aa r X i v : . [ m a t h . AG ] M a r PERIODS FOR FLAT ALGEBRAIC CONNECTIONS
MARCO HIEN
Abstract.
In previous work, [10], we established a duality between the algebraic de Rham cohomologyof a flat algebraic connection on a smooth quasi-projective surface over the complex numbers and therapid decay homology of the dual connection relying on a conjecture by C. Sabbah, which has beenproved recently by T. Mochizuki for algebraic connections in any dimension. In the present article, weverify that Mochizuki’s results allow to generalize these duality results to arbitrary dimensions also. Introduction
In [10], we proved a duality theorem for the algebraic de Rham cohomology of a flat algebraic connection( E, ∇ ) on a smooth quasi-projective surface over the complex numbers assuming a conjecture of C. Sabbah([23], Conjecture I.2.5.1) on the existence of the so-called good formal structure after birational pull-back of meromorphic connections on surfaces. Lately, a proof of Sabbah’s Conjecture in the case ofalgebraic connections (which lie in the main focus of our work) has been achieved by T. Mochizuki (see[19]). Moreover, in another astonishing work [20], T. Mochizuki is able to prove a far reaching higher-dimensional generalization of this result, namely the existence of a good lattice again after a suitablebirational pull-back (loc.cit., Theorem 19.5). As in the one- and two-dimensional case, the canonical nextstep into structural examinations of flat meromorphic connections consists in the question of lifting thegood formal properties to asymptotic analogues. This step is also carried out in Mochizuki’s paper [20] andresults in the construction of Stokes structures for flat meromorphic connections in arbitrary dimensions.With these tools in hand, we are now able to construct the period pairing of a flat algebraic connectionbetween the algebraic de Rham cohomology and the rapid decay homology which we introduced in [10]and prove the perfectness of this period pairing unconditionally in all dimensions. As a consequence,assuming rational structures of the given geometric data over a subfield k ⊂ C as well as of the localsystem associated to the flat connection over another subfield F ⊂ C , we deduce a well-defined notionof the period determinant arising from the comparison of these rational structures by means of theperfect period pairing. This determinant will be an element in the quotient C × /k × F × . One of themotivations to study this determinant lies in the mysterious analogies between flat connections overthe complex numbers and ℓ -adic sheaves on varieties over finite fields, according to which the perioddeterminant is expected to behave as the ε -factor in the latter theory. If one restricts to the subcategoryof regular singular connections, corresponding to the tamely ramified sheaves over finite fields, the perioddeterminant and its analogies to the ε -factor has been extensively studied by T. Saito and T. Terasomain [26]. With the present work, we hope to contribute to a generalization of these lines of thought to theirregular singular case.Let us give a short summary of the contents of this paper. The main object under consideration is theperiod pairing between the algebraic de Rham cohomology of the given flat algebraic connection and therapid decay homology – analytic in nature – of the dual connection. We will discuss this construction insection 5. The main result Theorem 5.2, the perfectness of the period pairing, relies on a local dualitybetween the algebraic de Rham complex and an analytic de Rham complex of the dual bundle withasymptotically flat coefficients. This duality will be proved in Theorem 4.4 and Proposition 5.1 comparesthe asymptotically flat de Rham cohomolgy with the rapid decay homology creating the link between thelocal duality and the period pairing. Mathematics Subject Classification.
Key words and phrases.
Period integrals, meromorphic connections, algebraic de Rham cohomology. Formal completion and real oriented blow-up
We consider the following geometric situation. Let U be a smooth quasi-projective variety over C and X a smooth projective variety such that D := X r U is a divisor with normal crossings. Introducingreal polar coordinates around each irreducible component of D leads to the real oriented blow-up π : e X → X an . Locally at some p ∈ D , we can choose complex coordinates x , . . . , x d such that p = 0 and D = { x · · · x k = 0 } . Then π reads as π : (cid:0) [0 , ε ) × S (cid:1) k × Y → X an , (cid:0) ( r ν , e iϑ ν ) kν =1 , y (cid:1) (cid:0) ( r ν · e iϑ ν ) kν =1 , y (cid:1) , where Y is a small analytic neighbourhood of 0 ∈ C d − k , y = ( x k +1 , . . . , x d ) and ε > π − ( U an ) the real oriented blow-up is a homeomorphism by which we read U an as a subspace of e X . We write e : U an ֒ → e X for the inclusion.On e X , we consider the following sheaves of functions (cp. [23], II.1.). Firstly, e X carries the structureof a real manifold with boundary. Additionally, the logarithmic differential operators x ν ∂ x ν act on thesheaf C ∞ e X of C ∞ functions. Let A e X be the sheaf of functions which in the local siuation as above is givenby A e X := k \ ν =1 ker( x ν ∂ x ν ) ∩ d \ ν = k +1 ker ∂ x ν ⊂ C ∞ e X . Local sections in A e X for some open Ω ⊂ e X are differentiable functions on Ω which are holomorphicon Ω ∩ U an and admit an asymptotic development in the higher-dimensional analogue of Poincar´e’sasymptotic developments due to Majima (cp. Proposition B.2.1 in [23] and [13]).Next, if P
Deligne-Malgrange lattices
Let there be given a flat meromorphic connection ∇ on the locally free O X an ( ∗ D )-module E or rank r ,where D ⊂ X is a divisor with normal crossings. After chioce of a local trivialization E ∼ = ( O X an ( ∗ D )) r ,the connection reads as ∇ = d + A with the connection matrix A ∈ M ( r × r, Ω X an ( ∗ D )). A change T ∈ GL r ( O X an ( ∗ D )) of the trivialization transforms the connection matrix due to the formula A ′ := T − dT + T − AT .
The local classification of these connections, i.e. of the connection matrices up to this transformation, isa difficult task in general.The first major subdivision of flat meromorphic connections lies in the distinction between regularsingular connections and irregular singular ones. Let us recall this notion in the given geometric situation,i.e. with D being a normal crossing divisor. Then, a flat meromorphic connection ( E, ∇ ) is regularsingular , if there is a trivialization E ∼ = ( O X ( ∗ D )) r such that the resulting connection matrix haslogarithmic poles along D at most, i.e. A can be written as A = P ki =1 A i ( x ) dlog x i + P dj = k +1 A j ( x ) dx j with holomorphic matrices A i ( x ) ∈ M ( r × r, O X an ). The structure of regular singular connections is wellunderstood (see [8]), in particular we know that each regular singular connection is a succesive extensionof rank one connections. For the latter, one finds a basis vector e over O X an such that ∇ e = e ⊗ (cid:0) λ dlog x + . . . + λ k dlog x k (cid:1) for some λ ∈ C k . Such a connection will be denoted by x λ .Next, the most elementary irregular singular connections are rank one connections which in a suitablebasis vector e read as ∇ e = e ⊗ d a for some a ∈ O X an ( ∗ D ), which up to isomorphism depends on a mod O X an only. Such a connection will be denoted by e a .Now, let Y ⊂ D be a stratum in the natural stratification of the normal crossing divisor D . Passingfrom O X an to the formal completion O d X | Y of O X an along Y leads to the problem of formal classification,to which a extensive answer is given in the case dim X = 1 by the Levelt-Turrittin Theorem (see [14],Chapter III). In [23], C. Sabbah investigated the two-dimensional situation leading to a precise conjectureas well as partial results in this direction. Recently, T. Mochizuki was able to give a proof Sabbah’sConjecture and a higher-dimensional generalization using a different apporach by exmamining the so-called good lattices which go back to Malgrange (see [16]). We will now explain Mochizuki’s result aboutthese formal properties of meromorphic connections. For more details, we refer to [20], chapter 5.Let us choose local coordinates such that D = { x · · · x k = 0 } with the irreducible components D i = { x i = 0 } and consider the local situation, i.e. we put X = ∆ k × Y where ∆ k is s small poly-discin C k centered at the origin and Y is a small neighbourhood of the origin in the remaining variables.Accordingly, we will denote the first k variables by z , . . . , z k and the others by y , . . . , y d − k . Let ≤ be thepartial order on Z k given by m ≤ n if m i ≤ n i for all i . Now, any meromorphic function f ∈ O X an ( ∗ D )admits a Laurent expansion f = P m ∈ Z k f m ( y ) z m with holomorphic functions f m ∈ O Y . The order of f will be the minimum ord( f ) = min { m ∈ Z k | f m = 0 } , assuming that this minimum with respect to ≤ exists. T.Mochizuki defines the notion of a good set ofirregular values on ( X, D ) to be a finite set S ⊂ O X an ( ∗ D ) / O X an , such thati) ord( a ) and ord( a − b ) exist for all a = b in S , M.HIEN ii) the set { ord( a − b ) | a , b ∈ S } is totally ordered with respect to ≤ on Z k ,iii) the leading terms a ord( a ) ( p ) and ( a − b ) ord( a − b ) ( p ) of the Laurent expansions are non-vanishingfor all p ∈ Y .For any subset I ⊂ { , . . . , k } , let I c be its complement and furthermore D I := T i ∈ I D i and D ( I ) := S i ∈ I D i . The completion of X along D I resp. D ( I ) will is denoted by b D I resp. b D ( I ). For a given subset S ⊂ O X an ( ∗ D ) / O X an let S ( I ) := { a mod O X an ( ∗ D ( I c )) | a ∈ S } .Let E be a lattice in E , i.e. a locally free O X an -module such that E ⊗ O X an ( ∗ D ) = E . Then E is calledan unramified good lattice at p ∈ D if there is a good set of irregular values S as above such that foreach I we have a formal decomposition(3.1) ( E , ∇ ) | b D I = M a ∈ S ( I ) ( I c E a , I c ∇ a ) , such that (cid:0) c ∇ a − d a (cid:1) ( I c E a ) ⊂ I c E a ⊗ (cid:0) Ω X an (log D ( I )) + Ω X an ( ∗ D ( I c )) (cid:1) . The set S of irregular values is uniquely determined by the given connection and hence it is denoted by S := Irr( ∇ ).The given connection ( E, ∇ ) is said to admit a good lattice at p ∈ D , if for some coordinateneighbourhood U = ∆ d of p as above, there exists a ramification map ρ e : ∆ d → U , ( t , . . . , t d ) ( t e , . . . , t ek , t k +1 , . . . , t d )for some e ∈ N such that ρ ∗ e ( E, ∇ ) admits an unramified good lattice e E . With these notions, T. Mochizukiproves the following far reaching result Theorem 3.1 ( T. Mochizuki , [20], Theorem 19.5) . Let ( E, ∇ ) be a flat meromorphic connection on ( X, D ) . Then there exists a regular birational map ϕ : X ′ → X such that the pull-back ϕ ∗ ( E, ∇ ) admitsa good lattice at any p ∈ ϕ − ( D ) . Actually, the theorem in [20] is more precise in the sense that due to Malgrange’s work ([16]) oneknows that generically there is a ”canonical lattice”, which Mochizuki calls the Deligne-Malgrange lattice.Mochizuki proves that its extension to the whole X is again locally free over O X an and a good lattice.3.2. Good decomposition in multisectors
If we apply Mochizuki’s Theorem 3.1 to the given flat meromorphic connection ( E, ∇ ) on ( X, D ), weknow that the pull-back with respect to some birational map and some finite ramification admits a goodlattice E . It will be important for our purposes to see that the resulting formal decomposition can belifted to an asymptotic one on small multisectors. In dimension two, this is the ingredient of [23], II.2, inhigher dimensions it can be derived from Mochizuki’s approach of good lattices and can be found in [20]also, see the remark after the theorem below.Let A de X | Z denote the formal completion of A e X along π − ( Z ) for a closed subset Z ⊂ X and T Z the natural morphism A e X → A de X | Z . We will consider mainly the case where Z ⊂ D is a union oflocal irreducible components of D . Note that A de X | Z | π − ( Z ) = π − O d X | Z in such a case. The followinggeneralization of the one-dimensional Borel-Ritt theorem tells us that the sequence 0 → A The proof of Theorem 3.2 is completely contained in [20]. However, since Mochizuki develops thetheory of Stokes structures in a more general setting, adapted to the examination of wild twistor D -modules, and in much more detail than needed here, the proof of the above theorem is not so easily foundin loc.cit. It occupies several steps and culminates in the statement to be found in Remark 7.73, [20].We plan to include a short overview of the necessary steps in a future version of this paper. The local duality pairing Our aim now is to study the meromorphic de Rham complex associated to the flat algebraic connection( E, ∇ ) by lifting to the real oriented blow-up π : e X → X an . To this end, we consider the following deRham complexes on e X : Definition 4.1. The asymptotically flat de Rham complex is defined to be the complex DR The local duality pairing is the natural pairing (4.1) DR mod D e X ( ∇ ∨ ) ⊗ C DR Let ϑ ∈ π − ( x ) ≃ ( S ) k be a direction in e D over x . Then the complex of stalks at ϑ which we have toconsider is given as(4.3) . . . −→ (cid:16) A ? D t e X ⊗ π − O X π − Ω pX (cid:17) ϑ ∇ −→ (cid:16) A ? D e X ⊗ π − O X π − Ω p +1 X (cid:17) ϑ −→ . . . , where ? stands for either < or mod. In degree p and with the usual basis dx I for I = { ≤ i < . . .
Theorem 4.4. The local duality pairing (4.1) is perfect in the sense that the induced morphisms DR mod D e X ( ∇ ∨ ) → R Hom e X (DR Proof of Theorem 4.4 According to Mochizuki’s results, ( E, ∇ ) admits a good lattice after some regular birational morphism b : Y → X . Let Z := b − ( D ) be the resulting divisor in Y and let π X : e X → X an and π Y : e Y → Y an denote the oriented real blow-ups. Let e b : e Y → e X be a lift of b . We will also use the following notationfor the embeddings e : X an r D ֒ → e X and e ı : Y an r Z ֒ → e Y . The de Rham complexes on the real orientedblow-ups behave functorial with respect to this situation: Lemma 4.5. We have natural isomorphisms R e b ∗ (DR mod Z e Y ( b ∗ ∇ )) ∼ = DR mod D e X ( ∇ ) and R e b ∗ (DR Via the projection formula – note that E is locally free – we obtain R e b ∗ (DR ? Z e Y ( b ∗ ∇ )) = R e b ∗ (cid:0) A ? Z e Y L ⊗ π − Y ( O Y an ) π − Y DR Y an ( b ∗ ∇ ) (cid:1) ∼ = R e b ∗ ( A ? Z e Y ) L ⊗ π − X ( O X an ) π − X DR X an ( ∇ ) , where ? stands for < or mod respectively. Hence, it suffices to prove that R e b ∗ ( A ? Z e Y ) = A ? D e X .Since the restiction of b : Y an → X an to Y an r Z is an isomorphism Y an r Z → X an r D = U an , we seethat the sheaves of flat C ∞ -functions on each space, i.e. all of whose derivations vanish on the boundary,are compatible with e b , i.e. e b ∗ P Definition of rapid decay homology We now want to interpret the local duality pairing as a pairing via period integrals. To this end, let usrecall the definition of rapid decay homology in [10].Note that for the mere definition in [10], we consider the following geometric set-up. We start withthe given flat algebraic connection ( E, ∇ ) on the smooth quasi-projective variety U . We then compactify U by some smooth projective X with a normal crossing divisor D := X r U as complement. Due toMochizuki’s Theorem, quoted as Theorem 3.1 above, we can pull-back the connection with respect tosome regular birational map b : Y → X and obtain a good lattice for the connection. In other words,by replacing X with Y as a different choice of compactification, we may assume that the givenconnection admits a good lattice on the chosen compactification ( X, D ).Let π : e X → X denote the real oriented blow-up of the normal crossing divisor D ⊂ X . We willwrite e D := π − ( D ) and denote by j : U → X and e : U → e X the inclusions. For any p ∈ N , we willwrite S p ( e X ) for the free Q -vector space over all piecewise smooth maps c : ∆ p → e X and we will considersingular homology with Q -coefficients and piecewise smooth chains in the following.Let C − p e X, e D denote the sheaf of relative p -chains of the pair e D ⊂ e X , i.e. the sheaf associated to thepresheaf V S p (cid:0) e X, ( e X r V ) ∪ e D (cid:1) . Let E := ker( ∇| U ) be the local system on U of flat sections of ∇ . According to [10], Definition 2.3, alocal section c ⊗ ε ∈ Γ( V, C − · e X, e D ⊗ Q e ∗ E ) is called a rapid decay chain , if for any point y ∈ c (∆ p ) ∩ e D ∩ V the following holds: Let x , . . . , x d be local coordinates of X around y = 0 such that D = { x · · · x k = 0 } for some 1 ≤ k ≤ d . We chose a meromorphic basis e : E ∼ = ( O X ( ∗ D )) r of E at y and require that if wedevelop ε = P di =1 f i · e i in this basis with analytic functions f i ∈ j ∗ O U , these coefficients f i ( x ) decreasefaster than any monomial for x approahcing D , i.e. that for all N ∈ N k there is a C N > x we have | f i ( x ) | ≤ C N · | x | N · · · | x k | N k . For chains c ⊗ ε inside U , we do not impose any condition. The sheaves of all these rapid decay p -chains will be denoted by C rd , − p e X ( ∇ ). Together with the usual boundary operator ∂ of singular chainsthey define the complex of rapid decay chains C rd e X ( ∇ ). The rapid decay homology of ( E ( ∗ D ) , ∇ ) is thehypercohomology H rdk ( U an , E, ∇ ) := H − k ( e X, C rd e X ( ∇ )) , which can be computed as the cohomology of the global sections, since the usual barycentric subdivisionoperator can be defined on the rapid decay chains and thus one deduces that the complex of rapid decaychains, similar to the sheaf complex of singular chains (cp. [27], p. 87) is homotopically fine. For moredetails, we refer to [10], section 2.3.5.2. The local duality pairing and periods In order to obtain the desired interpretation of the local duality pairing (4.1) in terms of period integrals,we will examine the relation between the asymptotically flat de Rham complex of the dual connection( E ∨ , ∇ ∨ ) and its rapid decay complex. Let E ∨ denote the local system of the dual connection.Recall that we chose ( X, D ) in a way such that ( E, ∇ ) admits a good lattice on ( X, D ). According toProposition 4.3, the natural inclusion(5.1) DR The subspace V a is characterized by the property that V a ∩ e D consists of those directions along which e a ( x ) has rapid decay for x approaching e D . In particular, c ⊗ e a is a rapid decay chain on V if and onlyif the topological chain c in e X approaches e D ∩ V in V a at most. Hence C rd e X ( e − a ) | V = C − · e X, e D ⊗ ( e a ) ! ( e a C U ) = C − · e X, e D ⊗ ∨ S We remain in the situation given above, namely ( E, ∇ ) admitting a good lattice on ( X, D ). Due toProposition 4.3, the local duality pairing reduces to the pairing(5.3) ∨ S The period pairing (5.4) induces a perfect pairing H rdp ( U, E ∨ , ∇ ∨ ) ⊗ C H pdR ( U ; E, ∇ ) −→ C betweem the algebraic de Rham cohomology of ( E, ∇ ) and the rapid decay homology of the dual connection. The period determinant Let k ( C be a subfield of the field of complex numbers and assume that the given geometric data, thevariety U as well as the vector bundle and the connection are defined over k already. Then the de Rhamcohomology inherits this k -structure, i.e. H dR ( U, E, ∇ ) is naturally a k -vector space.Now, let F ( C denote another subfield of C and let us assume that the local system E on U an comesequipped with a given F -structure. In analogy to [26] (for the regular singular case), we consider thecategory W k,F ( U ) – which we already defined in [10], 2.5 – of triples M = (( E, ∇ ) , E F , ρ ) with:i) a vector bundle E on U with rank r together with a flat connection ∇ : E → E ⊗ O U Ω U ,ii) a local system E F of F -vector spaces on the analytic manifold U an ,iii) a morphism ρ : E F → E an of sheaves on U an inducing an isomorphism E F ⊗ F C ∼ → ker( ∇ an ) oflocal systems of C -vector spaces on U an .A morphism between (( E, ∇ ) , E F , ρ ) and (( E ′ , ∇ ′ ) , E F ′ , ρ ′ ) is given by a morphism E → E ′ respectingthe connections together with a morphism E F → E F ′ of F -local systems with the natural compatibilitycondition with respect to ρ and ρ ′ .Let (( E, ∇ ) , E F , ρ ) ∈ W k,F ( U ) be an object in this category for given subfields k, F ( C . Then thelocal system E ∨ inherits an F -structure from the given F -structure on E . Consequently, we can considerthe F -lattice C rd , − p e X ( ∇ ∨ ) F ⊂ C − p e X, e D ⊗ Q e ∗ E ∨ F of all rapidly decaying chains in C − p e X, e D ⊗ Q e ∗ E ∨ F and end up with a natural F -lattice H rdp ( U an , E, ∇ ) F inside the rapid decay homology: H rdp ( U an , E, ∇ ) F ⊗ F C ∼ = −→ H rdp ( U an , E, ∇ ) , the isomorphism induced by ρ . The duality between the algebraic de Rham cohomology and the rapid de-cay homology via the period pairing enables us to compare these lattices and to generalize [10], Definition2.7, unconditionally to the case of arbitrary dimension of U : Definition 5.3. For (( E, ∇ ) , E F , ρ ) ∈ W k,F ( U ) , we define its period determinant to be the element det(( E, ∇ ) , E F , ρ ) := Y p ≥ det( h γ ( p ) j , ω ( p ) i i ) ( − p i,j ∈ C × /k × F × , where ω ( p ) i denotes a basis of H pdR ( U, E, ∇ ) over k and γ ( p ) j a basis of the F -vector space H rdk ( U an , E ∨ , ∇ ∨ ) F . 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