# On a reconstruction theorem for holonomic systems.

Introduction. Let X be a complex manifold. The Riemann-Hilbert correspondence of [2] establishes an anti-equivalence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

between regular holonomic D-modules and C-constructible complexes. Here, [[PHI].sup.0] ([Laplace]) = [RHom.sub.DX] ([Laplace], [O.sub.X]) is the complex of holomorphic solutions to C, and [[PSI].sup.0] (L) = THom(L, [O.sub.X]) = RHom(L, [O.sup.t.sub.X]) is the complex of holomorphic functions tempered along L. Since [Laplace][equivalent][[PSI].sup.0]([[PHI].sup.0](Laplace])), this shows in particular that C can be reconstructed from [[PHI].sup.0](Laplace]).

We are interested here in holonomic D-modules which are not necessarily regular.

The theory of ind-sheaves from [6] allows one to consider the complex [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of tempered holomorphic solutions to a holonomic module M. The basic example [[PHI].sup.t] ([D.sub.c][e.sup.1/x]) was computed in [7], and the functor [[PHI].sup.t] has been studied in [10,11]. However, since [[PHI].sup.t]([D.sub.C][e.sup.1/x])[equivalent][[PHI].sup.t]([D.sub.C][e.sup.2/x]), one cannot reconstruct M from [[PHI].sup.t](M).

Set [PHI](M) = [[PHI].sup.t](M [??] [D.sub.P][e.sup.t]), for t the affine variable in the complex projective line P. This is an ind-R-constructible complex in X x P. The arguments in [1] suggested us how M could be reconstructed from [PHI](M) via a functor [PSI], described below ([section]3).

We conjecture that the contravariant functors

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

between the derived categories of [D.sub.X]-modules and of ind-sheaves on X x P, provide a RiemannHilbert correspondence for holonomic systems.

To corroborate this statement, we discuss the functoriality of [PHI] and [PSI] with respect to proper direct images and to tensor products with regular objects ([section]4). This allows us to reduce the problem to the case of holonomic modules with a good formal structure.

When X is a curve and M is holonomic, we prove that the natural morphism M [right arrow] [PSI]([PHI](M)) is an isomorphism ([section]6). Thus M can be reconstructed from [PHI](M).

Recall that irregular holonomic modules are subjected to the Stokes phenomenon. We describe with an example how the Stokes data of M are encoded topologically in the ind-R-constructible sheaf [PHI](M) ([section]7).

In this Note, the proofs are only sketched. Details will appear in a forthcoming paper. There, we will also describe some of the properties of the essential image of holonomic systems by the functor [PHI] Such a category is related to a construction of [13].

1. Notations. We refer to [3-6].

Let X be a real analytic manifold.

Denote by [D.sup.b]([C.sub.X]) the bounded derived category of sheaves of C-vector spaces, and by

[D.sup.b.sub.R-c]([C.sub.X]) the full subcate gory of objects with R-constructible cohomologies. Denote by [cross product], RHom, [f.sup.-1], [Rf.sub.*], [Rf.sub.!], [f.sup.!] the six Grothendieck operations for sheaves. (Here f: X [right arrow] Y is a morphism of real analytic manifolds.)

For S [subset] X a locally closed subset, we denote by [C.sub.S] the zero extension to X of the constant sheaf on S.

Recall that an ind-sheaf is an ind-object in the category of sheaves with compact support. Denote by [D.sub.b]([IC.sub.X]) the bounded derived category of ind-sheaves, and by [D.sup.b.sub.IR-c]([IC.sub.X]) the full subcategory of objects with ind-R-constructible cohomologies. Denote by [cross product], RIHom, [f.sup.-1], [Rf.sub.*], [Rf.sub.!!], [f.sup.!] the six Grothendieck operations for ind-sheaves.

Denote by [alpha] the left adjoint of the embedding of sheaves into ind-sheaves. One has [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Denote by [beta] the left adjoint of [alpha].

Denote by [Db.sup.t.sub.X] the ind-R-constructible sheaf of tempered distributions.

Let X be a complex manifold. We set for short [d.sub.X] = dim X.

Denote by [O.sub.X] and [D.sub.X] the rings of holomorphic functions and of differential operators. Denote by [[OMEGA].sub.X] the invertible sheaf of differential forms of top degree.

Denote by [D.sup.b]([D.sub.X]) the bounded derived category of left [D.sub.X]-modules, and by [D.sup.b.sub.hol]([D.sub.X]) and [D.sup.b.sub.r-hol]([D.sub.X]) the full subcategories of objects with holonomic and regular holonomic cohomologies, respectively. Denote by [[cross product].sup.D], [Df.sub.-1,] [Df.sub.*] the operations for D-modules. (Here f: X [right arrow] Y is a morphism of complex manifolds.)

Denote by DM the dual of M (with shift such that [DO.sub.X][equivalent] [O.sub.X]).

For Z [subset] X a closed analytic subset, we denote by R[[GAMMA].sub.[Z]M and M(*Z) the relative algebraic cohomologies of a [D.sub.X]-module M.

Denote by ss(M) [subset] X the singular support of M, that is the set of points where the characteristic variety is not reduced to the zero-section.

Denote by [O.sup.t.sub.X] [member of] [D.sup.b.sub.IR-c] ([IC.sub.X]) the complex of tempered holomorphic functions. Recall that [O.sup.t.sub.X] is the Dolbeault complex of [Db.sup.t.sub.X] and that it has a structure of [beta][D.sub.X]-module. We will write for short [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] instead of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2. Exponential D-modules. Let X be a complex analytic manifold. Let D [subset] X be a hypersurface, and set U = X \ D. For [phi] [member of] [O.sub.X](*D), we set

[D.sub.X][e.sup.[phi]] = [D.sub.X] / {P: P[e.sup.[phi]] = 0 on U},

[[epsilon].sup.[phi].sub.D|X] = ([D.sub.X][e.sup.[phi]])(*D).

As an [O.sub.x](*D)-module, [[epsilon].sup.[phi].sub.D|X] is generated by [e.sup.[phi]]. Note that ss([[[epsilon].sup.[phi].sub.D|X]) = D, and [[epsilon].sup.[phi].sub.D|X] is holonomic. It is regular if [phi] [member of] [O.sub.x], since then [[epsilon].sup.[phi].sub.D|X] [equivalent] [O.sub.x](*D).

One easily checks that (D[[epsilon].sup.[phi].sub.D|X]) (*D)[equivalent] [[epsilon].sup.-[phi].sub.D|X].

Proposition 2.1. If dim X = 1, and [phi] has an effective pole at every point of D, then (D[[epsilon].sup.[phi].sub.D|X] [equivalent] [[epsilon].sup.-[phi].sub.D|X].

Let P be the complex projective line and denote by t the coordinate on C = P \ {[infinity]}.

For c [member of] R, we set for short

{Re[phi]< c } = {x [member of] U: Re[phi](x) < c},

{Re(t + [phi]) < c} = {(x, t): x [member of] U, t [member of] C,

Re(t + [phi](x)) < c}.

Consider the ind-R-constructible sheaves on X and on X x P, respectively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The following result is analogous to [1, Proposition 7.1]. Its proof is simpler than loc. cit., since [phi] is differentiable.

Proposition 2.2. One has an isomorphism in [D.sup.b]([D.sub.X])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for q and p the projections from X x P.

The following result is analogous to [7, Proposition 7.3].

Lemma 2.3. Denote by (u, v) the coordinates in [C.sup.2]. There is an isomorphism in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 2.4. There is an isomorphism in [D.sub.b]([IC.sub.X])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. As [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Lemma 2.3 gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Write [phi] = a / b for a, b [member of] [O.sub.x] such that [b.sup.-1](0) [subset] D, and consider the map

f = (a,b): X [right arrow] [C.sup.2].

As [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [6, Theorem 7.4.1] implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. []

3. A correspondence. Let X be a complex analytic manifold. Recall that P denotes the complex projective line. Consider the contravariant functors

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for q and p the projections from X x P.

We conjecture that this provides a Riemann-Hilbert correspondence for holonomic systems:

Conjecture 3.1.

(i) The natural morphism of endofunctors of [D.sub.b]([D.sub.X])

(3.1) id [right arrow] [PSI] [omicron] [PHI]

is an isomorphism on [D.sup.b.sub.hol]([D.sub.x]).

(ii) The restriction of [PHI]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is fully faithful.

Let us prove some results in this direction.

4. Functorial properties. The next two Propositions are easily deduced from the results in [6].

Proposition 4.1. Let f: X [right arrow] Y be a proper map, and set [f.sub.P] = f x [id.sub.P]. Let M [member of] [D.sup.b.sub.hol]([D.sub.x]) and F [member of] [D.sup.b.sub.IR-c]([IC.sub.XxP]). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For [Laplace] [member of] [D.sup.b.sub.r-hol] ([D.sub.X]), set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Recall that [[PHI].sup.0]([Laplace]) is a C-constructible complex of sheaves on X.

Proposition 4.2. Let [Laplace] [member of] [D.sup.b.sub.r-hol]([D.sub.X]), M [member of] [D.sup.b.sub.hol]([D.sub.X]) and F [member of] [D.sup.b.sub.IR-c]([IC.sub.XxP]). Then

[PHI](D([Laplace] [[cross product].sup.D] DM)) [equivalent] RIHom([q.sup.-1][[PHI].sup.0]([Laplace]), [PHI](M)), [PSI](F [cross product] [q.sup.-1][[PHI].sup.0]([Laplace]) [equivalent] [PSI](F) [[cross product].sup.D] [Laplace].

Noticing that

[PHI]([O.sub.X]) [equivalent] [C.sub.X] [??] RIHom([C.sub.{t[not equal to][infinity]}], [C.sub.{Re t < ?}]),

one checks easily that [PSI]([PHI]([O.sub.X])) [equivalent] [O.sub.X]. Hence,

Proposition 4.2 shows:

Theorem 4.3.

(i) For [Laplace] [member of] [D.sup.b.sub.r-hol]([D.sub.X]), we have

[PHI]([Laplace]) [equivalent] [q.sup.-1][[PHI].sup.0]([Laplace]) [cross product] [PHI]([O.sub.X]) [equivalent] [[PHI].sup.0]([Laplace]) [??] RIHom ([C.sub.{t[not equal to][infinity]}], [C.sub.{Re t < ?}]).

(ii) The morphism (3.1) is an isomorphism on [D.sup.b.sub.r-hol]([D.sub.X]).

(iii) For any [Laplace], [Laplace]' [member of] [D.sup.b.sub.r-hol]([D.sub.X]), the natural morphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an isomorphism.

Therefore, Conjecture 3.1 holds true for regular holonomic D-modules.

5. Review on good formal structures.

Let D [subset] X be a hypersurface. A flat meromorphic connection with poles at D is a holonomic [D.sub.X]-module M such that ss(M) = D and M [equivalent] M(*D).

We recall here the classical results on the formal structure of flat meromorphic connections on curves. (Analogous results in higher dimension have been obtained in [8,9,12].)

Let X be an open disc in C centered at 0.

For F an [O.sub.X]-module, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[??].sub.X,0] is the completion of [O.sub.X,0].

One says that a flat meromorphic connection M with poles at 0 has a good formal structure if

(5.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where I is a finite set, [[Laplace].sub.i] are regular holonomic [D.sub.X]-modules, and [[phi].sub.i] [member of] [O.sub.X](*0).

A ramification at 0 is a map X [right arrow] X of the form x [??] [x.sup.m] for some m [member of] N.

The Levelt-Turrittin theorem asserts:

Theorem 5.1. Let M be a meromorphic connection with poles at 0. Then there is a ramification f: X [right arrow] X such that [Df.sup.-1] M has a good formal structure at 0.

Assume that M satisfies (5.1). If M is regular, then [[phi].sub.i] [member of] [O.sub.X] for all i [member of] I, and (5.1) is induced by an isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

However, such an isomorphism does not hold in general.

Consider the real oriented blow-up

(5.2) [pi]:B = R x [S.sup.1] [right arrow] X, ([rho], [theta]) [??] [rho][e.sub.i[theta]].

Set V = {[rho] > 0} and let Y = {[rho] [greater than or equal to] 0} be its closure. If W is an open neighborhood of (0, [theta]) [member of] [partial derivative] Y [pi] W [intersection] V) contains a germ of open sector around the direction [theta] centered at 0.

Consider the commutative ring

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [bar.X] is the complex conjugate of X.

To a [D.sub.X]-module M, one associates the [A.sub.Y]-module

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The Hukuara-Turrittin theorem states that (5.1) can be extended to germs of open sectors:

Theorem 5.2. Let M be a flat meromorphic connection with poles at 0. Assume that M admits the good formal structure (5.1). Then for any (0, [theta]) [member of] [partial derivative]Y one has

(5.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where [m.sub.i] is the rank of [[Laplace].sub.i].

(Note that only the ranks of the [[Laplace].sub.i]'s appear here, since [x.sub.[lambda]][(log x).sup.m] belongs to [A.sub.Y] for any [lambda] [member of] C and m [member of] [Z.sub.[greater than or equal to]0].)

One should be careful that the above isomorphism depends on [theta], giving rise to the Stokes phenomenon.

We will need the following result:

Lemma 5.3. If M is a flat meromorphic connection with poles at 0, then

R[[pi].sub.*] ([[pi].sup.*] M) [equivalent] M.

6. Reconstruction theorem on curves. Let X be a complex curve. Then Conjecture 3.1 (i) holds true:

Theorem 6.1. For M [member of] [D.sup.b.sub.hol]([D.sub.X] there is a functorial isomorphism

(6.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Sketch of proof. Since the statement is local, we can assume that X is an open disc in C centered at 0, and that ss(M) = {0}.

By devissage, we can assume from the beginning that M is a flat meromorphic connection with poles at 0.

Let f: X [right arrow] X be a ramification as in Theorem5.1,sothat [Df.sub.-1] M admits a good formal structure at 0.

Note that [Df.sub.*][Df.sup.-1] M [equivalent] M [direct sum] N for some N. If (6.1) holds for [Df.sup.-1] M, then it holds for M [cross product] N by Proposition 4.1, and hence it also holds for M.

We can thus assume that M admits a good formal structure at 0.

Consider the real oriented blow-up (5.2).

By Lemma 5.3, one has M [equivalent] [R[pi].sub.*][[pi].sup.*] M. Hence Proposition 4.1 (or better, its analogue for [pi]) implies that we can replace M with [[pi].sup.*] M.

By Theorem 5.2, we finally reduce to prove

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Set D' = {x = 0} [union]{t = [infinity]} and U' = (X x P) \ D'. By Proposition 2.1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By Proposition 2.4, we thus have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Noticing that [PHI]([epsilon].sup.[phi].sub.0|X]) [cross product] [C.sub.D'] [member of] [D.sup.b.sub.C- c]([C.sub.XxP]), one checks that [PSI]([PHI]([epsilon].sup.[phi].sub.0|X]) [cross product] [C.sub.D']) [equivalent] 0.

Hence, Proposition 2.2 implies

[PSI]([PHI]([epsilon].sup.[phi].sub.0|X])) [equivalent] [PSI](C{Re(t+[phi])<?}) [equivalent] [epsilon].sup.[phi].sub.0|X]. []

Example 6.2. Let X = C, [phi](x) = 1/x and M = [epsilon].sup.[phi].sub.0|X]. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

7. Stokes phenomenon. We discuss here an example which shows how, in our setting, the Stokes phenomenon arises in a purely topological fashion.

Let X be an open disc in C centered at 0. (We will shrink X if necessary.) Set U = x \ {0}.

Let M be a flat meromorphic connection with poles at 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assume that [psi] - [phi] has an effective pole at 0.

The Stokes curves of [epsilon].sup.[phi].sub.0|X] [direct sum] [epsilon].sup.[psi].sub.0|X] are the real analytic arcs [l.sub.i], i [member of] I, defined by

{Re([psi] - [phi]) = 0} = [[??].sub.i[member of]I] [l.sub.i].

(Here we possibly shrink X to avoid crossings of the [l.sub.i]'s and to ensure that they admit the polar coordinate [rho] > 0 as parameter.)

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the Stokes curves are not invariant by isomorphism.

The Stokes lines [L.sub.i], defined as the limit tangent half-lines to [l.sub.i] at 0, are invariant by isomorphism.

The Stokes matrices of M describe how the isomorphism (5.3) changes when [theta] crosses a Stokes line.

Let us show how these data are topologically encoded in [PHI](M).

Set D' = {x = 0} [union] {t = [inifinity]} and U' = (X x P) \ D'. Set

[F.sub.c] = [C.sub.{Re(t+[phi])<c}], [G.sub.c] =[C.sub.{Re(t+[psi])<c}],

F = C{Re(t+[phi])<?}, G = [C.sub.{Re(t+[psi]>)<?}].

By Proposition 2.4 and Theorem 5.2,

[PHI](M) [equivalent] RIHom([C.sub.U'], H),

where H is an ind-sheaf such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any sufficiently small open sector S.

Let [b.sup.[+ or -]] be the vector space of upper/lower triangular matrices in [M.sub.2](C),and let t = [b.sup.+] [intersection] [b.sup.-] be the vector space of diagonal matrices.

Lemma 7.1. Let S be an open sector, and [??] a vector space, which satisfy one of the following conditions:

(i) [??] = [b.sup.[+ or -]] and S [subset] {[+ or -]Re([psi] - [phi]) > 0},

(ii) [??] = t, S [contains] [L.sub.i] for some i [member of] I and S [intersection] [L.sub.j] = [??] for i [not equal to] j.

Then, for c' [much greater than] c, one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In particular,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This proves that the Stokes lines are encoded in H. Let us show how to recover the Stokes matrices of M as glueing data for H.

Let [S.sub.i] be an open sector which contains [L.sub.i] and is disjoint from [L.sub.j] for i [not equal to] j. We choose [S.sub.i] so that [[union].sub.i[member of]I] [S.sub.i] = U.

Then for each i [member of] I, there is an isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Take a cyclic ordering of I such that the Stokes lines get ordered counterclockwise.

Since [{[S.sub.i]}.sub.i[member of]I] is an open cover of U, the ind-sheaf H is reconstructed from F [direct sum] G via the glueing data given by the Stokes matrices

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

10.3792/pjaa.88.178

Acknowledgment. The first author expresses his gratitude to the RIMS of Kyoto University for hospitality during the preparation of this paper.

References

[1] A. D'Agnolo, On the Laplace transform for temperate holomorphic functions, arXiv:1207.5278.

[2] M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci. 20 (1984), no. 2, 319 365.

[3] M. Kashiwara, D-modules and microlocal calculus, translated from the 2000 Japanese original by Mutsumi Saito, Translations of Mathematical Monographs, 217, Amer. Math. Soc., Providence, RI, 2003.

[4] M. Kashiwara and P. Schapira, Sheaves on manifolds, Crundlehren der Mathematischen Wissenschaften, 292, Springer, Berlin, 1990.

[5] M. Kashiwara and P. Schapira, Moderate and formal cohomology associated with constructible sheaves, Mem. Soc. Math. France (N.S.) No. 64 (1996), iv+76 pp.

[6] M. Kashiwara and P. Schapira, Ind-sheaves, Asterisque 271 (2001), 136 pp.

[7] M. Kashiwara and P. Schapira, Microlocal study of ind-sheaves. I. Micro-support and regularity, Asterisque 284 (2003), 143 164.

[8] K. S. Kedlaya, Cood formal structures for flat meromorphic connections, II: excellent schemes, J. Amer. Math. Soc. 24 (2011), no. 1, 183 229.

[9] T. Mochizuki, Wild harmonic bundles and wild pure twistor D-modules, Asterisque 340 (2011), x+607 pp.

[10] C. Morando, An existence theorem for tempered solutions of D-modules on complex curves, Publ. Res. Inst. Math. Sci. 43 (2007), no. 3, 625 659.

[11] C. Morando, Preconstructibility of tempered solutions of holonomic D-modules, arXiv:1007.4158.

[12] C. Sabbah, Equations dinerentielles a points singuliers irreguliers et phenomene de Stokes en dimension 2, Asterisque 263 (2000), viii+190 pp.

[13] D. Tamarkin, Microlocal condition for nondisplaceablility, arXiv:0809.1584.

By Andrea D'AGNOLO * and Masaki KASHIWARA **, *** (Contributed by Masaki KASHIWARA, M.J.A., Nov. 12, 2012)

2010 Mathematics Subject Classi?cation. Primary 32C38, 35A20, 32S60, 34M40.

* Dipartimento di Matematica, Universita di Padova, via Trieste 63, 35121 Padova, Italy.

** Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.

*** Department of Mathematical Sciences, Seoul National University, Seoul, Korea.