Commutative rings of differential operators connected with two-dimensional Abelian varietie

a rX iv:mat h /11291v1[mat h.AG ]26Oct21Commutative Rings of Differential Operators Connected with Two-Dimensional Abelian Varieties A.E.Mironov ∗Abstract In this paper we find the explicit formulas of two dimensional com-muting (2×2)-matrix differential operators which were introduced by Nakayashiki.The common eigen functions and eigen values of these operators are parametrized by the points of principally polar-ized Abelian varieties.1Introduction The main result of this paper is construction of two-dimensional (2×2)-matrix differential operators with doubly periodic coefficients from spectral functions and the vector Baker–Akhiezer eigenfunction that was introduced by Nakayashiki in [1].A remarkable property of these operators is the fact that they are finite-gap at every energy ly,their Blˆo ch func-tions,the eigenfunctions of both the differential operator and the operators of translation by periods,are parametrized by the points of a two-dimensionalprincipally polarized Abelian variety X 2.Moreover,the eigenvalues (making the spectrum)are given by some meromorphic function λ(z )on X 2with pole on a theta divisor.We indicate a procedure for constructing a commutative ring of such operators from an Abelian variety X 2(with an irreducible theta divisor)and a spectral function λ.The multidimensional inverse problem was first solved by B.A.Dubrovin,I.M.Krichever,and S.P.Novikov for a periodic Schr¨o dinger operator witha magneticfield[2].They described a procedure for constructing such oper-ators that arefinite-gap at one energy level,i.e.,their Blˆo ch functions with afixed eigenvalue E0are parametrized by the points of a Riemann surface offinite ter,A.P.Veselov and S.P.Novikov[3]distinguished potential operators(i.e.,operators with zero magneticfield)among them.Thefinite-gap property of the Schr¨o dinger operator at afixed energy level is an exceptional phenomenon:Feldman,Knorrer,and Trubowitz[4] demonstrated that the two-dimensional Schr¨o dinger operator with a smooth real potential can befinite-gap only at one energy level.As wasfirst observed by Sato,for matrix operators it is possible that the Bloch functions are parametrized by surfaces offinite genus(in our case by the level surfacesλ= const)for all energy levels.This idea was implemented by Nakayashiki[1,5].In[1],using the Fourier–Mukai transform method[6],Nakayashiki con-structed the Baker–Akhiezer module over a ring of differential operators in g space variables,where g is the dimension of a principally polarized Abelian variety X g with a nonsingular theta divisor.To this module there corre-sponds(up to conjugation)a commutative ring of(g!×g!)-matrix partial differential operators whose coefficients are,in general,defined locally.Each operator corresponds to some meromorphic function on X g with pole on the theta divisor(a spectral function)which parametrizes the eigenvalues of the operator(these operators are now referred to as the Nakayashiki opera-tors).The vector-function whose components are the elements of the basis for the Baker–Akhiezer module parametrizes the general eigenfunctions of these operators.Observe that this construction requires nonsingularity of the theta divisor. As was demonstrated by Andreotti and Mayer[7],a theta divisor of a general Abelian variety is a nonsingular subvariety;however,a theta divisor of the Jacobian varieties of Riemann surfaces has singularities for g>3and in the case of hyperelliptic surfaces also for g=3.In Section2we describe the Fourier–Mukai transform[6]in the needed situation,recall Nakayashiki’s construction of the Baker–Akhiezer module[1], and indicate connection between Krichever’s construction[8]and the Fourier–Mukai transform[6].In Section3we give a new short analytical proof of Nakayashiki’s theorem of freeness of the Baker–Akhiezer module in dimension2.It essentially uses the fact that the Abelian variety X2in question is two-dimensional unlike the proof of the general case in[1]and requires a minimal apparatus of algebraic geometry.Here we also introduce a basis for the Baker–Akhiezer module2for g=2in which we manage tofind the coefficients of the Nakayashiki operators.In Section4we describe an effective procedure for constructing the Nakayashiki operators for g=2.In Propositions1–3we obtain explicit formulas for op-erators generating,as proven in Lemma9,the whole ring of the Nakayashiki operators.In Proposition1we introduce commuting operators Z1,...,Z g such that,for every Nakayashiki operator L,the commutator[L,Z j]isa Nakayashiki operator too.In Proposition2wefind explicit formulas for the second-order Nakayashiki operators and in Proposition3,explicit formulasfor the operators Z j.The author is grateful to S.P.Novikov and I.A.Taimanov for posingthe problem and to I.A.Taimanov for useful discussions and remarks.2Nakayashiki’s ConstructionLet X g=C g/(Z g+ΩZ g)be a principally polarized Abelian complex variety, whereΩis a symmetric(g×g)-matrix with ImΩ>0.Denote by Pic0(X g)the Picard variety of X g.If X g is principally polarized then Pic0(X g)is isomorphic to X g.The isomorphism is given by the mapping z→x,wherez∈X g and x∈Pic0(X g).The sections of a bundle x,lifted to the universal covering C g,are given by functions h(z)satisfying the periodicity conditionsh(z+Ωm+n)=exp(−2πi m,x )h(z),where m,n∈Z g and m,x =m1x1...+m g x g.Denote by P the Poincar´e bundle over X g×Pic0(X g).It is determinedby the following properties.The bundle corresponding to x∈Pic0(X g)is isomorphic to P|X g×{x}and the bundles P|X g×{0}and P|{0}×Pic0(X g)are trivial.The sections of P,lifted to the universal covering C g×C g,are givenby functions f(z,x)such thatf(z+Ωm1+n1,x+Ωm2+n2)=exp(−2πi( m1,x + m2,z ))f(z,x),(1) where m j,n j∈Z g.Denote by z=(z1,...,z g)⊤∈C g the coordinates of points on the universal covering of X g and by x=(x1,...,x g)⊤∈C g,the coordinates of points on the universal covering of Pic0(X g).We identify the sections of P with the functions on C g×C g satisfying(1).3The theta function with characteristic[a,b]is defined by the series θ[a,b](z,Ω)= n∈Z g exp(πi Ω(n+a),(n+a) +2πi (n+a),(z+b) ),where a,b∈C g.For short,the functionθ(z)=θ[0,0](z,Ω)is referred to as a theta function.The functionθ[a,b](z,Ω)possesses the periodicity propertiesθ[a,b](z+m,Ω)=exp(2πi a,m )θ[a,b](z,Ω),θ[a,b](z+Ωm,Ω)=exp(−2πi b,m −πi mΩ,m −2πi m,z )θ[a,b](z,Ω), m∈Z g.Denote byΘthe zero set of a theta function(a theta divisor)which is a subvariety in X g.Given a subvariety Y⊂X g of codimension1,we let A Y stand for the space of meromorphic functions on X g with pole in Y.We denote by F Y(U)the space of meromorphic sections of P over subsets of the form X g×U⊂X g×Pic0(X g)which have pole in Y×U,where U is an open subset in Pic0(X g).The set F Y= U F Y(U)is called the Fourier–Mukai transform of A Y[6].In[1]Nakayashiki constructed“covariant differentiation”operators(a con-nection on P)∇j:F Y(U)→F Y(U),∇k∇j=∇j∇k,k,j=1,...,g,that give F Y(U)a module structure over the ring O U[∇1,...,∇g],where U⊂Pic0(X g)is an open subset and O U is the ring of analytic functions on U.By construction,F Y(U)is also an A Y-module.Nakayashiki introduced the Baker–Akhiezer functionsf(z,x)exp −g j=1x j∂z j logθ(z) ,f(z,x)∈FΘ.The operators∇j are defined by the formula∇j=∂xj −∂zjlogθ(z).Let M c denote the space of the Baker–Akhiezer functionsf(z,c+x)exp −gj=1x j∂zjlogθ(z) |f(z,c+x)∈ U FΘ(c+U) , 4where U⊂C g is a neighborhood of zero,c∈C g,and f(z,c+x)belongs to the range of the Fourier–Mukai transform FΘ.Each function f(z,c+x)is defined on C g×U and has pole only inΘ×U.Roughly speaking,f(z,c+x) is a germ of a function at x=0with respect to x.It follows from(1)thatf(z+Ωm+n,c+x)=exp(−2πi m,(c+x) )f(z,c+x).We denote by M c(k)the subset of functions in M c such that f(z,c+x)has a pole of order≤k inΘ×U.The definition implies that M c is a module overthe ring D=O[∂x1,...,∂xg]of differential operators(a D-module),where Ois the ring of analytic functions in x1,...,x g in a neighborhood of0∈C g. The D-module M c is referred to as the Baker–Akhiezer module.From the definition of the Fourier–Mukai transform we infer that M c also possesses the structure of an AΘ-module.The D-module M c can be described by means of theta functions[1].Lemma1.The equality holds:M c=∞n=1 a∈Z g/n Z g O·θ[aθn(z)exp −g k=1x k∂z k logθ(z) .We also need the following[1]: Lemma2.The identity holds:∂xj θ[aθn(z)exp −gk=1x k∂zklogθ(z)=1n,0](nz+c+x,nΩ)Consequently,LΦc(λ)Φc=λΦc,(2)with[LΦc (λ)]kj the entries of the matrix differential operator LΦc(λ)andλΦc=(λφ1c,...,λφg!c)⊤.Since LΦc(λ)are differential operators in x j while λdepends only on z,from(2)we obtain the commutation conditionLΦc (λµ)=LΦc(λ)LΦc(µ)=LΦc(µ)LΦc(λ),whereµ(z)∈AΘ.We arrive at the followingCorollary[1].There is a ring embeddingLΦc:AΘ→Mat(g!,D),where Mat(g!,D)is the ring of(g!×g!)-matrix differential operators.The image of the embedding is a commutative ring of differential operators.As indicated in[5],Nakayashiki’s construction generalizes Krichever’s construction[8]which can be interpreted in terms of the Fourier–Mukai transform as follows:Recall the construction of the Baker–Akhiezer function[8].LetΓbe a Riemann surface of genus g,let D=p1+...+p g be a nonspecial positive divisor onΓ,and let∞be a point onΓother than the points of the divisor.Take a local parameter k−1at∞so that k−1(∞)=0.The one-point Baker–Akhiezer function with spectral data{Γ,∞,p1,...,p g,k−1}is a functionψ(z,x),z∈Γ,defined to within a factor depending only on x,by the following properties:(1)ψ(z,x)is meromorphic onΓ\∞and the poles ofψare independent of x and coincide with{p1,...,p g};(2)the functionψ(z,x)exp(−kx)is analytic in a neighborhood of∞.This function has the formψ(z,x)=θ A(z)−g j=1A(p j)−∆+V xThe functionθ A(z)−g j=1A(p j)−∆+V xθ ˜z−g j=1A(p j)−∆which belongs to the range of the Fourier–Mukai transform FΘ′,whereΘ′⊂X g is the subvariety defined by the equationθ ˜z−g j=1A(p j)−∆ =0.Denote by D(ψ(z,x))the D-module{dψ(z,x)|d∈ D},where D is the ring of differential operators in x.As shown in[8], D(ψ(z,x))is a free D-module;moreover,for every meromorphic function f(z)onΓwith a sole pole at∞,there is a unique differential operator L(f)such thatL(f)ψ(z,x)=f(z)ψ(z,x).Hence,we obtain a correspondence between the spectral data of commutative rings of scalar differential operators in the Burchnall–Chaundy–Krichever theory and the spectral data of commutative rings of the Nakayashiki matrix differential operators:{Γ,∞,D,f}←→{X g,Θ,c,λ}.3Proof of Nakayashiki’s Theorem for g=2Here we give a new proof of Nakayashiki’s theorem for g=2.Introduce the following functions in M c:θ(z+c+x)ψ=ψc′=θ(z+c+c′+x)θ(z−c′)2πi ∂S d log(aθ1(A(p)−∆)+bθ2(A(p)−∆))=1θ(z)−θ(z+c+x)θ1(z)+α2(x) θ2(z+c+x)θ2(z) +α3(x)θ(z+c+x)α2(x)=−θ2(p)θ1(p)∂x1logθ(z+c+x)+∂x2logθ(z+c+x)=−θ2(p)θ(z)+θ2(z)α2(x).The pole(in x)of the left-hand side depends on z∈Θ,whereas the pole of the right-hand side does not;a contradiction.Lemma4is proven.Lemma5.D(ψ,ψc′)is a free D-module of rank2.Proof.Suppose that D(ψ,ψc′)is not a free D-module.Then there exist operators d1,d2∈D such thatd1ψc′+d2ψ=0.(3) Consider the case in which ord(d1)>ord(d2)−1,where ord is the order of the operator.Suppose that ord(d1)=n.The operator d1looks liked1=f n(x)∂n x1+f n−1(x)∂n−1x1∂x2+...+f0(x)∂n x2+...,where f j(x)∈O,j=0,...,n.Divide(3)byexp(−x1∂z1logθ(z)−x2∂z2logθ(z)),multiply byθn+2(z),and assume z∈Θ.We obtainθ(z+c+c′+x)θ(z−c′)(f n(x)θn1(z)+f n−1(x)θn−11(z)θ2(z)+...+f0(x)θn2(z))=0.(4)By Lemma3,there is a point p∈C2such that p∈Θandθ1(p)=0. Put z=p in(4).We obtain f0=0.Divide(4)byθ1(z)and put again z=p.We obtain f1=0.Proceeding similarly,we obtain f n=f n−1=...=9f0=0.Consequently,the inequality ord(d1)>ord(d2)−1is impossible. Similarly,we show that the inequality ord(d1)+1<ord(d2)is impossible either.Consider the case in which ord(d1)+1=ord(d2)=n.Let the operators d1and d2look liked1=f n−1(x)∂n−1x1+f n−2(x)∂n−2x1∂x2+...+f0(x)∂n−1x2+...,d2=g n(x)∂n x1+g n−1(x)∂n−1x1∂x2+...+g0(x)∂n x2+....Divide(3)byexp(−x1∂z1logθ(z)−x2∂z2logθ(z)),multiply byθn+1(z),and assume z∈Θ.Thenθ(z+c+c′+x)θ(z−c′)(f n−1(x)θn−11(z)+f n−2(x)θn−21(z)θ2(z)+...+f0(x)θn−12(z))−θ(z+c+x)(g n(x)θn1(z)+g n−1(x)θn−11(z)θ2(z)+...+g0(x)θn2(z))=0. Decomposef n−1(x)θn−11(z)+f n−2(x)θn−21(z)θ2(z)+...+f0(x)θn−12(z),g n(x)θn1(z)+g n−1(x)θn−11(z)θ2(z)+...+g0(x)θn2(z)into factors to obtainθ(z+c+c′+x)θ(z−c′)(a n−1(x)θ1(z)+b n−1(x)θ2(z))...(a1(x)θ1(z)+b1(x)θ2(z))−θ(z+c+x)(a′n(x)θ1(z)+b′n(x)θ2(z))...(a′1(x)θ1(z)+b′1(x)θ2(z))=0,(5) where a′j(x),b′j(x),a k(x),and b k(x)are some functions,j=1,...,n,k= 1,...,n−1.Fix a point x.Observe that if two functions a′jθ1(z)+b′jθ2(z)and a kθ1(z)+ b kθ2(z)have a common zero onΘthen they are proportional.Indeed, suppose the contrary.Then the ratio of these functions has a single simple pole onΘ⊂X2(by Lemma3).Consequently,the mapping(a′jθ1(z)+b′jθ2(z):a kθ1(z)+b kθ2(z)):Θ→CP1,where CP1is the one-dimensional projective space,is an isomorphism.Since the genus of the Riemann surfaceΘ⊂X2equals2,this is a contradiction.10The zeros of a kθ1(z)+b kθ2(z)andθ(z+c+x)onΘcannot coincide,since these functions can be regarded as sections of the line bundle overΘ⊂X2 and their ratioa kθ1(z)+b kθ2(z)k,0 (kz+c+x,kΩ)|βa∈C .Consequently,dim C M c(k)=k2(see,for instance,[9]).Since the dimension of the space of differential operators of order≤k with constant coefficients equals1+...+(k+1)=(k+1)(k+2)2+k(k+1)4A Commutative Ring of (2×2)-Matrix Dif-ferential OperatorsDenote by D the ring of differential operators C [∂z 1,...,∂z g ]with constant coefficients.By definition,M c is a D -module.Let Φc =(φ1c ,...,φg !c )⊤be a basis for M c .Then we have the ring embeddingL Φc :D →Mat(g!,D )defined by the formulaL Φc (d )Φc =d Φc ,where d ∈D and d Φc =(dφ1c ,...,dφg !c )⊤.The range of L Φc is a commuta-tive ring of matrix differential operators which is isomorphic to D .Denote the operator L Φc ∂z jby Z j ,j =1,...,g.Find the commutator [L Φc (λ),Z j ]for λ∈A Θ:Z j L Φc (λ)Φc =Z j (λΦc )=λ(Z j Φc )=λ(∂z j Φc ),L Φc (λ)Z j Φc =L Φc ∂z j Φc =∂z j (L Φc (λ)Φc )=∂z j (λΦc )=(∂z j λ)Φc +λ(∂z j Φc ).Thus,[L Φc (λ),Z j ]Φc =(∂z j λ)Φc ,which proves the followingProposition 1.[L Φc (λ),Z j ]=L Φc (∂z j λ)for λ∈A Θ,j =1,...,g .We now clarify how the matrix operators change when we replace a ba-sis for the D -module M c .Suppose that Φc =(φ1c ,...,φg !c )⊤and Ψc =(ψ1c ,...,ψg !c )⊤are two bases for the D -module M c .By Nakayashiki’s the-orem,there exist unique operators A ,B ∈Mat(g!,D )such that A Φc =Ψc and B Ψc =Φc ;hence,AB =BA is the identity in Mat(g!,D ).We denote the operator B by A −1.It is easy to verify thatA L Φc (λ)A −1Ψc =λΨc ,AL Φc (d )A −1Ψc =d Ψc ,where λ∈A Θand d ∈D .Consequently,we have the following Lemma 7The equalities hold:A L Φc (λ)A −1=L Ψc (λ),AL Φc (d )A −1=L Ψc (d ).12Henceforth we suppose that g=2.In Proposition2wefind the second-order operators LΦc(λ).They correspond to the following spectral functions:λ=∂2logθ(z)θ(z)exp(−x1∂z1logθ(z)−x2∂z2logθ(z))=∂zk ∂zj θ(z+c+x)θ(z)exp(−x1∂z1logθ(z)−x2∂z2logθ(z)),where k,j=1,2.Let c′be in a general position.Denote by p1,p2∈C2the intersection points ofΓc′withΘ(p1and p2are defined to within elements of the lattice Z2+ΩZ2).Denote by L c,c′the Nakayashiki operators for the basisψ,ψc′. For brevity,we introduce the notationsH kjc,c′=[L c,c′(∂zk∂zjlogθ(z))]11,F kjc,c′=[L c,c′(∂zk∂zjlogθ(z))]12.Proposition2.The equality holds:[L c,c′(∂zk ∂zjlogθ(z))]11=−∂xk∂xj+f kjc,c′∂x1+g kj c,c′∂x2+h kj c,c′,withf kj c,c′=θ2(p1)θ2(p2)θ2(p1)+∂xjlogθ(p1+c+x)θk(p1)θ2(p1)−∂xk logθ(p2+c+x)θj(p2)θ2(p2)+θkj(p2)g kj c,c′=θ1(p1)θ1(p2)θ1(p1)+∂xklogθ(p1+c+x)θj(p1)θ1(p1)−∂xj logθ(p2+c+x)θk(p2)θ1(p2)+θkj(p2)θ(∆+c′+c+x)(∂zk∂zj−f kjc,c′∂z1−g kj c,c′∂z2+2∂zk∂zjlogθ(z))× θ(z+c+x)θ(c+c′+x)θ(c′)×(∂zk ∂zj−f kjc,c′∂z1−g kj c,c′∂z2−h kj c,c′+2∂zk∂zjlogθ(z)) θ(z+c+x)θ2(z)=α11∂2z1logθ(z)+α12∂z1∂z2logθ(z)+α22∂2z2logθ(z)+α.Proof of Proposition2.It is easy to verify that the function∂xk ∂xjψ+∂zk∂zjlogθ(z)ψHence,the operator H kjc,c′looks likeH kjc,c′=−∂xk∂xj+f kjc,c′∂x1+g kj c,c′∂x2+h kj c,c′,and F kjc,c′is the operator of multiplication by the functionH kjc,c′ψ+F kjc,c′ψc′=∂zk∂zjlogθ(z)ψ.(7)Rewrite(7)for z∈Γc′as follows:H kjc,c′ψ=∂zk∂zjlogθ(z)ψ.By Lemmas2and8,the last equality amounts to(∂zk ∂zj−f kjc,c′∂z1−g kj c,c′∂z2−h kj c,c′+2∂zk∂zjlogθ(z)) θ(z+c+x)θ(p1+c+x),f kj c,c′θ1(p2)+g kj c,c′θ2(p2)=θj(p2+c+x)θk(p2)+θk(p2+c+x)θj(p2)−θkj(p2)θ(z)+F kjc,c′θ(z+c+c′+x)θ(z−c′)θ(z).Putting z=0in this equality,we obtain F kjc,c′.Wefind the remaining entries of the operator L c,c′(∂zk ∂zjlogθ(z)).Re-place c′with−c′and c with c+c′in(7)and multiply both sides byθ(z−c′)θ2(z)ψ=∂zk∂zjlogθ(z)ψc′.(9)15Letθ(z−c′)θ(z+c′)θ1(p2)θ2(p1)−θ1(p1)θ2(p2)× θj(p1−c′)θ(p1+c+x+c′)θ2(p2)θ(p2+c+x) ,k j2=1θ(p1+c+x)−θj(p2−c′)θ(p2+c+x+c′)θ1(p1)h j c,c ′(x )=θj (∆)θ(∆+c +x +2c ′)θ(∆+c +x +c ′)−θ1(∆+c ′)θ(∆+c ′+c +x )−θ2(∆+c ′)θ(c ′)+θj (c +x +c ′)θ(c ′)θ(c +x +c ′)(k j 1∂z 1+k j2∂z 2+h j c,c ′)θ(z +c +x )θ(z )exp(−x 1∂z 1log θ(z )−x 2∂z 2log θ(z ))−(x 1∂z 1∂z j log θ(z )+x 2∂z j ∂z 2log θ(z ))θ(z +c +x )θ(z )θ(z +c +x +c ′)θ(z )∂z jθ(z +c +x +c ′)θ2(z )×exp(−x 1∂z 1log θ(z )−x 2∂z 2log θ(z )).(10)17Immediate calculations show that∂zj θ(z−c′)θ(z)exp(−x1∂z1logθ(z)−x2∂z2logθ(z))−∂x jψc′=θj(z−c′)θ(z+c+x+c′)−θ(z−c′)θj(z+c+x+c′)θ(z) θ(z+c+x+c′)θ(p1+c+x),k j1θ1(p2)+k j2θ2(p2)=−θj(p2−c′)θ(p2+c+x+c′)θ3,...,f8where f i is not divisible byθ,i=1,...,8.As we show below,we can take the basis of the functions1,∂zk ∂zj∂zslogθ(z),∂3z1logθ(z)+∂zj∂zslogθ(z),(∂z1∂z2logθ(z))2−∂2z1logθ(z)∂2z2logθ(z),(12)where k,j,s=1,2.By the Lefschetz theorem,the mapping(θ3:f1:...: f8)defines an embedding F:X2→CP8into the projective space.Let (y0:...:y8)be homogeneous coordinates in CP8.Then the restrictions ofthe functions y1y0to F(X2)generate the coordinate ring of the affinealgebraic variety F(X2\Θ).Consequently,the functionsf1θ3generate AΘ.We are left with demonstrating that the functions(12)are linearly independent over C.This follows from the fact that the operatorsL c,c′(1),L c,c′(∂zk ∂zj∂zslogθ(z)),L c,c′(∂3z1logθ(z)+∂zj∂zslogθ(z)),(L c,c′(∂z1∂z2logθ(z))2−∂2z1logθ(z)∂2z2logθ(z)),where k,j,s=1,2,are linearly independent,because the leading symbols of the11-entries of these operators are equal respectively to1,−2∂xk ∂xj∂xs,−2∂3x1−∂xj∂xs,f22c,c′∂3x1+(g22c,c′−2f12c,c′)∂2x1∂x2+(f11c,c′−2g12c,c′)∂x1∂2x2+g11c,c′∂3x2.Lemma9is proven.References.[1]Nakayashiki A.,Structure of Baker–Akhiezer modules of principally polarized Abelian varieties,commuting partial differential operators and as-sociated integrable systems,Duke Math.J.,1991,vol.62,N.2,315–358.[2]Dubrovin B.A.,Krichever I.M.,and Novikov S.P.,The Schroedinger equation in a periodic magneticfield and Riemann surfaces,Soviet Math. 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