Hitchin system on singular curves I

a rXiv:h ep-th/03369v18M ar23ITEP-TH-76/02Hitchin system on singular curves I A.Chervov 1,D.Talalaev 23Institute for Theoretical and Experimental Physics 4Abstract.In this paper we study Hitchin system on singular curves.Some examples of such system were first considered by N.Nekrasov (hep-th/9503157),but our methods are different.We consider the curves which can be obtained from the projective line by gluing several points together or by taking cusp singularities.(More general cases of gluing subschemas will be considered in the next paper).It appears that on such curves all ingredients of Hitchin integrable system (moduli space of vector bundles,dualizing sheaf,Higgs field etc.)can be explicitly described,which may deserve independent interest.As a main result we find explicit formulas for the Hitchin hamiltonians.We also show how to obtain the Hitchin integrable system on such a curve as a hamiltonian reduction from a more simple system on some finite-dimensional space.In this paper we also work out the case of a degenerate curve of genus two and find the analogue of the Narasimhan-Ramanan parameterization of SL(2)-bundles.We describe the Hitchin system in such coordinates.As a demonstration of the efficiency of our approach we also rederive therational and trigonometric Calogero systems from the Hitchin system on cusp and node with a marked point.Contents1Introduction21.1Constructing Hitchin system (3)1.2Narasimhan-Ramanan parameterization (4)2Algebraic-geometric background52.1Curves defined by gluing points with multiplicities (5)2.2Canonical(dualizing)sheaf on curves defined by gluing points (6)2.2.1Description of the dualizing sheaf (6)2.2.2Serre’s pairing in theˇCech description of H1(F)on a singular curve.72.3Holomorphic bundles on singular curves (9)2.3.1Projective modules over an affine part (9)2.3.2Vector bundles over the projectivization (11)2.4Endomorphisms of MΛ (12)2.4.1Endomorphisms of the module MΛover an affine chart (12)2.4.2Endomorphisms of the bundle MΛover the projectivization (13)2.5Description of End(MΛ)⊗K (14)2.5.1Examples of node and cusp curves (14)2.5.2Curves obtained by gluing points without multiplicities (15)2.6Description of H1(End(MΛ)) (16)2.6.1Curves obtained by gluing points without multiplicities (16)2.6.2The cusp curve (19)2.7Canonical1-form on the cotangent bundle to the moduli space of vector bundles in terms ofΦ,Λ2.7.1Curves obtained by gluing points without multiplicities (20)2.7.2Curves with many cusps (24)2.8Hitchin system (25)3Trigonometric and rational Calogero-Moser systems253.1Node (25)3.2Cusp (28)4Curves with two cusps294.1Construction (29)4.2Hitchin Hamiltonians (31)4.3Degenerated Narasimhan-Ramanan parameterization (32)5Conclusion331IntroductionHitchin system was introduced in[1]as an integrable system on the cotangent bundle of the moduli space T∗M of stable holomorphic bundles on an algebraic curveΣ.This phase space can be obtained by the Hamiltonian reduction by the gauge group action from the space of pairs d′′A,Φ,where d′′A is the operator defining the holomorphic structure on the bundle V andΦis an endomorphism of this bundle,more preciselyΦ∈Ω0,1(Σ,End(V)) where the gauge group is the group of GL N-valued functions onΣ.The invariant sym-plectic structure on the“big”space can be written as:ω= ΣT rδΦ∧δd′′A.(1)The zero level of the moment map is described by the condition d′′AΦ=0which means thatΦis holomorphic with respect to the induced holomorphic structure on the bundle End(V).It turns out that the system of quantities T rΦk,treated as vector functions on the phase space,Poisson-commute and their number is exactly half the dimension of the phase space.The importance of Hitchin system and its generalizations[2,3,4]in modern math-ematical physics cannot be overestimated.Many well-known systems can be obtained as particular cases.Automatically they inherit the universal construction of a family of commuting hamiltonians as well as the geometric description of the hamiltonianflows, the Lax representation,and the“action-angle”variables.This domain is also connected with important questions in mathematical physics like the geometric Langlands correspondence[5,6,7],conformalfield theory(in a sense Hitchin system is a Knizhnik-Zamolodchikov-Bernard equation on the critical level) [6,8],non-linear partial differential equations such as KP[9],Davey-Stewartson equation [10],Nahm’s equations describing monopoles[11],and other problems(see for example [12],[13]).Despite its importance Hitchin system is far from being fully investigated.One of the reasons for such a situation is that the moduli space of vector bundles is a compli-cated manifold and it is difficult to choose“good”coordinates on it to write down the Hamiltonians explicitly.Several attempts have been done in[2]and in[14].Nevertheless such descriptions appear to be complicated and do not answer many questions(at least yet).So it is important to work out some examples of Hitchin system which on the one hand are sufficiently simple and on the other hand are rich enough tofind out general methods for solving Hitchin system and to understand such phenomena as the separation of variables and the geometric Langlands correspondence.The approach elaborated in this paper can be applied in rather specific cases,namely when the base algebraic curve is singular and its normalization is a rational curve.Its richness is proved by the number of nontrivial examples.For such curves all ingredients of Hitchin systems(vector bundles,their endomorphisms,the moduli space of vector bundles,the dualizing sheaf,Higgsfields)can be described very explicitly and in a quite simple way.So we hope that the understanding of such systems will shed light on the general case.We proceed by formulating the main results of this paper.1.1Constructing Hitchin systemConsider the curve Σproj which results from gluing N distinct points P i on C P 1to one point (i.e.the curve which is obtained by adding the smooth point ∞to the curve Σaff =Spec {f ∈C [z ]:∀i,j f (P i )=f (P j )}.•A rank r vector bundle on such a curve corresponds to a rank r module M Λover the affine part given by the subset of vector-valued functions s (z )on C i.e.s (z )∈C [z ]r which satisfy the conditions:s (P 1)=Λi s (P i ).The moduli space of vector bundles on Σproj is the factor by GL r of the set of invertible matrices Λi ,∀i =2,...,N where GL r acts by conjugation.(See section 2.3.2,theorem 1).•The basis of global sections of the dualizing sheaf on Σproj can be described asmeromorphic differentials on C given by dzz −P i ,∀i =2,...,N (see section 2.2.1,example 3).•The endomorphisms of the module M Λare matrix valued polynomials Φ(z )such that Φ(P 1)=Λi Φ(P i )Λ−1i ,∀i =2,...,N (see section 2.4.1,proposition 6).The action of Φ(z )on s (z )is:s (z )→Φ(z )s (z ).The space H 1(End (M Λ))can be described as the space gl [z ]of matrix valued polynomials factorized by the subspaces:End out ={χ(z )∈gl [z ]|χ(z )=const }and End in ={χ(z )∈gl [z ]|χ(P 1)=Λi χ(P i )Λ−1i ,∀i =2,...,N }.The elements of H 1(End (M Λ))are the tangent vectors to Λi ,the element χ(z )gives the following tangent vector to Λi :δχ(z )Λi =χ(P 1)Λi −Λi χ(P i )(2)•The global sections of H 0(End (M Λ)⊗K )(”Higgs fields”)are described asΦ(z )= i =2,...,N −Λi Φi Λ−1i z −P i dz,(3)where i =2,...,N −Λi Φi Λ−1i +Φi =0(see section 2.5.2,proposition 9).Let us mention that precisely this condition arises as the zero moment level condition (see section2.7,formula 23).The symplectic form on the cotangent bundle to the moduli space can be described as the reduction of the form on the space Λi ,Φi ,∀i =2,...,N given by− i =2,...,NT rd (Φi Λ−1i )∧d Λi(4)(see section 2.7,proposition 15).Result 1:The Hitchin system on the curve Σproj can be described as the system with a phase space which is the hamiltonian reduction of the space Λi ,Φi with the symplectic form 4;the reduction is taken by the group GL (r ),which acts by conjugation,the Lax operator is given by 3.Remarks:For the case of gluing two points the same Lax operator has been proposed by N.Nekrasov([2]),though his methods are different from ours,and the explicit descrip-tion of bundles,dualizing sheaf,endomorphisms etc is absent in his approach.When one glues several groups of points:P i=P j,Q i=Q j...it is obvious how to modify all propositions above,for example the Lax operator becomes:i=2,...,N−ΛiΦiΛ−1iz−P idz+ i=2,...,N−˜Λi˜Φi˜Λ−1i z−Q i dz.Actually one can easily guess the Lax operator above from the case of gluing two points: one mustfirst consider the gluing of N−1pairs of points together P2=R2,P3=R3..., then take R k=P1.Analogously we obtain all propositions for the case of a curve with several cusps atpoints P i on C P1.•The curve:Σaff=Spec{f∈C[z]:∀if′(P i)=0}.•The modules:s(z)∈C[z]r:s′(P i)=Λi s(P i).•The basis of global sections of the dualizing sheaf:dz(z−P i)2+[Λi,Φi]dz4.(7)The Hitchin Hamiltonians in this case areH1=T rΦ21=4p21t1+p22t3+4p1p2t2;H2=2T rΦ1Φ2+(z1−z2)2T r[Λ1,Φ1]2=4p1p2t1+4p2p3t3+(8p1p3+2p22)t2−2(z1−z2)2p22(t1t3−t22);H3=T rΦ22=4p23t3+p22t1+4p2p3t2,wheret1=T rΛ21,t2=T rΛ1Λ2,t3=T rΛ22and p i are the corresponding conjugated variables.This paper is organized as follows:thefirst section contains all algebraic-geometric preliminaries.In the second section we work out the case of a rational curve with double point and cusp and show that the arising systems are the trigonometric and rational Calogero-Moser system with spin.In the third section we treat the case of a rational curve with two cusps,which is a curve of algebraic genus2.We consider the moduli space of holomorphic SL2-bundles on it and construct the analog of the Narasimhan-Ramanan parameterization in the singular case.In conclusion we state some open problems for future work.Acknowledgements.The authors are grateful for their friends and colleagues for use-ful and stimulating discussions:N.Amburg,Yu.Chernyakov,V.Dolgushev,V.Kisunko, A.Kotov,D.Osipov,S.Shadrin,G.Sharygin,A.Zheglov,A.Zotov.The authors are thankful to B.Machet for careful reading of the manuscript.2Algebraic-geometric background2.1Curves defined by gluing points with multiplicities. Let us consider a curveΣand some effective divisor D= i n i P i(n i>0)such that degD>1.One defines a new curveΣD by,roughly speaking,gluing all points P i with multiplicities n i to one point P;formally speaking we define the structure sheaf O(ΣD) to be a subsheaf of O(Σ)with the properties:f(P i)=f(P j);f k(P i)=0,k=1,...,n i−1. In Serre’s terminology this is“the curve defined by the module D”(see[16]ch.4sect.4).The new curveΣD obviously has one more singular point P,the normalization ofΣD isΣ(of course,ifΣis a smooth curve).Example1Main example to keep in mind.If we considerΣ=C1and D=P1+P2 we obtain the curve Spec{f∈C[z]:f(P1)=f(P2)}which is called node(or doublepoint in another terminology),it is an affine curve which can be defined by the equation√y2=x2(x+a),z=y a,P2=−Example2If we considerΣ=C1and D=2P we obtain the curve Spec{f∈C[z]: f′(P)=0}which is called cusp,it is an affine curve which can be defined by the equation y2=(x−a)3,z=yline bundle on a singular curve).The dualizing sheaf is not always locally free.It is true for complete intersections and arbitrary plane curves(see[16]for the discussion). Example3Consider the node curveΣ,i.e.,C P1with two points P1,P2glued together, so the affine part of this curve is Spec{f∈C[z]:f(P1)=f(P2)}.The sections ofthe sheaf K node on the chart without infinity are described as cdzz−P2+f(z)dz,where f(z)is holomorphic.On the other charts one obtains the sections of K node by the usual localization procedure:on the charts,which do not contain the singular point,the sections of K node are the usual holomorphic1-forms.So the only global section of K nodeis cdzz−P2.One can easily guess what is going on for the case when we glue n points P1,...,P n on C P1together:for example the basis of global holomorphic differentials can be given by dzz−P i,for i=2,...,n.Example4Consider the cusp curveΣ,i.e.,C P1with the point P glued with multi-plicity2,so the affine part of this curve is Spec{f∈C[z]:f′(P)=0}.The sections of the sheaf K cusp on the chart without infinity are described by cdz(z−P)2is the only global section of K cusp.The description of the canonical class on an n-cusp curve when we glue one point P on C P1with multiplicity n is analogous.For example the basis of global holomorphic differentials can be given by dzx -is the coordinate on the normalization of this curve.Theholomorphic differential on an elliptic curve is given by the formula:dxz(z2−a)=2dza−2dza,so we obtain a differential which satisfies Serre’s conditions:the orders of the poles are1 and the sum of its residues is equal to zero.If one puts a=0which corresponds to the cusp curve we obtain as a limit the differential2dzdoes not work,at least naively,so we prefer theˇCech description of H1(F),which works perfectly even for singular curves.Let us cover the curveΣD by the two charts U P=ΣD\∞and U∞=ΣD\P,where we denote by∞an arbitrary point inΣD,distinct from P(recall that P is the only singular point obtained by gluing the points P i together).This choice of covering is the most convenient for our calculation and will be used throughout the paper.One knows that a curve minus any point is an affine curve,so this covering is sufficient to calculate the cohomology of the coherent sheaves:H1(F)=F(U∞ U P)/(F(U P) F(U∞)). Proposition3The Serre’s pairing between f∈H1(F)and h⊗w∈H0(F∗⊗KΣD)can be described as follows:consider˜f∈F(U∞ U P)the representative of the element f, then the pairing is given by:<f,h⊗w>= i Res P i<˜f,h>w(9)This pairing is well-defined(i.e.it does not depend on the choice of the representative˜f) and non-degenerate.Corollary2So one obviously obtainsdimH1(OΣD )=dimH0(KΣD)=H1(OΣ)+degD−1.Let us sketch why the pairing is well-defined and nondegenerate.In order to see that the pairing is well-defined one needs to check that it is zero for˜f∈F(U P)and for˜f∈F(U∞).Indeed,if˜f∈F(U P)then g=<˜f,h>belongs to OΣD(U P)and hence itspullback˜g has the same values at the preimages of the point P˜g(P i)=˜g(P j)=:g(P) and satisfies the conditions˜g(k)(P i)=0,k=1,...,n i−1,∀i.Hence:<˜f,h⊗w>= i Res P i(˜g⊗w)=g(P) i Res P i w=0,where we used the fact that the sum of the residues of a meromorphic differential is zero.For an element˜f∈F(U∞)one has<˜f,h⊗w>= i Res P i(˜g⊗w)=−Res∞(˜g⊗w)=0,because both w and˜g are regular at∞.To show that this paring is nondegenerate it is sufficient to prove that there is no meromorphic differential w of the prescribed type such that<f,w>=0∀f∈O(U∞ U P).(10) We can present n−1functions f i on the normalization with their only pole of sufficiently high order at∞and with nondegenerate matrix of their derivatives up to n i−1order at the points P i.The condition of orthogonality<f i,ω>=0,i=1,...,n−1is a system of linear homogeneous equations on the negative coefficients ofωat the points P i.The only solution of this system is zero vector and one obtains that the pairing is nondegenerate due to the absence of holomorphic differentials on C P1.2.3Holomorphic bundles on singular curves2.3.1Projective modules over an affine partHolomorphic bundles on a non-singular manifold can be described by sheaves of its sec-tions.Such sheaves are locally free or equivalently(by a general theory)they are sheaves of projective modules over the structure sheaf.The geometric description of a holomor-phic bundle on a singular manifold is problematic in contrast with the algebraic side which is unambiguous.Definition1The holomorphic bundle on a singular curveΣis the sheaf of projective modules over O(Σ).(It is known that a projective module is locally free(in Zariski’s topology)also for singular manifolds,so it is equivalent to speak about projective or locally free modules).First let us describe the projective modules over the affine curveΣD aff which is obtained by gluing the points P i on C with multiplicities n i to one point.As usual we denote by D the effective divisor i n i P i.We will describe such modules as submodules of the trivial module on the normalization.Proposition4Consider the curveΣD aff given by Spec{f∈C[z]:∀i,j f(P i)= f(P j);f′(P i)=...=f n i−1(P i)=0}.Consider the following set of matrices:invertible ma-tricesΛ2,...,ΛN and arbitrary matricesΛl i∈Mat(r),where i=1,...,N;l=1,...,n i−1. The subset of vector-valued polynomials s(z)on C i.e.s(z)∈C[z]r such that they satisfy the conditions:s(P1)=Λi s(P i);s(l)(P i)=Λi l s(P i)is a projective module of rank r over the algebra{f∈C[z]:∀i,j f(P i)=f(P j);f′(P i)=...=f n i−1(P i)=0}.All projective modules can be obtained in this way.Notation Let us denote the module and the bundle described above by MΛ.(We will use the same notation MΛfor the vector bundles on the projectivizationΣD proj of the curveΣD aff,we hope that it will not be confusing).Remark3One can easily show(calculating the divisor for example)that in the case of rank r=1all these modules are non isomorphic for differentΛ’s.For the r>1it is certainly not true,but we will see below that the vector bundles on the projectivization ΣD proj of the curveΣD aff corresponding to these modules are isomorphic iffallΛ’s are conjugated by the same constant matrix C.Remark4Let us mention that even in the case r=1if one considers the analytic topology then all bundles MΛbecome isomorphic,because from the exponential sequence one can easily see that H1(O∗)=H1(O)and H1(O)=0on any affine curve.But,for projective curves the GAGA principle guarantees the same results for both the algebraic and analytic setups.We will be interested in projective curves so one must not pay too much attention to the remarks above.Sketch of Proof.This proposition is quite simple so let us only sketch out its proof. Letπ:C→ΣD aff be the normalization map.Consider any torsion free module F of rank r.Soπ∗F is a torsion free rank r module,but all such modules over C[z]are trivial, soπ∗F=C[z]r.Considerπ∗π∗F.It is isomorphic to C[z]r considered as a module over O(ΣD)={f∈C[z]:∀i,j f(P i)=f(P j);f′(P i)=...=f n i−1(P i)=0},it is a torsionfree,but not a projective module (the fiber at the singular point P jumps).We have the exact sequence F →π∗π∗F →C rg ,where C rg is a skyscraper sheaf at the point P ,r is the rank,and g =degD −1.So we see that any torsion free module F can be described as the kernel of the map φ:C [z ]r →skyscraper at P .Such maps φbijectively correspond to the maps ˜φ:fiber at P of module C [z ]r →C rg .The finite dimensional linear space fiber at P of module C [z ]r in our case is the space ⊕P i C r (n i +1).So in general the kernel of such a map can be described by the maps:Λi :C r P i →C r P i +1and Λj i :C r 0,P i →C r j,P i .One can easily see that if we are not in the general case or if Λi ’s are not invertible the modules will not be projective.Let us recall that the fiber at a point P of a module M over a ring R is defined as M loc /I loc M loc ,where I is the maximal ideal of the point P and loc means “localization at point P ”.Example 5Consider the node (or double point)curve:Σ=Spec {f ∈C [z ]:f (P 1)=f (P 2)}.Then the rank 1modules (line bundles or rank one torsion free sheaf)are parameterized by one complex number λ∈C .They are given by the condition {s (z )∈C [z ]:s (P 1)=λs (P 2)}.Obviously M Λis a torsion free module.For λ=0one can see that it is not a projective module,because the fiber at the point zero jumps and becomes two-dimensional,which is impossible for locally free modules.It is a nice exercise to calculate the divisor of the line bundle M Λ.For λ=0one can see that this module is locally free (hence projective).(A rank 1projective module becomes free on any open set which does not contain any representative of its divisor).This example illustrates also that the moduli space of line bundles (the so called generalized Jacobian)on a singular curve is non compact.In this case Jac ∼=C ∗and it is also an isomorphism of groups,where as usually one considers the tensor product as a group operation on line bundles.Jac can be compactified by the torsion free modules.In this case one should add one module corresponding to λ=0,(it coincides with the module λ=∞,i.e.the module {s ∈C [z ]:0=s (P 2)}).It can be shown that if one constructs properly the algebraic structure on the set of torsion free sheaves of rank 1as a manifold it coincides with the curve Σproj itself.This result is related to the fact that the Jacobian of an elliptic curve is isomorphic to this curve.This can be done by constructing the Poincar´e line bundle on the product of the curve with itself.Given the divisor D =P 1+P 2+...+P N one obtains the curve ΣD =Spec {f ∈C [z ]:f (P 1)=f (P 2)=...=f (P N )}by gluing the points P i ∈C together.The rank r modules can be described as subsets of vector-valued polynomials s (z )on C i.e.s (z )∈C [z ]r such that they satisfy the conditions:s (P 1)=Λi s (P i ),i =2,...,N ,where Λi are arbitrary invertible matrices.Example 6Consider the cusp curve:Σ=Spec {f ∈C [z ]:f ′(P )=0},recall that it means that we glued the point P with itself with multiplicity 2.The modules can be described by {s (z )∈C [z ]:s ′(P )=λ1s (P )}.In this example for all λ1∈C these modules are projective.So C is the moduli space of line bundles.It can be compactified by adding one point λ1=∞(i.e.the module {s ∈C [z ]:0=s (0)},which is the same as the maximal ideal of the singular point z =0,and the same as the direct image ofO norm and the same as just C [z ]considered as a module over our algebra).A properly introduced algebraic structure will show that this moduli space is the curve Σproj itself,not CP 1as one might think from a naive point of view.Analogously,for the curve ΣD =Spec {f ∈C [z ]:f ′(P )=f ′′(P )=...=f N (P )=0}corresponding to the divisor D =NP the rank r modules can be described as subsets of the vector-valued polynomials s (z )on C i.e.s (z )∈C [z ]r such that they satisfy the conditions:s (i )(P )=Λi s (P ),i =1,...,N ,where Λi are arbitrary matrices.2.3.2Vector bundles over the projectivizationThe modules M Λand M ˜Λare equivalent,if there exists an invertible map of modules K (z ):M Λ→M ˜Λ.Recall that we denoted by ΣD proj the projective curve which we obtain from the affine curve Spec {f ∈C [z ]∀i,j :f (P i )=f (P j ),f k (P i )=0,k =1,...,n i }by adding one smooth point at infinity.The modules M Λgive a vector bundle over Σproj in an obvious way:we define the sheaf which is a trivial rank r module over the chart containing infinity and not containing the singular point and which is the module M Λ(or more precisely its localization)over the chart which contains the singular point.Let us denote these bundles by M Λ(we hope that it will not be too confusing to denote by the same M Λthe projective module over the affine chart and the corresponding vector bundle on the projectivization).The degree of such bundles equals zero.The vector bundles are equivalent if there exists an invertible map of modules K (z ):M Λ→M ˜Λover each chart.So we see that K (z )is a matrix polynomial which must be regular both at infinity and on the affine part ΣD .The only such function K (z )is a constant.So we obtain:Proposition 5The vector bundles M Λover Σproj are isomorphic if there exist a constant matrix K such that ∀i,j Λi =K ˜Λi K −1;Λj i =K ˜Λj i K −1.We obtain the following corollary:Theorem 1The open subset in the space of semistable vector bundles of degree zero and rank r over the curve Σproj can be described asM =Λ/GL rwhere Λis the set of matrices {Λi ,Λjk };i =2,...,N ;k =1,...,N ;j =1,...,n i ,with Λi invertible and Λj k arbitrary,and the GL r -action is defined by the common conjugation byconstant matrices.Remark 5Let us also note that for the bundle M Λthe pullback π∗M Λis the trivial bundle on C P 1,where π:C P 1→Σproj is the normalization map.Obviously there are lots of bundles F of degree zero on Σproj such that π∗F are not trivial bundles but some bundles of the type ⊕k =1,...r O (t k )such that t k =0.So by no means we obtain all bundles on Σproj as bundles M Λfor some Λ.But nevertheless the general stable and possibly semistable bundles satisfy the property that π∗F is a trivial bundle on C P 1,and so it is easy to see from our previous description of projective modules that the general semistable bundles can be obtained as the bundles M Λfor some Λ.2.4Endomorphisms of MΛ2.4.1Endomorphisms of the module MΛover an affine chartIn this section we will describe endomorphisms of the bundles over the curves obtainedby gluing distinct points P i together and for the cusp curve;the case of gluing points with multiplicities is more complicated and will be treated in[20].Recall that the module MΛover the algebra{f∈C[z]:∀i,j f(P i)=f(P j)},is definedas the subset of the vector-valued polynomials s(z)on C,i.e.s(z)∈C[z]r,which satisfy the conditions:s(P1)=Λi s(P i),i=2,...,N.It is natural to look for endomorphisms ofMΛas endomorphisms of C[z]r which preserve the submodule MΛ.Proposition6An endomorphism of the module MΛcan be described as a matrix poly-nomialΦ(z):s(z)→Φ(z)s(z),which satisfy the conditionΦ(P1)=ΛiΦ(P i)Λ−1i(11) The condition above implies thatΦ(z)s(z)satisfies:Φ(P1)s(P1)=ΛiΦ(P i)s(P i),so Φ(z)s(z)is again an element of MΛandΦ(z):s(z)→Φ(z)s(z)is an endomorphismof MΛ.Example7In the abelian case(i.e.rank1modules over any manifold)the condi-tion above is empty and any elementΦ(z)defines an endomorphism i.e.the sheaf ofendomorphisms of any rank1coherent sheaf is just O as in the regular case.Example8Consider the node(or double point)curve Spec{f∈C[z],f(1)=f(0)}. An endomorphisms of the module MΛ(which is defined as{s∈C[z]r,s(1)=Λs(0)},forsome matrixΛ)is given by a matrix-valued polynomialΦ(z)=Φ0+Φ1z+Φ2z2+...such thatΦ(1)=ΛΦ(0)Λ−1.HenceΦ(z)=Φ0+(ΛΦ0Λ−1−Φ0)z+z(z−1)˜Φ(z), where˜Φ(z)is arbitrary.When one considers the projectivization of our curve and thebundle corresponding to MΛon it,we see that in order to be regular at infinity one mustonly consider constant endomorphismsΦ(z)=Φ0.So in order to satisfy the condition Φ(1)=ΛΦ(0)Λ−1one must request that the matrixΦ0commutes withΛ.As a corollary we see that there is only r-dimensional space of global endomorphisms for a general module MΛ.Example9Consider the node(or double point)curve Spec{f∈C[z],f(A)=f(B)}.An endomorphism of the module MΛis given by a matrix-valued polynomialΦ(z)=Φ0+Φ1z+Φ2z2+...such thatΦ(A)=ΛΦ(B)Λ−1.HenceΦ(z)=Φ0+Φ1z+(z−A)(z−B)(˜Φ(z)),whereΦ0,Φ1must satisfyΦ0+AΦ1=Λ(Φ0+BΦ1)Λ−1and˜Φ(z)is arbitrary.It is moreconvenient to rewrite this expression as follows:Φ(z)=Θ(z−A)−ΛΘΛ−1(z−B)+(z−A)(z−B)˜Θ(z),whereΘis an arbitrary constant matrix.So global endomorphisms are given byΦ(z)=Θ(B−A),withΘcommuting withΛ.Example10Consider the triple point curve Spec{f∈C[z],f(P1)=f(P2)=f(P3)}. So an endomorphism of module MΛis given by a matrix-valued polynomialΦ(z)=Φ0+Φ1z+Φ2z2+...such thatΦ(P1)=Λ2Φ(P2)Λ−12,Φ(P1)=Λ3Φ(P3)Λ−13.HenceΦ(z)=Φ(P1)(z−P2)(z−P3)(P2−P1)(P2−P3)++Λ−13Φ(P1)Λ3(z−P1)(z−P2)2.5Description of End (M Λ)⊗K 2.5.1Examples of node and cusp curvesExample 11Consider the node curve Spec {f ∈C [z ],f (A )=f (B )}.An endomor-phism of the module M Λis given (see example 9)byΘ(z )=Θ(z −A )−ΛΘΛ−1(z −B )+(z −A )(z −B )˜Θ(z ),where Θ,˜Θ(z )are arbitrary.The sections of the dualizing module are given by ω=cdzz −B+holomorphic in z.So the sections of End (M Λ)⊗K can be described as Φ(z )=(B −A )(ΛΘΛ−1dzz −B )+holomorphic in z.(13)Hence the global sections H 0(End (M Λ)⊗K )over the projectivization are Φ(z )’s which are regular at infinity.The condition that Φ(z )has no pole of order greater than 2gives that there is no holomorphic term in the expression (13).The condition that the residue at infinity is zero is equivalent to ΛΘΛ−1−Θ=0.Hence the global sections are:Φ(z )=(B −A )(Θdzz −B )where ΛΘ=ΘΛ.We can see in this case thatH 0(End (M Λ)⊗K )=H 0(End (M Λ))⊗H 0(K ).Example 12Consider the cusp curve Spec {f ∈C [z ],f ′(P )=0}.An endomorphisms of the module M Λis given byΘ(z )=Θ+[Λ1,Θ](z −P )+(z −P )2˜Θ(z )where Θ,˜Θ(z )are arbitrary.The sections of the canonical module are given by c 1dz(z −P )2+holomorphic in z termsThe global sections H 0(End (M Λ)⊗K )are Φ(z )’s which are regular at infinity.Hence[Θ,Λ]=0,and the global sections are:Φ(z )=(B −A )Θdz。