Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles
aa r X i v : . [ m a t h . AG ] D ec VARIATIONS OF HODGE STRUCTURES FORHYPERGEOMETRIC DIFFERENTIAL OPERATORS ANDPARABOLIC HIGGS BUNDLES
ROMAN FEDOROV
Abstract.
Consider the holomorphic bundle with connection on P −{ , , ∞} corresponding to the regular hypergeometric differential operator h Y j =1 ( D − α j ) − z h Y j =1 ( D − β j ) , D = z ddz . If the numbers α i and β j are real and for all i and j the number α i − β j isnot integer, then the bundle with connection is known to underlie a complexpolarizable variation of Hodge structures. We calculate some Hodge invariantsfor this variation, in particular, the Hodge numbers. From this we derive aconjecture of Alessio Corti and Vasily Golyshev. We also use non-abelianHodge theory to interpret our theorem as a statement about parabolic Higgsbundles. Introduction
Fix two sequences of real numbers0 ≤ α ≤ . . . ≤ α h < ≤ β ≤ . . . ≤ β h < . (1)Assume that for all i, j we have α i = β j . The regular hypergeometric differentialoperator is the following complex differential operator(2) Y j ( D − α j ) − z Y j ( D − β j ) , D = z ddz . This differential operator has been studied in details in [BH] as the differentialoperator annihilating the generalized hypergeometric function n F n − .Expanding, we see that the leading term of this operator is z h (1 − z )( d/dz ) h , sothe operator gives rise to a connection ∇ on the trivial rank h holomorphic vectorbundle V over P − { , , ∞} . The bundle with connection ( V, ∇ ) is known to be irreducible and physically rigid . The latter means that a bundle with connection( V ′ , ∇ ′ ) is isomorphic to ( V, ∇ ), provided its monodromy at each of the singularpoints is conjugate to that of ( V, ∇ ) (see Section 2.1 below). Rigid bundles withconnections form, in a sense, the simplest class of bundles with connections. In par-ticular, the Katz algorithm allows to reduce such to a rigid bundle with connectionof a smaller rank by tensoring with a line bundle followed by a middle convolution(see [Sim1, Sect. 5] and also [Kat, Ari, DS, DR, BE]).It was conjectured by Alessio Corti and Vasily Golyshev [CG] that ( V, ∇ ) under-lies a real polarizable variation of Hodge structures , provided that for m = 1 , . . . , h we have α m + α h +1 − m ∈ Z , β m + β h +1 − m ∈ Z . More importantly, they gave conjectural formulas for the Hodge numbers. We provethese conjectures below, see Theorem 2.To prove the conjectures, we note first that by [Sim2, Cor. 8.1] every rigid irre-ducible bundle with connection underlies a complex polarizable variation of Hodgestructures (this structure is essentially unique by [Del2, Prop. 1.13(i)]). This no-tion will be recalled in Section 2.2. Our first main result is the calculation of thecorresponding Hodge numbers for ( V, ∇ ). Theorem.
Set ρ ( k ) := { j : α j < β k } − k . Then we have up to a shift h p = ρ − ( p ) = { k : 1 ≤ k ≤ h, ρ ( k ) = p } . This is the content of Theorem 1 below. Our proof of this theorem is based onthe technique of middle convolution and on the paper [DS], where it is explainedhow Hodge data changes under middle convolution. In fact, we calculate not onlyHodge numbers but also the numerical invariants of the limits of Hodge structuresas z tends to 0 or ∞ ; see Theorem 3.Finally, according to the results of [Sim2], complex variations of Hodge structurescorrespond under non-abelian Hodge theory to certain decompositions of Higgsbundles. We translate our Theorem 1 into a statement about Higgs bundles: thenumbers α i and β i give a stability condition on the moduli of Higgs bundles andwe describe the unique stable Higgs bundle, see Theorem 4.2. Main results
We keep the notation of the introduction. Thus α i and β i are real numberssatisfying (1) and such that α i = β j for all i, j ∈ { , . . . , h } . Next, ( V, ∇ ) denotesthe holomorphic bundle with connection over P C − { , , ∞} corresponding to theregular hypergeometric operator (2). Thus, ∇ : V → V ⊗ Ω is a C -linear map.We denote by mult( α i ) the multiplicity of α i , that is, the number of elements inthe set { j : α j = α i } . We fix a square root of − x ) := exp(2 π √− x ).We denote by { α } the fractional part of α , that is, { α } is uniquely defined by theconditions that 0 ≤ { α } < α − { α } ∈ Z .2.1. Monodromy, irreducibility, and rigidity.
We would like to recall someresults from [BH]. First of all, we recall the local monodromy of ( V, ∇ ). Let A be a regular linear operator acting on an h -dimensional space such that the listof eigenvalues of A with multiplicities is α , . . . , α h . In other words, for each i , A has a unique Jordan block with eigenvalue α i of the size mult( α i ). Similarly,let B be a regular operator of acting on an h -dimensional vector space whose list ofeigenvalues with multiplicities is − β , . . . , − β h . Recall that the local monodromyof a bundle with connection around a singular point is defined up to conjugation. Proposition 2.1. ( a ) The local monodromy of ( V, ∇ ) at z = 0 is e( A ) . ( b ) The local monodromy of ( V, ∇ ) at z = ∞ is e( B ) . ( c ) The local monodromy of ( V, ∇ ) at z = 1 is a pseudo-reflection (that is, thesum of the identity operator and a rank one operator).Proof. Combine Prop. 3.2 and Theorem 3.5 of [BH]. (cid:3)
HS FOR HYPERGEOMETRIC DIFFERENTIAL OPERATORS 3
We note that the conjugacy class of the monodromy at z = 1 is uniquely deter-mined by the proposition. Indeed, the conjugacy class of a pseudo-reflection is deter-mined by its determinant. However, this determinant is equal to (det(e( A )e( B ))) − .Next, we have Proposition 2.2.
The bundle with connection ( V, ∇ ) is irreducible.Proof. Combine Prop. 3.2 and Prop. 3.3 of [BH]. (cid:3)
Remark . The condition α i = β j is necessary for the bundle with connection tobe irreducible, see [BH, Prop. 2.7]. On the other hand, the conditions 0 ≤ α i < ≤ β i < α i or β j , we get anisomorphic bundle with connection, see [BH, Cor. 2.6].The bundle with connection ( V, ∇ ) is physically rigid . Precisely, this means thatthe following holds. Proposition 2.4.
Let ( V ′ , ∇ ′ ) be a bundle with connection over P C − { , , ∞} such that for the local monodromy of ( V ′ , ∇ ′ ) all the statements of Proposition 2.1hold. Then ( V ′ , ∇ ′ ) is isomorphic to ( V, ∇ ) .Proof. This is Theorem 3.5 of [BH]. (cid:3)
Complex polarizable variations of Hodge structures.
Let X be a non-singular complex manifold. (It will be the complement of three points in P C inour applications.) Let V be a holomorphic vector bundle with a flat holomorphicconnection ∇ : V → V ⊗ Ω X . We can extend ∇ to a flat smooth connection(3) D ∇ : A ( V ) → A , ( V ) ⊕ A , ( V ) , where A i,j ( V ) is the sheaf of smooth complex V -valued differential forms of type( i, j ). Thus D ∇ = D ′∇ + D ′′∇ , where D ′∇ : A ( V ) → A , ( V ) is induced by ∇ and D ′′∇ = ¯ ∂ : A ( V ) → A , ( V ) is induced by the holomorphic structure on V .A complex variation of Hodge structures is a decomposition of V into a directsum of smooth complex vector bundles(4) V = M p ∈ Z H p such that D ′∇ ( A ( H p )) ⊂ A , ( H p ⊕ H p − ) , D ′′∇ ( A ( H p )) ⊂ A , ( H p ⊕ H p +1 ) . Denote by h p the Hodge number rk H p . Thus we have P p h p = rk V .This variation of Hodge structures is polarizable , if there is a D ∇ -flat hermitianform Q on V such that the decomposition (4) is Q -orthogonal and the restriction of( − p Q to H p is positive definite. We abbreviate a complex polarizable variationof Hodge structures as CPVHS, and denote it by ( V, ∇ , H • ). We also say that thedecomposition H • is a CPVHS on ( V, ∇ ). Remark . Let ( V, ∇ , H • ) be a CPVHS. Then for any c ∈ Z we have a shiftedCPVHS ( V, ∇ , H • + c ).Now let X = P C − { , , ∞} and let the hypergeometric bundle with connection( V, ∇ ) be as in Section 2.1. Proposition 2.6.
The bundle with connection ( V, ∇ ) is a part of a unique up toa shift CPVHS. ROMAN FEDOROV
Proof.
Since the bundle with connection is rigid and irreducible (Propositions 2.4and 2.2), the existence of CPVHS follows from [Sim2, Cor. 8.1], see also [DS,Thm. 2.4.1]. Alternatively, it follows by induction on the rank h from Lemma 3.1below and [DS, Prop. 3.1.1]. The uniqueness follows from [Del2, Prop. 1.13(i)]. (cid:3) Let h p be the corresponding Hodge numbers (defined up to a shift h p h p +const ,see Remark 2.5). Our first main result is Theorem 1.
Set ρ ( k ) := { j : α j < β k } − k . Then we have up to a shift h p = ρ − ( p ) = { k : 1 ≤ k ≤ h, ρ ( k ) = p } . This theorem follows immediately from Theorem 3(c) below.
Remark . It seems plausible that this theorem can be proved using results ofTerasoma [Ter1, Ter2]. On the other hand, our proof is very natural from theperspective of the theory of bundles with connections on curves.
Corollary 2.8.
The bundle with connection ( V, ∇ ) admits a hermitian flat metricof signature P hk =1 ( − ρ ( k ) .Proof. By definition of CPVHS, ( V, ∇ ) has a flat hermitian metric of signature P p ( − p h p . Now apply the theorem. (cid:3) The automorphism of P C given by ϕ ( z ) = 1 /z takes ( V, ∇ ) to a similar bundlewith connection corresponding to α m = {− β m } , β m = {− α m } . Using this, one cancheck that the previous corollary is equivalent to [BH, Thm. 4.5]. Corollary 2.9.
The following are equivalent ( a ) The numbers α j and β j interlace, that is, either α < β < α < β < . . . < α h < β h or β < α < β < α < . . . < β h < α h . ( b ) The trivial complex variation of Hodge structures H = V , H p = 0 if p = 0 ispolarizable. ( c ) ( V, ∇ ) admits a positive-definite flat hermitian metric. That is, the monodromyrepresentation is unitary.Proof. The implication (a) ⇒ (b) follows from the theorem immediately. Conversely,if (b) is satisfied, then, using uniqueness of CPVHS up to a shift and the theorem,we see that ρ ( k ) is constant. Since ρ ( h ) ≤ ρ (1) ≥ −
1, this constant is either 0or −
1. One easily derives that α j and β j interlace.The equivalence of (b) and (c) follows from the definition of CPVHS. (cid:3) Note that the equivalence of (a) and (c) above is the content of [BH, Cor. 4.7].2.3.
Real variations of Hodge structures and a conjecture of Corti andGolyshev.
Assume now that for m = 1 , . . . , h we have(5) α h +1 − m + α m ∈ Z , β h +1 − m + β m ∈ Z . In this case ( V, ∇ ) underlies a real variation of Hodge structures as we show below.Let us recall the definition. Let V R be a local system of real vector spaceson a smooth complex manifold X . Let V R ⊗ R C be its complexificaiton, and let HS FOR HYPERGEOMETRIC DIFFERENTIAL OPERATORS 5 ( V, ∇ ) be the holomorphic bundle with connection corresponding to this local sys-tem under the Riemann–Hilbert correspondence. Let k be an integer. Let H • , • bea decomposition into the sum of smooth complex vector bundles(6) V = M p + q = k H p,q such that for all p, q we have H p,q = H q,p and D ∇ ( A ( H p,q )) ⊂ A , ( H p,q ⊕ H p − ,q +1 ) ⊕ A , ( H p,q ⊕ H p +1 ,q − ) , where, as before, D ∇ is the smooth connection corresponding to ∇ . In this case wesay that ( V R , H • , • ) is a real variation of Hodge structures of weight k .Let Q R be a bilinear form of parity ( − k on V R , denote by Q the induced bilinearform on V . We say that Q R is a polarization of ( V R , H • , • ) if the Hodge decompo-sition (6) is Q -orthogonal and for x ∈ H p,q , x = 0 we have (cid:0) √− (cid:1) p − q Q ( x, ¯ x ) > V R , H • , • ) is called polarizable, if a polar-ization exists. Note that in this case, setting H p := H p,k − p , we get a CPVHS( V, ∇ , H • ). ( V and ∇ are as above.)Let ρ be as in Theorem 1, let ( V, ∇ ) be the hypergeometric local system from Sec-tion 2.1. The following has been conjectured by Alessio Corti and Vasily Golyshev;see Conjecture 1.4 of [CG]. Theorem 2.
Let p + = max ρ ( k ) , p − = min ρ ( k ) . Then there is a real variation ofHodge structures ( V R , H • , • ) of weight p + − p − such that the bundle with connectioncorresponding to V R ⊗ R C is isomorphic to ( V, ∇ ) , and rk (cid:0) H k − p − , − k + p + (cid:1) = ρ − ( k ) . This theorem will be proved in Section 6.
Remark . If ( V R , H • , • ) is a real polarizable variation of Hodge structures ofweight k , then setting ′ H p,q = H p + c,q + c we get a real polarizable variation ofHodge structures of weight k + 2 c with the same V R . Using this operation, one canalways assume that H ,k = 0, H k, = 0, H p,q = 0 if p < q <
0. This is exactlythe normalization of the above theorem.2.4.
Local Hodge invariants.
Let ( V, ∇ ) be as in Section 2.1. Recall from Propo-sition 2.6 that ( V, ∇ ) can be extended to a CPVHS on P C − { , , ∞} . We are goingto describe the limits of this CPVHS at z = 0 and z = ∞ . Let us give the corre-sponding definitions in the general case. We follow [DS, § V, ∇ ) be an arbitrary holomorphic bundle with connection ona smooth complex curve X . Let ¯ X ⊃ X be a curve such that ¯ X − X = { x } is asingle point. Assume that the eigenvalues of the monodromy of ( V, ∇ ) around x lieon the unit circle. The following is known as Deligne’s canonical extension. Lemma 2.11.
For all α ∈ R there is a unique extension ( V α , ∇ α ) of ( V, ∇ ) to ¯ X such that • V α is a holomorphic vector bundle on ¯ X ; • ∇ α : V α → V α ⊗ Ω ¯ X ( x ) is a connection with a first order pole at x ; • The eigenvalues of the residue ∗ of ∇ α at x belong to [ α, α + 1) . ∗ Our convention is that res( d − A/z ) = res( A ). ROMAN FEDOROV
Proof.
We may assume that ¯ X is a disc and x is its center. In this case the statementis a particular case of [Del1, Ch. II, Prop. 5.4]. (cid:3) Replacing [ α, α + 1) by ( α, α + 1], we get a definition of a similar extension( V >α , ∇ >α ) such that the eigenvalues of res ∇ >α are in ( α, α + 1]. Let : X → ¯ X be the open embedding, consider the pushforward ∗ V . Identifying V α with itssheaf of sections, we can view V α as a subsheaf of ∗ V . It is not difficult to seethat V α ⊃ V β if α < β , V >α = ∪ β>α V β , and the quotients V α /V β , V α /V >α areset theoretically supported at x . Let ψ α ( V ) = ψ αx ( V, ∇ ) := Γ( ¯ X, V α /V >α ) . be the space of moderate nearby cycles. It is equipped with a nilpotent endo-morphism N α induced by z ∇ ddz − α , where z is any coordinate near x such that z ( x ) = 0. Remark . We have V α +1 = V α ( − x ), so V α /V α +1 = V αx is the fiber of V α at x . It is not difficult to see that for β ∈ [ α, α + 1) we can identify ψ β ( V ) with thegeneralized eigenspace of the residue of ∇ α corresponding to the eigenvalue β .Next, we have a general statement (see [Sch, Lemma 6.4]). Lemma 2.13.
Let L be a finite-dimensional vector space, let N be a nilpotentendomorphism of L . Then ( a ) There is a unique increasing filtration W • L indexed by integers such that forall k we have N ( W k L ) ⊂ W k − L and for all k > N k induces an isomorphism gr kW L ≃ gr − kW L . ( b ) For l ≥ set P l L := (Ker ¯ N l +1 ) ∩ gr lW L, where ¯ N is the endomorphism of gr W L induced by N . Then for k ∈ Z we havea Lefschetz decomposition gr kW L = M l ≥ , ¯ N l P k +2 l L. Applying this to ψ α ( V ) and N α , we get a filtration W k ψ α ( V ) and the primitivesubspaces P l ψ α ( V ) ⊂ gr lW ψ α ( V ) for l ≥ V, ∇ , H • ) bea CPVHS on X = ¯ X − x . Note that F p V := L q ≥ p H q is preserved by D ′′ , so F p V is a holomorphic subbundle of V . We get a decreasing filtration F • V on V . Thisfiltration uniquely extends to a filtration F • V α on V α by vector subbundles for α ∈ R because partial flag varieties are compact. Moreover, it is easy to see that F p V α = ∗ F p V ∩ V α . Similarly, we get a filtration F • V >α on V >α .Setting F p ψ α ( V ) := F p V α /F p V >α , we get filtrations on the spaces of moderatenearby cycles. These filtrations induce filtrations on gr lW ψ α ( V ) and, in turn, on P l ψ α ( V ). Definition 2.14.
The local Hodge invariants of ( V, ∇ , H • ) at x are ν pα,l := dim Gr pF P l ψ α ( V ) = dim F p P l ψ α ( V ) − dim F p +1 P l ψ α ( V ) , where l ∈ Z ≥ , p ∈ Z , α ∈ [0 , HS FOR HYPERGEOMETRIC DIFFERENTIAL OPERATORS 7
Remarks . (a) The above formulas define the invariants ν p,lα for all α ∈ R and it is not difficult to see that they only depend on α mod Z (cf. Remark 2.12).However, it will be convenient for us to have them defined only if α ∈ [0 , ν pα,l is denoted by ν p exp(2 π √− α ) ,l in [DS] (cf. the previousremark). The “additive” notation is more convenient for us, we hope it will notlead to confusion.(c) It is easy to see that ν pα,l = 0 unless e( α ) is an eigenvalue of the monodromy.Later, we will also need similar ‘vanishing cycles’ invariants(7) µ pα,l := ( ν pα,l if α = 0 ν pα,l +1 if α = 0 . Now let us return to CPVHS from Proposition 2.6 (recall that it is unique up toa shift). So, let ( V, ∇ ) be the bundle with connection on P C − { , , ∞} constructedin the beginning of Section 2. Theorem 3.
There is a CPVHS ( V, ∇ ) such that ( a ) The local Hodge invariants at z = 0 are given by ν pα m ,l = ( if p = { i : α i ≤ α m } − { i : β i ≤ α m } , l = mult( α m ) − otherwise. ( b ) The local Hodge invariants at z = ∞ are given by ν p − β m ,l = ( if p = { i : α i < β m } − { i : β i < β m } , l = mult( β m ) − otherwise. ( c ) The Hodge numbers are given by h p = ρ − ( p − . This theorem will be proved in Section 5.2.5.
Stability conditions on the moduli of Higgs bundles.
The results ofSimpson [Sim2] allow us to translate Theorem 1 into a statement about Higgsbundles as we presently explain. For a vector bundle E we denote its fiber ata point z by E z .Let H iggs h,δ denote the moduli stack of collections(8) ( E, E , . . . , E h , E ∞ , . . . , E h ∞ , E , Φ) , where • E is a rank h degree δ vector bundle over P C ; • E ⊂ E ⊂ . . . ⊂ E h = E is a full flag in the fiber of E at z = 0; • E ∞ ⊂ E ∞ ⊂ . . . ⊂ E h ∞ = E ∞ is a full flag in the fiber of E at z = ∞ ; • E ⊂ E is a 1-dimensional subspace in the fiber of E at z = 1; • Φ : E → E ⊗ Ω P C (0 + 1 + ∞ ) is a Higgs field possibly with singularities at0, 1, and ∞ ; • for z = 0 or ∞ the residue of Φ at z is a nilpotent transformation compatiblewith the flag: (res z Φ)( E iz ) ⊂ E i − z ; • (res Φ)( E ) ⊂ E , (res Φ)( E ) = 0. ROMAN FEDOROV
We will abuse notation by writing (
E, E i , E i ∞ , E , Φ) instead of (8).We would like to define some stability conditions on H iggs h,δ . Consider twosequences of real numbers 0 < a < . . . < a h < < b < . . . < b h < . Let c ∈ [0 ,
1) be the unique number such that δ := − c − X i a i − X i b i ∈ Z . Let (
E, E i , E i ∞ , E , Φ) ∈ H iggs h,δ . For a subbundle E ′ ⊂ E define the set ofjumps I ( E ′ ) := { i : E ′ ∩ E h − i = E ′ ∩ E h − i +10 } ⊂ { , . . . , h } . Define I ∞ ( E ′ ) similarly. Setdeg a,b ( E ′ ) := ( deg E ′ + P i ∈ I ( E ′ ) a i + P i ∈ I ∞ ( E ′ ) b i if E E ′ c + deg E ′ + P i ∈ I ( E ′ ) a i + P i ∈ I ∞ ( E ′ ) b i if E ⊂ E ′ . We say that (
E, E i , E i ∞ , E , Φ) is ( a, b )-stable, if for every subbundle E ′ ⊂ E such that E ′ is preserved by Φ and E ′ = 0 , E , we have deg a,b ( E ) < a,b ( E ) = 0). Theorem 4.
Assume that for all pairs i and j we have a i + b j = 1 . Then there isa unique ( a, b ) -stable point ( E, E i , E i ∞ , E , Φ) in H iggs h,δ . Also, there is a decom-position E = L p E ( p ) such that ( a ) Φ( E ( p ) ) ⊂ E ( p − ⊗ Ω P C (0 + 1 + ∞ ) ; ( b ) This decomposition is compatible with the flags in the sense that for z ∈{ , , ∞} and all i we have E iz = M p (cid:0) E ( p ) z ∩ E iz (cid:1) . ( c ) rk E ( p ) is given by the formula of Theorem 1 with α i = 1 − a h +1 − i , β i = b i .Proof. The data (
E, E i , E i ∞ , E , a i , b i ) gives rise to a filtered vector bundle, thatis, to a vector bundle E := E | P C −{ , , ∞} with a family E ( α ) of extensions to P C parameterized by α ∈ R , see [Sim2, Synop-sis]. Comparing our definitions with that of [Sim2], one sees that an ( a, b )-stablepoint ( E, E i , E i ∞ , E , Φ) gives rise to a stable filtered regular Higgs bundle of degreezero. Using [Sim2, p. 718, Theorem], we see that this Higgs bundle corresponds toa stable filtered regular D-module over P C − { , , ∞} . The local monodromy ofthis D-module is calculated using the table on page 720 of [Sim2], so we see that thecorresponding bundle with connection is isomorphic to ( V, ∇ ) from Section 2.1 with α i = 1 − a h +1 − i , β i = b i (use rigidity). Moreover, the corresponding extensionsof V to P C at each of the points 0, 1, and ∞ are the extensions V > − α defined afterLemma 2.11. Indeed, it follows from the same table that the jumps of the filtrationare opposite to the residues of the connection (note that our convention for theresidue of the connection is opposite to that of Simpson).Let H • be a CPVHS on ( V, ∇ ) from Theorem 1 and let F p V be the correspondingholomorphic filtration on V . Then E ( α ) = gr F V > − α , Φ = gr F ∇ . Now one HS FOR HYPERGEOMETRIC DIFFERENTIAL OPERATORS 9 easily checks all the statements of the theorem with E ( p ) = gr pF V > (cf. [DS,Thm. 2.2.2]). (cid:3) Remark . One can also consider the case when some of the a i or b i coincide.This corresponds to the case of partial flags at z = 0 and z = ∞ . We leave theformulation to the reader. 3. Middle convolution
In this section we recall the notion of the Katz middle convolution. Every rigidvector bundle with connection can be obtained from a rank one bundle with con-nection by iterated application of Katz middle convolution and tensoring with rankone bundle with connection (see [Sim1, Sect. 5] and also [Kat, Ari, DS, DR, BE]).In Lemma 3.1 we show how this applies to the hypergeometric bundle with con-nection ( V, ∇ ) considered above. This seems to be well-known but the author havebeen unable to find a reference.We will use the language of D-modules, so let us give a dictionary (all D-modulesare assumed algebraic below). Let ( V, ∇ ) be a holomorphic vector bundle withconnection on a subset U of A C such that Σ := A C − U is finite. By [Mal, Ch. 2, (1.5)]we can uniquely extend ( V, ∇ ) to an algebraic bundle with meromorphic connectionon U such that the connection has regular singularities at Σ ∪∞ . Denote this bundlewith connection by ( V alg , ∇ alg ). Then the minimal extension M = ! ∗ ( V alg , ∇ alg ),where : U → A C is the inclusion, is a holonomic D-module on A C with regularsingularities at Σ ∪ ∞ . This construction gives a bijection between irreducibleholomorphic vector bundles with connection on U and irreducible not supportedat single point holonomic D-modules on A C whose singularities are regular andcontained in Σ ∪ ∞ . We will often use this identification, sometimes implicitly. Bya CPVHS on M we mean a CPVHS on ( V, ∇ ).Let us recall the notion of middle convolution with Kummer D-modules. For α / ∈ Z , let K α be the Kummer D-module ( C [ z, z − ] , d − α dzz ). (We are usingadditive notation, so K α is denoted by K e( α ) in [DS, Sect. 1.1].) Then, for aholonomic D-module M on A C , its middle convolution with K α (denoted M C α ( M ))is uniquely defined by the condition that its Fourier transform F M C α ( M ) is theminimal extension at the origin of F M ⊗ L − α , see [DS, Prop. 1.1.8]. Note that M C α ( M ) is also a holonomic D-module on A C . Again, we are using additivenotation, so M C α is denoted by M C e( α ) in [DS, (1.1.7)].Now we want to explain how to obtain the hypergeometric bundle with con-nection ( V, ∇ ) from a similar bundle with connection of smaller rank via a middleconvolution and tensoring with line bundles. Since z = ∞ is a special point forthe middle convolution, we need to move the singularity of ( V, ∇ ) from z = ∞ toanother point (cf. [DS, Assumption 1.2.2(2)]). Let ϕ : P C → P C be the projectivetransformation sending (0 , ,
2) to (0 , , ∞ ). Then ϕ ∗ ( V, ∇ ) is a bundle with con-nection on P C − { , , } . Let us restrict it to A C − { , , } and then extend to anirreducible holonomic D-module M on A C with regular singularities at z = 0, 1, 2,and ∞ as explained in the beginning of this section.We see from Proposition 2.1 that the local monodromy of M at z = 0 is conjugateto e( A ), its local monodromy at z = 2 is conjugate to e( B ), and its monodromy at z = 1 is a quasi-reflection. The monodromy at z = ∞ is trivial. Let the irreducible D-module M k,j on A C be constructed in the same way as M from the differential operator Y m = k ( D − α m ) − z Y m = j ( D − β m ) , D = z ddz . Now Proposition 2.1 describes the local monodromy of M k,j . In particular, themonodromies at z = 0 and z = 2 are regular operators, the monodromy at z = 1 isa pseudo-reflection, and the monodromy at z = ∞ is trivial.Let L k,j be (the D-module corresponding to) a line bundle with connection on P C − { , , ∞} with monodromy e( α k ) at z = 0, monodromy e( − β j ) at z = 2and monodromy e( β j − α k ) at z = ∞ . Similarly, let L ′ k,j be a line bundle withconnection on P C − { , , ∞} with monodromy e( − β j ) at z = 0, monodromy e( α k )at z = 2 and monodromy e( β j − α k ) at z = ∞ . Lemma 3.1.
For any k, j ∈ { , . . . , h } we have (9) M ≃ (cid:0) M C β j − α k ( M k,j ⊗ L ′ k,j ) (cid:1) ⊗ L k,j . Proof.
First of all, the generic rank of
M C β j − α k ( M k,j ⊗ L ′ k,j ) is(10) ( h −
1) + ( h −
1) + 1 − ( h −
1) = h by [DS, (1.4.2)].The only non-zero µ -invariants of M k,j ⊗ L ′ k,j at z = 0 are (cf. [DS, (1.2.3) and(1.2.5)]) given by µ α m − β j , mult( α m ) − = 1 if α m = α k µ α k − β j , mult( α m ) − = 1 if mult( α k ) ≥ . (We are using that α m = β j .) The monodromy at z = ∞ is e( β j − α k ) · Id.By [DS, Prop. 1.3.5] the only non-zero µ -invariants of M C β j − α k ( M k,j ⊗ L ′ k,j ) at z = 0 are µ α m − α k , mult( α m ) − = 1 if α m = α k µ , mult( α m ) − = 1 if mult( α k ) ≥ . Since the rank of
M C β j − α k ( M k,j ⊗ L ′ k,j ) is h , one derives that the monodromy of M C β j − α k ( M k,j ⊗ L ′ k,j ) at z = 0 is a regular operator whose list of eigenvalues ise( α − α k ) , . . . , e( α h − α k ). Similarly, the monodromy of M C β j − α k ( M k,j ⊗ L ′ k,j ) at z = 2 is a regular operator whose list of eigenvalues is e( β j − β ) , . . . , e( β j − β h ).Next, since the monodromy of M k,j ⊗ L ′ k,j at z = 1 is a pseudo-reflection,there is a unique non-zero µ -invariant µ γ, = 1 at z = 1. By [DS, Prop. 1.3.5] M C β j − α k ( M k,j ⊗ L ′ k,j ) also has a unique non-zero µ -invariant µ γ ′ , = 1 at z = 1.It follows that the corresponding local monodromy is also a pseudo-reflection.Finally, the monodromy of M C β j − α k ( M k,j ⊗ L ′ k,j ) at z = ∞ is e( α k − β j ) · Id;this follows from [DS, Cor. 1.4.1].We see that the left and the right hand sides of (9) have the same local mon-odromy (see the remark after the proof of Proposition 2.1). Since M is physicallyrigid, the statement follows. (cid:3) HS FOR HYPERGEOMETRIC DIFFERENTIAL OPERATORS 11 Behavior of local Hodge data under middle convolution
Let α, β, γ ∈ [0 ,
1) be distinct numbers. The notation α → β → γ will meanthat e( α ), e( β ) and e( γ ) appear on the unit circle in the counterclockwise order.Recall that we have fixed sequences α i and β j satisfying (1). In this section weassume that α > . Fix k, j ∈ { , . . . , h } . Let M k,j be as in the previous section.Let us put a CPVHS on M k,j (existing by Proposition 2.6). Then, according toLemma 3.1, [DS, Prop. 3.1.1], and [DS, Sect. 2.3], we get an induced CPVHSon M . Let µ pα,l denote the local Hodge invariants at z = 0 for M , while k,j µ pα,l denote similar invariants for M k,j . Proposition 4.1. ( a ) For all α m = α k , l ≥ we have µ pα m ,l = ( k,j µ pα m ,l if α m → α k → β jk,j µ p − α m ,l if α k → α m → β j . ( b ) For l ≥ we have µ pα k ,l = k,j µ p − α k ,l − . Proof.
In the notation of Lemma 3.1, let ′ µ pα,l denote the local Hodge invariantsat z = 0 for M ⊗ L − k,j , while ′′ µ pα,l denote the local Hodge invariants at z = 0 for M k,j ⊗ L ′ k,j . Recall that we assumed that α >
0. Thus, according to [DS, (2.2.13)],part (a) of the proposition is equivalent to(11) ′ µ pα m − α k ,l = ( ′′ µ pα m − β j ,l if α m → α k → β j ′′ µ p − α m − β j ,l if α k → α m → β j .Similarly, according to [DS, (2.2.14)], part (b) is equivalent to(12) ′ µ p ,l − = ′′ µ p − α k − β j ,l − . To prove (11), we may apply [DS, Thm. 3.1.2(2)] with α = { α k − β j } , α = { α k − α m } . We get ′ µ pα m − α k ,l = ( ′′ µ pα m − β j ,l if { α k − α m } < { α k − β j } ′′ µ p − α m − β j ,l otherwise.It remains to check that { α k − α m } < { α k − β j } if and only if α m → α k → β j .Finally, (12) follows from [DS, Thm. 3.1.2(2)] with α = 0. (cid:3) Proof of Theorem 3
Let ( V, ∇ ) the hypergeometric bundle with connection from the statement ofthe theorem. As explained before, it is more convenient for us to work with thebundle with connection ( ˜ V , ˜ ∇ ) := ϕ ∗ ( V, ∇ ) on P C −{ , , } . Recall that M denotesthe D-module on A C corresponding to ( ˜ V , ˜ ∇ ). Clearly, it is enough to prove the(analogue of the) theorem for a CPVHS on ( ˜ V , ˜ ∇ ).The theorem is obvious for h = 1. Thus we may assume that h ≥
2. We mayalso assume that the theorem is proved for all ranks smaller than h .The statement of Theorem 3 is invariant under subtracting the same numberfrom all α i and β i as explained below. Set α ′ i = { α i − γ } , β ′ i = { β i − γ } . Let M ′ be the D-module constructed in the same way as M from α ′ i and β ′ i . Then theD-module M ′ is isomorphic to M ⊗ K ′ γ , where K ′ γ is a line bundle with connection on P C − { , } with monodromy e( − γ ) around zero. Thus given a CPVHS on M ,we get a CPVHS on M ′ . Lemma 5.1.
Assume that the local Hodge invariants at z = 0 and z = 2 for aCPVHS on M are given by formulas of Theorem 3 (a) and Theorem 3 (b) respec-tively. Then the local Hodge invariants for the corresponding CPVHS on M ′ aregiven by the same formulas with α i and β i replaced by α ′ i and β ′ i up to a shift.Proof. First of all, the local Hodge invariants of M and M ′ are related via [DS,(2.2.13)].If the lemma holds for certain γ and all sequences (1), then it holds for anymultiple of γ . Thus we may assume that γ is small in the sense that γ < min( α k , β j ),where α k is the smallest positive element of the set { α , . . . , α h } , β j is the smallestpositive element of the set { β , . . . , β h } .Assume that Theorem 3 holds for some sequences α i and β i . Consider the case α = 0 (then γ < β ). For a number α we use notation α ′ := { α − γ } . Let α ∈ { α , . . . , α h } . By [DS, (2.2.13)] the only non-zero invariant of the form ν pα ′ ,l for M ′ corresponds to p = { i : α i ≤ α } − { i : β i ≤ α } = { i : α ′ i ≤ α ′ } − { i : β ′ i ≤ α ′ } + mult( α ) l = mult( α ′ ) = mult( α ) . This coincides with the required formula up to a shift by mult( α ). One checkssimilarly, that the Hodge invariants at z = 2 undergo a shift by mult( α ) as well.Now use Remark 2.5.The case β = 0 is completely similar. The case α > β > (cid:3) Using the lemma, we may assume that 0 < α < β and that the mult( α ) isthe largest among mult( α j ) through the end of the section. In particular, we havean equality of local Hodge invariants at z = 0 ν pα m ,l = µ pα m ,l for all m , l and p by (7). We start with the part (a) of Theorem 3. Proposition 5.2.
The local Hodge invariants at z = 0 for a CPVHS on M satisfythe formulas of Theorem 3 (a) up to a shift. Our strategy is as follows. Since we are assuming that the theorem is true forall ranks smaller than h , the proposition is also true for all ranks smaller than h .First, we prove the proposition in the case of the low rank h = 2 and α < α .Then we prove the proposition in the case when the rank is arbitrary but we have α < α < . . . < α h . Finally, we prove the proposition in the general case.If the eigenvalues are distinct, we can only have ν pα m ,l and µ pα m ,l non-zero, if l = 0. Thus, we skip the index l in the next two subsections.5.1. Rank two case—distinct eigenvalues.
According to Lemma 3.1, we have M ≃ M C β − α ( M , ⊗ L ′ , ) ⊗ L , . Using Lemma 5.1, we may assume that we are in the one of two cases0 < α < α < β ≤ β (Case I)0 < α < β < α < β (Case II) . (13) HS FOR HYPERGEOMETRIC DIFFERENTIAL OPERATORS 13
We put on M , the unique CPVHS such that h = 1, h p = 0 if p = 1. For thisCPVHS the only non-zero local Hodge invariant at z = 0 is , ν α = 1 (notationof Proposition 4.1). As before, this induces a CPVHS on M , and Proposition 4.1gives ν pα = ( p = 10 otherwise,which agrees with formulas of Theorem 3(a).Next, we will calculate the Hodge numbers h p for the CPVHS on M . Note that M , ⊗ L ′ , is (a D-module corresponding to) a line bundle with connection on P C − { , , , ∞} . Let ( V , ∇ ) be the extension of this line bundle with connectionto a line bundle with a singular connection on P C such that the residues of theconnection belong to [0 ,
1) (see Lemma 2.11). It is easy to see that( V , ∇ ) = (cid:18) O P ( − δ ) , d + { β − α } dzz + { α − β } dzz − { β − α } dzz − (cid:19) , where δ is chosen so that the residue at z = ∞ is { α − β } (recall that the degreeof the line bundle is opposite to the sum of the residues of a connection).In Case I of (13), we have δ = ( β − α ) + ( α − β + 1) + ( β − α ) + ( α − β + 1) = 2 . Similarly, in Case II we have δ = 3.The CPVHS on M , induces on M , ⊗ L ′ , a CPVHS, whose invariants at z = 0we denote by ′ h p , ′ ν pα etc. Clearly, ′ h = 1, ′ h p = 0 if p = 1.By [DS, Def. 2.3.1] we have ′ δ = − δ = ( − − ′ δ p are zero.Next, note that the local Hodge invariants ′ ν pα for M , ⊗ L ′ , at z = 0, 1, and 2are zero unless p = 1. Thus by [DS, (2.3.5*)] for the CPVHS on M we have h = δ − ′ h = ( , and h p = 0 for p = 1 ,
2. Since P p h p = rk( M ) = 2, we get h = ( . Finally, by [DS, (2.2.2**)], we have P m ν pα m = h p . Thus we get ν α = 1 in Case I ν α = 1 in Case II , all other ν pα being zero. This proves the proposition in the rank two case. Distinct eigenvalues.
Here we assume that h ≥ < α < . . . < α h < . We need to prove that up to a shift(14) µ pα m = ( p = m − { i : β i < α m } h . Thus for all k, j = 1 , . . . , h , we equip M k,j with a CPVHS such that for m = k we have(15) k,j µ pα m = ( p = { i : α i < α m , i = k } − { i : β i < α m , i = j } M with a CPVHS. Using uniqueness of CPVHS up to a shift andProposition 4.1, we see that there are constants c k,j ∈ Z such that we have for all k, j and m = k (16) µ pα m = ( k,j µ p + c k,j α m if α m → α k → β jk,j µ p − c k,j α m otherwise.Note that “+ c k,j ” is necessary because the CPVHS is defined up to a shift. Onthe other hand, these constants are the same for all m and p . Lemma 5.3.
The conditions (16) determine the numbers µ pα m and c k,j up to asimultaneous shift c k,j c k,j + c , µ pα m µ p + cα m .Proof. Assume that we have k, j, k ′ , j ′ ∈ { , . . . , h } . Since h ≥
3, we may find m such that 1 ≤ m ≤ h and m = k, k ′ . Applying (16) first to k, j, m and then to k ′ , j ′ , m , we see that c k,j − c k ′ ,j ′ are uniquely defined by (16). Thus the collection c k,j is uniquely defined up to simultaneously adding a constant. Now the lemma isobvious. (cid:3) Next, let ∗ µ pα m be defined by formula (14). In view of the previous lemma, itremains to prove Lemma 5.4.
The conditions (16) are satisfied with ∗ µ pα m instead of µ pα m and c k,j = ( if α k < β j if α k > β j . Proof.
Consider six cases.(a) α m < α k < β j . In this case ∗ µ pα m = k,j µ pα m c k,j = 0(b) α m < β j < α k . In this case ∗ µ pα m = k,j µ pα m c k,j = 1(c) α k < α m < β j . In this case ∗ µ pα m = k,j µ p − α m c k,j = 0(d) α k < β j < α m . In this case ∗ µ pα m = k,j µ pα m c k,j = 0 HS FOR HYPERGEOMETRIC DIFFERENTIAL OPERATORS 15 (e) β j < α k < α m . In this case ∗ µ pα m = k,j µ pα m c k,j = 1(f) β j < α m < α k . In this case ∗ µ pα m = k,j µ p +1 α m c k,j = 1We see that in each case (16) is satisfied. (cid:3) End of proof of Proposition 5.2.
Recall that we assumed that the propo-sition is proved for all ranks smaller than h . We also assumed that mult( α ) islargest among mult( α k ), so we may assume that mult( α ) ≥
2. Recall also that0 < α < β . By Proposition 4.1, after shifting the filtration, we have for α m = α ( µ pα m ,l = , µ pα m ,l if β < α m µ pα m ,l = , µ p − α m ,l if β > α m and for l ≥ µ pα ,l = , µ p − α ,l − . Now, using the induction hypothesis, it is easy to prove the proposition for the localHodge invariants at z = 0 except for µ pα , . However, we see that for some p wehave µ pα , mult( α ) − = 1. Since we have (see [DS, (1.2.3)])mult( α ) = µ α = X l ( l + 1) µ α ,l = X p,l ( l + 1) µ pα ,l , we see that µ pα , mult( α ) − = 1 is the only non-zero invariant. It follows that µ pα , = 0 and Proposition 5.2 is proved.5.4. Hodge invariants at z = 2 . Let ( V, ∇ ) be the bundle with connection on P C − { , , ∞} constructed from sequences (1) as above and let ( ˜ V , ˜ ∇ ) := ϕ ∗ ( V, ∇ )be the corresponding bundle with connection on P C − { , , } .Let ψ be a projective transformation taking (0 , ,
2) to (2 , , ψ takes ( ˜ V , ˜ ∇ ) to a similar bundle with connection ( ˜ V ′ , ˜ ∇ ′ ) corresponding to α ′ m =1 − β h − m +1 and β ′ m = 1 − α h − m +1 (we are using that α > β > V ′ , ˜ ∇ ′ ) underlies a CPVHS whose local invariants at z = 0 are given by Theorem 3(a) with α m , β m replaced by α ′ m , β ′ m . This CPVHSon ( ˜ V ′ , ˜ ∇ ′ ) induces a CPVHS ˜ V = L p ′ H p on ( ˜ V , ˜ ∇ ) via a pullback by ψ .We see that the non-zero local Hodge invariant ′ ν p − β m ,l of ( ˜ V , ˜ ∇ ) at z = 2 withrespect to the latter CPVHS correspond to l = mult( β m ) − p = { i : 1 − β i ≤ − β m } − { i : 1 − α i ≤ − β m } =( h − { i : β i < β m } ) − ( h − { i : α i < β m } ) = { i : α i < β m } − { i : β i < β m } . These are exactly the formulas of Theorem 3(b).Let ˜ V = L p H p be the CPVHS on ( ˜ V , ˜ ∇ ) satisfying formulas of Theorem 3(a)(it exists by Proposition 5.2). A priori, the Hodge decompositions ˜ V = L p H p and˜ V = L p ′ H p can differ by a shift of gradation. We will see below that, in fact, theycoincide, thus proving that the local Hodge invariants for ˜ V = L p H p at z = 2satisfy the formulas of Theorem 3(b). End of proof of Theorem 3.
According to [DS, (2.2.2**) and (2.2.3)] wehave for the CPVHS ˜ V = L p H p (18) h p = X α ∈{ α ,...,α h } ν pα = X α ∈{ α ,...,α h } X l ≥ l X k =0 ν p + kα,l . In particular,(19) max { p : h p = 0 } = max { p : ∃ α, l such that ν pα,l = 0 } =max m (cid:0) { i : α i ≤ α m } − { i : β i ≤ α m } (cid:1) . For t ∈ [0 , f ( t ) = { i : α i ≤ t } − { i : β i ≤ t } . The graph of f ( t ) is a union of horizontal intervals closed on the left and open onthe right. Now it is easy to see that f ( t ) attains its maximum on the union ofintervals of the form [ α k , β l ); fix one of such intervals. From (19) we obtainmax { p : h p = 0 } = f ( α k ) = max f ( t ) . Similarly, for the CPVHS ˜ V = L p ′ H p we have(20) ′ h p = X β ∈{ β ,...,β h } ′ ν p − β = X β ∈{ β ,...,β h } X l ≥ l X k =0 ′ ν p + k − β,l . and we show as above using (17) thatmax { p : ′ h p = 0 } = max m (cid:0) { i : α i < β m } − { i : β i < β m } (cid:1) = f ( β l ) + 1 = max f ( t ) . It follows that the two CPVHS are the same and we are done with part (b) of thetheorem.Now consider β ∈ { β , . . . , β h } . Assume that mult( β ) = j and β = β m = β m +1 = . . . = β m + j − (then β m − < β m and β m + j − < β m + j ). We have by Theorem 3(b) ν p − β,l = ( p = ρ ( m ) + 1 , l = j −
10 otherwise , where, as before, ρ ( m ) = { j : α j < β m } − m .Now we see from (20) that ν p − β,l contributes to h ρ ( m )+1 , h ρ ( m ) ,. . . , h ρ ( m ) − j +2 .Since ρ ( m + k ) = ρ ( m ) − k for 0 ≤ k ≤ j −
1, we see that β m = . . . = β m + j − contributes to h ρ ( m )+1 , h ρ ( m +1)+1 , . . . , h ρ ( m + j − . Therefore h p = ρ − ( p − Proof of Theorem 2
Let V be the local system of horizontal sections of ( V, ∇ ). Lemma 6.1.
There is an anti-linear automorphism θ of V with the property that θ = Id V . HS FOR HYPERGEOMETRIC DIFFERENTIAL OPERATORS 17
Proof.
We claim that the complex conjugate local system ¯ V is isomorphic to V . In-deed, by Proposition 2.1 and (5) both local systems have the same local monodromy,so the statement follows from rigidity (Proposition 2.4). Such an isomorphism givesan anti-linear automorphism θ of V .Then θ is a scalar endomorphism of V because V is irreducible (Proposition 2.2);write θ = λ · Id V . We claim that λ ∈ R > . Indeed, choose any x ∈ P C − { , , ∞} and consider the restriction θ x of θ to the fiber V x of V . Since the monodromyof V around z = 1 is a pseudo-reflection, θ x commutes with a pseudo-reflectionand, thus, with a rank one operator. Let v be any non-zero vector in the image ofthis rank one operator, then θ x v = ξv for ξ ∈ C . We have λ = ξ ¯ ξ and the claim isproved. It remains to scale θ by √ λ . (cid:3) Thus θ is a real structure on V , so we can write V = V R ⊗ C . This real structureinduces a real structure on V .Next, let V = L p H p be the Hodge decomposition from Theorem 1. Set ′ H p := H − p , then ( V, ∇ , ′ H • ) is another variation of Hodge structures on ( V, ∇ ).If Q is a polarization of ( V, ∇ , H • ), then ¯ Q given by ¯ Q ( x, y ) := Q (¯ x, ¯ y ) is apolarization of ( V, ∇ , ′ H • ). Thus ( V, ∇ , ′ H • ) is a CPVHS so, by uniqueness, itdiffers from ( V, ∇ , H • ) by a shift of gradation: there is c ∈ Z such that for all p wehave H p = ′ H p − c = H c − p . Since the smallest p such that H p = 0 is p = p − , andthe largest such p is p = p + , we see that c = p + + p − .For integers p and q such that p + q = p + − p − , set H p,q := H p + p − . Then V = M p + q = p + − p − H p,q , and we get a real variation of Hodge structures ( V R , H • , • ) of weight k = p + − p − .Since ( V, ∇ ) is irreducible, a horizontal non-degenerate hermitian form is uniqueup to a scaling by a real factor. Thus ¯ Q = λQ . Since ¯ Q = Q , we have λ = ± Q and ¯ Q are polarizations, we see that λ = ( − k , so Q gives rise to a( − k -symmetric form Q R on V R . Now it is easy to see that Q R or − Q R is a realpolarization; Theorem 2 is proved. Acknowledgements.
The author is grateful to Dima Arinkin, Mikhail Mazin,Anton Mellit, Vladimir Fock, Vladimir Rubtsov, and Masha Vlasenko for the inter-est to the subject and discussions. He also wants to thank Michael Dettweiler forbringing to author’s attention [Ter1, Ter2] and Claude Sabbah for useful remarkson an early draft. The author is partially supported by NSF grant DMS-1406532.Special thanks go to Vasily Golyshev who introduced the author to the problem.A part of this work was done, while the author was a member of Max PlanckInstitute in Bonn.
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