Comptes Rendus
Mathématique
Anoop Singh
Moduli space of rank one logarithmic connections over a compact Riemann surface
Volume 358, issue 3 (2020), p. 297-301.
<https://doi.org/10.5802/crmath.41>
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2020, 358, n3, p. 297-301
https://doi.org/10.5802/crmath.41
Algebraic Geometry, Analytic Geometry /Géométrie algébrique, Géométrie analytique
Moduli space of rank one logarithmic
connections over a compact Riemann surface
Anoop Singha
aHarish-Chandra Research Institute, HBNI, Chhatnag Road, Jhusi, Prayagraj 211 019, India.
E-mail:[email protected].
Abstract.LetMXdenote the moduli space of rank one logarithmic connections singular over a finite subset Sof a compact Riemann surfaceXwith fixed residues. We study the rational functions intoMX. We prove that there is a natural compactification ofMXand the Picard group ofMXis isomorphic to the Picard group of Picd(X).
2020 Mathematics Subject Classification.14D20, 14C22, 14E05.
Manuscript received 9th May 2019, revised and accepted 20th March 2020.
1. Introduction
The moduli space of logarithmic connection has been constructed in [4]. LetXbe a compact Rie- mann surface. In [1], several properties, like Picard group, algebraic functions and compactifica- tion have been studied for the moduli space of logarithmic connections in a holomorphic vector bundle overXwith singularity exactly over one point. LetSbe a finite subset ofXandMXdenote the moduli space of rank one logarithmic connection singular overSwith fixed residues. In [6], it has been proved that there is a natural symplectic structure onMXand there are no nonconstant algebraic functions onMX. In the present article, we reconsider the moduli spaceMXand try to study the biholomorphic class of theMX, see Section 3. We prove that any rational map from a normal variety toMXis defined on the whole of the normal variety, see Section 4. Also, we show, by explicit manipulation that there is a natural compactification ofMXand compute the Picard group ofMX, see Section 5.
2. Preliminaries
LetXbe a compact connected Riemann surface andS={x1, . . . ,xm} be a finite subset consisting of distinct points ofX. The setS will be fixed throughout this article. We denote byS =x1+
· · · +xmthe reduced effective divisor onXassociated to the finite setS. LetΩ1X(logS) denote the sheaf of logarithmic differential 1-forms alongS, see [5]. Notion of logarithmic connection was
298 Anoop Singh
introduced by P. Deligne in [2]. LetLbe a holomorphic line bundle overX. We will denote the fibre ofLover any pointx∈XbyL(x). A logarithmic connection onLsingular overSis aC-linear map
D:L→Ω1X(logS)⊗L (1)
which satisfies the Leibniz identity
D(f s)=f D(s)+df⊗s, (2)
where,f is a local section ofOXandsis a local section ofL.
Let Res(D,xβ) denote theresidueof the logarithmic connectionDat pointxβ∈S(see [2] for the details). IfDandD0are two logarithmic connections onLsingular overSwith
Res(D,xβ)=Res(D0,xβ), (3)
thenD0=D+θ, whereθ∈H0(X,Ω1X). Thus, the space of all logarithmic connections on a given holomorphic line bundleL, singular overS, and satisfying (3) is an affine space for H0(X,Ω1X).
3. Biholomorphic class of the moduli space
In this section, we will assume that genus(X)=g≥2. For each pointxj∈S, fix a complex number rj∈C, forj=1, . . . ,m. Letdbe a fix integer which denotes the degree of a line bundle overX. By a pair (L,D) overX, we mean thatLis a line bundle overX of degreed andDis a logarithmic connection inLsingular overSwith residues to be given complex numbersrj∈Cfor eachxj∈S.
LetMXdenote the moduli space of pairs (L,D) overX(see [4]).
LetX0=X\Sand fix a pointx0∈X0. Now, consider the fundamental group
π1(X0,x0)= (
a1,b1, . . . ,ag,bg,γ1, . . . ,γm
¯
¯
¯
¯
¯
g
Y
i=1
[ai,bi]γ1. . .γm=1 )
ofX0atx0, whereai,bi fori=1, . . . ,gare generators of fundamental group of genusgcompact Riemann surface X andγj are the homotopy class of loops atx0around xj’s. LetBg ={ρ: π1(X0,x0)→C∗|ρ(γj)=exp(2πp
−1rj)}. Now, the representation space Hom(π1(X0,x0),C∗) is complex algebraic variety in a natural way andBgis a closed subvariety of representation space.
Moreover,Bg is smooth, because the representations are irreducible. The spaceBg does not depend on the pointx0ofX0.
We have a map
Φ:MX→Bg (4)
sending any pair (L,D)∈MX to its monodromy representationρ :π1(X0,x0)→C∗such that ρ(γj)=exp(2πp
−1rj). Note that both the spacesMX andBg have underlying structure of complex manifold.
Therefore, we have
Proposition 1. The mapΦ:MX →Bg defined in(4)is a biholomorphism. Further, if(Y,T = {y1, . . . ,ym})is another compact connected Riemann surface of genus g ≥2, with finite subset T consisting of m elements andMY is the correspoding moduli space of logarithmic connections singular over T with same set of residues, then the two complex manifoldsMX and MY are biholomorphic.
C. R. Mathématique,2020, 358, n3, 297-301
4. Rational functions into the moduli space
LetMXbe the moduli space described in Section 3. Let
p:MX→Picd(X) (5)
be the projection defined by sending any pair (L,D) toL, where Picd(X) is a Picard variety of line bundles of degreed. Fix a pair (L0,D0)∈MX. Let L∗0 be the dual line bundle ofL0 and D∗0 denote the logarithmic connection inL∗0, induced fromD0. ThenD∗0 is sigular overSwith Res(D0∗,xj)= −rjfor allj=1, . . . ,m.
Now, for any (L,D)∈MX,L∗0⊗Lis a holomorphic line bundle with holomorphic connection
∇ =D∗0⊗1L+1L∗0⊗D. Note that Res(∇,xj)=0, for allj=1, . . . ,m.
LetMXh denote the moduli space of pairs (L,D), whereL is a holomorphic line bundle of degree 0, andDis a holomorphic connection inL. Then we have a map
Ψ(L0,D0):MX→MXh (6) sending (L,D) to (L∗0⊗L,D∗0⊗1L+1L∗0⊗D), which is an isomorphism. Therefore, the structure theory for both the moduli spaces are same. Note that the moduli spaceMXis an algebraic group.
We have an extension of group schemes
0→H0(X,Ω1X)→MX→J(X)→0 (7) From the above extension (7) ofJ(X) by H0(X,Ω1X), it is clear that there are no rational curves onMX. Now, we have following Theorem, which can be proved either using Hartogs’ Theorem or [3, Theorem 9.4, p. 103].
Theorem 2. Let f :Z99KMX be a rational map from a normal variety Z to the moduli space MX. Suppose that f is defined in a complement of a subvariety of Z of codimension≥2. Then f is defined on whole of Z .
Proof. From Proposition 1,MX is biholomorphic to the affine varietyBg. So, we getf :Z99K Bg. By hypothesis,f is defined in a complement of a subvariety ofZof codimension≥2, so the
conclusion follows from Hartogs’ Theorem.
Remark 3. Alternatively, given a rational mapf :Z99KMX, defines a rational mapZ99KMX→ J(X) by composition. Now, using [3, Theorem 9.4, p. 103], which says that any rational map from a normal variety to an abelian variety is defined on whole of the domain, and hence we are done.
Corollary 4. Every rational map f:CPn99KMXfrom projective space toMX is constant.
Proof. AsCPnis a normal variety, from Theorem 2f is defined on whole ofCPn. Moreover,CPn isCP1connected andMXdoes not contain any rational curve, which is equivalent to the fact that any rational map fromCP1toMXis constant. Thusf is constant.
5. Compactification and the Picard group of moduli space
The following proposition can be proved in more general context of any extension of an abelian variety by an additive group scheme using similar technique. Here, we restrict ourselves to the moduli spaceMX.
Proposition 5. There exists an algebraic vector bundleπ:E→Picd(X)such thatMXis embedded inP(E)withP(E) \MXas the hyperplane at infinity.
300 Anoop Singh
Proof. Let p :MX →Picd(X) be the map as defined in (5). Then for any L∈ Picd(X), the fiberp−1(L) is an affine space modelled on H0(X,Ω1X). In fact,p:MX →Picd(X) is aΩ1Picd(X)- torsor. Since the dual of an affine space is a vector space, so the dualp−1(L)∨={ϕ:p−1(L)→ C|ϕis an affine linear map} is a vector space overC. Define a sheafEon Picd(X) as follows;
For every Zariski open subsetUof Picd(X), sections ofEoverUis the following set E(U)=n
f :p−1(U)→Cis a regular function :f|p−1(L)∈p−1(L)∨o .
Clearly,Eis anOPicd(X)-module. MoreoverEis a locally free sheaf over Picd(X), and hence we get an algebraic vector bundleπ:E→Picd(X)
Let (L,D)∈MX, and define a mapΦ(L,D):p−1(L)∨→C, by Φ(L,D)(ϕ)=ϕ[(L,D)], which is nothing but the evaluation map. Now, the kernel Ker(Φ(L,D)) defines a hyperplane inp−1(L)∨ denoted byH(L,D). LetP(E) be the projective bundle defined by hyperplanes in the fiber ofπ. Then, we have natural projection
πe:P(E)→Picd(X) (8) induced fromπ. Define a map
ι:MX→P(E) (9)
by sending (L,D) to the hyperplaneH(L,D), which is clearly an open embedding. SetY =P(E) \ MX. Thenπe−1(L)∩Y is a linear hyperplane inπe−1(L) for everyL∈Picd(X), and henceY is a
hyperplane at infinity. This completes the proof.
Thus, from the Proposition 5,P(E) is the natural compactification of the moduli spaceMX. Further,pdefined in (5), induces a homomorphism of Picards groups
p∗: Pic(Picd(X))→Pic(MX) (10)
that sends a line bundleξover Picd(X) to a line bundlep∗ξoverMX. We have
Theorem 6. The homomorphism p∗: Pic(Picd(X))→Pic(MX)defined in(10)is an isomorphism.
Proof. First we show thatp∗in (10) is injective. LetΞ→Picd(X) be a line bundle such thatp∗Ξ is a trivial line bundle overMX. Giving a trivialization ofp∗Ξis equivalent to giving a nowhere vanishing section ofp∗ΞoverMX. Fixζ∈H0(MX,p∗Ξ) a nowhere vanishing section. Take any pointz∈Picd(X). Then,
ζ|p−1(z):p−1(z)→Ξ(z)
is a nowhere vanishing map. Notice thatp−1(z)∼=CgandΞ(z)∼=C. Now, any nowhere vanishing algebraic function on an affine spaceCg is a constant function, that is, ζ|p−1(z) is a constant function and hence corresponds to a non-zero vectorαz∈Ξ(z). Sinceζis constant on each fiber ofp, the trivializationζofp∗Ξdescends to a trivialization of the line bundleΞover Picd(X), and hence giving a nowhere vanishing section ofΞover Picd(X). Thus,Ξis a trivial line bundle over Picd(X). It remains to show thatp∗is surjective.
LetΘ→MX be an algebraic line bundle. SinceMX ,→P(E) follows from (9), in the proof of above Proposition 5, we can extendΘto a line bundleΘ0overP(E). Further, from the morphism πe:P(E)→Picd(X) in (8) in the above Proposition 5 we have
Pic(P(E))∼=πe∗Pic(Picd(X))⊕ZOP(E)(1). (11) Therefore,
Θ0=πe∗L⊗OP(E)(l) (12) whereLis a line bundle over Picd(X) andl∈Z. SinceY =P(E) \MX is the hyperplane at infinity, again from (11) the line bundleOP(E)(Y) associated to the divisorY can be expressed as
OP(E)(Y)=πe∗L1⊗OP(E)(1) (13)
C. R. Mathématique,2020, 358, n3, 297-301
for some line bundleL1over Picd(X). Now, from (12) and (13), we get
Θ0=πe∗(L⊗(L∨1)⊗l)⊗OP(E)(l Y). (14) Since, the restriction of the line bundleOP(E)(Y) to the complimentP(E) \Y =MX is the trivial line bundle and restriction ofπetoMXis the mappdefined in (5), therefore, we have
Θ=p∗(L⊗(L∨1)⊗l). (15)
This completes the proof.
Corollary 7. The homomorphism p∗:J(Picd(X))→J(MX)is an isomorphism of Jacobians.
Acknowledgements
The author would like to thank anonymous referee for detailed and helpful comments. The author is also deeply grateful to Prof. Indranil Biswas for useful discussions.
References
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