The irregularity of differential equations and of D-modules.

by Yves Laurent – University of Grenoble(France)

The aim of this talk is to show how the definitions of irregularity for differential equations may be extended in several variables. In a first part, we will present the theory of Malgrange and Ramis for differential equations. Then we will recall briefly the definitions and first properties of D-modules which is a quite natural framework where to define finite dimensional sets of solutions and we will finish with the irregularity of holonomic D-modules. In this talk, we will consider only linear (partial) differential equations.

1 The irregularity of ordinary differential equations.

Let P be a linear differential operator with holomorphic coefficients on a neighborhood of 0 in ℂ and F a ℂ vector space of functions, P has a finite index on F if the kernel and cokernel of P as a linear map from F to itself are finite dimensional ℂ vector spaces and the index of P in F is

χ(P, F) = dim_{ℂ} Ker P − dim_{ℂ} Coker P

Malgrange defined the irregularity of P as the difference between the index in formal series and in convergent series:

Irr(P) = χ(P, ℂ [[t]]) − χ(P, ℂ {t})

As is well known, the irregularity is 0 if and only if the formal solutions of P are convergent and this is equivalent to P of Fuchsian type or as we also say P regular at 0.

An interesting property of the irregularity is its positivity. This property has two interpretations, analytic and algebraic. Analytic means from the point of view of solutions. As shown by Malgrange the cokernel of P in the space ℂ [[t]]/ℂ {t} is always 0, thus the irregularity of P is the dimension of the kernel of P in ℂ [[t]]/ℂ {t} hence is positive.

From the algebraic point of view, the irregularity is calculated from the coefficients of the operator: if P (t, ∂_{t}) = a_{m}(t)∂_{t}^{m} + … a_{0}(t), and v_{j} the valuation of a_{j} at 0, one shows that Irr(P) = max_{j=0...m}(j − v_{j}) − (m − v_{m}) which is positive by definition.

Ramis [5] has shown how to split this positive number in a finite sum. He considers, for r rational number greater than 1, the spaces ℂ [[t]]_{r} of formal series u(t) = ∑ u_{k}t^{k} such that ∑ u_{k}t^{k}/(k!)^{r−1} is convergent. For r = 1, it is the space of convergent series ℂ {t} and for r = ∞ we define it as ℂ [[t]]. To study the solutions in theses spaces, Ramis considers the Newton Polygon of P which is defined in the following way:

If P and v_{j} are defined as above, the Newton Polygon of P is the convex hull of the sets V_{j} = {(x, y) ∈ ℝ^{2} | x ≤ j, y ≥ v_{j} − j} for j = 0, …, m. The boundary of this Newton Polygon looks like shown in figure 1 with two half lines and a finite number of segments. The slopes s_{j} of these segments are called the slopes of the Newton Polygon of P and the corresponding numbers 1 + 1/s_{j} the critical values of P.

The result of Ramis is the following:

[figure: j; k; m]

Figure 1: Newton Polygon

Theorem 1. 1)The index of P is finite in ℂ [[t]]_{r} for all r.

2)If u is a formal solution of P u = 0 then there exists a critical value r_{0} of P such that u ∈ ℂ [[t]]_{r_{0}} but u ∉ ℂ [[t]]_{r_{0}−ε} for ε > 0.

3)Let the irregularity of type r of P be Irr_{r}(P) = χ(P, ℂ [[t]]_{r}) − χ(P, ℂ [[t]]_{r−ε}) for small ε > 0. Then Irr_{r}(P) is equal to the height of the segment of slope 1/(r − 1) of the Newton polygon of P.

In particular, this irregularity is always positive, is 0 for all r except the critical values and Irr(P) = ∑_{r} Irr_{r}(P).

2 Definition of the index in higher dimensional space.

To work with finite dimensional spaces of solutions , we will have to consider systems of linear partial differential equations. If X is a complex analytic manifold (e.g. ℂ^{n}) and D_{X} the sheaf of differential operators with holomorphic coefficients on X, a system of partial differential equations is a matrix of D_{X}. If A is a (M, N)-matrix, we consider the D_{X}-module M = D_{X}^{N}/D_{X}^{M}A and the spaces of solutions will not depend on the choice of A but only of M. The theory of coherent modules shows that there exists (locally on X) an exact sequence:

0 → D_{X}^{N_{n}} ⟶^{A_{n−1}} D_{X}^{N_{n−1}} ⟶^{A_{n−2}} … ⟶^{A_{1}} D_{X}^{N_{1}} ⟶^{A} D_{X}^{N_{0}} ⟶ M ⟶ 0

Then if F is a sheaf of functions on which acts the differential operators (e.g. the sheaf of holomorphic functions O_{X} or the sheaf of C^{∞} functions we may consider the complex

0 → F^{N_{0}} ⟶^{A} F^{N_{1}} ⟶^{A_{1}} … ⟶^{A_{n−2}} F^{N_{n−1}} ⟶^{A_{n−1}} F^{N_{n}} ⟶ 0

Then, by definition, the sheaves of solutions of M in F are the cohomology groups Ext^{j}(M, F) = Ker A_{j}/ Im A_{j−1}. They depend only on M and the first one Ext^{0}(M, F) is the set of usual solutions of A.

We say that M has a finite local index in F at x ∈ X if the groups Ext^{j}(M, F)_{x} are finite dimensional and define the index as:

χ (M, F) = ∑_{i}(−1)^{i} dim_{ℂ} Ext^{j}(M, F)_{x}

3 Characteristic cycle and holomorphic index

If P is the differential operator ∑_{|α|≤m} a_{α}(x)∂_{x}^{α}, its principal symbol is the function σ(P)(x, ξ) = ∑_{|α|=m} a_{α}(x)ξ^{α} which is well defined as a function on the cotangent bundle T^{∗}X.

If M is the quotient of D_{X} by a coherent ideal I = D_{X}P_{1}+… D_{X}P_{N} (this corresponds to a system with one unknown) we define σ(I) = {σ(P)|P ∈ I} and the characteristic variety Ch(M) of M as the set of points in T^{∗}X where all functions of σ(I) vanish. The ideal σ(I) defines also multiplicities which are positive integers attached to each irreducible components of Ch(M), the set Ch(M) with these multiplicities is the characteristic cycle of M designed by C̃h(M). This cycle is positive which means that all multiplicities are positive. In the general case of a coherent D_{X}-module, there is also a definition of the characteristic variety and the characteristic cycle of M which we do not give here.

This characteristic variety is involutive and M is said to be holonomic if it is lagrangian. In this case, the characteristic cycle defines a function on X, the Euler function.

Then Kashiwara [1] proved the following theorem:

Theorem 2. If M is a holonomic D_{X}-module and O_{X} is the sheaf of holomorphic functions on X, then the sheaves of solutions Ext^{j}(M, O_{X}) are constructible (i.e. locally constant on a stratification of X and of finite dimension) and the index χ(M, O_{X}) is equal to the Euler function of C̃h(M).

4 Microcharacteristic cycles and irregularity

In this section, we will consider a submanifold Y of X of codimension 1 (analogous results are true if the codimension is greater than 1).

We choose local coordinates (x_{1}, …, x_{n−1}, t) of X such that Y is equal to {t = 0}. If a differential operator P is defined in a neighborhood of Y, its coefficients may be developed in Taylor series in t and P may be written as:

P (x, t, ∂_{x}, ∂_{t}) = ∑ a_{α,β,γ}(x)t^{γ}∂_{x}^{α}∂_{t}^{β}

Then if r > 1 is a rational number which we write r = p/q with (p, q) relatively prime positive integers we define the function:

σ_{r}(P)(x, t, ξ, τ) = ∑_{q(|α|+|β|)+(p−q)(|β|−|γ|)=M} a_{α,β,γ}(x)t^{γ}ξ^{α}τ^{β}

where M is maximum such that the function is not 0. Let Λ be the normal bundle T_{Y} X to Y in X and T^{∗}Λ be its cotangent bundle. It may be proved [2] that for any r > 1 the function σ_{r}(P) is well defined on T^{∗}Λ independently of coordinates. With the same procedure than in the previous section for defining the characteristic variety and cycle, we use these functions to define the microcharacteristic variety of type r of a coherent D_{X}- module M, they are denoted by Σ_{Λ}(r)(M), and in the same way the microcharacteristic cycle of type r denoted by Σ̃_{Λ}(r)(M).

Let H_{r} be the action of ℂ^{∗} given by H_{r}(λ)(x, t, ξ, τ) = (x, λ^{q−p}t, λ^{q}ξ, λ^{p}τ), then it is clear from the definition that Σ_{Λ}(r)(M) is conic for H_{r}. An important property of these varieties [2] is that for any D_{X}-module M, there exists a finite number of critical values r_{0} = 1 < r_{1} < ⋯ < r_{N} = +∞ such that Σ_{Λ}(r)(M) does not depend on r in the open intervals ]r_{i}, r_{i+1}[. These numbers are the generalization of the slopes of the Newton Polygon in dimension 1. If r is not one of those numbers, the corresponding variety is conic for all H_{r}, we say that they it is biconic.

Assume now that the module is holonomic, then the microcharacteristic varieties are lagrangian. But if Σ is a biconic lagrangian subvariety of T^{∗}Λ, there exist two conic lagrangian subvarieties S_{1} and S_{2} of T^{∗}Y such that Σ = {(x, t, ξ, τ)|t = 0, (x, ξ) ∈ S_{1}} ⋃ {(x, t, ξ, τ)|τ = 0, (x, ξ) ∈ S_{2}}. So, if r is not one of the critical values r_{i}, the microcharacteristic variety splits and we may associate to the microcharacteristic cycle two positive cycles of T^{∗}Y which we denote by S_{1,2}(r)(M). Now we define the irregularity of M as:

Irr_{Λ}(r)(M) = S_{1}(r − ε)(M) − S_{2}(r − ε)(M) − S_{1}(r + ε)(M) + S_{2}(r + ε)(M)

where ε > 0 is small so that there is no critical value except r in [r − ε, r + ε].

This defines a cycle of T^{∗}Y which vanishes if r is not a critical value and we define the irregularity as Irr_{Λ}(M) = ∑_{r>1} Irr_{Λ}(r)(M). These definitions are justified first by the fact that in the 1-dimensional case we retrieve the previous definitions and second by the fact that the Euler function of Irr_{Λ}(M) calculates the difference of indexes between formal and convergent solutions while Irr_{Λ}(r)(M) calculates the difference of indexes between solutions of growth r − ε and r + ε.

In this context, the positivity of the irregularity is the fact that the cycles Irr_{Λ}(r)(M) are positive. It is a non-trivial theorem which may be proved using the geometric properties of biconic lagrangian manifold [3].

From the analytic point of view, it may be shown [4] that the solutions in the quotient of formal series by convergent series define a perverse sheaf hence a positive cycle.

References

[1] M. Kashiwara, Systems of microdifferential equations, Progress in Mathematics, vol. 34, Birkhäuser, 1983.

[2] Y. Laurent, Polygone de Newton et b-fonctions pour les modules microdifférentiels, Ann. Ec. Norm. Sup. 4e série 20 (1987), 391–441.

[3] Y. Laurent and Z. Mebkhout, Pentes algébriques et pentes analytiques d’un D-module, Ann. Ec. Norm. Sup. 4e série 32 (1999), 39–69.

[4] Z. Mebkhout, Le théorème de positivité de l’irrégularité pour les D-modules, The Grothendieck Festschrift III, Progress in Mathematics, vol. 88, 1990, pp. 83–132.

[5] J.-P. Ramis, Théorèmes d’indices Gevrey pour les équations différentielles ordinaires, Memoirs of the AMS 48 (1984), no. 296.