Exponents of Some One-Dimensional Gauss–Manin Systems

EXPONENTS OF SOME ONE-DIMENSIONAL GAUSS-MANIN SYSTEMS ALBERTO CASTA˜ NO DOM´ INGUEZ Abstract. In this paper we provide a purely algebraic characterization of the exponents of onedimensional direct images of a structure sheaf by a rational function, related to the vanishing of the cohomologies of a certain Koszul complex associated with such a morphism. This can be extended to a more general family of Gauss-Manin systems. As an application, we calculate a set of possible exponents of the Gauss-Manin cohomology of some arrangements of hyperplanes with multiplicities, relevant to Dwork families and mirror symmetry. 1. Introduction Let kbe an algebraically closed field of characteristic zero. An algebraic variety, or just variety, will mean for us an equidimensional quasi-projective separated finite type scheme over k, reducible or not. For any smooth variety X, Db(DX) will denote the category of bounded complexes of DX-modules. For an open subvariety of the affine line, the exponents of a D-module over it are strongly related to the monodromy of its solutions. This notion is topological in nature when k=C, but we can manage to work in an algebraic way with a similar concept, and because of that, we will usually use both names, monodromy and exponents, to denote the phenomenon and the object of study. Although this theory can be constructed in any dimension thanks to the formalism of the V-filtration, the Bernstein-Sato polynomial and the vanishing cycles of Malgrange and Kashiwara (cf. [6, 9, 10] or the appendix by Mebkhout and Sabbah at [12, Ch. III, §4]), it is defined in a much more simple way in dimension one. In fact, we will follow the approach of [7, §2.11]. The main aim of this paper is to prove the following result: Theorem 1.1. Let nbe a fixed positive integer, and let g∈k[x1, . . . , xn]be a nonzero polynomial. Now let R=k((t))[x1, . . . , xn, g−1],f∈k[x1, . . . , xn, g−1], and denote by f0 ithe partial derivatives of fwith respect to the variables xi, i = 1, . . . , n, and by Gthe closed subvariety {g(x)=0} ⊆ An. Let α∈kand let ϕαbe the endomorphism of k((t)) given by ∂t−αt−1. Denote also by fthe associated morphism An−G→A1. Then, αmod Zis not an exponent at the origin of any of the DA1-modules Hif+OAn−Gif and only if the morphism Φ : Rn+1 −→ R, (a, b1, . . . , bn)7−→ (f−t)a+ (∂1+f0 1ϕα)b1+· · · + (∂n+f0 nϕα)bn is surjective. If it is not surjective, the number of Jordan blocks associated with αof the zeroth cohomology is the dimension of the cokernel of Φas a k-vector space. As the reader can check, the statement and its proof are completely independent of the choice of k, providing a purely algebraic way of dealing with the exponents of a morphism for any such field, apart from their definition itself. 2010 Mathematics Subject Classification. Primary 14F10, Secondary 13F25, 14D05. The author is partially supported by FQM218, P12-FQM-2696, MTM2013-46231-P, ERDF and the project SISYPH: ANR-13-IS01-0001-01/02 and DFG grants HE 2287/4-1 and SE 1114/5-1. 1 2 ALBERTO CASTA˜ NO DOM´ INGUEZ As indicated in the abstract, we will apply this result to a more general context, regarding Gauss-Manin systems over an open subset of the affine line. However, the notion of exponent is really local in nature, so that an open subvariety actually plays an irrelevant role and in the following we will consider everything over the whole affine line. In the final section we provide two examples illustrating the usefulness of the result. More concretely, we start with a well-known fact about quasi-homogeneous singularities and then we give a result about the exponents of a Gauss-Manin system associated to a special family of arrangements of hyperplanes, which is our second main result: Theorem 1.2. Let (w0, . . . , wn)∈Zn+1 >0be an (n+ 1)-tuple of positive integers, and let λ= xw1 1·. . . ·xwn n(1 −x1− · · · − xn)w0. Then α∈kis an exponent at the origin of some cohomology of λ+OAnonly if wiαis an integer for some i= 0, . . . , n. Moreover, the multiplicity of every exponent of the form j/wiis the same, without counting coincidences among some j/wifor different values of i= 0, . . . , n and j= 1, . . . , wi. This calculation does not belong only to the realm of hyperplane arrangements, but appears in other interesting contexts. Namely, it can appear when we study the Gauss-Manin cohomology of a generalized Dwork family (cf. [2, §3] and [8] to know more, respectively, about that relation or about Dwork families in general). That was in fact the main motivation for overcoming this problem. In addition, in [2, §3] we explain how the Gauss-Manin system associated with the morphism λis strongly related to the restriction of a linear form to a torus in An+1 (in fact the latter is just an inverse image by an ´etale covering of the former), a setting already treated in [3], for instance, and of importance in mirror symmetry, for it gives a description of the quantum cohomology of a weighted projective space. Acknowledgements. This work evolved from a part of the doctoral thesis of the author, advised by Luis Narv´aez Macarro and Antonio Rojas Le´on. The author wants to thank the former for suggesting the idea that motivated the main result to him, and both of them for their support and encouragement. He also want to thank them and Christian Sevenheck for their useful comments and discussions about this work, Jean-Baptiste Teyssier for his interesting questions regarding the main result and the anonymous referees for their many useful comments to improve this text. 2. Preliminaries In this section we will recall the basic concepts from D-module theory that we will need in the following. Definition 2.1. Let f:X→Ybe a morphism of smooth varieties. The direct image of complexes of DX-modules is the functor f+: Db(DX)→Db(DY) given by f+M:= Rf∗DY←X⊗L DXM, where DY←Xis the f−1DY,DX-bimodule DY←X:= ωX⊗f−1OYf−1HomOY(ωY,DY). The inverse image of complexes of DY-modules is the functor f+: Db(DY)→Db(DX) given by f+M:= DX→Y⊗L f−1DYf−1M, where DX→Yis the DX, f−1DY-bimodule DX→Y:= OX⊗f−1OYf−1DY. EXPONENTS OF SOME ONE-DIMENSIONAL GAUSS-MANIN SYSTEMS 3 Remark 2.2.When f:X=Y×Z→Zis a projection, DZ←X⊗L DXMis nothing but a shifting by dim Yplaces to the left of the relative de Rham complex of M DRf(M) := 0 −→ M −→ M ⊗OXΩ1 X/Z −→ . . . −→ M ⊗OXΩn X/Z −→ 0, so we will have f+∼ =Rf∗DRf(•)[dim Y] ([12, Ch. I, Lem. 5.2.2]). We will be interested in the case in which Xis an open subvariety of the affine line. From now on, we will denote by Dxthe product x∂x, omitting the variable as long as it is clear from the context. Definition 2.3. AKummer D-module is the quotient Kα=DGm/(D−α), for any α∈k. Remark 2.4.Note that any two Kummer D-modules Kαand Kβare isomorphic if and only if α−βis an integer. Then Kα∼ =OGmfor any α∈Z. Proposition 2.5. Let Mbe a holonomic DX-module, let pbe a point of X, and fix a formal parameter xat psuch that b OX,p ∼ =k[[x]]. The tensor product M ⊗OXk((x)) can be decomposed as the direct sum of its regular and purely irregular parts. Now assume that M ⊗OXk((x)) ∼ =k((x))[D]/(L), where L=X i xiAi(D)∈k[[x]][D], with degDL=g≥g0= degDA0(so that A06= 0). As a consequence, the rank of (M ⊗OXk((x)))reg is g0, and if this last degree is positive and A0(t) = γQi(t−αi)ni, its composition factors are Kαi,p with multiplicity ni, where Kβ,p is the tensor product k((x))⊗k[x±]Kβ∼ =k((x))[D]/(D−β). Moreover, if the roots of A0(t)are not congruent modulo Z, then (M ⊗OXk((x)))reg ∼ =k((x))[D]/(A0(D)) ∼ =M i k((x))[D]/(D−αi)ni. Proof. The decomposition into regular and purely irregular parts of the tensor product M ⊗OX k((x)) is a well-known fact of the theory of integrable connections (cf. [11, Ch. III, Thm. 1.5, Cor. 1.7]). The rest is analogous to [7, Cor. 2.11.7]. Although that result and those on which it depends at [7, §2.11] are stated over C, their proofs are purely algebraic and, in fact, can be generalized for any algebraically closed field of characteristic zero.  Proposition 2.6 (Formal Jordan decomposition lemma).Let M,pand xbe as before, and suppose that Mis regular at p. Then, (i) M ⊗OXk((x)) is the direct sum of regular indecomposable k((x))[D]-modules; (ii) writing Loc(α, nα) := k((x))[D]/(D−α)nα, then, for any two k((x))[D]-modules Loc(α, nα)and Loc(β, nβ), and i= 0,1, the vector space Exti DX(Loc(α, nα),Loc(β, nβ)) has dimension min(nα, nβ)if α−β∈Zand zero otherwise; (iii) any regular indecomposable k((x))[D]-module is isomorphic to Loc(α, nα), where αis unique modulo the integers; (iv) given α∈k, the number of indecomposables of type Loc(α, m)in the decomposition of M ⊗OXk((x)) is dimkHomDX(M,Loc(α, 1)). 4 ALBERTO CASTA˜ NO DOM´ INGUEZ Proof. When k=C, there is a topological proof as in [7, Lem. 2.11.8]. However, we can give a purely (linear) algebraic one. Since Mis holonomic, it is a finitely generated torsion DX-module, and so will M ⊗ k((x)) be over k((x))[D]. This ring is a noncommutative principal ideal domain, so by the structure theorem for finitely generated modules over such a ring (cf. [5, Ch. 3, Thm. 19]) we obtain that M ⊗ k((x)) is the direct sum of indecomposable k((x))[D]-modules. They must be regular since Mis, and that proves point (i). Now let Loc(α, nα) and Loc(β, nβ) be as in point (ii). We can suppose that both αand βbelong to the same fundamental domain (exhaustive set of representatives without repetitions) of k/Z, up to isomorphism. Since Loc(α, nα) is a flat k((x))-module, we can assume that α= 0. Now note that the vector spaces Exti DX(Loc(α, nα),Loc(β, nβ)) are just the kernel and the cokernel of Dnαover Loc(β, nβ). In general, each of the Loc(γ, m), for any γ∈k/Zand any m > 0, is a successive extension of Loc(γ, 1). Thus if β6= 0, since the operator Dis clearly bijective on them, both Ext spaces vanish. If β= 0, then the statement is easy to check. Let us go now for point (iii). Thanks to the discussion of the first point, we can affirm that M ⊗ k((x)) ∼ = r M i=1 k((x))[D]/(Ai(x, D)), where Ai(x, D) = Pj≥0xjAij(D). By Proposition 2.5, k((x))[D]/(Ai(x, D)) is isomorphic to a successive extension of the Loc(αi,1), the αibeing the roots of each Ai0. Now we just need to invoke the previous point; such an extension must be a direct sum of some Loc(β, nβ), with, possibly, some nβ>1 if the roots of Ai0were congruent modulo the integers. Point (iv) is just an easy consequence of the two preceding ones.  These two propositions show that the equivalence classes modulo Zof the numbers αappearing in the decomposition of the tensor product of a holonomic DX-module with k((x)), and their associated nα, are intrinsic to the DX-module and it is quite important, actually, to know its behaviour at a point, so that motivates the following definition. Definition 2.7. Let M,pand xbe as in Proposition 2.5. The exponents of Mat pare the values αi∈ksuch that (M ⊗OXk((x)))reg ∼ =M i Loc(αi, ni), seen as elements of k/Z. For each exponent αiwe define its multiplicity as ni. Remark 2.8.For the sake of simplicity, we will usually denote both an exponent and some of its representatives in kin the same way. Exponents are considered unordered and possibly repeated. Note that, when k=C, this notion of multiplicity of an exponent αis related to the size of the Jordan blocks of the local monodromy associated with the eigenvalue e2πiα, and not to its multiplicity as a root of the characteristic polynomial of the monodromy. However, these two notions are the same under some special conditions (cf. [7, Cor. 3.2.2, Lem. 3.7.2]). Nevertheless, in our algebraic setting, whenever we mention “Jordan block” we will mean a regular indecomposable Loc(α, nα), in analogy with the complex analytic case. Although, as we have seen, the exponents at the origin of a DA1-module can be defined even if it has an irregular singularity there, in the following we will deal with complexes of regular holonomic DA1-modules, since direct image preserves regularity (cf. [12, Ch. II, Thm. 9.3.1]). EXPONENTS OF SOME ONE-DIMENSIONAL GAUSS-MANIN SYSTEMS 5 3. Gauss-Manin systems, main result and Laurent series Let us recall now the basic setting of one-dimensional Gauss-Manin systems, seen from the point of view of D-module theory. Fix a positive integer n, some variables x1, . . . , xnand a special one called λ. Consider an open set U⊆An= Spec (k[x1, . . . , xn]) and a smooth variety X⊂U×A1=U×Spec(k[λ]), together with the second projection π2:X→A1. In terms of D-modules, the Gauss-Manin cohomology, or system, of X, is just the direct image of the structure sheaf π2,+OX. It is a complex of DA1- modules, so we could be interested in knowing its behaviour at the origin, and in particular its exponents. In this paper we will focus on the case where Xis a hypersurface. Going back to Theorem 1.1, the direct image f+OAn−Gcan be seen as the Gauss-Manin cohomology of the graph of fin (An−G)×A1. However, that is a rather concrete and simple example of a family of hypersurfaces. We will explain after proving our main result how to relate this setting to a broader family of Gauss-Manin systems. Proof of Theorem 1.1. Let us first deal with the zeroth cohomology of f+OAn−G. After finishing with it we will justify the extension of the statement to all of them. Let K=f+OAn−G. By Proposition 2.6 we can claim that H0(K)⊗OA1k((t)) ∼ = r M i=1 k((t))[D]/(D−βi)mi. From the second paragraph of the same proposition we can deduce that αwill not be an exponent of H0(K) if and only if the endomorphism D−α·:H0(K)⊗OA1k((t)) −→ H0(K)⊗OA1k((t)) is bijective, or even surjective, for the Exti k((t))[D]k((t))[D]/(D−α),k((t))[D]/(D−β)k do not vanish or do vanish at the same time, whenever αis or is not congruent to βmodulo Z, respectively, for i= 0,1 and any k. Now let us decompose the morphism fas the closed immersion into its graph iΓfollowed by the projection πon the first coordinates. By Kashiwara’s equivalence and the properties of local cohomology (cf. [4, Thm. 1.6.1, Prop. 1.7.1]), the complex iΓ,+OAn−Gis concentrated in degree zero and, furthermore, is O(An−G)×A1(∗Γ)/O(An−G)×A1, for the graph of fis smooth in An+1. Therefore, the fact that αis not an exponent of H0(K) is equivalent to the surjectivity of D−α on π+O(An−G)×A1(∗Γ)/O(An−G)×A1⊗OA1k((t)). Note that we are always dealing with affine morphisms and quasi-coherent O(An−G)×A1-modules and we are taking tensor products with k((t)), so it suffices (cf., for example, [4, Prop. 1.4.4]) to work from now on with the global sections of the objects involved in the proof. Write M=k[x, g−1, t](t−f)−1/k[x, g−1, t]. Recall that we are interested in the top cohomology of DRx(M). Since k((t)) is flat over k[t], tensor products with the former over the latter commute with cohomology, and thus we are going to deal with Mloc := k((t))[x, g−1](t−f)−1/k((t))[x, g−1], which is a module over R=k((t))[x1, . . . , xn, g−1] and b D:= Rh∂t, ∂1, . . . , ∂ni. Let us introduce just a bit more of notation that we are going to use. We will write Dt:= k((t))h∂ti,Dx:= k[x, g−1]h∂1, . . . , ∂niand b Dx:= k((t))[x, g−1]h∂1, . . . , ∂ni. 6 ALBERTO CASTA˜ NO DOM´ INGUEZ Summing everything up, αmod Zis not an exponent of the DA1-module H0Kat the origin if and only if R1HomDtDt/(D−α),RnHom b Dx(R, Mloc)= 0. Note that R∼ =k((t)) ⊗kk[x, g−1]∼ =k((t)) ⊗kDx⊗Dxk[x, g−1]∼ =b Dx⊗Dxk[x, g−1], so by extension of scalars, RnHom b Dx(R, Mloc)∼ =RnHomDxk[x, g−1], Mloc. Now applying the derived tensor-hom adjunction, R1HomDtDt/(D−α),RnHomDxk[x, g−1], Mloc ∼ =Rn+1 HomDxDt/(D−α)k[x, g−1], Mloc ∼ =Rn+1 Hom b Db D/(D−α, ∂1, . . . , ∂n), Mloc, the last isomorphism being by extension of scalars again. (Note that the first isomorphism is of k-vector spaces.) Now Mloc is a self-dual b D-module, being the direct image by a closed immersion of the self-dual object k[x, g−1] (cf. [12, Ch. I, Cor. 5.3.13]), so by duality αis not an exponent of the DA1-module H0Kat the origin if and only if Rn+1 Hom b DMloc,b D/(D+1+α, ∂1, . . . , ∂n)= 0. The second b D-module above is nothing but R·t−1−α, where t−1−αshould be understood as a symbol. The actions of the partial derivatives are the usual ones in Rof ∂1, . . . , ∂n, and regarding ∂t, ∂ta·t−1−α=∂t(a)·t−1−α+ (−1−α)t−1a·t−1−α. In order to finish all this construction, take into account that the annihilator of the class of (t−f)−1in Mloc is the left ideal (f−t, ∂1+f0 1∂t, . . . , ∂n+f0 n∂t); indeed, each of its generators make it vanish and the ideal is maximal. Therefore, Mloc can be presented as Mloc ∼ =b D/(f−t, ∂1+f0 1∂t, . . . , ∂n+f0 n∂t). Recall that the exponents are equivalence classes in k/Z, so we could replace αby α−1 in this whole procedure, taking another representative of the same exponent without affecting the validity of the proof. Then we can claim that αis not an exponent of the DA1-module H0f+OAn at the origin if and only if the k-linear homomorphism Φ : Rn+1 −→ Rgiven by Φ = f−t, ∂1+f0 1ϕα, . . . , ∂n+f0 nϕα is surjective. The statement on the dimension of the cokernel follows easily by reversing the isomorphisms and equivalences and applying point (ii) of Proposition 2.6. Note that the operators f−t, ∂1+f0 1ϕα, . . . , ∂n+f0 nϕαcommute pairwise, so the Koszul complex K•(R;f−t, ∂1+f0 1ϕα, . . . , ∂n+f0 nϕα) is well defined. Moreover, thanks to the same choice of the representative of αmod Zas two paragraphs above we can see that its cohomologies are just the vector spaces RkHom b DMloc,b D/(D+1+α, ∂1, . . . , ∂n), whose duals are in turn an extension of the (1) RiHomDtDt/(D−α),RjHom b Dx(R, Mloc) EXPONENTS OF SOME ONE-DIMENSIONAL GAUSS-MANIN SYSTEMS 7 with j=kand j=k−1. Now we claim that the surjectivity of Φ is equivalent to the vanishing of all the cohomologies of such a Koszul complex. One implication is trivial; the other is [1, §9, Cor. 1]. Then, Φ is surjective if and only if all of the vector spaces (1) vanish, which, following an argument analogous to the case of the zeroth cohomology H0K, is equivalent to the fact that αis not an exponent of any of the cohomologies of f+OAn−G. This ends the proof of the theorem.  We might have that some αis an exponent at the origin of some cohomologies of f+OAn−G. In that case, reviewing the final argument of the proof, one can still have a partial result when dealing with the vanishing of the cohomologies of the whole Koszul complex. More concretely, we have the following: Corollary 3.1. Under the same conditions as before, were the Koszul complex K•(R;f−t, ∂1+ f0 1ϕα, . . . , ∂n+f0 nϕα)acyclic in degrees d0to d1(possibly equal to 0 or n+ 1, respectively), then αmod Zis not an exponent at the origin of any of the cohomologies Hkf+OAn−Gfor d0−1≤k+n≤d1. Proof. As we noted at the end of the proof of the theorem, if K•(R;f−t, ∂1+f0 1ϕα, . . . , ∂n+f0 nϕα) is acyclic in degree k, then RkHom b DMloc,b D/(D+1+α, ∂1, . . . , ∂n)= 0, whose dual is the extension of the RiHomDtDt/(D−α),RjHom b Dx(R, Mloc) with j=kand j=k−1. As a consequence, for every iand jwith d0−1≤j≤d1such an object must vanish, and in conclusion, the endomorphism D−α:Hj(K)⊗k((t)) −→ Hj(K)⊗k((t)) is surjective for d0−1≤j+n≤d1, so αmod Zis not an exponent at the origin of any of the cohomologies Hjf+OAn−Gfor such values of j. The following corollary, despite being an easy consequence of both the theorem and the previous corollary, seems to us interesting enough to be written explicitly: Corollary 3.2. Using the same notation as in the theorem, if αmod Zis an exponent at the origin of some cohomology Hif+OAn−Gwith i < 0, then it is also an exponent of H0f+OAn−G. Proof. If αmod Zwere not an exponent of H0f+OAn−G, then its associated morphism Φ as in the theorem would be surjective, so as in the end of the proof of our main result, by [1, §9, Cor. 1] we could claim that every cohomology of the Koszul complex K•(R;f−t, ∂1+f0 1ϕα, . . . , ∂n+f0 nϕα) would vanish. Then by the corollary above, αmod Zwould not be an exponent at the origin of any of the other cohomologies of f+OAn−G. We return now to the context of Gauss-Manin systems in the form of the following statement: Proposition 3.3. Keep the notation as at the beginning of this section. Let r(x)be a polynomial of k[x], and let Ube the basic open set {r(x)6= 0}. Assume there exist two regular functions on U,p(x)and q(x), such that Xis defined by the equation p(x)−λdq(x) = 0, for certain d > 0. Write f=p/q,q= ˜q/rmand G={r˜q= 0}. Then, the noninteger exponents and the integer ones with multiplicity greater than one of the Gauss-Manin system π2,+OXare dtimes those of f+OAn−G. 8 ALBERTO CASTA˜ NO DOM´ INGUEZ Proof. Let us show how to reduce ourselves to consider d= 1. Indeed, form the Cartesian diagram X  π2  id ×[d] //˜ X π2  A1[d] //A1, where [d] just means taking the dth power of the argument. It is easy to check that if Xis smooth, so is ˜ X. Therefore, by the base change formula [4, Thm. 1.7.3], π2,+OX∼ =[d]+π2,+O˜ X, so as written above, we could find the exponents of the Gauss-Manin cohomology of Xby finding those of ˜ X; the former will just be dtimes the latter. Rename, for the sake of clarity, ˜ Xas X, assuming that d= 1 throughout the rest of the proof. Write both pand qas fractions with the same denominator ¯p/rNand ¯q/rN, respectively. Then Xis the vanishing locus of ¯p(x)−λ¯q(x) in U×A1. Let Zbe the hypersurface of Xwith equation ¯q(x) = 0. Then Zis contained in Xand is defined by {¯p= ¯q= 0}in the whole of U, so it is the product of a subvariety Z0⊂Uwith A1. Now we can form the excision triangle (cf. [12, Ch. I, §6.1]) RΓ[Z]OX−→ OX−→ OX(∗Z). Let us see what happens when we apply π2,+to the triangle. Let ibe the closed immersion X→U×A1. Thanks to [12, Ch. I, Prop. 6.4.1], we can affirm that i+RΓ[Z]OX∼ =RΓ[Z](i+OX)∼ = RΓ[Z]RΓ[X]OU×A1[1], for Xis smooth in U. But RΓ[Z]RΓ[X]OU×A1∼ =RΓ[Z]OU×A1because Zis contained in X(cf. [12, Ch. I, Prop. 6.2.4] and beware the typo). Now the latter local cohomology module is nothing but π+ 1RΓ[Z0]OU, where π1is the first projection U×A1→U, by [12, Ch. I, Prop. 6.3.1]. Thus abusing a bit of the notation so that π2represents two different projections onto A1, π2,+RΓ[Z]OX∼ =π2,+π+ 1RΓ[Z0]OU. And now it is easy to see that this complex has only copies of OA1among its cohomologies; it is simply a consequence of applying the base change formula to the Cartesian square U×A1  π1  π2//A1 πA1  UπU//{∗}, where πUand πA1are the projections from the variety in the subscript to a point. In conclusion, we can claim that π2,+RΓ[Z]OXis just a bunch of copies of the structure sheaf OA1. Then apart from the purely constant part, the information about the exponents of π2,+OX can be found within π2,+OX(∗Z). But X−Zcan be seen as the graph of ¯p/¯q=p/q in U×A1, so now this complex can be realized in the form of our Theorem 1.1 just by taking g=r¯qand f= ¯p/¯q. Remark 3.4.We have provided in the end a way of computing the noninteger exponents of π2,+OX. In fact, we could have thought of a slightly broader family of Gauss-Manin systems, namely, those associated with a family of the form X={p(x)−γ(λ)q(x)=0} ⊂ U×A1, for some polynomial γ∈k[λ]. The reduction of the beginning of the proof by base change would still be possible, but the calculation of γ+π2,+O˜ Xwould not at all be as direct as with γ(λ) = λd. We finish this section by providing several results or notions regarding the field of formal Laurent series that will be of interest later when we tackle a particular example. EXPONENTS OF SOME ONE-DIMENSIONAL GAUSS-MANIN SYSTEMS 9 Lemma 3.5. Let ϕ:k((t)) −→ k((t)) be a k-linear automorphism of k((t)) such that ϕ(k[[t]] · tk) = k[[t]] ·tkfor every k∈Z. Then, for any k-linear endomorphism ψof k((t)) such that ψ(k[[t]] ·tk)⊆k[[t]] ·tk+1, the sum ϕ+ψis another automorphism of k((t)). Proof. Multiplying by ϕ−1we can assume that ϕ= id. We will write the elements of k((t)) as a=Pkaktk. Then let bbe a fixed formal Laurent series and let us see if there exists an a∈k((t)) such that (id +ψ)(a) = b. Evidently, the exponents of the least powers of t(which are called the orders) of both of aand bwill be the same, so let us write a=X k≥m aktk, ψ(a) = X k≥m+1 a0 ktkand b=X k≥m bktk. From the equation (id +ψ)(a) = bwe deduce that am=bm. Now call a1=a−amtmand b1=b−(id +ψ)(amtm); both of them have order m+ 1. We have (id +ψ)a1= (id +ψ)a−(id +ψ)(amtm) = b1. Thus we can start the same process over again with a1and b1. Since this can be continued for every power of t, we can deduce the surjectivity of id +ψ. Moreover, if we take bk= 0 for every k∈Z, it follows that every akvanishes too, so id +ψis also injective.  Definition 3.6. Let rbe an element of k. Then we can define the operators Dt,r =t∂t+rand analogously Di,r =xi∂i+r, for i= 1, . . . , n. We will write ϕr=∂t+rt−1=t−1Dt,r (note the sign change with respect to the previous notation). They are k-linear endomorphisms of k((t)), so we can also consider them to be operating within any k((t))-algebra by extension of scalars. Remark 3.7.It is easy to see that Dt,r (and so ϕr) is an automorphism of k((t)) for every rnot an integer and only for them, for Dt,r sends a power tkof tto (k+r)tk. In this case we can define another family of operators that will play a fairly main role in the next section: Definition 3.8. Fix an element αof k, and let rand sbe two other elements of ksuch that α+sis not an integer. Then we can define the operator Ar,s =t+rϕ−1 α+s. Let Rn=k((t))[x1, . . . , xn] and β∈k. We can also define the k-linear endomorphisms of Rn given by AβDi,r,s =t+βDi,rϕ−1 α+s, where i= 1, . . . , n. In the following, for the sake of simplicity, we will denote by Ar,AβDi,r and Drthe operators Ar,0,AβDi,r ,0and Dt,r, respectively. Remark 3.9.As before, Ar,s is not always an automorphism of k((t)), as AβDi,r,s is of Rn. Since Ar,sϕα+s=Dt,α+r+s, the former is bijective whenever α+r+sis not an integer. Analogously, AβDi,r,sϕα+s=βDi,r +Dt,α+s. It sends tkxuto (β(ui+r) +α+k+s)tkxu, so AβDi,r,s is bijective if and only if, for every integer l, we have that β(l+r) + α+sis not an integer. Now we could wonder about the commutativity of those operators that we have just defined. We will use the following lemma, whose proof is easy and left to the reader (for each relation, use some of the ones proved before and the Leibniz rule): Lemma 3.10. Let αand βbe two elements of k, and r,r0,sand s0four other elements of k such that neither α+snor α+s0are integers. Then, the following relations hold: (1) tϕα=ϕα−1t , ϕαϕβ=ϕβ+1ϕα−1. (2) Ar,st=tAr,s+1 , AβDi,r ,sxi=xiAβDi,r+1,s , Ar,sAr0,s0=Ar0,s0−1Ar,s+1. 16 ALBERTO CASTA˜ NO DOM´ INGUEZ system of homogeneous polynomials of degree m (12)                            dAm+d+n−1 d Fm+d−2+ (−1)w0w0˜ F= 0, . . . dAm+d+n−r d Fm+d−r−1−(d−w0)Am+d+n−r d−w0 Fm+d−r +(−1)w0−r+1w0w0−1 w0−r˜ F= 0, r = 2, . . . , w0−1, . . . ΥAm+n d−w0˜ F−(d−w0)Am+d+n−w0 d−w0 Fm+d−w0= Υcm. This system is the one that we will prove to have a solution, so that we finally show that Φ is surjective. The Aβdo not commute pairwise, so in principle we cannot deal with the determinant of the matrix of the system. However, under an easy change of variables, we can see the Aβas elements of a commutative subring of the ring of endomorphisms of R. By our assumption on α, the endomorphism Dαof k((t)) is invertible, so we can define a new operator Bβas Aβαt−1=α(1 + βD−1 α), which is an element of k[D−1 α], a commutative ring whose action on k((t)) is defined by D−1 αtl= (l+α)−1tl. Now αt−1is an isomorphism of R, so we can rename the Fkto mean tα−1Fk, for each k=m+d−w0, . . . , m +d−2. Now every coefficient of system (12) is of the form of some Bβand thus lives in k[D−1 α], except for ΥBm+n din the final equation, which has degree 1 in t. Nevertheless, note that this operator goes together with −w0, so by Lemma 3.5 its sum is an automorphism of R. Moreover, regarding just the existence of solutions to the system and not their actual form, we can restrict ourselves to working only with −w0. If w0= 1, then we have only a single equation, from which we can solve ˜ Fand thus the system, showing the existence of solutions. In the following we will assume that w0≥2. We have finally arrived at a point where we have a matrix of coefficients in k[D−1 α], so we just need to show that its determinant is an invertible endomorphism of k((t)). If we manage to do so, we will have proved that system (12) has a solution, but that implies the existence of a solution to system (11) and this, in turn, implies the existence of a preimage to our c. Expanding it along the last column, the determinant is  dBm+d+n−1 d 0··· 0 (−1)w0w0 −(d−w0)Bm+d+n−2 d−w0 dBm+d+n−2 d ··· 0 (−1)w0−1w0(w0−1) 0−(d−w0)Bm+d+n−3 d−w0 .... . .. . . . . .. . ....dBm+d+n−w0+1 d w0(w0−1) 0 0 · · · −(d−w0)Bm+d+n−w0 d−w0 −w0  = =−w0 w0 X r=1 w0−1 w0−r(−1)w0−r r−1 Y k=1 dr−1Bm+d+n−k d w0−1 Y k=r (d−w0)w0−rBm+d+n−k−1 d−w0 . Call the determinant above −w0∆α. Recall that, up to now, we have used that wiαis not an integer for any i= 1, . . . , n. Now is when we will use that w0αis not an integer either. Since every operator Bβpreserves powers of tup to a coefficient in k, so does ∆α; say ∆αtl=dα,ltl. But then there will be some power of tin its kernel if and only if ∆αis not bijective. Therefore, we need to show just that, for the values of αunder consideration, dα,l does not vanish for every EXPONENTS OF SOME ONE-DIMENSIONAL GAUSS-MANIN SYSTEMS 17 l∈Z. It is easy to see that dα,l is w0 X r=1 w0−1 w0−r(−1)w0−r r−1 Y k=1 dα +αm+d+n−k l+αw0−1 Y k=r(d−w0)α+αm+d+n−k−1 l+α =qw0−1 α,l w0 X r=1 w0−1 w0−r!(−1)w0−r r−1 Y k=1 (d(α+l) + m+d+n−k) w0−1 Y k=r ((d−w0)(α+l) + m+d+n−k−1), where qα,l is the quotient α/(l+α). Up to the factor qw0−1 α,l , the expression above is a polynomial in αof degree w0−1, so there will be at most w0−1 values of αso that it vanishes. In fact, for a fixed l, they are −l−a/w0, for a= 1, . . . , w0−1. Indeed, let abe as above. Then q−(w0−1) −l−a/w0,ld−l−a/w0,l can be written as w0−a Y k=1 −d w0 a+m+d+n−kw0 X r=1 w0−1 w0−r!(−1)w0−r a−1 Y j=1 −d w0 a+m+n+d−r+j =Ca w0 X r=1 w0−1 w0−r(−1)w0−rpa(r), where Caand paare, respectively, a constant and a polynomial of degree a−1≤w0−2. Now thanks to the following lemma, we can deduce the vanishing of the determinant, so we have finally found that for every αsuch that wiαis not an integer, the original system has a solution. All of this process could be made independently of the choice of m,n, all of the wiand cm, so it finally proves the surjectivity of Φ. The claim about the multiplicity of the exponents follows from two facts: the order of the wi does not play any role in the complex λ+OAnas we commented in the introduction to the context of the proposition, and within each set of possible exponents {1/wi,...,(wi−1)/wi}, the proof shows that whichever we use, the outcome is the same because of the different possible values of m. Remark 4.3.Note that we already knew before the proposition that by the monodromy theorem, the exponents were to be rational. What this result provides, with respect to that fact, is a much shorter set of possible exponents. The converse of the last proposition holds in a stronger way: every cohomology of λ+OAnis constant, except the last one, H0λ+OAn, whose exponents are exactly those of the form j/wi, for j= 1, . . . , wiand i= 0, . . . , n. However, it needs much more preparatory work; it can be found in [2, §5]. Lemma 4.4. Let m,nbe two integers such that 0≤m<n−1. Then, n X k=1 n−1 n−k(−1)n−kkm= 0. Proof. First of all rewrite the formula above as n−1 X k=0 n−1 k(−1)k(n−k)m, which is obviously true if n= 2, so assume n > 2. We will be done as long as we can show that Pn−1 k=1 n−1 k(−1)kkm= 0 for every nand m < n −1. Let us do it by induction on nand m. If n= 3 or m= 0 it is also very easy to prove it. 18 ALBERTO CASTA˜ NO DOM´ INGUEZ Let us go therefore for a general n, take some n−1>m>0 and assume the validity of Pa−1 k=1 a−1 k(−1)kkb= 0 for every a < n and b < min(m, a −1). Now note that n−1 X k=1 n−1 k(−1)kkm= (−1)n−1(n−1)m+ n−2 X k=1 n−2 k−1+n−2 k(−1)kkm = n−2 X k=1 n−2 k(−1)k(km−(k+ 1)m). Since km−(k+ 1)mis a polynomial of degree m−1 in k, we just need to apply the induction hypothesis to finish.  References 1. N. Bourbaki, ´el´ements de math´ematique. Alg`ebre. Chapitre 10. Alg`ebre homologique, Springer-Verlag, Berlin, 2007, Reprint of the 1980 original. 2. A. Casta˜no Dom´ınguez, Dwork families and D-modules, preprint arXiv:1512.08390 [math.AG], 2015. 3. 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