; if n is odd then there is a natural trivialization up to sign. This indeterminacy of sign can be removed by using the ordering of factors in the Cartesian product X × X. There is a natural trivialization of M^{⊗n} over 2Δ which is an extension of the trivialization over Δ. A description of this trivialization using coordinate charts is given in (2.4). This trivialization over 2Δ is determined by the condition that when n is even (resp. odd), then it is invariant (resp. anti-invariant) under the involution of X × X defined by switching coordinates. Thus, if the involution and the trivialization are denoted by τ and s respectively, then s coincides with τ^{∗}s (resp. −τ^{∗}s) if n is even (resp. odd).

Denoting by q_{i}, i = 1, 2, the projection of ℂℙ^{1} × ℂℙ^{1} onto the ith factor, the line bundle q_{1}^{∗}O(n) ⨂ q_{2}^{∗}O(n) ⨂ O_{ℂℙ^{1}×ℂℙ^{1}}(−nΔ_{0}), where Δ_{0} ⊂ ℂℙ^{1} × ℂℙ^{1} is the diagonal divisor, has a canonical trivialization given by the section

s(n) := (z_{1} − z_{2})^{n}(\frac{∂}{∂z_{1}} )^{n/2} ⊗ (\frac{∂}{∂z_{2}} )^{n/2}  (2.4)

where (z_{1}, z_{2}) is the coordinate function on ℂℙ^{1}×ℂℙ^{1} obtained from the obvious coordinate function z on ℂℙ^{1} = ℂ ∪ \{∞\}, and (\frac{∂}{∂z} )^{1/2} denotes

a section of O(1) such that ((\frac{∂}{∂z} )^{1/2})^{⊗2} coincides with the section \frac{∂}{∂z} by the natural isomorphism between O_{ℂℙ^{1}}(2) and T_{ℂℙ^{1}}. Note that though there are two choices for the section (\frac{∂}{∂z} )^{1/2}, the section in (2.4) does not depend upon the choice.

For any biholomorphism ϕ : U ⟶ U′, between open sets of ℂℙ^{1}, the two sections, s(n) and (ϕ, ϕ)^{∗}s(n) respectively, defined over U × U, actually coincide when they are restricted to 2Δ_{0} ∩ (U × U). Thus s(n) gives a trivialization of M^{⊗n} over 2Δ for any Riemann surface X. Indeed, the above property of s(n) ensures that for any two coordinate charts on X, the two pullbacks of s(n) using the two coordinate functions actually coincide on the first order infinitesimal neighborhood of the diagonal.

We noted earlier that for a projective structure 𝔓 on X, the element

ρ ∈ H^{1}(X, P SL(2, ℂ))

corresponding to 𝔓, lifts to a cohomology class

ρ̄ ∈ H^{1}(X, SL(2, ℂ))

using L. The line bundle on X associated to ρ̄, for the natural action of SL(2, ℂ) on O_{ℂℙ^{1}}(1), is canonically identified with L.

The diagonal action of SL(2, ℂ) on ℂℙ^{1} × ℂℙ^{1} naturally lifts to the line bundle q_{1}^{∗}O(n) ⨂ q_{2}^{∗}O(n) ⨂ O(−nΔ_{0}) preserving its trivialization.

For a covering of X by coordinate charts compatible with a given projective structure, the pullbacks of O_{ℂℙ^{1}}(1) by coordinate maps patch together to produce the line bundle L on X. The trivialization of q_{1}^{∗}O(n) ⨂ q_{2}^{∗}O(n) ⨂ O(−nΔ_{0}) gives a trivialization of M^{⊗n} over an analytic neighborhood of Δ. The association between projective structures and trivializations, mentioned in Theorem 2.2, is obtained by restricting this trivialization to the completion of X × X along Δ.

Theorem 2.2 has been proved in [BR] with n = 2. The proof in any other case is identical to it.

Restricting M^{⊗n} to 2Δ and 3Δ respectively, and using the trivialization of M^{⊗n} over Δ, we have the following exact sequence

0 ⟶ K_{Δ}^{⊗2} ⟶ (M^{⊗n})|_{3Δ} ⟶ (M^{⊗n})|_{2Δ} ⟶ 0

Hence, if \overline{𝔓} and \overline{𝔓′} denote the trivializations of (M^{⊗n})|_{3Δ} corresponding to any two projective structures 𝔓 and 𝔓′ on X,

\overline{𝔓′} − \overline{𝔓} ∈ H^{0}(X, K_{X}^{⊗2})

since \overline{𝔓} and \overline{𝔓′} agree over 2Δ.

On the other hand, the space of all projective structures on X is an affine space for H^{0}(X, K_{X}^{⊗2}) [Gu]. Thus

𝔓′ − 𝔓 ∈ H^{0}(X, K_{X}^{⊗2})

The relationship between these two observations is provided by the following identity ([BR], Lemma 3.6)

\overline{𝔓′} − \overline{𝔓} = \frac{n}{12}(𝔓′ − 𝔓)  (2.5)

6 INDRANIL BISWAS AND A. K. RAINA

2b. Decomposition of a differential operator. For a holomorphic vector bundle E on X and a positive integer n, the n-th order jet bundle of E, denoted by J^{n}(E), is defined to be the following direct image on X:

J^{n}(E) := p_{1∗}(\frac{p_{2}^{∗}E}{p_{2}^{∗}E ⊗ O_{X×X}(−(n + 1)Δ)})

where, as before, p_{i} is the projection of X × X onto the i-th factor. Since Δ is an effective divisor, p_{2}^{∗}E ⊗ O_{X×X}(−(n + 1)Δ) is a subsheaf of p_{2}^{∗}E. There is a natural exact sequence

0 ⟶ K_{X}^{⊗(n+1)} ⊗ E ⟶ J^{n+1}(E) ⟶ J^{n}(E) ⟶ 0

The inclusion K_{X}^{⊗(n+1)} ⊗ E ⟶ J^{n+1}(E) is constructed by using the inclusion

K_{X}^{⊗(n+1)} ⟶ J^{n+1}(O_{X})

which is defined at x ∈ X by (df)^{⊗(n+1)} ⟼ f^{(n+1)}/(n + 1)! , where f is any function with f(x) = 0.

The sheaf of differential operators Diff_{X}^{n}(E, F) coincides with Hom(J^{n}(E), F). The homomorphism

σ : Diff_{X}^{n}(E, F) ⟶ Hom(K_{X}^{⊗n} ⊗ E , F)

obtained by restricting a homomorphism from J^{n}(E) to F to the subsheaf K_{X}^{⊗n} ⨂ E, is known as the symbol map.

Furthermore, for any two integers r, s ∈ ℕ, the natural projection

J^{r+s}(E) ⟶ J^{s}(E)

admits a canonical lift

μ : J^{r+s}(E) ⟶ J^{r}(J^{s}(E))

which is an injective homomorphism of vector bundles.

Now consider the following commutative diagram:

0 → L^{−n−1} = K_{X}^{⊗n} ⊗ L^{⊗(n−1)} →  J^{n}(L^{⊗(n−1)})  → J^{n−1}(L^{⊗(n−1)}) → 0
↓^{μ}  ∥
0 → K_{X} ⊗ J^{n−1}(L^{⊗(n−1)}) → J^{1}(J^{n−1}(L^{⊗(n−1)})) → J^{n−1}(L^{⊗(n−1)}) → 0
(2.6)

The differential operator D_{𝔓}(n) constructed in (B.4) of Appendix B gives a homomorphism

D_{n} : J^{n−1}(L^{⊗(n−1)}) ⟶ J^{n}(L^{⊗(n−1)})  (2.7)

which is a splitting of the top exact sequence in (2.6). The composition μ ∘ D_{n} is a splitting of the bottom exact sequence in (2.6). In other words, μ ∘ D_{n} is a holomorphic connection on J^{n−1}(L^{⊗(n−1)}). Thus we have the following lemma [Bi, Theorem 4.1]:

Lemma 2.8. For a Riemann surface X with a projective structure, for any n ∈ ℕ, the jet bundle J^{n}(L^{⊗n}) is equipped with a natural flat connection.

Using the splitting D_{n} in (2.7) as the base point, the space of all splittings of the top exact sequence in (2.6) is identified with

H^{0}(X, Diff_{X}^{n−1}(L^{⊗(n−1)}, L^{⊗(−n−1)}))

Similarly, the space of all splittings of the bottom exact sequence in (2.6) is identified with

H^{0}(X, End(J^{n−1}(L^{⊗(n−1)})) ⊗ K_{X})

The composition of a splitting of the top exact sequence in (2.6) with the inclusion μ (in (2.6)) is a splitting of the bottom exact sequence in (2.6). Thus we have a natural homomorphism

Φ : H^{0}(X, Diff_{X}^{n−1}(L^{⊗(n−1)}, L^{⊗(−n−1)})) ⟶ H^{0}(X, End(J^{n−1}(L^{⊗(n−1)}))⊗K_{X})

Using the GL(n, ℂ) invariant polynomials A ⟼ trace(A^{i}), 1 ≤ i ≤ n, on M(n, ℂ) we have a homomorphism

H : H^{0}(X, End(J^{n−1}(L^{⊗(n−1)})) ⊗ K_{X}) ⟶ ⨁_{i=1}^{n} H^{0}(X, K_{X}^{⊗i})

(2.9)

which is known as the Hitchin map [Hi].

Theorem 2.10. Let X be a compact Riemann surface equipped with a projective structure. The homomorphism

H ∘ Φ : H^{0}(X, Diff_{X}^{n−1}(L^{⊗(n−1)}, L^{⊗(−n−1)})) ⟶ ⨁_{i=1}^{n} H^{0}(X, K_{X}^{⊗i})

is an isomorphism.

8 INDRANIL BISWAS AND A. K. RAINA

Proof. We shall first consider the case where the genus of X is at least 2. The remaining cases will be treated separately.

Our first step will be to establish the injectivity of the homomorphism H ∘ Φ.

For D ∈ H^{0}(X, Diff_{X}^{n−1}(L^{⊗(n−1)}, L^{⊗(−n−1)})), let σ ∈ H^{0}(X, K_{X}^{⊗i}) be the component of H ∘ Φ(D) of lowest degree. In other words, σ ≠ 0, and furthermore, if j < i, then the component of H ∘ Φ(D) in H^{0}(X, K_{X}^{⊗j}) vanishes identically. It is a simple computation to check that the symbol of the differential operator D coincides with σ. (This implies that D is actually of order n−i.) Thus H ∘Φ must be injective.

We shall complete the proof by showing that the dimensions of the domain and the target of H ∘ Φ coincide.

Using the long exact sequence of cohomologies for the following exact sequence of vector bundles on X

0 ⟶ Diff_{X}^{i}(L^{⊗(n−1)}, L^{⊗(−n−1)}) ⟶ Diff_{X}^{i+1}(L^{⊗(n−1)}, L^{⊗(−n−1)}) ⟶ K_{X}^{⊗(n−i−1)} ⟶ 0

it can be deduced that

H^{1}(X, Diff_{X}^{j}(L^{⊗(n−1)}, L^{⊗(−n−1)})) = 0  (2.11)

for all j ≤ n − 2. Indeed, if n ≥ 2, then

H^{1}(X, Diff_{X}^{0}(L^{⊗(n−1)}, L^{⊗(−n−1)})) = H^{1}(X, K_{X}^{⊗n}) = 0

since the genus of X is at least two. Now (2.11) follows from the exact sequence

H^{1}(X, Diff_{X}^{j}(L^{⊗(n−1)}, L^{⊗(−n−1)})) ⟶ H^{1}(X, Diff_{X}^{j+1}(L^{⊗(n−1)}, L^{⊗(−n−1)}))

⟶ H^{1}(X, K_{X}^{⊗(n−j−1)}) ⟶ 0

by induction on j.

Finally, (2.11) implies that

dim H^{0}(X, Diff_{X}^{j}(L^{⊗(n−1)}, L^{⊗(−n−1)})) = ∑_{i=n−j}^{n} dim H^{0}(X, K_{X}^{⊗i})

for all j ≤ n − 1. This can be seen by considering the long exact sequence

0 ⟶ H^{0}(X, Diff_{X}^{j}(L^{⊗(n−1)}, L^{⊗(−n−1)})) ⟶ H^{0}(X, Diff_{X}^{j+1}(L^{⊗(n−1)}, L^{⊗(−n−1)}))

⟶ H^{0}(X, K_{X}^{⊗(n−j−1)}) ⟶ H^{1}(X, Diff_{X}^{j}(L^{⊗(n−1)}, L^{⊗(−n−1)}))

PROJECTIVE STRUCTURES ON A RIEMANN SURFACE, II 9

and using induction on j. This completes the proof of the theorem in the situation where the genus of X is at least 2.

If X = ℂℙ^{1}, then the dimension of both the domain and the target is

0. To see that the domain has dimension 0, observe that J^{n−1}(L^{⊗(n−1)}) is a trivial bundle, as it has a flat connection [Lemma 2.8].

If X is an elliptic curve, then Diff_{X}^{n−1}(L^{⊗(n−1)}, L^{⊗(−n−1)}) is actually isomorphic to ⨁_{i=1}^{n} H^{0}(X, K_{X}^{⊗i}). This follows from the observation that J^{n−1}(L^{⊗(n−1)}) is isomorphic to the symmetric power S^{n−1}(J^{1}(L)) [Bi]. This completes the proof of the theorem. □

The isomorphism H ∘ Φ depends upon the projective structure that was used in its construction. The homomorphisms Φ or H are not individually isomorphisms. In fact if g > 1 then,

dim H^{0}(X, End(J^{n−1}(L^{⊗(n−1)})) ⊗ K_{X}) > ∑_{i=1}^{n} dim H^{0}(X, K_{X}^{⊗i})

Now, any element D ∈ H^{0}(X, Diff_{X}^{n}(L^{⊗(n−1)}, L^{⊗(−n−1)})) decomposes uniquely as

D = cD_{𝔓}(n) + D_{0}

where c ∈ ℂ and D_{0} ∈ H^{0}(X, Diff_{X}^{n−1}(L^{⊗(n−1)}, L^{⊗(−n−1)})). Hence Theorem 2.10 has the following corollary:

Corollary 2.12. For a Riemann surface X with a projective structure, the following decomposition

H^{0}(X, Diff_{X}^{n}(L^{⊗(n−1)}, L^{⊗(−n−1)})) = ⨁_{i=0}^{n} H^{0}(X, K_{X}^{⊗i})

is valid.

The differentials appearing on the right hand side may be viewed as the Laguerre-Forsyth invariants of the differential operator [W].

In the next section we shall establish an identification between a certain subspace of the space of differential operators H^{0}(X, Diff_{X}^{n}(L^{⊗(n−1)}, L^{⊗(−n−1)})) and the space of trivializations of M over an infinitesimal neighborhood of the diagonal.

10 INDRANIL BISWAS AND A. K. RAINA

3. Differential operators on a Riemann surface

We continue with the notation of the previous section, but in the present section we shall not assume that the Riemann surface X is equipped with a projective structure.

For a nonnegative integer n and an integer d ≥ 3, let

S(n, d) ∈ H^{0}(X × X, \frac{M^{⊗n}}{M^{⊗n} ⊗ O_{X×X}(−dΔ)})  (3.1)

be a trivialization of M^{⊗n} over dΔ, such that its restriction to Δ is the constant function 1. Furthermore, denoting the involution of X × X by τ, the section S(n, d) is required to coincide with τ^{∗}S(n, d) (resp. −τ^{∗}S(n, d)) if n is even (resp. odd). Hence the restriction of S(n, d) to 2Δ is the canonical trivialization referred to in Theorem 2.2.

Consider the restriction of the section S(n, d) to 3Δ. We shall denote by 𝔓_{S(n,d)} the natural projective structure on X associated to it by Theorem 2.2. Let

S′(n, d) ∈ H^{0}(X × X, \frac{M^{⊗n}}{M^{⊗n} ⊗ O_{X×X}(−dΔ)})  (3.2)

be the trivialization associated to the projective structure 𝔓_{S(n,d)} using Theorem 2.2.

Let ξ denote the direct image

ξ := p_{1∗}(\frac{S(n, d)}{S′(n, d)}) ∈ H^{0}(X, J^{d−1}(O_{X}))

on X of the function over dΔ given by the quotient of the two trivializations defined in (3.1) and (3.2) respectively. The image of ξ in J^{2}(O_{X}) coincides with the image of the constant function 1.

Setting d = n + 2, let

M_{ξ} : J^{n+1}(L^{⊗n}) ⟶ J^{n+1}(L^{⊗n})

be the homomorphism defined by multiplication with ξ. The multiplication in question is the natural surjective homomorphism

J^{n}(L) ⨂ J^{n}(L′) ⟶ J^{n}(L ⨂ L′) ,

where L and L′ are any two line bundles on X, which sends any two local sections s and s′ of L and L′ respectively, to s ⊗ s′.

PROJECTIVE STRUCTURES ON A RIEMANN SURFACE, II 11

Notation 3.3 : For any integer n ≥ 1, let T (n) denote the space of all trivializations S(n, n + 2) of M^{⊗n} over (n + 2)Δ which restrict to the canonical trivialization over 2Δ.

Note that there is a one-to-one correspondence between T (n) and the space of all trivializations of M over (n + 2)Δ which restrict to the canonical trivialization over 2Δ. The map is defined by simply taking the n-th tensor power of a section of M over (n + 2)Δ.

For any S(n, n + 2) ∈ T (n), let D = D_{𝔓_{S(n,n+2)}}(n + 1) be the differential operator defined in (B.4) for the projective structure 𝔓_{S(n,n+2)} constructed earlier. Since D is a homomorphism from J^{n+1}(L^{⊗n}) to L^{⊗(−n−2)}, the composition

D_{S} := D ∘ M_{ξ}  (3.4)

is an element of H^{0}(X, Diff_{X}^{n+1}(L^{⊗n}, L^{⊗(−n−2)})). The earlier observation that the image of ξ in J^{2}(O_{X}) coincides with the image of the constant function 1, implies that the difference

D_{S} − D ∈ H^{0}(X, Diff_{X}^{n−2}(L^{⊗n}, L^{⊗(−n−2)}))

Let 𝔓(X) denote the space of all projective structures on the Riemann surface X. The following theorem identifies H^{0}(X, Diff_{X}^{n−2}(L^{⊗n}, L^{⊗(−n−2)})) with the space of trivializations.

Theorem 3.5. For a compact Riemann surface X, the map

W : T (n) ⟶ 𝔓(X) × H^{0}(X, Diff_{X}^{n−2}(L^{⊗n}, L^{⊗(−n−2)}))

defined by S(n, n + 2) ⟼ (𝔓_{S(n,n+2)} , D_{S} − D), where 𝔓_{S(n,n+2)} was constructed earlier and D_{S} was defined in (3.4), is bijective.

Proof. To construct the inverse of W, for any

(𝔓 , D) ∈ 𝔓(X) × H^{0}(X, Diff_{X}^{n−2}(L^{⊗n}, L^{⊗(−n−2)}))

consider the differential operator D′ := D_{𝔓}(n+1)+D, where D_{𝔓}(n+1) is the operator constructed in (B.4). Let

M : J^{n+1}(L^{⊗n}) ⟶ J^{n+1}(L^{⊗n})

be the homomorphism defined by the identity D′ = D_{𝔓}(n + 1) ∘ M.

12 INDRANIL BISWAS AND A. K. RAINA

Let S ∈ T (n) be the trivialization associated to the projective structure 𝔓 by Theorem 2.2. Since M ∈ H^{0}((n + 2)Δ, O), the product M.S ∈ T (n).

Associating M.S to (𝔓 , D), the inverse of the map W is obtained. It is rather straight-forward to check that this map is indeed the inverse of W. □

From Theorem 2.10, together with the fact used in the proof of Theorem 2.10 that for any

D′ ∈ H^{0}(X, Diff_{X}^{n−1}(L^{⊗(n−1)}, L^{⊗(−n−1)}))

the lowest order term of H ∘ Φ(D′) is the symbol of D′, it immediately follows that if X is equipped with a projective structure then

H^{0}(X, Diff_{X}^{n−2}(L^{⊗n}, L^{⊗(−n−2)})) = ⨁_{i=3}^{n+1} H^{0}(X, K_{X}^{⊗i})

Now the following theorem is a consequence of Theorems 2.10 and 3.5.

Theorem 3.6. For a compact Riemann surface X, the space of trivializations T (n) is identified, in a canonical fashion, with the Cartesian product

𝔓(X) × ( ⨁_{i=3}^{n+1} H^{0}(X, K_{X}^{⊗i}))

Let

Σ(n + 1) ⊂ H^{0}(X, Diff_{X}^{n+1}(L^{⊗n}, L^{⊗(−n−2)}))  (3.7)

be the subset consisting of all differential operators with symbol 1. The vector space

V := H^{0}(X, Diff_{X}^{n−1}(L^{⊗n}, L^{⊗(−n−2)}))

acts on Σ(n + 1) simply by addition.

It is somewhat surprising (at least to the authors) that the action of V on Σ(n + 1) has a distinguished orbit. This is the orbit containing the differential operator D_{𝔓}(n + 1), defined in (B.4), where 𝔓 is a projective structure on X. The observation noted down in (B.6) that the difference of two such operators differ by an operator of degree n − 1, implies that the above defined orbit does not depend upon the choice of the projective structure.

PROJECTIVE STRUCTURES ON A RIEMANN SURFACE, II 13

Let

Σ_{0}(n + 1) ⊂ Σ(n + 1)  (3.8)

denote the distinguished orbit for the action of V.

Remark 3.9. Using the exact sequence

0 ⟶ K_{X}^{⊗(i+1)} ⊗ L^{⊗(n−1)} ⟶ J^{i+1}(L^{⊗(n−1)}) ⟶ J^{i}(L^{⊗(n−1)}) ⟶ 0

and the isomorphism between L^{⊗2} and T_{X}, it readily follows that

⋀^{n}J^{n−1}(L^{⊗(n−1)}) = O_{X}  (3.10)

Hence, the top exterior product ⋀^{n}J^{n−1}(L^{⊗(n−1)}) has a canonical trivialization.

We shall describe a property of the class of differential operators Σ_{0}(n) defined in (3.8), in terms of the above trivializations. For a D ∈ Σ_{0}(n) let ∇ be the holomorphic connection on J^{n−1}(L^{⊗(n−1)}) constructed using the map μ in (2.6) and the splitting given by D. It is straight-forward to check that the connection on ⋀^{n}J^{n−1}(L^{⊗(n−1)}) induced by ∇ coincides with the natural connection on O_{X} using the isomorphism (3.10). □

Consider the map

𝔓(X) × H^{0}(X, Diff_{X}^{n−2}(L^{⊗n}, L^{⊗(−n−2)})) ⟶ Σ_{0}(n + 1)

defined by (𝔓 , D) ⟼ D_{𝔓}(n + 1) + D. The Lemma B.9 implies that this map is an isomorphism.

Now the following theorem is easily derived from Theorem 3.5.

Theorem 3.11. For any compact Riemann surface X, the space of trivializations T (n) is canonically identified with the space of differential operators Σ_{0}(n + 1).

For any a trivialization t ∈ T (n), its restriction to 3Δ gives a projective structure 𝔓 on X. If D is the differential operator that corresponds to t by Theorem 3.11, then 𝔓 is the unique projective structure such that

D − D_{𝔓}(n + 1) ∈ H^{0}(X, Diff_{X}^{n−2}(L^{⊗n}, L^{⊗(−n−2)}))

Remark 3.12. Following Notation 3.3 we noted that the space T (n) is identified in a canonical fashion with the space of all trivializations

14 INDRANIL BISWAS AND A. K. RAINA

of M over (n + 2)Δ which restrict to the canonical trivialization of M over 2Δ. Thus, denoting the space of such trivializations of M by T_{n}, Theorems 3.6 and 3.11 remain valid if T (n) is replaced with T_{n} in their statements. □

Let

ℙ_{X}(n) := ℙ(J^{n+1}(L^{⊗n})) ⟶ X

denote the projective bundle over X of dimension n + 1 consisting of all lines in the vector bundle J^{n+1}(L^{⊗n}). Let

ℙ_{X} := ℙ(J^{n}(L^{⊗n})) ⊂ ℙ_{X}(n)

denote the projective bundle over X for the natural surjection of J^{n+1}(L^{⊗n}) onto J^{n}(L^{⊗n}).

A differential operator D ∈ H^{0}(X, Diff_{X}^{n+1}(L^{⊗n}, L^{⊗(−n−2)})), with the constant function 1 as the symbol, naturally gives a section of the fiber bundle

A_{X}(n) := ℙ_{X}(n) − ℙ_{X}

by simply considering D as a homomorphism of J^{n+1}(L^{⊗n}) onto L^{⊗(−n−2)}.

Conversely, for any section of A_{X}(n) there is a differential operator such that the section arises from it in the above fashion. Indeed, any section s of ℙ_{X}(n) is given by a surjective homomorphism, say H, of J^{n+1}(L^{⊗n}) onto a line bundle ζ over X. If s is a section of A_{X}(n) then the restriction of H to L^{⊗(−n−2)} is nowhere zero. In other words, ζ is isomorphic to L^{⊗(−n−2)} by H.

Thus we have a one-to-one bijective correspondence between the space of sections of A_{X}(n) and the subspace of of the space of differential operators

H^{0}(X, Diff_{X}^{n+1}(L^{⊗n}, L^{⊗(−n−2)}))

consisting of operators with symbol 1.

Using the above one-to-one correspondence, the class of operators Σ_{0}(n + 1) defined in (3.8) gives a class of sections of A_{X}(n). Let S denote this class of sections. It will be interesting to be able to characterize geometrically the class of sections S. Also, it will be interesting to be able to directly find the element in T (n), given by Theorem 3.11, corresponding to any given element of S.

PROJECTIVE STRUCTURES ON A RIEMANN SURFACE, II 15

4. Embeddings of the higher order infinitesimal neighborhoods of the diagonal in a projective bundle

The structure sheaf of the subscheme nΔ ⊂ X × X is isomorphic to the ringed space

(X , p_{1∗}O_{nΔ})

where p_{1} is the projection of X × X onto the first factor. The O_{X}- algebra p_{1∗}O_{nΔ} on X coincides with the jet bundle J^{n−1}(O_{X}) with its natural algebra structure.

Another natural embedding of X in a fiber bundle over X is obtained from the exact sequence

0 ⟶ K_{X} ⊗ L ⟶ J^{1}(L) ⟶ L ⟶ 0

In this case X is embedded in the projective bundle ℙ(X) := ℙ(J^{1}(L)). Denoting the image of X by X̄, the structure sheaf of nX̄ is the ringed space

(X , q_{∗}O_{nX̄})  (4.1)

where q is the projection of ℙ(X) onto X.

Lemma 4.2. For any n ≥ 0, the vector bundle q_{∗}O_{(n+1)X̄} on X, defined in (4.1), is canonically isomorphic to

O_{X} ⊕ J^{n−1}(L^{⊗(n−1)}) ⊗ L^{⊗(−n−1)}

Proof. For x ∈ X, let ℙ_{x} := q^{−1}(x) be the projective line. The (n − 1)-th order infinitesimal neighborhood of the point

x̄ := ℙ_{x} ∩ X̄

in ℙ_{x} coincides with the fiber (q_{∗}O_{nX̄})x̄.

In order to determine the infinitesimal neighborhood of x̄, consider the following exact sequence on ℙ_{x}:

0 ⟶ O_{ℙ_{x}}(−(n + 1)x̄) ⟶ O_{ℙ_{x}} ⟶ J^{n}(O_{ℙ_{x}})x̄ ⟶ 0

Since H^{1}(ℙ_{x}, O_{ℙ_{x}}) = 0, the n-th order infinitesimal neighborhood of x̄, namely J^{n}(O_{ℙ_{x}})x̄, coincides with

ℂ ⊕ H^{1}(ℙ_{x}, O_{ℙ_{x}}(−(n + 1)x̄))  (4.3)

16 INDRANIL BISWAS AND A. K. RAINA

Using Serre duality, and the canonical isomorphisms O_{ℙ_{x}}(−x̄)x̄ = (K_{X})_{x} and L^{⊗2} = T_{X}, and the natural self-duality of H^{0}(ℙ_{x}, O_{ℙ_{x}}(n − 1)) induced by the natural trivialization of the line ⋀^{2}J^{1}(L)_{x}, the vector space in (4.3) gets identified with the fiber at x of the vector bundle

O_{X} ⊕ J^{n−1}(L^{⊗(n−1)}) ⊗ L^{⊗(−n−1)}

The details of the last part of argument can be found in Section 3 of [Bi]. This completes the proof of the lemma. □

Given a projective structure on X, for any n ∈ ℕ, there is a natural compatible embedding of the n-th order infinitesimal neighborhood of Δ into the projective bundle ℙ(X) [De]. The compatibility of embeddings means that the embedding of the (n − 1)-th order infinitesimal neighborhood is obtained by restricting the embedding of the n-th order infinitesimal neighborhood. To obtain these embeddings, first observe that if X is ℂℙ^{1}, then ℙ(X) is X × X and so the embedding is the (tautological) diagonal embedding. Then observe that this tautological embedding commutes with the diagonal action of Aut(ℂℙ^{1}) on ℂℙ^{1} × ℂℙ^{1}. This immediately implies the earlier statement on the existence of canonical compatible embeddings for any Riemann surface with a projective structure.

The embedding of the second order infinitesimal neighborhood 3Δ of Δ into ℙ(X) actually does not depend upon the choice of the projective structure. A direct construction of this embedding of the second order infinitesimal neighborhood of Δ into ℙ(X) is given below.

In view of Lemma 4.2, giving an identification of the second order infinitesimal neighborhoods of the above type is equivalent to giving an isomorphism

J_{0}^{2}(O_{X}) ⟶ J^{1}(L) ⊗ L^{⊗−3}  (4.4)

where the fiber J_{0}^{2}(O_{X})_{x} ⊂ J^{2}(O_{X})_{x} at any x ∈ X is given by the jets of functions vanishing at x. Since the map defined by f ⟼ df identifies J_{0}^{2}(O_{X}) with J^{1}(K_{X}), we have a natural isomorphism of the type (4.4) as soon as we have established the following statement:

For any nonzero integer i, there is a nondegenerate pairing between J^{1}(T_{X}) and J^{1}(L^{⊗i}) with values in L^{⊗i}.

Now, the operation of taking the Lie derivative with respect to a vector field is exactly such a pairing. Indeed, the Lie derivative

(V , s) ⟼ L_{V} s

where s (resp. V) is a local section of L^{⊗i} (resp. T_{X}), defines an isomorphism of J^{1}(T_{X}) with J^{1}(L^{⊗i})^{∗} ⨂ L^{⊗i}. Using the isomorphism Ψ in (2.1) and the Leibniz identity

L_{V} (s ⊗ t) = (L_{V} s) ⊗ t + s ⊗ L_{V} t

the Lie derivative of sections of any tensor power of L is defined. The canonical isomorphism of the type (4.4) that we are seeking is obtained by setting i = 1 and −2.

To equip X with a projective structure is equivalent to extending the above identification, of the second order infinitesimal neighborhoods of Δ and X̄, to an identification of their third order infinitesimal neighborhoods ([De], Definition 5.6 bis).

The following lemma will be needed in our study of embeddings of the higher order infinitesimal neighborhoods of Δ.

Lemma 4.5. For a Riemann surface with a projective structure the space of all differential operators of order n−1 from L^{⊗(n−1)} to L^{⊗(1−n)} admits a natural decomposition of the following form

H^{0}(X, Hom(J^{n−1}(L^{⊗(n−1)}), L^{⊗(−n+1)})) = ⨁_{i=0}^{n−1} H^{0}(X, K_{X}^{⊗i})

Proof. Let V/ℂ be a vector space of dimension 2 equipped with a symplectic form ω. Denoting by L the line bundle O_{ℙ}(1) on the projective line ℙ = ℙ(V), the vector bundle J^{n}(L^{⊗k}) on ℙ is canonically isomorphic to J^{n}(L^{⊗n}) ⨂ L^{⊗(k−n)} in the following two cases:

1. when k < 0,

2. when k ≥ n. The first step to construct the isomorphism is to consider the long exact sequence of cohomologies for the exact sequence

0 ⟶ L^{⊗k} ⊗ O_{ℙ}(−(n + 1)x) ⟶ L^{⊗k} ⟶ J^{n}(L^{⊗k})_{x} ⟶ 0

where x ∈ ℙ. Note that in the first case H^{0}(ℙ, L^{⊗k}) = 0 and in the second case H^{1}(ℙ, L^{⊗k} ⊗ O_{ℙ}(−(n + 1)x)) = 0. A choice of 0 ≠ w ∈ V

18 INDRANIL BISWAS AND A. K. RAINA

representing x provides an isomorphism between O_{ℙ}(x) and L. Now after noting that H^{0}(ℙ, L^{⊗i}) = S^{i}(V^{∗}) = S^{i}(V) (using ω), and also using Serre duality when k < 0, the key point in the construction is that for any 0 ≠ v ∈ V, the following two vector spaces

1. the kernel of the homomorphism S^{n−1−k}(V) ⟶ S^{−k−2}(V) defined using the contraction with (v^{∗})^{⊗(n+1)}, where k < 0 and v^{∗} ∈ V^{∗} is the dual of v with respect to ω,

2. the cokernel of S^{k−n−1}(V) ⟶ S^{k}(V) defined using the multipli-

cation with v^{⊗(n+1)}, where k ≥ n, are both canonically isomorphic to S^{n}(V). Indeed, in the first case the isomorphism is defined using the multiplication with v^{⊗(−k−1)} on S^{n}(V), and in the second case it is defined using the contraction with (v^{∗})^{⊗(k−n)}.

Since the isomorphism is equivariant for the actions of SL(V) on ℙ and L, it induces an isomorphism

ψ(n, k) : J^{n}(L^{⊗k}) ⟶ J^{n}(L^{⊗n}) ⊗ L^{⊗(k−n)}

where X is a Riemann surface equipped with a projective structure together with a choice of a square-root of the canonical bundle, whenever n and k satisfy either of the above two conditions. (The choice of a theta characteristic provides a lift of the transition functions to SL(V).)

Now consider

ψ(i, n − 1) ⊗ ψ(i, −2(n − 1 − i)) : J^{i}(L^{⊗(n−1)}) ⊗ J^{i}(K_{X}^{⊗(n−1−i)})

⟶ J^{i}(L^{⊗i}) ⊗ J^{i}(L^{⊗i}) ⊗ L^{⊗(−n+1)}  (4.6)

where i ≤ n−1. The symplectic form ω on V induces a nondegenerate form on any S^{i}(V) (which is symmetric if i is even and skew-symmetric if i is odd), and hence on J^{i}(L^{⊗i}) over ℙ. This is because any fiber of J^{i}(L^{⊗i}) is canonically isomorphic to S^{i}(V) [Bi]. Since this form on J^{i}(L^{⊗i}) is equivariant under the action of SL(V), we have a nondegenerate form on J^{i}(L^{⊗i}) using the coordinate charts on X compatible with the projective structure. Thus (4.6) gives the pairing

ψ̄(i, n − 1) : J^{i}(L^{⊗(n−1)}) ⊗ J^{i}(K_{X}^{⊗(n−1−i)}) ⟶ L^{⊗(−n+1)}

Now for a section s ∈ H^{0}(X, K_{X}^{⊗(n−1−i)}), consider the homomorphism ψ̄(i, n − 1)(−, s) which sends any vector v ∈ J^{i}(L^{⊗(n−1)})_{y} to ψ̄(i, n − 1)(y)(v, s(y)). The differential operator

s̄ ∈ H^{0}(X, Diff_{X}^{i}(L^{⊗(n−1)}, L^{⊗(−n+1)}))

defined by this homomorphism has (−1)^{i}s ∈ H^{0}(X, K_{X}^{⊗(n−1−i)}) itself as its symbol. This lifting of the symbol of a differential operator that maps any s to (−1)^{i}s̄ immediately yields the decomposition (4.5) of differential operators. This completes the proof of the lemma. □

We note that the decomposition in Lemma 4.5 depends on the choice of projective structure.

For an integer n ≥ 3, let

ρ : J_{0}^{n}(O_{X}) ⟶ J^{n−1}(L^{⊗(n−1)}) ⊗ L^{⊗(−n−1)}

be an isomorphism of algebra bundles giving an embedding of the n-th order infinitesimal neighborhood of Δ into ℙ(X) such that the restriction of ρ to the second order infinitesimal neighborhood 3Δ of Δ is the canonical one defined above. Restricting ρ to the third order infinitesimal neighborhood 4Δ of Δ we get a projective structure 𝔓 on X [De].

Let

ρ(𝔓) : J_{0}^{n}(O_{X}) ⟶ J^{n−1}(L^{⊗(n−1)}) ⊗ L^{⊗(−n−1)}

be the isomorphism giving the embedding corresponding to the projective structure 𝔓.

It can be checked that any automorphism of J^{n−1}(L^{⊗(n−1)}), which as an automorphism of J^{n−1}(L^{⊗(n−1)}) ⊗ L^{⊗(−n−1)} preserves its algebra structure, is actually conjugate to a unique automorphism of the form c.Id + A, where c ∈ ℂ^{∗} and A ∈ H^{0}(X, Hom(J^{n−1}(L^{⊗(n−1)}), L^{⊗(−n+1)})). The inclusion

L^{⊗(n−1)} ⊗ K_{X}^{⊗(n−1)} = L^{⊗(−n+1)} ⟶ J^{n−1}(L^{⊗(n−1)})

enables H^{0}(X, Hom(J^{n−1}(L^{⊗(n−1)}), L^{⊗(−n+1)})) to be realized as a subspace of the space of endomorphisms of J^{n−1}(L^{⊗(n−1)}).

Now consider the automorphism

ρ ∘ (ρ(𝔓))^{−1} : J^{n−1}(L^{⊗(n−1)}) ⟶ J^{n−1}(L^{⊗(n−1)})

where ρ and (ρ(𝔓))^{−1} are defined above. Let

ρ̄ ∈ H^{0}(X, Hom(J^{n−1}(L^{⊗(n−1)}), L^{⊗(−n+1)}))

correspond to ρ ∘ (ρ(𝔓))^{−1} by the above decomposition.

Consider the decomposition of the differential operator ρ̄ according to Lemma 4.5 for the projective structure P on X. Since, by definition, ρ(𝔓) and ρ agree on the third order infinitesimal neighborhood of Δ, it follows immediately that the components of ρ̄ in H^{0}(X, K_{X}^{⊗i}) vanish for all i ≤ 2.

Let

F (ρ) ∈ ⨁_{i=3}^{n−1} H^{0}(X, K_{X}^{⊗i})  (4.7)

be the element corresponding to ρ̄.

Let G(n) denote the space of all embeddings of the n-th order infinitesimal neighborhood of Δ into ℙ(X) whose restriction to the second order infinitesimal neighborhood of Δ is the canonical one. Let 𝔓(X) denote the space of all equivalence classes of projective structures on X.

Lemma 4.8. The map from the space of embeddings G(n) to

𝔓(X) × ⨁_{i=3}^{n−1} H^{0}(X, K_{X}^{⊗i})

which sends any ρ ∈ G(n) to the pair (𝔓 , F (ρ)), constructed in (4.7), is a bijective identification.

Proof. The above lemma follows from Theorem 2.10. We omit the details. □

The following theorem is an immediate consequence of Theorem 3.6.

Theorem 4.9. For any compact Riemann surface X, the space of trivializations T (n), defined in (3.3), is canonically identified with G(n+2), the space of embeddings of the (n + 2)-th order infinitesimal neighborhood of Δ into ℙ(X) restricting to the canonical embedding on the second order infinitesimal neighborhood 3Δ.

In the rest of the paper we shall investigate some natural differential operators on Riemann surfaces with a projective structure.

PROJECTIVE STRUCTURES ON A RIEMANN SURFACE, II 21

Appendix A : A differential operator

For a holomorphic function g on an open set of ℂ, its n-th derivative, namely \frac{∂^{n}g}{∂z^{n}}, will be denoted by g^{(n)}. Also, often g′ will be used instead of g^{(1)}. Let

γ : U ⟶ U_{1}  (A.1)

be a biholomorphism between two simply connected analytic open sets of ℂ. Fix on U a square-root (γ′)^{1/2} of the function γ^{(1)}. For an integer n ≥ 1 and a holomorphic function f ∈ Γ(U_{1}, O) on U_{1}, define D(γ, n)f ∈ Γ(U, O) by

(D(γ, n)f)(z) = (f^{(n)} ∘ γ)(z)(γ′(z))^{(n+1)/2} − (\frac{f ∘ γ}{(γ′)^{(n−1)/2}} )^{(n)}(z)

(A.2)

where z ∈ U.

Evidently the map f ⟼ D(γ, n)f is a differential operator of order n from Γ(U_{1}, O) to Γ(U, O). Alternatively, the ℂ-linear homomorphism

ψ ⟼ D(γ, n)(ψ ∘ γ^{−1}) ,  (A.3)

where ψ ∈ Γ(U, O), is a section over U of the sheaf of differential operators of order n on the trivial line bundle. Since the tangent bundle T U has a natural trivialization, the symbol of a differential operator of order i from Γ(U, O) to Γ(U, O) is an element of Γ(U, O). It is a simple calculation to check that the symbol of this order n differential operator D(γ, n) vanishes identically. It involves a slightly more nontrivial computation to check that the symbol of the order n − 1 differential operator also vanishes identically. The conclusion is that the operator D(γ, n) is actually of order n − 2.

Another simple computation yields that if M is Möbius transformation, which means that

M(z) = \frac{az + b}{cz + d}

where ad − bc = 1, then

D(M, n) = 0  (A.4)

for all n ≥ 1. In fact, more generally, for a Möbius transformation M as above, denoting the function w ⟼ (cw + d)^{k} by M^{[k]}, where k ∈ ℤ, the following identity is valid:

D(M ∘ γ, n)(f)(z) = D(γ, n)(M^{[n−1]}f ∘ M)(z)  (A.5)

where f ∈ Γ(M(U_{1}), O) and z ∈ U. Since D(Id, k) = 0 for any k ≥ 1, the equality (A.5) immediately gives (A.4). Similarly, the identity

D(γ ∘ M, n)(f)(z) = D(γ, n)(f)(M(z))M^{[−n−1]}(z) ,  (A.6)

where z ∈ M^{−1}(U), is valid.

Clearly D(γ, 1) = 0. To calculate D(γ, 2), for a holomorphic function g on an open subset of ℂ, let

SD(g) := \frac{2g′(z)g′′′(z) − 3(g′′(z))^{2}}{2(g′(z))^{2}}

denote the Schwarzian derivative of g (page 164 of [Gu]). The following identity relating D(γ, 2) with SD(γ)

(D(γ, 2)f)(z) = \frac{f(γ(z))SD(γ)(z)}{2(γ′)^{1/2}(z)}  (A.7)

is valid. To calculate the symbol of the higher order operators, let

\overline{D(γ, n)} ∈ Γ(U, Diff_{U}^{n−2}(O, O))

be the differential operator of order n − 2 defined in (A.3), which operates on the sheaf Γ(U, O). The symbol of this operator \overline{D(γ, n)} is calculated to be

symbol (\overline{D(γ, n)}) = \frac{n(n^{2} − 1)(γ′)^{(n−3)/2}}{12} SD(γ)  (A.8)

The equality (A.7) follows from (A.8).

For any simply connected open set W ⊆ ℂ, let (L_{W} , Ψ_{W}) be a pair consisting of a line bundle L_{W} on W and a section

Ψ_{W} ∈ Γ(W, L_{W}) .

Furthermore, there is a given isomorphism A_{W} : L_{W}^{⊗2} ⟶ T_{W}, such that A^{⊗2}(Ψ_{W}^{⊗2}) coincides with the section \frac{∂}{∂z}, of T_{W}.

Let

D_{W} (n) ∈ Γ(W, Diff_{W}^{n} (L_{W}^{⊗(n−1)}, L_{W}^{⊗(−n−1)}))  (A.9)

be the differential operator of order n on W defined by

D_{W} (n)(fΨ_{W}^{⊗(n−1)}) = \frac{∂^{n}f}{∂z^{n}} Ψ_{W}^{⊗(−n−1)}

where f is a holomorphic function on W. Note that the operator D_{W} (n) remains unchanged if the isomorphism Ψ_{W} is replaced by −Ψ_{W}.

Let γ be as in (A.1), and let (L_{U}, Ψ_{U}) and (L_{U_{1}}, Ψ_{U_{1}}) be as defined above. Fix an isomorphism

F_{γ} : γ^{∗}L_{U_{1}} ⟶ L_{U}

such that (A_{U}^{⊗2}) ∘ F_{γ}^{⊗2} ∘ (A_{U_{1}}^{⊗2})^{−1}(z) = dγ^{−1}(γ(z)), for all z ∈ U, where A_{U} and A_{U_{1}} were defined before (A.9); the homomorphism dγ is the differential of γ mapping T_{U} to γ^{∗}T_{U_{1}}. The above condition determines F_{γ} up-to the sign. This ambiguity of sign is removed by imposing the condition that

(γ′)^{1/2}F_{γ}(γ^{∗}Ψ_{U_{1}}) = Ψ_{U}

The isomorphism between (γ^{∗}L_{U_{1}})^{⊗i} and L_{U}^{⊗i}, induced by F_{γ}, will be denoted by F_{γ}^{⊗i}. Note, using the identity

(γ′)^{1/2}F_{γ}(Ψ_{U_{1}}) = Ψ_{U},

that F_{γ} automatically fixes a square-root (γ′)^{1/2}.

The following lemma is follows from a simple computation.

Lemma A.10. Let γ be as in (A.1), and let (L_{U}, Ψ_{U}) and (L_{U_{1}}, Ψ_{U_{1}}) be as defined above. Then for any holomorphic function f ∈ Γ(U_{1}, O), the identity

F_{γ}^{⊗(−n−1)}D_{U_{1}}(n)(fΨ_{U_{1}}^{⊗(n−1)}) − D_{U}(n)((F_{γ}(Ψ_{U_{1}}))^{⊗(n−1)}f∘γ) = Ψ_{U}^{⊗(−n−1)}D(γ, n)f

is valid, where D(γ, n)f as in (A.2).

Note that the differential operator, which maps fΨ_{U_{1}}^{⊗(n−1)} to Ψ_{U}^{⊗(−n−1)}D(γ, n)f, does not depend upon the choice of the square-root (γ′)^{1/2}. Indeed, if (γ′)^{1/2} is replaced by −(γ′)^{1/2} then F_{γ} changes to −F_{γ}.

In Appendix B we shall derive some implications of the operator D(γ, n) in the context of a Riemann surface equipped with a projective structure.

24 INDRANIL BISWAS AND A. K. RAINA

Appendix B : A Differential operator on a Riemann surface with a projective structure

Let X be a Riemann surface equipped with a projective structure, which we shall denote by 𝔓. Fix a pair (L , Ψ), where

Ψ : L^{⊗2} ⟶ T_{X}  (B.1)

is an isomorphism of line bundles.

Take a coordinate chart (U_{α}, ϕ_{α}) compatible with 𝔓. Recall the pair (L_{W} , Ψ_{W}) constructed in Section 2 for any open set W of ℂ. Substituting ϕ_{α}(U_{α}) for W, fix an isomorphism

θ_{α} : L|_{U_{α}} ⟶ ϕ_{α}^{∗}L_{ϕ_{α}(U_{α})}  (B.2)

such that θ_{α}^{⊗2}∘Ψ^{−1} = dϕ_{α}. The isomorphism from L^{⊗i}|_{U_{α}} to (ϕ_{α}^{∗}L_{ϕ_{α}(U_{α})})^{⊗i}, induced by θ_{α}, is denoted by θ_{α}^{⊗i}. Note that there is a natural isomorphism between L_{W}^{⊗2} and T_{W}. Let D_{α}(n) be the differential operator on U_{α} obtained from the operator D_{W} (n) defined in (A.9), using the isomorphism θ_{α}, namely

D_{α}(n) := (θ_{α}^{⊗(−n−1)})^{−1}∘D_{ϕ_{α}(U_{α})}(n)∘θ_{α}^{⊗(n−1)} ∈ Γ(U_{α}, Diff_{U_{α}}^{n}(L^{⊗(n−1)}, L^{⊗(−n−1)}))

(B.3)

which maps sections of L^{⊗(n−1)} to sections of L^{⊗(−n−1)}. Note that although there are two choices of the homomorphism θ_{α}, differing by multiplication with −1, the operator D_{α}(n) does not depend upon the choice of θ_{α}.

Let (U_{β}, ϕ_{β}) be another coordinate chart compatible with the projective structure 𝔓. Since ϕ_{β} ∘ϕ_{α}^{−1} is a Möbius transformation, Lemma A.10 and the identity (A.4) combine together to imply that the restrictions of the two differential operators D_{α}(n) and D_{β}(n) to the open set U_{α} ∩ U_{β} actually coincide. Thus patching these operators we have a global differential operator

D_{𝔓}(n) ∈ H^{0}(X, Diff_{X}^{n}(L^{⊗(n−1)}, L^{⊗(−n−1)}))  (B.4)

on X which depends on the projective structure 𝔓. This differential operator is called the Bol’s operator [Bo]. (An alternative construction of this operator can be found in [Bi].)

Take another projective structure 𝔓_{1} on X. Let ϕ and ϕ_{1} be two coordinate functions on U_{0} ⊆ X compatible with 𝔓 and 𝔓_{1} respectively. As in (B.2), let θ and θ_{1} be two isomorphisms from L|_{U_{0}} to

PROJECTIVE STRUCTURES ON A RIEMANN SURFACE, II 25

ϕ^{∗}L_{ϕ(U_{0})} and ϕ_{1}^{∗}L_{ϕ_{1}(U_{0})} respectively. With a slight abuse of notation, the isomorphisms of tensor powers induced by θ (and θ_{1}) will also be denoted by θ (and θ_{1}).

Consider the differential operator D(γ, n) defined in (A.2), after substituting U = ϕ(U_{0}) and γ = ϕ_{1} ∘ ϕ^{−1}. Finally, let

D_{U_{0}}^{n}(𝔓, 𝔓_{1}) ∈ Γ(U_{0}, Diff_{U_{0}}^{n−2}(L^{⊗(n−1)}, L^{⊗(−n−1)}))  (B.5)

be the differential operator of order n − 2 defined by

D_{U_{0}}^{n}(𝔓, 𝔓_{1})(θ_{1}^{−1}(fΨ_{ϕ_{1}(U_{1})}^{⊗(n−1)})) = θ^{−1}(Ψ_{ϕ(U)}^{⊗(−n−1)}D(γ, n)f)

where f ∈ Γ(ϕ_{1}(U_{0}), O) and Ψ_{U} as in Appendix A. The identities (A.4), (A.5) and (A.6) together imply that for any two open sets U_{0} and U_{1} of X, the two corresponding differential operators, defined in (B.5), actually coincide over U_{0} ∩ U_{1}. Thus, patching up these locally defined operators we get a global differential operator

D_{X}^{n}(𝔓, 𝔓_{1}) ∈ H^{0}(X, Diff_{X}^{n−2}(L^{⊗(n−1)}, L^{⊗(−n−1)}))  (B.6)

of order n − 2 on X.

The following lemma is immediate from Lemma A.10.

Lemma B.7. Let D_{𝔓}(n) and D_{𝔓_{1}}(n) be the two differential operators for 𝔓 and 𝔓_{1} respectively, as constructed in (B.4). Then the identity

D_{𝔓_{1}}(n) − D_{𝔓}(n) = D_{X}^{n}(𝔓, 𝔓_{1})

is valid.

The space of all projective structures on the Riemann surface X is an affine space for H^{0}(X, K_{X}^{⊗2}) [Gu]. Let ω ∈ H^{0}(X, K_{X}^{⊗2}) be such that

𝔓_{1} = 𝔓 + ω  (B.8)

Note that D_{𝔓_{1}}(2) − D_{𝔓}(2) ∈ H^{0}(X, K_{X}^{⊗2}). The following equality

D_{𝔓_{1}}(2) = D_{𝔓}(2) + \frac{ω}{2}

is valid [HS, CMZ].

Lemma B.7 and the equality (A.8) together immediately imply the following lemma:

Lemma B.9.

symbol (D_{𝔓_{1}}(n) − D_{𝔓}(n)) = symbol (D_{X}^{n}(𝔓, 𝔓_{1})) = \frac{n(n^{2} − 1)ω}{12}

References

[Bi] I. Biswas : A remark on the jet bundles over the projective line. Math. Res. Lett. 3 (1996) 459–566.

[Bo] G. Bol : Invarianten linearer Differentialgleichungen. Abhandlungen Math. Seminar der Universität Hamburg. Bd. 16, Heft 3/4, (1949) 1–28.

[BR] I. Biswas and A. K. Raina : Projective structures on a Riemann surface. International Math. Research Notices. No.: 15 (1996) 753–768.

[CMZ] P. Cohen, Yuri I. Manin and D. Zagier : Automorphic pseudodifferential operators. Algebraic aspects of integrable systems, (Ed. A.S. Fokas and I.M. Gelfand). Progr. Nonlinear Differential Equations Appl., 26, Birkhäuser Boston, Inc., (1997), 17–47.

[De] P. Deligne : Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Math. 163. Springer-Verlag, Berlin (1970).

[Gu] R. C. Gunning : Lectures on Riemann surfaces. Mathematical Notes 2. Princeton University Press, Princeton, New Jersey (1966).

[Ha] R. Hartshorne : Algebraic Geometry. Graduate Texts in Math. 52, Springer-Verlag, New-York-Berlin-Heidelberg (1977).

[Hi] N. Hitchin : Stable bundles and integrable systems. Duke Math. Jour. 54 (1987) 91–114.

[HS] N. Hawley and M. Schiffer : Half-order differentials on Riemann surfaces. Acta Math. 115 (1966) 199–236.

[K] V. Kac : Vertex algebras for beginners. American Math. Soc., Providence, RI

PROJECTIVE STRUCTURES ON A RIEMANN SURFACE, II 27

[R1] A.K. Raina : An algebraic geometry view of a model quantum field theory on a curve. In: Geometry from the Pacific Rim. Eds.: J. Berrick., B. Loo and H. Wang, Walter de Gruyter, Berlin, New York (1997).

[R2] A.K. Raina : An algebraic geometry view of currents in a model quantum theory on a curve. C. R. Acad. Sci. Paris Sér.I Math. 318 (1994) 851-856.

[Ty] A. Tyurin : On periods of quadratic differentials. Russian Math. Surveys 33 (1978) 169–221.

[W] E.J. Wilczynski : Projective differential geometry of curves and ruled surfaces. Chelsea, New York (1961).

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, INDIA

  E-mail address: indranil@math.tifr.res.in

Theoretical Physics Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, INDIA

  E-mail address: raina@theory.tifr.res.in

current address of AKR: Max-Planck-Institut für Mathematik, Gottfried-Claren Strasse 26, 53225 Bonn, GERMANY

  E-mail address: raina@mpim-bonn.mpg.de