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I'm currently trying to have a better understanding of the concepts of characteristic variety and holonomic $D$-modules (let us assume that they are coherent) on a holomorphic manifold $X$. I know that for a system of differential equations $P$, the holonomicity of the $D$-module $D_X / D_X P$ means that the system $P$ is maximaly overdetermined. In order to see that, we compute the characteristic variety $\text{char}(D_X/D_X P)$ and if it is lagrangian in $T^* X$, the $D$-module is holonomic by definition.

I'd like to compute explicitely the charactistic variety of two "easy" coherent $D$-modules in order to see the holonomicity. So to be precise, my question is

Find two systems of differential equations $P$ and $Q$ on, let's say $\mathbb{C}^2$, such that $\text{char}(D_X/D_X P)$ and $\text{char}(D_X/D_X Q)$ can be explicitely computed in a short time and such that $\text{char}(D_X/D_X P)$ is lagrangian and $\text{char}(D_X/D_X Q)$ is not.

For example, I've also studied the notion of regularity and for that concept, the examples are easy : $z\partial_z -z$ is regular and $z\partial_z -1$ is not. But I couldn't find such easy examples for holonomicity. Perhaps it is linked with the complexity of the computation of characteristic varieties. (I know it is linked to Gröbner basis, but I don't know very well this theory)

Any examples or literature recommendations will be highly appreciated.

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  • $\begingroup$ If you choose coordinates $z_1,z_2$ for $\mathbb{C}^2$ then $Q=0$ and $P=\{\mathrm{d/d}z_1,\mathrm{d/d}z_2\}$ seem to satisfy your requirements. Then $\mathrm{char}(D_X/D_XP)$ is just the zero section of $T^\ast X$ and $\mathrm{char}(D_X/D_XQ)$ is the whole of $T^\ast X$. Do you want to apply further constraints or do these satisfy your requirements? $\endgroup$
    – Simon Wadsley
    Commented Jan 16, 2017 at 12:16
  • $\begingroup$ Mmm, yes I would prefer less trivial examples. But, I have nonetheless a question. How could $\text{char}(D_X/D_X Q)$ be $T^*X$ since we know in general that $\text{dim}\text{char}(A) \leq \text{dim} X.$ ? $\endgroup$
    – C. Dubussy
    Commented Jan 16, 2017 at 12:31
  • $\begingroup$ Can you be more precise about how non-trivial? With regards the inequality, it goes the other way. The dimension of the characteristic variety is at least dim $X$ not at most. $\endgroup$
    – Simon Wadsley
    Commented Jan 16, 2017 at 12:38
  • $\begingroup$ Oh yes, sorry, you're right. It is difficult to precise how complex must be the system, but perhaps it would be very helpful to compute the characteristic variety of something like $$\{f(z,w)\partial_z + g(z,w)\partial_w + h(z,w), f'(z,w)\partial_z + g'(z,w)\partial_w + h'(z,w)\}$$ with $f,g,h,f',g',h'$ polynomials of low degree.Of course some of these polynomials can be constant but not all, if possible. $\endgroup$
    – C. Dubussy
    Commented Jan 16, 2017 at 12:44

1 Answer 1

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(These examples are shamelessly pilfered from Gröbner Deformations of Hypergeometric Differential Equations by Saito, Sturmfels and Takayama, which is perhaps the place to learn computational D-module stuff.)

Holonomic: $D\cdot\{z_1\partial_2,z_2\partial_1\}$. This left ideal has characteristic ideal of dimension 2, so it is holonomic.

Non-holonomic: let $f=(z_1^3-z_2^2)$; $D\cdot\{f\partial_1+\partial f/\partial z_1, f\partial_2+\partial f/\partial z_2\}=D\cdot\{\partial_1 f,\partial_2 f\}$. This left ideal has char.ideal of dimension 3, so it isn't holonomic.

(Edit: if you want to actually compute stuff, I recommend the Dmodules package for Macaulay2; it has tools to do most D-module things, including characteristic variety/ideal, gröbner bases etc.)

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  • $\begingroup$ @KetilTveiten Isn't the term holonomic rank usually reserved for the dimension of the (holimorphic) solution space near a generic point? I've only ever heard the dimension of the characteristic variety of $M$ referred to as the dimension of $M$. $\endgroup$
    – Avi Steiner
    Commented Jan 17, 2017 at 15:24
  • $\begingroup$ @AviSteiner Well, yes, I'm being sloppy. The holonomic rank is the dimension of the localization of the char.variety at the generic point, which as you rightly point out is the dimension of the solution space; in these examples the numbers happen to coincide (I was sort of skimming through the book for the examples and left my brain behind). Answer edited to fix this cock-up. $\endgroup$
    – Ketil Tveiten
    Commented Jan 18, 2017 at 10:47

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